AOK History: Thoughts on Systemic Racism in North America

The recent killings of African-Americans by the police in a number of cities in the USA has prompted a great social uprising demanding change captured in the slogans “I Can’t Breathe”, “Black Lives Matter”, and “Defund the Police”. It is believed that these police killings are a result of a “systemic racism” inherent in all aspects of African-American lives, and there is a demand for change in these systems and institutions with the elimination of their oppression of  the day-to-day lives of African-Americans, as well as women and members of the LGBTQ community. Here we will attempt to give some thought to how this “systemic racism” arose and from what and where are its origins.

For someone such as myself who remembers the social upheavals of the 1960s in the USA, I had, at first, perceived the current protests as simply another event in a long line of events where African-Americans protested against their oppression and that these protests would be either brutally crushed or simply allowed to wallow and eventually fade away due to the short memories of those of us who dwell within the technological society.

But these protests appear to be different from the protests of the 60s. For one, they appear to have the support of the white majority in the country. In the 60s, the protests were fragmented with the whites protesting the war in Vietnam while the African-Americans expressed their anger and outrage over the assassination of Dr. Martin Luther King. This fragmentation is not present in the current protests going on not only in North America but around the world. They appear to have evolved into a “movement” rather than merely a protest and this movement has gathered a significant amount of momentum..

Dutch Slave Ship Arrives In Virginia
African Slave Trade

To begin to give thought to the history of North America is to note two basic facts: the history of North America begins with the genocide of its Native aboriginal peoples; and secondly, North America itself, for whites, has no history from before the Age of Progress. While the genocide against the Natives peoples was already well underway, the first 19 or so African slaves reached what were then the British colonies in Point Comfort, Virginia, near Jamestown, in 1619, brought by British privateers who had seized them from a captured Portuguese slave ship. This was over 150 years before the American Revolutionary War and the Constitution which resulted from it. In a somewhat incredible irony (based on superstition, perhaps), the slaves were usually baptized in Africa before embarking, the irony being that enslaving a human being is perhaps the most un-Christian action that a human being can do to another human being.

The “systemic racism” that is seen not only in North America but around the world wherever white Western Europeans be they English, Spanish, Portuguese, French or Dutch, through their imperial adventures, was established when they arrived at the various shores of lands that were alien to them. Their subjugation of the aboriginal inhabitants of those lands required a “morality”, a racism, that was the product of a perceived “superiority” either in their Christian faith or the result of what they perceived as the superiority of their “civilization” which found its concrete realization in the superiority of their weapons. The views of those conquered peoples by their conquerors were those of “savages” and “barbarians”. One can update this racism with a look at how America has treated those who have come under the oppression of its imperialism and its building of its empire. It should not be forgotten that the price to be paid for the realization of the American Dream at home is in the human blood shed by the victims of American imperialism abroad.

The view of Nature held by the Native peoples of North America, for instance, was quite different from that held by the white conquerors who came with Hobbes and Locke and the Protestant or Roman Catholic Christianity embedded in their consciousness. The vastness and intractability of the land created a fear that could only be overcome through a meeting with it being a relationship of conquest. This innate fear remains present even today and manifests itself in multivarious ways in the North American psyche.

The early settlers of North America were unique Europeans. They brought with them the Calvinist Protestantism (Puritanism) which was a break from the traditional Christianity of Europe, and they also brought with them the new revolutionary philosophies of Hobbes and Locke as well as those of Rousseau from France which were breaks from the contemplative tradition of ancient Greece given to Medieval Europeans from the writings of Aristotle and Plato.

Rousseau’s conflict with the English philosophers remains embedded within the consciousness of North Americans even today. Many of the commentaries on the need to change American society from today’s protesters speak of the USA’s failure to uphold the “social contract” with regard to its African-American communities and peoples. There is no questioning of the goals of the overall deeper drives that provide the stimulus for the calculating technological reasoning and its conquering of the necessities of Nature,  but rather, for a just participation in the society of which this conquering relationship is a primordial given. The desire is for the upholding of the promise held in the originating liberalism that would provide the equity, justice and liberty to allow participation in that drive and the benefits that result from technological mastery.

To understand North America it is necessary to understand the connections between the new physical and moral sciences of Newton and others and their acceptance by the Protestants that first came to what was a new land. The differences between ancient and modern science can be found in the writings on The Natural Sciences in this blog. The Natural Sciences: Historical Background Both Max Weber and the Marxist historians, for instance, have demonstrated the practical connection between the early Protestants and property as primarily due to the “worldly asceticism” of those Protestants. But the deeper connection lies in the metaphysical connections between the new sciences and the new Christianity of those Protestants.

jefferson_bible_1170
Jefferson’s Bible

The new physical sciences of Bacon, Galileo and Newton were accepted by the Calvinist Protestants because these sciences were a critique of Medieval Aristotelianism and thus of the Roman Catholicism which based some of its doctrine on the principles of Aristotle’s understanding of nature. The new sciences critiqued the  teleology of Aristotle’s science as causing human beings to view the world in a way in which it was not. The theologians criticized Aristotle’s science as a misleading road to “natural theology” that led human beings away from the Divine Revelation in the person of Jesus Christ and the reality of His Crucifixion. The tension between these two views existed within the framers of the American Constitution with Deists such as Jefferson, Washington and Franklin on the one hand and the practicing Protestant Founders on the other. The picture of Jefferson’s Bible illustrates that his Christianity would not sit well with most of today’s Christians in the USA.

How Locke made the Hobbesian view of nature compatible with the English speaking Protestantism of the early days of America is a subject that requires too much detail for this post. Suffice it to say that his doctrine of “comfortable self-preservation” as the highest end for human beings is hardly compatible with any notion of Christianity. The idea of “comfortable self-preservation” became re-worded as “the pursuit of happiness” in the final Declaration of Independence replacing Locke’s original word “property”.

Modern African-Americans have chosen the Rousseauian side of the tension between “natural law”” and “positive law” that was present in America from its beginnings, but the atheism of Rousseau would hardly find a place for the majority of them at the present time. At the heart of the current protests is the cry for the fulfillment of the “social contract” realized in the American Dream for all the citizens of the USA regardless of race.

To try to explore the reality of “systemic racism” and to provide some notes on its origins and its ultimate flowerings, it is necessary to speak of “liberalism”. In liberalism, freedom and reliance on technique are indissolubly linked, such that technology becomes the very ontology of American lives and defines who and what they are. This ontology itself is prior to any “-isms” and determines how those “-isms” are understood and interpreted by the people who hold them up as “ideals”. This ideal of what human beings are is encapsulated in the word “freedom”. This technological world-view is the common horizon that embraces both sides of what is currently understood as “the left” and “the right”. The “theory” and the “practice” within these “-isms” are indistinguishable and this must be understood if one is to gain access to the roots of who, what and how North America has become what and how it is.

The “systemic thinking” is prior to the systems and the institutions which are created from it and we must try, in this particular case, to understand how “racism” has become embedded in the systems and institutions that have been created in North America and that have since come to prominence around the world through the English-speaking and European empires and their victories in past historical wars and in the two great Wars of the 20th century. This is difficult for white people because that systemic thinking, in its commandeering, controlling and dominating stance towards the environment as “object” and in the novelty which it creates, prevents any reflection on its roots because it is primarily whites who have benefited from that thinking and its results and they have come to perceive that they must somehow be given the evolving truth of things. African-Americans wish to be a part of that hope and that truth, and to also benefit from that technological dominance that has made human beings the masters of nature. But the acceptance of the viewing comes at a cost.

The two most important documents relating to the establishment of American society and its institutions are its “Declaration of Independence” and its “Constitution”. The American Constitution begins: “We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defence, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution for the United States of America.” Clearly, “the People” referred to “ourselves and our Posterity” i.e. the white founders. The African slaves, the Native Peoples, and women were not considered to be “the People”. The desire for “a more perfect Union” indicates the divisiveness present from the very beginnings of what is known as the USA.

The clarification of who gets to be a “person” and who doesn’t was at the core of the establishment of the “systemic racism” that is the bulwark of white societies and their economies. Determining a subservient order for those with darker skin allowed the American founding generation (and the generations after) to define “all men” and “the people” as “white men.” As a result, they guaranteed white men the rights and liberties promised by the Constitution while preserving a thriving economy based on racial oppression. It remains a matter of debate whether or not the American Civil War was due to “economic” factors rather than the freeing of the slaves of the South or whether the freeing of the slaves was itself an “economic factor”. Subsequent American history would suggest the former rather than the latter, and that the War was not undertaken with such “noble” motives as the subsequent mythology provides. It was a war over the price of commodities.

In “The Declaration of Independence”, the attempt to hold together the permanence of “natural law” with the changeableness of “positive law” is clearly in evidence: “”We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain inalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.” The ‘men’ who are ‘created equally’ are, of course, the bourgeois landowners and slave owners who framed the Declaration. The “pursuits” were meant to be enjoyed by a “leisure class” founded upon the labour of the slaves.

The key to how the Declaration and the Constitution came to be written was in the replacing of Locke’s concept of “property” with “the pursuit of happiness”. Locke’s original concept of property related to the body: each human being possessed a body through which it undertook “work” and in doing so made “worthless nature” amenable to human needs. If all “men” had a right to property, and primarily the property that was their own bodies, then slave-holding would, prima facie, be “unconstitutional”. White slaveholders and the States that benefited from slave holding would not agree to this, so as the Montpelier Organization notes: “The answer lies in the idea of compromise: the founders compromised their morals (many were recorded as being opposed to slavery), and power (in some cases, states bowed to slaveholding counterparts in order to ensure the Constitution would be ratified), in the name of economics. Slavery, when all was said and done, was both profitable and convenient for many white Americans—and not just in the South.” https://www.montpelier.org/learn/slavery-constitution-lasting-legacy The entry continues: “As lifelong bondage of enslaved African Americans became more financially viable, the indentured servitude of whites (their terms only lasted five to seven years), was phased out. The system proved itself so lucrative that law and legal precedent began to leave future governments leeway for prioritizing economy over morality.” Current events in American politics continue this compromise of morality to economy both with regard to the suppression of African-Americans and to the Covid-19 pandemic.

Two examples from literature illustrate how difficult it is for whites to gain some illumination of self-knowledge to recognize how they have benefited from systemic racism. The Nigerian writer, Chinua Achebe, accused the writer,  Joseph Conrad, of being “racist” in his great novella “Heart of Darkness”: https://polonistyka.amu.edu.pl/__data/assets/pdf_file/0007/259954/Chinua-Achebe,-An-Image-of-Africa.-Racism-in-Conrads-Heart-of-Darkness.pdf, and also https://www.newyorker.com/magazine/1995/11/06/the-trouble-with-heart-of-darkness. Today, statues of King Leopold II of Belgium are being torn down because of his imperial racist past. While there is no doubt that Conrad speaks out against, and condemns, this imperial racism in the novella, Achebe points to Conrad’s almost unconscious racism in Conrad’s use of symbols, motifs and metaphors in the novella which de-humanize the Africans that are presented there. A small point in Conrad’s defense, the descriptions Conrad uses are for all human beings, regardless of color, and of the thin veneer that is “civilization” separating us from the brutes; but the Africans are, nevertheless, still portrayed as brutes.

A second example is Mark Twain’s The Adventures of Huckleberry Finn, a source and inspiration for Conrad’s “Heart of Darkness”. Twain’s novel is now the most censored book in North America (although this censoring is done with the best of intentions) as it uses language and terms denigrating African-Americans (the use of the “n-word”, for instance). But the novel, in illustrating Huck’s education and growth, demonstrates the need to overcome what had become “sivilization” as it was understood in the America of Twain’s time (and remains in our time), and that included the recognition of the “humanity” of African-Americans in the character of Jim. The core theme of the book is the search for a ‘higher morality” than that present in the America of his day and our day. Twain’s warning that “Those attempting to find a moral” in his book “would be shot” is part of his effort through his humour. If these two literary geniuses, Conrad and Twain, are to be labelled “racist”, it is difficult to see how we mere mortals can possibly avoid being called the same. It represents the long journey ahead for those of us who must attempt to overcome the “systemic racism” in our views of the world.

Under the future “technology of the helmsman”, the skin color of the “human resources” and “human capital” will be a matter of indifference. The corrosiveness of the nihilism that is at the heart of our technological calculative reasoning embraces both the American “Right” and the American “Left” within a common horizon greater than either. The American “Right”, those who would probably call themselves “Republicans” and the roots of whose thinking reach back to Locke, appear to be longing for some lost “golden age” which they do not specify exactly, though its paraphernalia seems to relate to the “lost cause” of the American Confederacy and its symbols of white supremacy and which is prior to the 20th century’s various migrations to America of people of colour, These Rightists adhere to the freedom of the individual to hold property and for the enforcement of the laws that have currently been institutionalized even though those laws prevent individuals from other races from ever attaining that property and the sense of feeling ownership for those laws created for the community. But both sides do not doubt the central fact of the North American dream which is to be realized in progress through technological advance.

This leads to a number of questions: do not the institutions as they exist and produce those benefits come from the same calculative rationality? are the benefits possible without those stifling institutions? can those institutions exist as participatory democracies since centralization and uniformity are part of the essence of technology? The spontaneity of freedom is made possible through the conquering of the spontaneity of nature. Both share the deeper assumptions that have made technological society possible.

Nevertheless, as I have written elsewhere, at the present time liberalism and its “values” and “ideals” are all we have, and it is our duty to ensure that the institutions and their laws which have been and are being created are directed in such a way that all human beings can enjoy the benefits of the dynamic technology that were originally envisioned in the writings of the philosophers who were the founders of what we call “modernity”.

 

 

AOK History:

History: Is it a science or an art?

History is different from the other Human Sciences, or indeed other sciences in general, in that the knowers or researchers cannot directly observe the past in the same way that the object of research can be observed and studied in the Natural Sciences. “Historiology” is the study of history in general, the search for what its essence is, what its purpose is. “Historiography”, that is, a study of the writings of history, is not a study of all of the past, but rather a study of those traces or artifacts that have been deemed relevant and meaningful by historians; and this choosing of artifacts and evidence is the most important aspect of the study of history as it attempts to aspire to “scientific research”. This is where the importance of “shared knowledge” comes into play; what we call our “shared knowledge” is “history” and what we choose to select is determined beforehand by our culture.

We must distinguish between “shared knowledge” or culture, what is commonly called history, and “personal knowledge” as independently acquired knowledge. By shared knowledge we mean the scientific or philosophic knowledge that a human being takes over from former generations or from others, what we would call “history”; personal knowledge is that knowledge, whether it be scientific or philosophical in nature, that a mature scholar acquires in his unbiased discourse which is as fully enlightened as possible regarding its limits and horizons with an awareness of its presuppositions within any area of knowledge i.e. what you are attempting to learn to do here in TOK .

In the modern, this distinction between personal and shared knowledge tends to lose its crucial significance due to our belief in progress. In TOK, it appears that we tacitly assign the same cognitive status to both shared and personal knowledge and this impacts how we understand history and what we feel its importance is to our futures. What we deem to be “historical” first appears and coincides with ratio, calculation, and thought understood as ratio and calculation. What is chosen to be called “history” arises with a pre-determined understanding and definition of what human being is (the animale rationale) and this, in turn, determines what “will be held to account” in the selection of what is deemed to be important in relation to that understanding of human being.

The question of whether history is an art or a science is as old as “historiography” itself. Aristotle in his Poetics distinguishes between the poet and the historian and the philosopher and the historian. The historian presents what has happened while the poet is concerned with the kind of things that might, or could, happen: “therefore poetry is more philosophic and more serious than history, for poetry states rather the universals, history however states the particulars”. (Poetics 1451a36-b11) History might be called pre-philosophic in that it concerns itself with particular human beings, particular cities, individual kingdoms, or empires, etc. The historian must choose between the important and the unimportant things when writing her report, and in her choices illuminate the universal in the individual event so that the purpose of her recording is meant to be a possession for all times. You have done much the same in your Exhibition (if you have done it correctly). The presentation is analogical.

The availability of those relevant traces of the past and their relevance and meaning may be influenced in many ways by factors such as ideology, perspective or purpose, but this is a “modern” version of how we examine things. As knowers we seek to clarify the past and to determine whether or not what is claimed is true. In doing so, we will face problems of reliability and attitudes, and may consider the purpose of historical analysis and the issue of the nature of historical truth. “Historical truth” is bound together with our understanding of truth as “correctness” and “correspondence” arising as it does from ratio and calculation.

The spirit of historicism (the understanding of time as history) permeates every aspect of every text and every approach to the study of and knowledge of the things of our world, and it is particularly present in the IB program. Plato viewed time as “the moving image of eternity”, an infinite accretion of “nows”; we tend to view time as the “progress” of the species towards ever greater perfection, much like how we view the latest models of our technological devices and gadgets as being more “fitted” towards accomplishing our ends and purposes. Our “evolution” and “adaptation”, we believe, are signs of our progress and growth as a species as we move towards ever greater “perfection”, both moral and physical. It is sometimes called “the ascent of man”, but such a concept of human being, as an “ascending” creature, is only possible within the technological world-view.

When we speak of History as an area of knowledge, we are speaking of “human history” not the history of rocks or plants or other objects that are also part of our world. These are covered in the Group 4 subjects as part of the Natural Sciences. History as an area of knowledge deals with human actions in time whether by individuals or communities so it is considered a “human science” for the most part, and the approach to the study of it is a “scientific” one. This attempted approach to the study of history is the same as that carried out in the Natural Sciences wherein history is looked at “objectively” and demands are made of it to give us its reasons. We seek for the “causes” of events. This approach has given rise to one of the complaints against history and how it is studied nowadays: we can only learn about the past; we cannot learn from it. Nor do we today feel that we need to. This dearth of knowledge of history is most in evidence in America, and this is not surprising as America is the heartland of technological dynamism.

The spiritual crisis of our “civilization”, our “culture”, and thus our history, is that the historical moment of technological mastery of the earth comes forth from the same science which gives us the historical sense or historicism. In the past reason, virtue and happiness were united as giving to human beings purpose and meaning for their actions. The “age of progress” realized its goal of freedom in the democratic equality of all human beings. But what evidence is there for the equality of all human beings when the evidence from the biological sciences would clearly suggest that human beings are not equal when it comes to what are considered the most important matters and traits? Not all human beings are “fit” for the ends which our culture aspires to. The question of this “fittedness” is the dark question of “justice” in our time. Josef Stalin’s cynical statement that “Only the winners get to write the history” equates “winning” with the ever-evolving process of “truth” and its realization of the “empowerment” of those who can claim to be “victorious”.

Modern science has shown us that the “values” of rationalism are not finally sustained in the whole of the things that are; that is, Nature is finally not rational and it is only human beings that give to nature its “rationality” and, thus, its Being. Reason is only an instrument and it is used to provide meaning and purpose to our willing, to our desiring and creating, our knowing and making. The “happiness” sought for for all human beings is only achievable through the “lowering” of the understanding of what that “happiness” is; and its foundations are not to be found in “reason” and “virtue” (which only the few are capable of) but in the instincts and their “liberation” (which is a real possibility for the many). “Happiness” as it is understood in its modern form is only possible where “nobility” and “greatness” are forgotten or are not important as ends and have been replaced by “recognition” or “15 minutes of fame”. Human beings know that they create their own “values”, and this is upheld by both the nihilists of the political right and the democratic libertarians of the political left.

History: The Absolutist and Relativist Approaches:

Knowledge Questions: What methods do historians use to gain knowledge? What is unique about the methodology of history compared to other areas of knowledge?
On what criteria can a historian evaluate the reliability of their sources? If our senses are sometimes unreliable, does this mean that eyewitness testimony is an unreliable source of evidence? Have technological developments enabled us to observe the past more directly? What challenges does archive-based history emphasize about how knowledge is shared and preserved? Is there less emphasis on collaborative research in history than there is between researchers in other areas of knowledge? How do the methods and conventions of historians themselves change over time?

History deals with memory and time or temporality, the past, present and future. The knowledge questions and issues that arise in the study of history rest in two mutually exclusive positions with regard to the writing of history (historiography) and the “re-searching” or study of history (historiology). The two positions are commonly referred to as the absolutist position and the relativist position. The discussion below attempts to illustrate both positions.

According to relativism, all human thought is historical and hence unable to grasp anything eternal or “unhistorical”; there is no permanence to things or to thoughts. Plato views time as “the moving image of eternity”. According to Plato (an absolutist), philosophizing means to leave the cave where things may be viewed in their “absolute” truth beyond opinion. To we moderns, all philosophizing and thinking essentially belongs to the “historical world” or the cave, what we call our “culture”, “civilization”, and involves opinions based on these contexts. This belief is what is called historicism and it is a recent arrival on the historical scene (early 19th century) but it continues to gain preeminence in our thinking and viewing of the world as it erodes what we have come to believe during the age of progress. The two most prominent thinkers of historicism are the German philosophers Friedrich Nietzsche and Martin Heidegger; and while these thinkers are reviled in the English-speaking West, their thought permeates many aspects of the shared knowledge in the West through its interpretations and applications by lesser thinkers.

History always concerns “individuals” whether those individuals be individual groups, individual human beings, individual achievements, individual “civilizations” or the one “individual process” of human civilization from its beginnings to the present, and so on. In the IB, Group 3 subjects are called “Individuals and Societies” and History is listed as a Group 3 subject although it is given a special distinction as an Area of Knowledge in TOK. History can be our “personal history” or our “shared history”, and both provide knowledge of some type. The historical sense shows us that we create history, whether by “just doing it” as far as our own actions are concerned or by living in a society along with others and sharing their beliefs, customs, etc. The outcomes of our personal and social/political actions are matters of chance so we study history so as to control the outcomes making chance as ineffective as possible. History is determined by the technological and its rendering is “a giving an account of” or “giving an account for”.

History and the approach to it is most closely related to inductive argumentation similar to experimentation in the natural sciences. Things are explored through what is called research, and an attempt is made to arrive at the “timeless” philosophic questions regarding the incident, individual or event chosen in order to get at its “what”, “why”, and “how”. This method is possible because of the positivism that lies at the ground of how we view the world: we no longer discern any difference between historical and philosophical questions. The concepts which we use are viewed as entirely historical in that they are seen as products of our own individual societies and their historical backgrounds. Technology and The Human Sciences Pt 2: Rousseau, Kant, Hegel, Marx

Is the study of history relevant? What is the purpose of its study? Many people today hold the relativist view that the standards that we use to make judgements in history are nothing more than the ideals adopted by our society or our “civilization” that are embodied in its way of life or its institutions. But, according to this view, all societies have their ideals, cannibal societies (indigenous societies, if you like) no less than “civilized” ones, fascist societies as well as democratic ones. If the principles of historical choice are sufficiently justified by the fact that they are accepted by a society such as is understood by the pragmatists, are the principles of fascism or fanaticism or cannibalism as defensible or sound as those of democracy or “civilized” life?

If there is no standard higher than the ideal(s) of our society,  are we unable to take a critical distance from that ideal? But the mere fact that we can raise the question of the worth of the ideals of our own society shows that there is something in human beings that is not in slavery to society, call it “freedom” if you will, and that we are able (and obligated) to look for a standard with reference to which we can judge the ideals of our own as well as any other society (c.f. Plato and the Cave). All societies are caves. This standard that we are driven and obligated to search for, according to Plato, is the Good, the “best society” or regime, “the good life”, the “good human being”, etc. One of the purposes of the study of history is its aid in helping us to discover what these are through the “shared knowledge” that has been handed down and over to us.

Our modern study of History teaches us that we can become wise in all matters of secondary importance, but that we must remain ignorant in the most important matters: the historian cannot have any knowledge regarding the ultimate principles of his/her choices i.e. regarding their soundness or unsoundness other than blind preferences. Our inability to gain any genuine knowledge (of the absolutist type) of what is good or right or to recognize all preferences as equally respectable leads to the position that only unlimited tolerance is in accordance with reason; but this leads to an “absolutist” position from a position that rejects all “absolutist” positions.

Absolutist positions, so it is said, are based upon the false premise that human beings can know the good. The Chinese, for example, wish to tell the Japanese what needs to be included in their textbooks regarding Japanese behaviour and atrocities during WW 2. With the relativist position, the Japanese are correct in rejecting this intrusion. Japanese citizens cannot know what behaviours occur when societies become imperial, including their own. What, then, is the purpose for studying history? What “truth” can we learn from it? What standards need to be applied to it? Parallel studies can be made with regard to the wars in Iraq and Afghanistan or to the history of African-Americans. Clearly, Americans are being given different views of their imperial wars and their domestic oppression of their people than the views of those who are suffering from those imperial wars and that oppression. The current Covid-19 pandemic and the protests over police killings of African-Americans illustrate that the “truth” and the “facts” of science and history are now being put to the test as they clash with the desires and views of the political populists.

The relativist position has a respect for individuality and a respect for diversity. Tolerance is one ideal or “value” among many and is not intrinsically superior to its opposite: intolerance. But it is practically (in practice) impossible to leave this at the equality of all choices or preferences. If this equality of choices is the case, then genuine choice is nothing but resolute or deadly serious decision. Such decision is more akin to intolerance than to tolerance. One sees these outcomes of these decisions in the world’s daily news events or in the discussions that you may be having in your TOK classes.

The relativist position is a late product from the “age of progress” and it is also a consequence of the thinking contained in “logical positivism”. The “belief in progress” was the belief that the current age is superior to all previous ages in that the evolution of the “historical process” showed a “progression” to the current historical situation which was far superior to previous civilizations, much like the human species in its evolution is “superior” to the apes from which it evolved; this superiority rests in reason. The past was only a preparation for the present. The positivists’ approach began as an overturning of the idealism of Hegel in favour of a realism that looked at “facts” and “reality”, and that life itself delivers evidence of this progress so that the “winners of history” are somehow in touch with an  “evolving truth of history” and therefore get to write the history. That which is new is superior to that which is old.

We are in need of historical studies to familiarize ourselves with the complexity of these issues.

 

CT 1: Knowledge and the Knower: Historical Background

CT 1: The Knower: Ontological

We will approach the topic of the historical background of who we are as knowers from two different perspectives: the ontological, which defines what human beings are; and the ethical which illustrates how human beings behave or act or how human beings have acted historically. From these approaches we hope to get a better understanding of who we are and who we think we are regarding what we consider “knowledge”.

We have an understanding of ourselves as “persons”. From where do the concepts of “person”, “personal”, and “personality” arise? Originally, “person” comes from”persona” the Latin word for “mask” or “what is before the face”. It originates from Greek drama where actors wore masks to indicate their roles in the performance they were about to perform.  Personalitas originally designates the “role” that a persona indicates or illustrates, but it has the sense of the word “dignity” implied in it so that the designation of a “personality” was someone who was distinguished and dignified by the “role” that they played in events or in the society or community of which they were a member. It was considered an honour and a duty of a citizen to perform in the Greek dramas as these were more “religious” in nature and not merely as educational and a source of entertainment. A “role” is a particular way and manner of being a human being and it is very much related to its origins in Greek drama. In the words of T.S. Eliot in his “The Love Song of J. Alfred Prufrock”: it is “To prepare a face to meet the faces that you meet”. What “roles” do we think we play in our society today?

For the Romans, particularly Cicero, a persona is someone who possesses a high degree of the quality of what a human being is as such: the animal rationale. “Dignity” is thus grounded on ratio, the animal possessing reason and the capability of discourse. The concept of the persona and of the human being are closely linked and are grounded in the determination of human being. This concept of the animal rationale is still with us today in, for example, the Roe vs. Wade decision regarding abortion by the U.S. Supreme Court wherein the Supreme Court determined that foetuses were not “persons” in the whole sense.

St. Augustine
St. Augustine 5th Century

With the arrival of Christianity, human being comes to be determined as a “mixture of body and soul” in the writings of St. Augustine in the 5th century. This determination rested, too, on the notion of human being as the animal rationale. In the Christian determination, the human being, the persona, was determined as an individual soul whose goal and salvation  lie in gaining eternal life as an individual. God is determined as the essential unity of the three personae: God the Father, God the Son, and God the Holy Spirit. (Augustine) Here we find the beginnings of the shift in the concept of the persona in the direction of the individual, that he or she is their own goal and purpose in their search for certainty and surety regarding one’s individual salvation.

In the Middle Ages, Thomas Aquinas determines human being as the “person” who is “rational” by nature (in essence) and is incorporated in an individual body. Thomas’s understanding signified the individual self-sufficiency of a rational being: the independence of the human being, the persona, comes to the fore. The emphasis here stresses the “free will” and responsibility of the individual in their choices and decision-making, their morals and ethics.

Descartes
Rene Descartes

The French philosopher Rene Descartes takes up the concepts of human being that were handed over to him to develop an entirely new concept of the “personality”: ego cogito, ergo sum, the basic principle of modern philosophy: “I think, therefore, I am”. The ego cogito is essential for within it human being is determined through its self-certainty which corresponds to an understanding of truth as “certainty”.  The ratio that was historically involved in all determinations of human being receives the particular form of self-certainty (“I know because….) on the basis of which certainty about anything else first becomes possible. This means that the ego in Descartes’ principle is the subject lying at the root of everything. The human being determines itself now wholly in and from itself and no longer needs Church doctrine; the essence of human being is determined according to its capacity for self-determination, its “freedom” to choose, its freedom to make promises and to enter into contracts.

Kant
Immanuel Kant

The German philosopher, Immanuel Kant, determined the next stage historically in his distinguishing the difference between human beings and things (Groundwork of the Metaphysics of Morals) and with respect to the determination of the human being according to three elements, one of which is the “person”.  (Religion Within the Limits of Reason Alone). For Kant, a thing can certainly exist in itself but its independence, its “freedom”, is only ever a mere means. In contrast, a being that is rational —for Kant, reason is the power of the principles– can never be a mere means. Because it has reason, it is its own end. The three elements of human being in Kant, are the “animal” that is humanity, the “humanity” that is humanity (together with the “animal” giving the animal rationale), and “personality” as a rational and responsible being. Kant distinguishes between the reason that is thinking and apprehending, and the reason that is “accountability”, that is, responsibility, the ethical. This distinction is important for we all know of many human beings who think according to the principle of non-contradiction but who are not at all responsible. The human being, however, is responsible in that it is free to act according to principles. These principles are ethical or as Kant says “practical principles”, praxis. For Kant, the highest is the categorical imperative; the categorical imperative was an improvement on the Golden Rule:  Act as you would want all other people to act towards all other people. Kant’s categorical imperative demands us to act according to the maxim that you would wish all other rational people to follow, as if it were a universal law. For Kant, autonomy/freedom to decide constitutes the modern personality.

Kant further identifies “personality” with “character”. For Kant, “character” is the mode and manner that a cause is a cause (see the etymology of the word “character”). Kant distinguishes two types of causes: those of intelligible character and those of empirical character. This is in contrast to Hume. Kant needs these two types of character for his determination of human being as a responsible being. A “responsible” being must have free will as the cause of his actions. This free will is not found in the human world. In this world, the empirical world, human will is not free; it is conditioned, that is, its character is empirical. “Personality” is equivalent to character”. Personality is a being that is and acts according to its own responsibility. For Kant, all other determinations of personality derive from this.

The Knower: Ethical

From where does our emphasis on personal knowledge and “experience” arise in the West for it was not part of the thinking of the ancients at the beginning of Western thinking although phronesis was a way of knowing held in high esteem by both Plato and Aristotle? During the period which is called the Renaissance, a great paradigm shift occurs in what was called “knowing”, and this was the result of changes in how the world as Nature was understood, and thus how human beings were understood,  and how human beings understood themselves. The French philosopher Rene Descartes is primarily responsible for this change, and the change is based on what our understanding of knowledge and truth are.

With this paradigm shift brought about by Descartes and others before and after him, what is called “humanism” comes to the fore with its focus on human beings’ central place in Nature and in the whole of things. This change occurs during the 15th and 16th centuries with the change in the understanding of the “person”. Humanism could also be said to find one of its origins in the arrival and grounding of algebra in mathematical thinking. In the search for certainty and surety of human beings’ salvation and redemption as a “personal” event in the Protestant Reformation within Christianity, and in the arrival of modern science in the  experiments of Galileo, and in the philosophy of Rene Descartes where human “subjectivity” is grounded and where Nature itself is understood differently from previous interpretations, we have a great paradigm shift of how human beings understood themselves and their place within the world.

English-speaking teachers of philosophy and theory of knowledge have rarely paid attention to the two most comprehensive thinkers, the great anti-theological/atheist thinkers of the West: Jean Jacques Rousseau and Friedrich Nietzsche.  There are a number of reasons for this and to go into all of them would require far more writing than this post would or could bear. Many of them will be touched upon in other areas of these writings. How has the thought of Rousseau been received and understood by English-speaking teachers of philosophy and Theory of Knowledge?

bertrand_russell_smoking4
Bertrand Russell

If one is familiar with the English-speaking tradition of philosophy (I will say, for the moment, the materialist, empirical, “the analytical school”), Rousseau has been called an “unsystematic poet”, a man quite incapable of the sustained and disciplined thought necessary to the true philosopher. This account can be seen from the writings of Jeremy Bentham right up to the writings of Karl Popper. Bertrand Russell’s account of Rousseau in The History of Western Philosophy, where Rousseau is dismissed as a self-indulgent poet, is filled with Russell’s contempt and anger for the man ‘whose thought is so filled with contradictions of such an obvious nature that they could be discovered by any high school student of average ability’. These, shall I say, misreadings of Rousseau have caused a lack of serious attention to this thinker which has resulted in the darkening of our self-understanding and the dimming of our understanding of ourselves as knowers and the consequences of this dimming for life and thought.

The ascendancy of the English-speaking peoples (and the IB Diploma Program is but one outgrowth or flowering of this ascendancy) has been with us historically from the Battle of Waterloo to the victories in the two great wars of the 20th century. It was achieved under the rule of various species of “bourgeois”. The members of this elite class felt their right to rule was self-evident since it was not seriously questioned at home and they were successfully extending their empires around the world. The constitutional liberalism, empowered by technological progress, was justified by various permutations and combinations of John Locke’s contractualism and utilitarianism. English-speaking political philosophy, understood as the theory of living well within communities, has largely been concerned with emendations to Locke’s account. But why be concerned with Rousseau who in many respects agreed with Locke?

Rousseau is the primary instigator of that period which has come to be called the Romantic Period. Because of Rousseau’s influence, what we know as ‘German Idealism’, the philosophies of Immanuel Kant, Hegel, and Marx get their initiation. This is because within Rousseau we come upon the presence of the concept ‘history’: the temporal process in which beings are believed to have acquired their abilities. By History is not meant ‘historiography’. Historiography is our study of the written account of human history and is included in our Part 3 subjects, the Human Sciences, or as a distinct area of knowledge in itself here in TOK. The meaning of ‘history’ used here is ontological: it is a realm of being in which human beings dwell. We call this realm “time”.

In the writings of Kant, for instance, English-speaking philosophers were deflected from the true intent of his writings by his statement that David Hume, the British philosopher, had awoken him from his “dogmatic slumber” and so they looked at him from within their own philosophical tradition and have, up till now, tried to make him part of their own philosophical tradition. But Kant’s chief encounter was with the philosophy of Rousseau and there are far more references to Rousseau in his work than to Hume. (This is not to deny Rousseau’s debt to Hobbes and Locke, both of whom established the history of English philosophy, but Rousseau is profoundly critical of that debt).

Darwin
Charles Darwin

For the English-speaking peoples, ‘history’ becomes part of our ‘shared knowledge’ in the discoveries and writings of Charles Darwin. While the historical sense was present in English writings well before Darwin, the historical sense becomes central through the writings of Darwin because it was at the heart of the most important activity of the 19th century—Natural Science. It is said that Darwin’s main contribution to our shared knowledge was not ‘evolution’, but how evolution took place: through ‘natural selection’. Darwin’s chief concern, however, was not Natural Selection, but the question of Creation or Modification. (See Life and Letters, vol. II p. 371). “Modification”, in Darwin’s sense, is a synonym for History understood as the temporal process in which beings acquire their abilities, that beings ultimately have no essence. Darwin’s thinking is not possible without, first, the thought of Rousseau. Once History becomes part of our shared knowledge, what happens to the ahistorical political science of Locke who has provided the foundation of our English-speaking political and social institutions?

Thomas Jefferson
John Locke

Locke’s contractualism is ahistorical. The American statesman, Thomas Jefferson, reveals this when he says in the American Declaration of Independence: “We hold these truths to be self-evident: that all men are created equal; that they are endowed by their Creator with certain inalienable rights; that among these are life, liberty, and the pursuit of happiness”. Jefferson’s Constitution is an attempt to bring together both Locke and Rousseau. Being “endowed by one’s Creator” and possessing “inalienable rights” are ahistorical principles. Shifting Locke’s “right to property” to the right of the pursuit of happiness is possibly the result of Thomas Paine’s, a student of Rousseau’s, influence on Jefferson. Locke himself was an atheist even though he wrote a book entitled The Reasonableness of Christianity. While being a man of sobriety or seriousness, he was not without a sense of humour nor without a sense of irony. These contradictions are part of the everyday reality of American and other English-speaking political and social institutions today.

JohnRawls
John Rawls

The attempt to hold together history and ahistorical contractualism (that which is beyond time or permanent and that which is within time as motion or change) has made English-speaking political philosophy become thin to the point where it has become the sheer formalism of the analytical tradition. One can find an attempt at this formalism in John Rawls’ book A Theory of Justice. As a cautionary note, I would say that even though our academic curriculum and our manner of knowing is dominated by the perspectivism of historicism, our freedom from historicism in our practical affairs has preserved us, so far, from the great crimes of National Socialism and communism (I am referring to our ‘internal’ politics, our domestic politics, and not to our misguided imperial adventures of the 20th and 21st centuries nor to the behaviour of our corporate institutions abroad. That this preservation from the crimes of fascism and communism is slowly breaking down appears to be the chief concern of our new TOK guide for 2022).

The attempt to maintain contractualism, our being in societies, our politics, our ethics, freed from any ontological statements (our being-in-the-world and our understanding of ourselves as beings in this world), fails because it requires that science be taken in phenomenalist (empirical) and instrumentalist (the analytical school) senses. It may be possible to attempt this when discussing the small results of academic technological scientists (the attempt to make the Sapir-Whorf hypothesis substantive, for instance), but it is quite impossible to assume it about the results of a great synthetic scientist such as Darwin.  When Darwinism is taught at school, it is not taught as a useful hypothetical tool only of interest to those who are going to be specialists in the Group 3 and Group 4 subjects. As Darwin well knew, the discussion of Creationism and Modification is an ontological one, despite the clever chat by analytical philosophers. His Holiness, the Pope’s, acceptance of evolution and the Big Bang theories retain the sense of purpose in the “createdness of Nature” (what is called “teleology), that there is meaning to creation, but whether or not this is sufficient ontologically is quite another matter. That these theories ultimately clash and contradict with His Holiness’ beliefs in not discussed in depth (to my knowledge).

What is the issue: you cannot hope to successfully combine an ahistorical political philosophy, an ethical philosophy if you will, with a natural science which is at its heart historical.

With the idea of History/Modification we are led back to Rousseau. Science views Nature as non-teleological, that is, it is a product of accident, chance not purpose. Nature has no goal in and of itself. It seems that when there is a great outpouring of scientific activity—in the case spoken of here that of the 19th century—there is always a great philosopher who in his thought of the whole has made a breakthrough against all previous thought. By “breakthrough” I am not speaking about the “progress of truth”: breakthroughs can also lead into errors. This great breakthrough occurs in the thought of Rousseau. This is  what we mean by a “paradigm shift”. A true paradigm shift is very rare in history and it occurs within human being-in-the-world.

Rousseau first stated that what we are, our essence as human beings, is not given to us by what the Ancients understood as Nature but is the result of what human beings were forced to do to overcome chance or to change nature (in the modern sense of what we understand nature to be). Life is experienced as a problem to be solved. Human beings have become what they are and are becoming what they will be (the “empowerment” of human beings) through their solutions to “the problem that is life”. We are the free, undetermined animal, the perfectly malleable animal, that can be understood by a science which is not teleological (i.e. by a science that sees no final purpose in the things that are).

Rousseau understood the difficulties and the ambiguities of his thinking of human being as an historical animal far better than say, one of his followers, Karl Marx. Rousseau’s battling with the contradictions that appear in the discoveries of his thought is what has led English-speaking commentators to dismiss him, for the most part, in their tutorials at Oxford and Harvard. The contradictions are the result of Rousseau’s refusal to avoid the ambiguities which he was given to think.

Nietzsche
Friedrich Nietzsche

The greatest critic of Rousseau is the German philosopher Nietzsche. For Nietzsche, Rousseau is the epitome of the ‘last man’, the ‘secularized Christian’ who is responsible for the “decadence” of European thought over the last three hundred years. But Nietzsche accepts from Rousseau the belief in the fact that we are historical, that we acquire our abilities in the course of time in a way that can be explained without purpose. Nietzsche claims that he is the thinker who understood the ‘finality of becoming’ in an historical way. But one deeply wonders how Nietzsche failed to recognize how much of his thought on the finality of becoming had been worked through by Rousseau. Was Nietzsche moved by an anger that clouded his openness to the whole?

Karl_Popper
Karl Popper

The understanding that human beings acquire their abilities (their “empowerment”) through the course of time expresses itself in what we call ‘historicism’. Historicism is the fate of all Areas of Knowledge in our time. The attempts to refute historicism from within the tradition of English-speaking liberalism (Karl Popper’s The Poverty of Historicism, as an example) while well-intentioned are feeble. This is reason itself why we should read Rousseau carefully so that we can attempt to know what it is that behooves us to know when all thought is touched with the deadening hand of historicism.  This becomes even more pressing as we become enamored with the word “empowerment”, the word of Nietzsche, and how this “empowerment” will unfold in the nihilism that is our future.

OT 4: Knowledge, Religion and Mathematics

The Principles of Pythagorean Mathematics Applied to Ethics/Actions

We will be discussing how mathematics, in particular how Pythagorean geometry,  provides the principles for our actions; in other words, how mathematics determines our ethics. In doing so, we shall examine some considerations of the differences between what is called calculative thinking and what is called contemplative thinking.

The Greeks initially thought that mathematics is “what can be learned and what can be taught”. What we call mathematics is the axiomatical theoretical viewing and projection of the world which establishes the surety and certainty of the world through calculation. Axioms provide the principles or the archai, which in turn determine the laws which, in turn, lead to the calculative results. Calculative thinking based on the principle of reason (nothing is without reason) determines that the things of the world are disposables and are to be used by human beings in their various dispositions towards the world. This commandeering challenging of the world and the beings in it to demand of them to give us their reasons for being as they are is what we have come to call knowledge. This under-standing is that upon which all of our actions are based although we attempt to separate theory from practice. This surety or certainty that beings are as we say they are through calculation arises through the viewing and use of algebraic calculation in the modern world, the mathematical projection. The results of what is and has been achieved through this calculative thinking are what we have come to determine what knowledge is in our day and what is best to be known and how it is to be known. The axiomatic viewing or theory of mathematics, what provides the theory, is not separate or distinct from the praxis required to carry it out. The theory is not something left behind as the journey for results/knowledge becomes undertaken. The theory permeates each and every stage of the journey or search, what we call the methodology.

Ethics are based on what Aristotle called phronesis: our careful deliberation over what best actions will ultimately bring about the best end result. We call this end result our happiness or what Aristotle called our eudaimonia, our “good spirits”. But how can happiness be the end result of what is, essentially, a hubristic way of viewing and being in the world?

We shall reflect on this question by examining the passage below from Shakespeare’s King Lear.

CORDELIA
We are not the first
Who, with best meaning, have incurr’d the worst.
For thee, oppressed king, am I cast down;
Myself could else out-frown false fortune’s frown.
Shall we not see these daughters and these
sisters?
KING LEAR
No, no, no, no! Come, let’s away to prison:
We two alone will sing like birds i’ the cage:
When thou dost ask me blessing, I’ll kneel down,
And ask of thee forgiveness: so we’ll live,
And pray, and sing, and tell old tales, and laugh
At gilded butterflies, and hear poor rogues
Talk of court news; and we’ll talk with them too,
Who loses and who wins; who’s in, who’s out;
And take upon’s the mystery of things,
As if we were God’s spies: and we’ll wear out,
In a wall’d prison, packs and sects of great ones,
That ebb and flow by the moon.
EDMUND
Take them away.
KING LEAR
Upon such sacrifices, my Cordelia,
The gods themselves throw incense. Have I caught thee?
He that parts us shall bring a brand from heaven,
And fire us hence like foxes. Wipe thine eyes;
The good-years shall devour them, flesh and fell,
Ere they shall make us weep: we’ll see ’em starve
first. Come. 

Explication of the Passage from King Lear

To attempt a summary and explication of the whole of the greatest work in the English language is impertinent.  But a brief introduction is necessary to provide some context with which to understand the play as it appears in the scene above.

At this point in the play, Lear and Cordelia, supported by French troops, have lost the civil war for Britain to Edmund’s forces. Lear, as King, has been ultimately responsible for this civil war. At the beginning of the play, he has disowned his ‘truthful’ daughter, Cordelia, and fallen victim to the flattery and machinations of his two eldest daughters, Goneril and Regan. He has divided the kingdom in two giving each sister control of half, his intention being to avert future strife. Lear wishes to retain the appurtenances of a king, the appearances of a king, while retaining none of the responsibility: Lear is satisfied with the appearances rather than the realities of things. It is this satisfaction with the appearance of things that leaves Lear open to the machinations of his two daughters, Goneril and Regan.

Pythagorean circle
Pythagorean Circle

Lear’s responsibility is, chiefly, a moral one. Goneril and Regan soon work together to remove from Lear the power and possessions that he once held. Lear becomes an O, a nothing, and there is a pun here indicating both his reality and his suffering. In his nothingness, Lear becomes mad and rages against the ingratitude shown by his daughters and the injustice that he sees in the nature of things and in the created world as it is. This scene quoted above is Lear’s anagnorisis or moment of enlightenment, the moment in tragedies when all tragic heroes recognize the errors of their ways and the consequences of their hubris which is at the root of their previous actions.
Lear ends up houseless and homeless and wanders on a heath in the heart of a terrible storm, the inner and the outward man caught up in a terrible disunity. Lear’s physical, mental and spiritual sufferings soon drive him mad. The storm’s effect is a purification of Lear: Lear removes his clothing; his ego, his “I”, is destroyed in the madness; he no longer focuses on himself but is able to see the ‘otherness’ of human beings and to feel compassion and pity for them (in the characters of Edgar as Poor Tom and the Fool) because he sees himself and his humanity in them. Lear’s life as a king has been one of a tyrant for whom all sense of ‘otherness’ has been forgotten. Edgar, too, has become a ‘nothing’ due to the machinations of his bastard brother Edmund in the parallel plot of the play. Lear has gone from King to nothing and he is ready for re-birth. His ego has blinded him to understanding what his true relationship to his god is: initially he looked upon this god and his power as being something which he, Lear, himself possessed. Lear believed that only he himself possessed the truth and that that truth lay in his power.

The play King Lear is a play about the consequences of not knowing who we truly are, as individuals and as a species. Lear, focused as he is on his ego, his Self, is willingly duped by machination in the play; he is willingly duped by flattery as this flattery is recognition of his social prestige. His suffering and madness bring him to a true understanding of his relation to the god and to other human beings, and this relationship is Love. Love is, as Plato describes it, “fire catching fire”. It is recognition that in the most important things, all human beings are equal in that all are capable of the capacity for Love. It is not without reason that in our art and poetry, Love has been described as a homeless, houseless beggar.

Many critics suggest this play is atheistic; Lear has lost his faith in God. The above passage suggests that such is not the case: what Lear has come to understand is his true relationship to God, the true relationship of all human beings to God. Lear has lost the illusion of what he had once understood as God and what his relationship was to that God. It is this illusion that is the trap cast for those who believe that they are in possession of the truth or that truth is of their own creation or doing, their power. The God in King Lear is absent: He will not perform some miracle preventing the hanging of Cordelia by the Captain later in the play. Cordelia has been a symbol of truth throughout the play, and in her death we see the literal destruction of truth among human beings. Good does not triumph over the evil of human actions in this play and we, too, by our very silence, are made complicit in the death of Cordelia.

The play King Lear shows that the purpose of suffering is to allow the de-creation of our selves, the de-struction of ourselves. For the Pythagoreans, the study of geometry served an identical purpose: the purification of our selves through a contemplative understanding of the things that are. For the Pythagoreans, mathematics was a religious activity. When we stand on the outside of the sphere above (the circumference) and are subject to its spinning, we suffer the ups and downs of Fate. There is a wheel of fortune motif that runs throughout the play: Fortune is personified in the passage through the alliteration and personification of ‘false fortune’s frown’ to illustrate that it is, in this case, one of human making: even with the best of intentions one can incur the worst: good does not triumph over evil in this sphere but is subject to the same necessity as are rocks and stones. To decreate one’s Self is to have the Self replaced by an assimilation into the divine; it is to become one of ‘God’s spies’, to see all with God’s eyes and to see all for God. When a human being sacrifices the Self, his most treasured possession, for assimilation in God, “the gods themselves throw incense” upon this sacrifice. We believe our Self to be our most precious possession; the renouncing of this possession is not easy, nor is it pleasant: it is done through suffering.

The centre of the sphere is simultaneously both in time and space and out of time and space. The Self as center here is indifferent to the size of the prison, the size of the circle, what can be encountered through ‘experience’ and novelty. For Lear, imprisonment is a liberation, not a restriction.”Suffering (affliction), when it is consented to and accepted and loved, is truly a baptism” (Simone Weil, “The Love of God and Affliction”). Baptism is a spiritual re-birth. The spiritual rebirth for Lear is clear from this passage as well as from Act III onwards in the play. The attempted suicide of Gloucester due to his suffering is a counterpoint to this: suicide is a sin against the gods because we falsely believe that our self is our own and of our own making and that we can do away with it as we would with any other possessions that we may have. Gloucester’s realization of this results in his finding Edgar again and having ‘his heart burst smilingly’ when he meets his final end. Contrary to our view, in the world of Shakespeare some kinds of suffering have a purpose.

Our personal knowledge is our ‘sphere of influence’ on our world and on the other human beings who inhabit our world. It is what we have come to call our empowerment. That sphere should be seen as composed of wheels within wheels with our actions the spokes of the wheels. The spokes reach out to the circumferences of the wheels: from the diameter, the right angled triangle cannot exceed that circumference. The sphere created by the circumferences may be large or small; most of our lives are spent in our attempts to enlarge this sphere. In it we are ’empowered’ to carry out our activities, but the prison of ourselves is still a ‘prison’ beginning with our bodies and our egos which are placated by the social prestige and recognition which comes from this fulfillment. We become the ‘poor rogues’ and ‘gilded butterflies’ that Lear and Cordelia will chat with, the partisans and politicians of the court. The outer edges of the sphere in its spinning indicate the fates of those who are ruled by Fortune: ‘who loses and who wins; who’s in, who’s out’. It is the fate of all of us who are dominated by the wish for social prestige, recognition. This fate and our desire for this fate is part of the ‘mystery of things’: to see this we must remain at the centre of the sphere where we are not moved by the wheel’s or the sphere’s spinnings, nor are our desires dominated by the wish for social prestige and recognition.

Lear, through his madness and suffering, has been re-born (see other sections of the play particularly Lear’s awakening when he sees Cordelia as an angel, a mediator, and in the play she is, from the beginning, representative of truth). His self, ego, I has been destroyed. In this scene, Lear demonstrates the friendship that is the love between two unequal yet equal beings. Lear’s ‘kneeling down’ when asked for his blessing in order to ask for forgiveness is the recognition of this equality. It is no longer the view of the Lear who said “I am a man more sinned against than sinning”, a false view of Lear’s at the moment of its occurrence in the play for it is the view of most of us with regard to our own sufferings.

It is with a great and terrible irony that after this speech of Lear’s, the following occurs:

EDMUND

Come hither, captain; hark.
Take thou this note;

Giving a paper

Go follow them to prison:
One step I have advanced thee; if thou dost
As this instructs thee, thou dost make thy way
To noble fortunes: know thou this, that men
Are as the time is: to be tender-minded
Does not become a sword: thy great employment
Will not bear question; either say thou’lt do ‘t,
Or thrive by other means.

Captain

I’ll do ‘t, my lord.

EDMUND

About it; and write happy when thou hast done.
Mark, I say, instantly; and carry it so

As I have set it down.

Captain

I cannot draw a cart, nor eat dried oats;
If it be man’s work, I’ll do ‘t.

The Captain’s final words are a statement for all of us motivated by social prestige. Human crime or neglect is the cause of most suffering. On the orders of superiors, we carry out acts that we believe are “man’s work” i.e. they are not the work of Nature but we ascribe the moral necessity for our actions to Nature: “I cannot draw a cart, nor eat dried oats”. We believe that we are compelled to commit immoral actions because we believe Nature imposes its necessities upon us; and, at times, Nature does indeed do so. But if we live with a thoughtful recognition that there are simply acts which we cannot and must not do, we are capable of staying within these limits imposed by the order of the world upon our actions. Such words as the Captain’s have been used by human beings to justify to themselves and to others the reasons for their actions from the committing of petty crimes to genocides. They see their crimes as performing a duty, just following orders.

 
The root of all crimes is, perhaps, the desire for social prestige whether that is achieved through position, money or recognition. For the Captain, it is Edmund who will determine what ‘happy’ will become for him by his giving to the Captain ‘noble fortunes’; and the Captain believes it. He will achieve his noble fortunes through the committing of an ignoble act. One would need to look far across the breadth and depth of English literature to find two more contrasting views of humanity in a work than that which is presented here in these two brief, sequential scenes from King Lear. Human beings are capable and culpable of both forms of action: we have an infinite capacity for Love and forgiveness as well as a finite capacity for committing the most heinous crimes; only Love is both beyond and within the circle, and all human action is done within the circle (or the realm of Necessity).

Contemplation and Calculative Thinking: Living in the Technological World

The passages from King Lear give us an entry to understanding a practical alternative way of being-in-the-world to the current conditioning or ‘hard-wiring’ of our way of being under the technological world-view operating as it does under the principle of reason. This alternative way involves contemplative thinking as opposed to calculative thinking. This contemplative thinking is open to all human beings: it is not a special mental activity for the few. It is an attitude toward things as a whole and a general way of being-in- the-world. It is the attitude that Lear proposes for himself and Cordelia on how they will spend their time in prison: while they will still be in the world, they will not be of the world. While they will be involved with the “poor rogues” and “gilded butterflies”, the world of these rogues and butterflies will not be their world.

What does this mean for us? It suggests that we are in the technological world, but not of this technological world; we are here in body but not in spirit. We avail ourselves of technological things but we place our hearts and souls elsewhere. This detachment involves both a being-in and a withdrawal-from. Like Lear and Cordelia, we let the things of the technological world go by, but we also let them go on. Like Lear and Cordelia, the detachment is both a “no” to the social and its machinations, but it is also a “yes” to it in that it lets that world go on in their entertaining of it.

What is Calculative Thinking:

Calculative thinking is how we plan, research, organize, operate and act within our everyday world. This thinking is interested in results and it views things and people as means to an end. It is our everyday practical attitude towards things. Contemplative thinking is detached from ordinary practical interests.

H.L. Mencken-8x6
H. L. Mencken

Calculative thinking is not computational. It does not require computers or calculators and it is not necessarily scientific or sophisticated. It would be better understood in the sense of how we call a person “calculating”. When we say this we do not mean that the person is gifted in mathematics. We mean that the person is designing; he uses others to further his own self-interests. Such a person is not sincere: there is an ulterior motive, a self-interested purpose behind all his actions and relations. He is engaged with others only for what he can get out of them. He is an “operator” and his doings are machinations. His being-in-the-world may be said to rest on the principle attributed to H. L. Mencken: “No one ever went broke underestimating the intelligence of the American public.”

Calculative thinking is, then, more of a general outlook on things, our ‘way of life’. It is an attitude and approach that the things are there for what we can get out of them. People and things are there for us to exploit. This general outlook is determined by the disclosive looking of technology and its impositional attitude toward things.

There is no lack of calculative thinking in our world today: never has there been so much planning, so much problem-solving, so much research, so many machinations. TOK itself is a branch and flowering of this calculative thinking. But in this calculative thought, human beings are in flight from thinking. The thinking that we are in flight from is contemplative thinking, the very essence of our being human. In this flight, we are very much like Oedipus who, after hearing the omen from the oracle at Delphi and its prophecy, rashly flees in the hope that he can escape his destiny. As with Oedipus we, too, are blind and unable to see in our flight from thinking and our rash attempts to “change the world”.

What is Contemplative Thinking:

Contemplative thinking, on the other hand, is the attention to what is closet to us. It pays attention to the meaning of things, the essence of things. It does not have a practical interest and does not view things as a means to an end but, much like Lear and Cordelia, dwells on the things for the sake of disclosing what makes them be what they are. Contemplative thinking allows us to take upon ourselves “the mystery of things”, to be “God’s spies” in the two-way theoretical looking of Being upon us and of ourselves upon Being. To be God’s spies we must remove our own seeing and our own looking, that looking and seeing that we have inherited as our “shared knowledge”, and allow Being to look through us. This seeing and looking is not a redemption that is easily achieved. The pain-filled ascent in the release from the enchainment within the Cave to the freedom outside of the Cave or Lear’s suffering and de-struction on the heath in the storm are indications of just the kinds of exertions that are required. King Lear in his anagnorisis has arrived at the truth of what it means to be, as such, and of his place in that Being. Contemplative thinking is a paying attention to what makes beings be beings at all, but it is not a redemption which can be cheaply bought.

The word “con-templation” indicates that activity which is carried out in a “temple”. It is a communing with the divine. The temple is where those who gather receive messages from the divine. Lear and Cordelia’s prison is, as such, a “temple” to Lear. Within a temple, one receives auguries. An augury is an omen, a being which bears a divine message which must be heard by those to whom it is spoken. In and through this hearing one is given to see the essence of things and to “give back” those essences to Being. Contemplation is the observing of beings just as they exist and is an attending to their essence. It is a reserved, detached mode of disclosing that expresses itself in gratitude, the giving of thanks. This attention is available to all human beings who through their love, like Lear and Cordelia, are open to the otherness of beings without viewing those beings as serving any other purpose than their own being.  For human beings, it is the highest form of action directed by what the essence of human being is. As the highest form of human being itself, it must be available to all since it is our very nature as human beings.

 

 

 

 

 

 

 

AOK: The Mathematical Projection

The Mathematical Projection of Nature

What is the Mathematical Projection?

In discussing Mathematics as an AOK, we wish to explore what we understand as the “mathematical” and how “calculative thinking” has come to dominate our modern way of thinking. We tend to think that what is understood as the “mathematical” deals in numbers; but the use of numbers is but one aspect of what is meant by the “mathematical” and this view of mathematics as numbers has only come to dominate historically after the great change which erupts during that age we call the Renaissance.

In the following we will examine the arrival of the “mathematical projection” as the approach to what we have come to define as Human Being. It should be understood that this is not a criticism of the mathematical itself nor is it “anti-science”, but it is a reflection upon the implications and consequences of what this interpretation of human being, beings, and Being brings about. What we wish to show is that this understanding of ourselves and of what we think knowledge to be has great implications for our human being-in-the- world and our destiny or fate as beings as we totter towards the apogee of what and how we see through the technological world-view.

“Projection” is ‘to throw’; it suggests ‘throwing away, off’, and is thus related to ‘jacio’ (Lat. ‘to throw’) and subject/object. Projection originally meant ‘to form a picture, design’ in weaving by turning the shuttle to and fro. It then came to apply to literary and mental formation. It acquired the sense of provisional, preliminary drafting under the influence of the French projeter, ‘to plan, lit. throw before’. Today “projection” means ‘to sketch, design, draft, draw up, depict, outline’. Similarly, a “project” is a ‘sketch, outline, design, blueprint, draft’. The words are thus aptly translated as ‘project’ and ‘projection’, from the Latin proicere, ‘to throw forward’. Think of the steps of the Design Cycle which you learned in your MYP courses and how these are a “throwing forward”, or the projection you have made in the planning of your Exhibition. A projection is not a particular plan or project; it is what makes any plan or project possible. In TOK we have given various accounts of what is projected: a world; the being of beings or the ‘constitution of their being’; fundamental scientific conceptions of being such as the mathematical view of nature; Human Being itself. Human Being understood as the animale rationale is the projection of something onto something else: the understanding projects the being of Human Being onto its ‘For-the-sake-of’ and onto the significance of its world; understanding, or Human Being itself, projects Human Being onto its possibilities or onto a possibility; beings are projected onto their being (space); being is projected onto time.

A project (ion) is ‘free’. It is not determined by our prior knowledge or desires, since it is only in the light of a project that we can have any specific knowledge or desires. A project is not projected piecemeal, by gradual steps, but all at once, by a leap ahead so it is prior to reasoning and algorithmic thought. In Kant’s terms, the “projection” is the transcendental intuition and the transcendental imagination working in consort to give us a world in which we may live. There are three main types of project.

Any Human Being must project a world and have a pre-ontological understanding of being, i.e. project being, including its own being. Such a projection occurs at no definite time: it is an ‘original action’ of Human Being. This projection enables Human Being to understand, for example, what a tool is or what another person is, independently of the particular tools and persons it encounters. It is comparable to one’s overall understanding of what a town is and one’s general sense of direction which are prior to any creation and consultation of a map. The projection is how we can even conceive of the journey towards knowledge in the metaphor of a map. A science involves a project (ion) of the constitution of the entities/things it deals with, e.g. Galileo’s and Newton’s projection of being as mathematical which we shall discuss further. Such a project is not grounded in the experience of beings: the project decides in advance what counts as a being and as experience. Nor is it grounded in a previous project or in criticism of it: a new project is not commensurable with its predecessor; it alters our whole view of being and beings.  A mathematical physicist still needs a pre-ontological understanding of tools, people, time, etc. A scientific project is analogous to a selective map of a town; it cannot dispense with one’s overall pre-ontological understanding of beings any more than a map-user can do without a sense of direction. Think of this in relation to Thomas Kuhn’s The Nature of Scientific Revolutions and the paradigm shifts which he speaks of in that understanding.

As we attempt to think in TOK we acquire a conceptual, ontological understanding of being, which involves an understanding of the projects outlined above. It is not enough to simply painstakingly describe these projects without a prior specifically determined projection. The nature of being, of Human Being for example, is ‘covered up’, not open to unvarnished empirical inspection. We must project a being (e.g. Human Being) ‘onto its being and its structures’ which are given prior. We understand something, x, by projecting it onto something else, y, the ‘Upon-which’ of the projection and the ‘sense’ of x. There is thus a ‘stratification’ of projects. We might want to understand this as what we call the Reduction Thesis. We understand beings by projecting them onto Being. We understand Being by projecting it onto time. The regress ends with time: time is ‘self-projection’; it need not be projected onto anything else to be understood. Our projections proceed in the reverse direction to the projection they conceptualize, Human Being’s basic project. This agrees with Aristotle’s view: what is prior in itself is posterior for us. Time is prior to being and makes it possible; Being is prior to beings and makes them possible. But owing to the obscurity of these relationships, we proceed from beings to Being, and thence to time or what we would call “historicity”.

A project involves ‘anticipation’ and the ‘apriori‘. What a tool is such as a map; other people; that there is a world: these are apriori within the project, and thus for every Human Being. That things are exactly measurable: this is apriori for mathematical physics. That Human Being ‘exists’: this is apriori for us.

Apriori‘ comes from the Latin for ‘what comes before, earlier’; the apriori is ‘the earlier’. The apriori is not ‘true’ or ‘correct’ beyond the project which it helps to define, just as a map is not true or correct beyond that which it defines: ‘The apriori is the title for the essence of things, their “whatness”. The apriori and its priority are interpreted in accordance with our conception of the thinghood of the thing and our understanding of the being of beings in general. A project is more like a decision than a discovery (this is a response to the question “Is mathematics discovered or invented?”); it cannot be correct or incorrect: correctness, and the criteria for it, only applies within the light shed by the project. What the light of a project reveals are possibilities – for our everyday knowledge, but also for other everyday dealings with beings, the beings understood and delimited/defined by the project. Thus in projecting, human being always projects itself on its possibilities, though the range of possibilities varies depending on whether human being is resolute or not. In doing this it understands itself in terms of the possibilities open to it.

Human being projects itself in its own project – one of the meanings of the claim that a project is thrown forward. Human being does not have a constant, project-independent understanding of itself: it first understands itself, or understands itself anew, after the projection. 

The mathematical projection of nature is the broadest in scope, and it is at the core of the methodologies in the sciences and the conceptual tools used in the sciences. This projection predetermines the ontology or the Being of the things encountered in experience: it predetermines what and how things are, how we view a tree, a rock, a child or a road.  It pre-determines what we, in the West, have come to call our ‘knowledge’. This projection and its manner of seeing is based on the principle of reason, nihil est sine ratione “nothing is without reason”, “nothing is without a reason/cause”, the principle of non-contradiction, and the “I-principle” of Cartesian philosophy.

The mathematical projection does not occur out of nowhere or out of nothing. Newton’s “First Law of Motion”, for instance, is a statement about the mathematical projection the visions of which first began to emerge long before his Principia Mathematica. Newton’s First Law states that “an object will remain at rest or in uniform motion in a straight line unless acted upon by an external force”. It may be seen as a statement about inertia, that objects will remain in their state of motion unless a force acts to change that motion. But, of course, there is no such object or body and no experiment could help us to bring to view such a body. The law speaks of a thing that does not exist and demands a fundamental representation of things that contradicts our ordinary common sense and our ordinary everyday experience. The mathematical projection of a thing is based on the determination of things that is not derived from our experience of things. This fundamental conception of things is not arbitrary nor self-evident. It required a “paradigm shift” in the manner of our approach to things along with a new manner of thinking.

Galileo, for instance, provides the decisive insight that all bodies fall equally fast, and that differences in the time of fall derive from the resistance of the air and not from the inner natures of the bodies themselves or because of their corresponding relation to their particular place (contrary to how the world was understood by Aristotle and the Medievals). The particular, specific qualities of the thing, so crucial to Aristotle, become a matter of indifference to Galileo.

Galileo’s insistence on the truth of his propositions saw him excommunicated from the Church and exiled from Pisa. Both Galileo and his opponents saw the same “fact”, the falling body, but they interpreted the same fact differently and made the same event visible to themselves in different ways. What the “falling body” was as a body, and what its motion was, were understood and interpreted differently. None denied the existence of the “falling body” as that which was under discussion, nor propounded some kind of “alternative fact” here.

Galileo in his Discourses stated: “I think of a body thrown on a horizontal plane and every obstacle excluded. This results in what has been given in a detailed account in another place, that the motion of the body over this plane would be uniform and perpetual if the plane were extended infinitely.” In another place he states: “I think in my mind of something movable that is left entirely to itself”. This “to think in the mind” is that giving to oneself a cognition about the determination of things, of what the things are. Plato speaks of such thinking in his dialogue Meno and we must remain mindful of the Greeks’ understanding of the mathematical as “that which can be learned, and that which can be taught”.

There is a prior grasping in the mind, a representation of what should be uniformly determinative of the bodily as such, what the thing is. All bodies are alike. No motion is special. Every place is like every other place. Every force is determinable only by the change of motion which it causes, the change of motion being the change of place. This fundamental design of nature creates the blueprint wherein nature is everywhere uniform.

In Galileo, the mathematical becomes a “projection” of the determination of the thingness of things which skips over the things in their particularity. The project or projection first opens a domain, an area of knowledge, where the things i.e. facts, show themselves. What and how things or facts are to be understood and evaluated beforehand is what the Greeks termed axiomata i.e. the anticipating determinations and assertions in the project, what we would call the “self-evident”.

Galileo’s projection entailed six conclusions about the essence of “the mathematical”. First, it was a projection which “skips over the things”; 2. It was axiomatic, which is to say it prescribes certain features by which entities/things are to be understood before they are encountered; 3. This prescription regarding the being of beings goes to the very essence and structure of beings,, what they are and how they are; 4. It established a uniform field in which all entities will be encountered; 5. The “mathematical” realm requires that entities be accessed through experimentation; 6. And finally, it establishes measurement, in particular numerical measurement, as the uniform determinant of things. It is only through and along with this transition to the “mathematical” approach to nature that the analytical geometry of Descartes and the calculus developed by Newton and Leibniz could have been possible as well as necessary.

Newton entitles the section of his work in which things are fundamentally determined as moved “The Axioms or Laws of Motion”. The project or projection is “axiomatic” and it is what determines the laws. As what we call thinking and cognition in the sciences is expressed in propositions, the cognition (the way of seeing, the beholding) in the mathematical project is of such a kind as to set things upon their foundations in advance; they are defined and delimited in advance. The axioms are fundamental propositions, “a positing that is put forward”. Because the mathematical project is axiomatic, what things are as bodies is taken in advance and the mathematical project becomes the basic blueprint (schema, framework) of the structure of every thing and its relation to every other thing in advance. What the thing will be and can be is determined in advance. It is a priori. This is the result of Kant’s great effort in his three Critiques of Pure Reason, Practical Reason and Judgement.

The framework or blueprint provides in advance what we call “areas of knowledge” and how the things within those areas are to be determined, classified and defined and, thus, knowable beforehand. The more the numerical can be applied and the things brought to light through it, the more precise and correct the definitions are considered to be. Unlike in Aristotle, nature is no longer an inner capacity of a body determining its form of motion and its place; circular motion is of no greater dignity than rectilinear motion. With Galileo and Newton, Nature now becomes the realm of uniform space-time with regard to the context or place of uniform masses in motion which are outlined in the project and within which alone bodies can be bodies as part of it and anchored or positioned within it.

Nature as understood within the axiomatically pre-determined mathematical project requires a mode of access to the objects that have been thus determined. The mode of access and the manner of questioning and our cognitive determinations of nature (what we in TOK have called our “ways of knowing”) are no longer ruled by traditional opinions and concepts. A new form of questioning and conceptual thinking is required. Bodies have no concealed qualities, powers, and capacities. Natural bodies are only what they show themselves as within this projected realm i.e. masses in motion in relation to places and time points; and once they are determined as such, they then can be measured as masses and working forces.

The mathematical project determines the mode of taking in and studying what shows itself, what we call “experience”. Because inquiry is now pre-determined by the axiomatic outline of the project, how we question and inquire is determined in advance and nature must answer one way or another. Upon the basis of the mathematical project (“what can be learned and what can be taught”), “experience” becomes the modern “experiment”. The experiment is the setting up of the controlled environment that will allow us to gain access to the “facts”, the things. The experimental urge to the “facts” is a consequence of the initial mathematical skipping over of all facts and this has many consequences for our thinking in all areas of knowledge and our day-to-day lives. When the skipping ceases, mere facts are collected and we have what we know today as “positivism” where “knowledge” becomes mere “information”.

The Mathematical Project as Numerical

Because the mathematical project has established a uniformity of all bodies according to relations of space, time, and motion, it also makes possible and requires a universal uniform measure as an essential determinant of things i.e. numerical measurement. This numerical measuring is what we know as “mathematics” in its narrower sense. The new form of modern science of Galileo and Newton, Descartes and Leibniz did not arise because mathematics became an essential determinant within it. The particular type of mathematics had to come into play as a consequence of the mathematical projection, of how the things can be known and taught. The founding of analytic geometry by Descartes, infinitesimal calculus by Newton, and differential calculus by Leibniz are not the causes of the mathematical projection that is the paradigm shift from the ancient to the modern, but its necessary consequences. As Galileo himself said: “the book of nature is written in the language of mathematics”.

What is provided here is merely an outline within which unfolds the entire manner in which we pose questions and experiments, establish laws in our politics, and disclose new areas of things in order for us to have knowledge of them. The questions regarding space and time, motion and force, body and matter remain open and we are attempting to discuss them here in TOK. Every manner of thinking is a doing, a carrying out, that is a consequence of our manner of being-in-the-world, of the fundamental position that we take towards beings so that they show themselves and, thus, their truth. It is fundamentally ethical.

The mathematical projection of the world finds its apotheosis in current studies of the philosophy of science as the “Reduction Thesis”. It is the hypothesis that modern natural science, in all of its manifestations, is ontologically dependent on mathematical physics. This connection of mathematics to physics and of physics to mathematics is a limitation which both physics and mathematics cannot overcome. Experiments in Physics must report their results in the language of mathematics if they are to provide certainty.

The “Reduction Thesis” asserts a complex correspondence between science and the world. The world, in ascending order of complexity, is composed of elementary particles (states of energy), higher, more complex, structures such as those observed by chemistry, yet more complex ones such as organisms, and, lastly, human beings and their institutions. Analogously, the sciences can be rank-ordered in corresponding fashion with mathematical physics at one end (the Group 4 subjects) and, at the other, the sciences concerned with the human: anthropology, sociology, psychology, and political science, among others (the Group 3 subjects). This viewing impacts all AOKs and is what we have been calling the “mathematical projection”.

It is not just the new method of the physical sciences which warrants the scientific character of the modem science of politics, for instance. Just as ontologically, or in actuality, the world is in the final analysis “mathematical”, so the sciences (if the “Reduction Thesis” is a guide to modern convention or normative standards) make contact with the world through mathematical physics. And, as we have stated, Jacob Klein in his book Greek Mathematics and the Origin of Algebra takes us a long way in understanding a deep-seated conceptual connection between method and ontology in modem consciousness which reveals and discloses this dual authority of modern natural science in our Cave.

Distinctions Between Ancient and Modern Mathematics

GalileoModern: Galileo’s understanding of mathematics: Whereas ancient and medieval investigations sought out “the metaphysical essence and hidden causes of the appearances that impose themselves on us in immediate reality, Galileo’s science signifies something fundamentally new in its method. It seeks to gain mastery over the diversity of appearances by means of “laws.” Both the ontological (the essence of what Nature is in its Being), and the epistemological (the knowledge of the “how” that Nature happens to be the way it is) assumptions of modern mathematics are evident in Galileo’s famous mathematization of nature (“The book of nature is written in the language of mathematics”). Galileo’s new method posits measurable and comprehensible relationships between phenomena (epistemological knowledge claims), and admits only these knowable entities and their relationships to the plane of existence (ontological claims): “tracing all appearances back to the basic mathematically definable laws of a general dynamics, or motion.” The ground of Galileo”s view of Nature are the axiomatic principles of mathematics which through the principle of reason account for the “laws” of motion. Galileo’s mathematics will have a great influence on the science of Bacon and on the political philosophy of Thomas Hobbes. Technology and the Human Sciences Pt. 1

Aristotle_Altemps_Inv8575
Aristotle

Ancient Mathematics: Aristotle’s understanding of the mathematical: mathematics is the attempt to understand sophia (wisdom) and the opposition between sophia and immediate, everyday, pre-scientific “common sense” or phronesis.  According to Aristotle, sophia is distinctive as the “most rigorous” mode of inquiry because it “touches the foundations of beings in their Being”, what we call “metaphysics”. Moreover, inquiries characteristic of sophia are determined from their outset by archai, first principles, which “require the greatest acuity to be grasped…because they are the fewest”. Only “because the archai are limited is a determination of beings in their Being possible” at all. The examples Aristotle gives of this “rigorous science” are the mathematical disciplines of arithmetic and geometry both of which are axiomatic, or that which is worthy or self-evidently true in itself.

Mathematics is characterized as “that which shows itself by being withdrawn from something and specifically from what is immediately given. The mathematika are extracted from the physika onta, from what immediately shows itself.” It is important not to read Aristotle through Cartesian or Kantian lenses: for Aristotle, this withdrawal from the immediately given is not a givenness to a subject, but a withdrawal from the natural place of the object (“place belongs to beings themselves”), or what the Greeks referred to as the topos. The mathematical, however, does not have a place, a topos; this is what distinguishes it from natural reflection about objects or what we refer to as “experience”. Whereas “the natural man sees a surface as peras, as the limit of a body,” the mathematician “considers the mathematical objects purely in themselves.” Because the mathematician is not recasting her objects as providing some different peras or some alternate motion in our experience of them, she is not in danger of distortion. This is not to be conceived of as some kind of “subjectivity”. 

But within mathematics itself, the distinction between arithmetic and geometry will prove to be of crucial importance for the later development of modern mathematics. Whereas mathematical abstraction properly leaves behind the topos of its objects, including the kinoumena – or determinate relation to motion – which is the concern of natural observers, the physicist does not recognize topos and therefore kinesis or motion as natural aspects of the object in question which must then be left behind in the artificial (though not distorting) process of abstraction; this mis-recognition in turn allows him to make “of these archai genuine beings, among which finally even kinesis itself becomes one.” (Heidegger, Plato’s Sophist 71, hereafter referred to as PS) The German philosopher, Heidegger, thus opposes the mathematician, for whom kinesis is not another archai but rather “the topos itself whereby Being and presence are determined” (PS 71) to the (Platonic) physicist, who is guilty of insufficient abstraction who does not determine an object’s being by its kinesis or motion. If we regard topos as what we understand by “space” and kinesis as what we understand as “time”, we can understand the importance of these differentiations. We also may be able to grasp what Plato meant when he said that “Time is the moving image of eternity”.

The distinction between geometry and arithmetic clarifies the opposition between the two. Monas, unit, is the solitary element of arithmetic; the most basic concern of geometry, however, is stigme, the point, which is a monas with a thesis added to it. (PS 71) This thesis makes all the difference: while both monas and stigme “are ousia, (presence)(something that is for itself” (PS 72), the thesis operative in geometry signifies that the object in question has been wholly divorced from its natural place, and has acquired “an autonomy over and against the physical body.” (PS 76) For Aristotelian metaphysics proper, place is a natural, integral part of a being: “the place is constitutive of the presence of the being” (PS 73) – rocks fall because the ground is their natural topos, fire naturally goes up, etc.The difference between the kinds of abstraction taking place in geometry and arithmetic are exemplified in the ways each relates the basic units of its operation to one another: for Aristotle, neither number nor the line is merely a construction of ones or points. The first number is in fact two, and the line is comprised of more than its points: “number and geometrical figures are in themselves in each case a manifold. The ‘fold’ is the mode of connection of the manifold.” (PS 76) What is being spoken of here is the difference between geometrical and numerical relation. What is the connection between a one and a two? What is a “one”?

While geometrical objects retain some similarity to those physical objects from which they are derived, for example the quality of continuous extension, Aristotle derives his understanding of continuity not from geometry itself but from his reflections on physics.

The relation characteristic of geometry is synekhes, the continuum: “what is posited in this thesis is nothing else than the continuum itself. This basic phenomenon is the ontological condition for the possibility of something like extension, megethos.” (PS 81) The argument against Platonic theoretical construction – where a line would simply be the collection of its points – is that such a collection may still have something infinitely large or different between the points that would disrupt their succession (the paradox of Zeno, for instance). The addition of a thesis typical of geometry ensures that, in positing the continuum, the quality of extension can be understood. In absence of a thesis, the relation characteristic of arithmetic is therefore ephekses: “for there is nothing between unity and twoness” (PS 80), i.e. the nothing between 1 and 2 is of a different ontological nature than the numbers that bound it. Because geometry must posit a supplement, a pros-thesis, in order to constitute itself, whereas arithmetic requires no such thesis, Aristotle finds number to be ontologically prior: it characterizes being “free from an orientation toward beings” (PS 83) –which is why Plato’s radical ontological reflection starts with number. But although arithmetic is dependent on sufficiently few archai, Aristotle does not admit it as the science of beings because its genuine arche, monas, is itself no longer a number i.e. “one” is not a number. (PS 83) With that Aristotle, and Heidegger, turn to sophia as the genuine candidate for the science of being.

Descartes sees extension as “basically definitive ontologically for the world,”; he predetermines what kinds of beings will be encountered in or admitted to experience. The res corporea or bodies are characterized above all by extension – size, length, thickness, etc. – in space, and this is defined as the constitutive quality that enables things to express all of their other qualities. 

Heidegger asserts that Descartes’ interpretation is not only “ontologically defective,” but that he has failed to “securely grasp” the entities he was after. Since the only ontologically admissible entities are res extensa, “the only genuine access to them lies in knowing, intellectio, (noetic) in the sense of the kind of knowledge we get in mathematics and physics….That which is accessible in an entity through mathematics, makes up its Being.” (Being and Time 128) Heidegger’s objection to this understanding is that despite his claims, Descartes’ ontology “is not primarily determined by his leaning towards mathematics…but rather by his ontological orientation in principle towards Being as constant presence-at-hand, which mathematical knowledge is exceptionally well suited to grasp.” (BT 129) In other words, the mathematics half of the “mathematical physics” to which Descartes appeals is inessential. Heidegger’s counterexamples bear this out: they dispute the physical sense of Descartes’ claims rather than their mathematical validity. Against the famous example of the melting wax, Heidegger retorts that the continuation across time of the malleable substance tells us nothing ontologically interesting about it – thus being is either inaccessible as such (which neither party is prepared to accept) or extension itself does not reveal being. Likewise, in the example of a hard substance resisting pressure, Heidegger replies that in abandoning everything but the hardness or resistance-property of the entity under consideration, Descartes also abandons the possibility of distinguishing between the two entities in contact: the mere closeness of a thing “does not mean that touching and the hardness which makes itself known in touching consist ontologically in different velocities of two corporeal Things.” (BT 130) Only if a being has the kind of being which Human Being has will it be shown hardness or resistance. The first example seeks to undermine the certain grasp of entities in the world; the second undermines the self-knowledge of the subject. The overall impact of Descartes’ orientation is that he has “made it impossible to lay bare any primordial ontological problematic of Dasein (Human Being); this has inevitably obstructed his view of the phenomenon of the world”.

What “conditions implied in Dasein’s state of Being” (BT 408) are necessary for the theoretical attitude to emerge. Theory, Heidegger notes, is not a simple withdrawal from or absence of engaged physical praxis; rather, it has a kind of praxis all its own, (BT 409) whether highly specialized as in the preparation of archaeological experiments, or simplistic measurements of a hammer which seems too heavy. In fact, the simple assertion that “the hammer is heavy” already signifies a switch to the theoretical attitude, and this is not a minor variation but a modification in which “our understanding of Being is tantamount to a change-over.” (BT 413) Not only is the hammer’s readiness-to-hand as a tool abandoned, but an essential feature of its presence-at-hand, its place, is also overlooked. “[I]ts place becomes a spatio-temporal position, a ‘world-point,’ which is in no way distinguished from any other.” (BT 413) This sounds remarkably similar to Heidegger’s description of geometric thesis from the 1924-25 lectures, in which objects are no longer considered in their natural places but as points on a grid or as surfaces in space. The crucial historical example of the emergence of this theoretical attitude is in fact the prevalence of mathematical physics since Galileo, Newton, and Descartes: What is decisive for its development does not lie in its rather high esteem for the observation of ‘facts,’ nor in its ‘application’ of mathematics in determining that character of natural processes; it lies rather in the way in which Nature herself is mathematically projected. (BT 413-4)

Only when nature has been predetermined and projected as knowable can entities/things be encountered as inert matter ready for experimentation and measurement. The crucial feature of mathematical physics is that it “discloses something that is a priori…the entities which it takes as its theme are discovered in it in the only way in which entities can be discovered – by the prior projection of their state of Being.”

Newton’s obliteration of the distinction between earthly and celestial bodies; the removal of the ancient priority of circular over linear motion; the neutralization of natural place, inherent force and capacity, and motion; the relativization of the ancient notion of violence against nature into a notion of violence as simple change of motion; the abandonment of nature as an inner expressive principle in favor of nature understood as an aggregate of motion and forces; and therefore the establishment of a radically unjustified method for questioning nature (i.e. the scientific method). It would take Kant to ground these views.

The link between metaphysics and the mathematical is shown in the rise of the mathematical and marks the emergence of a self-grounding knowledge, a self-binding form of obligation, and a new experience of freedom as such as is demonstrated in the works of Descartes. Modern mathematics as “mathematical” coincides with the abandonment of the Church and faith as the grounds of knowledge: in the essence of the mathematical “lies a specific will to a new formation and self-grounding of the form of knowledge as such.” Thus modern science, mathematics, and metaphysics “sprang from the same root of the mathematical in the wider sense.” Insofar as Descartes participated in the widespread project of extending and developing what would become the “mathematical” orientation toward the knowledge of what is, with the elevation of the proposition – the positing, the asserting characteristic of “mathematical” thinking – to the status of the first and the only given principle, reason becomes the highest ground of inquiry.  The problems of Cartesian philosophy and modern metaphysics in general are not only philosophical problems, but ontological problems as well.

The essence of technology is called Framing or “En-framing [Ge-stell],” which means “the gathering together of that setting-upon which sets upon man, i.e., challenges him forth, to reveal the real, in the mode of ordering, as standing-reserve.” (Heidegger, “The Question Concerning Technology and other Essays” QT 20) En-framing corresponds fairly precisely to the concept of “the mathematical”. Heidegger says as much: “man’s ordering attitude and behavior display themselves first in the rise of modern physics as an exact science. Modern science’s way of representing pursues and entraps nature as a calculable coherence of forces.” (QT 21) En-framing, this distinctively modern attitude that approaches beings as “calculable in advance” (QT 21) displays how we determine the being of beings in advance and what we mean when we say that we have “knowledge” of those beings. 

 

 

OT 4: Religion, Knowledge and Idolatry

Simone Weil
Simone Weil

“Things of the senses are real if they are considered as perceptible things, but unreal if considered as goods.” (Simone Weil, Gravity and Grace, p. 45)

To understand the statement above, one must see it in the light of Plato. It has been said, with some justice, that every philosopher is either a Platonist or an Aristotelian, and there is no doubt that Simone Weil is a Platonist and was hostile to Aristotle. What can it mean to say that things such as health and fitness, food and drink, property and progeny, are illusory goods?

We wish to look for counterclaims to positions that we have been given in our social and cultural contexts, in our education, for our goal is to attempt to get beyond our Caves. The quest for knowledge is a moral impulse. The essence of education is liberation. We wish to stop saying silly Russellian things like ‘God is as incredible as a celestial teapot’, or some other such comments that issue from propagandist ‘scientists’ who in their public speaking have ceased to be scientists and have become sophists at best, or politicians at their worst. Human beings will have their gods whether they recognize them or not; the goal of liberation or education is to ensure that one is not worshiping false gods.

What we call our ‘personal knowledge’ is the adopting of a position where an ineluctable element of de-cision, a cutting off of reflection and an engaging of the will, has been made: one must decide (and, indeed, has decided) what one will believe and how one will live. These decisions are grounded in the choices provided by our our culture, our ‘shared knowledge’, from our Caves. They are the products of what that kind of thinking which the Greeks understood as phronesis establishes. There is no argument, or set of arguments, that definitively establishes or grounds the desired conclusion, or justifies one’s personal way of life; and if one thinks that one has found that argument or set of arguments, then one has decided in favor of that argument or set of arguments without, perhaps, realizing that one has done so. If nothing else, one has decided to leave off investigating the matter. One has chosen, like some of the prisoners in the Cave, to return to the realm of the shadows. In most cases, it is our social and cultural contexts, our shared or historical knowledge, which grounds our de-cisions and our ceasing to inquire and to reflect.

What does it mean to say that the world of the senses is the world of shadows in the Cave and what relation does this have to knowledge and religion?

First of all, to call the things of the senses ‘shadows’ does not mean that such things when conceived as goods have no reality whatsoever; the point is rather that they lack absolute reality, according to Plato. When Macbeth, for example, sees a dagger before him, it has a ‘reality’, but its reality is as a shadow; it is the construct of a mind that sees daggers. (The dagger could also be interpreted in a positive sense in that it is the “last warning” to Macbeth before he makes his decision and acts). It is a construction of Macbeth’s de-cision: he is going to kill Duncan. Because the shadows lack an absolute reality, they cannot satisfy us ultimately (as Macbeth’s crime will not ultimately satisfy him). This delusion of desires/needs is the foundation of consumerism and of those societies based on the appetites.

The Idea of the Good is that which imparts to things their goodness. For Plato, the Ideas determine the ‘essence’ or the ‘what-ness’ of some thing. Birches, oaks, and larches all share in the idea of ‘treeness’, but the individual tree is not the idea of the tree itself. So with all the things of the world: what is good in them is given by the idea of the Good, but is not the Good itself. Their ‘goodness’ is a shadow of the Good just as a photograph of a loved one is not the loved one themselves, but a ‘shadow’ or an image of them.

Because human beings are by nature the religious animal in that they are capable of being moved by gods, we can approach the question of what is most important with regard to knowledge and religion via the notion of idolatry. The essence of idolatry lies in the absolutizing of the relative, or of the universalizing of the particular. This is, in fact, what Aristotle does in his interpretation and understanding of Plato’s idea of the Good (agathon), and his interpretation of the ideas in general. A finite good becomes an idol when it is treated as if it were an infinite good, i.e., one capable of satisfying our infinite desire. That our desire is infinite is shown by the fact that it is never satisfied by any finite object or series of finite objects. Not even an infinite series of finite objects (novelties or ‘experiences’) could satisfy it since what we really want is not an endless series of finite satisfactions but, though we don’t know it, the absolute good which is the Good itself. This is why our releasement from the chains in the Cave must be done by “force”, and involves some “violence”, and why the experience of this releasement is a painful one. Our enchainment to the desire to experience “experiences” is one of the roots of the difficulties for us in understanding ourselves. Self-knowledge for the Greeks was called “wisdom” or sophia and this involved contemplating the eternal things.

Ultimately, all desire, all need is the desire or need for the Absolute. A desire or need that understood itself, that was transparent to itself, would understand this fact about itself. But our deluded desire thinks it can find satisfaction in the finite. Therein lies the root of idolatry. We give our love to that which is not deserving of our love. In the West, this need/desire was seen in eros whom the Greeks recognized as a god i.e. infinite. Yet Eros, and our experience of Eros is, curiously, both infinite and temporal.

In the East, the Buddha understood this very well: he saw that desire is infinite in that it desires its own ultimate quenching or extinguishing, its own nibbana (http://www.buddhanet.net/nutshell10.htm), but that finite quenchings are unsatisfactory in that they only exacerbate desire by giving birth to new desires endlessly. Contrary to the Buddhist belief that all being is suffering, in the West, this has been seen in the figure of eros or need. Both Plato and the Buddha see this desire in the element or metaphor of Fire, a fire that does not extinguish itself. No desire or need is finally sated; each is reborn in a later desire. (See, for example, the discussion of King Lear on the wheel and its relation to the Pythagorean doctrine under Mathematics and Religion). This wheel of cyclical desire in Buddhism is the wheel of Samsara (http://www.buddhanet.net/e-learning/buddhism/bs-s07b.htm). The more one is driven by the appetites looking for the ultimate satisfaction, the more frustrated one becomes. The desire to consume or possess the Beautiful has been understood mythologically as the ‘fall’ of human beings; it is in our nature to consume/possess because we are the needing beings. We believe that taking something into ourselves will somehow make us whole and our desire/need will find rest.

So the Buddha understood the nature of desire or need as infinite in the needing human being. But since he had convinced himself that there is no Absolute, no Atman (see the following link for a discussion of this difficult concept in Buddhism http://www.buddhanet.net/buddhism-self.htm,) nothing possessing self-nature, (in this he can be distinguished from both Plato and Aristotle who saw in physis a self-nature or essence of what something is) he preached salvation through the extirpation of desire/need itself. Desire as such is at the root of suffering, dukkha, not desire for the wrong objects; so the way to salvation is not via redirection of desire upon the right Object, but via an uprooting of desire itself. This uprooting is a ‘violence’ that must be present in detachment from the things of the world, and this detachment, again, is a painful experience.

In Buddhist terms, we could say that idolatry is the treating of something that is anatta, devoid of self-nature, as if it were atta, possessive of self-nature. Idolatry arises when some finite foreground object is falsely ascribed the power to provide ultimate satisfaction. This is the conception of knowledge in the sciences; but in our sciences, there is no conception and no place for the world to be seen as beautiful as the world is seen as ‘object’. This de-cision of our sciences is a closing down rather than an opening up of the world of perception.

The distinction between Buddhism and the thinking that originated in the West is that for Socrates and Plato the world is conceived as good. The drawing power of eros is necessary for us to be led to the Good, and this drawing power is the beauty of the world. The world itself is a souvenir, a remembrance or reminder of the ultimate Good of which it is a testimony. Again, think of it as a photograph of someone we love. The photo is a reminder of the being who draws our love, but is not the real person themselves. This world and all its goods are but a reminder of the ultimate Good itself. Our error lies in mistaking the two as identical.

It is not without reason that the peculiar madness of the lover (Shakespeare’s Romeo and Juliet as an example) is the taking of the finite for the infinite. For Plato, there is the presence of the Good in all things that are; and this good is given to us through the perception of the Beautiful which, in its erotic power, draws us towards the Good itself. We can mistake the Beautiful for the Good itself, and this is what creates our ‘values’: we value what we consider the beautiful and what we think the beautiful itself to be, as the Good, and we consider this good of our own making since it is we who impose values on things. Beauty is in the eye of the beholder, no?

According to the Pythagoreans, whether or not the absolute Good exists is not the question: reason suggests that we should love the finite as finite, that our love should be attuned to, and commensurate with, its object or its ‘otherness’. To love the finite as infinite is to go beyond the limits (to attempt to exceed the circumference of the circle) and is, essentially, hubristic. Romeo and Juliet love not ‘wisely’ but ‘too well’. The desire/need that is infinite is such because it is for the Infinite and can only be satisfied in the Infinite. Eros is both god and mediator, both finite and infinite as Christ Himself becomes in Christianity. As a young William Blake would conclude in his text “There is No Natural Religion”: Conclusion. If it were not for the Poetic or Prophetic Character the Philosophic & Experimental would soon be at the ratio of all things, and stand still, unable to do other than repeat the same dull round over again. Application. He who sees the Infinite in all things sees God. He who sees the Ratio only sees himself only. Therefore God becomes as we are, that we may be as he is.” What Blake had come to realize here is that “ratio” understood as “reason”, or the principle of reason, the metaphysics of the experimental sciences, gives the “eternal recurrence of the Same” (as understood by the German philosopher Nietzsche). To counteract this, the Prophetic character of the imagination was, for Blake, required.

 

CT 1: Our Knowledge and Reason

Georgegrant“When one contemplates the conquest of nature by technology one must remember that that conquest had to include our own bodies.”—George Grant, “In Defence of North America” (1969)

One of the most common words used today by students in TOK classes is “mindset”, but when asked what exactly this word means the users of the word are at a loss to explain it. “Mindset” is one of the words that we use without thinking, or hearing. This writing will attempt to explore the relationship of reason to what we understand as our knowledge (which the historicists preclude is the product of a ‘mindset’) and how reason is, actually, the ground of the ‘mindsets’ that we think we have chosen in our “freedom”, or what we call our “empowerment”. When we speak of ‘mindsets’, we are speaking of human cognition, how we think, perceive and understand the world around us, the language and the concepts that we use, and how the manner (methodology) of this thinking, perceiving and understanding has come about (historical background). We shall understand “cognition” as an (intellectual) processing of (intellectual) contents, the contents which are what we have come to understand as “data”. As we shall see, we have a relation to the self only insofar as we have a relation to others.

What we call reason as a way of knowing is grounded in the principle of reason: nihil est sine ratione, “nothing is without reason” or “nothing is without a reason”. The principle of reason holds that each and every thing that is, no matter how it is, has a reason. Whatever happens to be actual has a reason for its actuality. Whatever happens to be possible has a reason for its possibility. Whatever happens to be necessary has a reason for its necessity. 

We require reasons for the assertions that we make in knowledge claims. We insist upon a foundation for every attitude when we explore our emotions and how these emotions shape and determine, attune, our human cognition, our processing of contents. It is from within this principle of reason that we determine who among us is sane and who among us is not. In our search for reasons we begin with the immediate reasons for the things in front of us and then proceed to attempt to get to the bottom of, or ground of, the more remote reasons and, finally, ask about the ultimate reason.

The principle of reason is ubiquitous in all that we do, and it is so because it is “illuminating”. Nothing happens without a reason: nothing happens without a cause. Every cause is in some way a reason. Not every reason brings about something in the way of causation, however. For example, the universally valid statement “All men are mortal” contains the reason for seeing that Socrates is mortal, but the statement does not bring about, is not the cause for, the fact that Socrates dies. As we shall see, the principle of reason is not the same as the principle of causality; it is broader and encompasses the principle of causality.

The principle of reason requires that reasons must be rendered for all that is. The rendering of reasons is carried out through logos or language as a way of knowing. Logos is any type of rendering; it is not merely that which can be expressed in words. In fact, the dominant logos of our age is mathematics and in the sciences, the providing of sufficient reasons for propositions must occur mathematically.

We need to explain three questions that arise from this: 1. how come a reason is always a rendered reason? 2. How come a reason must be rendered in the first place, that is, explicitly brought forward? 3. to whom or to what is a reason rendered?

Leibniz_Hannover
Gottfried Leibniz: The Founder of Finite Calculus

The German philosopher Gottfried Leibniz was the first to formulate the principle of reason as a statement and as a principle in the 17th century. He insisted that it was the principle. What does it mean? Why did it take so long in the history of ideas and philosophy for this statement to be uttered and why was it written in Latin by Leibniz?

Leibniz answers our first question with the observation that a reason is a rendered reason “because a truth is only the truth if a reason can be rendered for it.” For Leibniz, truth is always a correct judgement. Judgement is the connection of what is stated with that about which the statement is made. We call this the correspondence theory of truth. As the philosopher Kant stated: “Judgement is the seat of truth”. What the statement indicates is that which, as the unifying unity of subject and predicate, supports their being connected is the basis, the ground of judgement: it gives a justification for the connection. Reason renders an account of the truth of judgement. “Account” in Latin is called ratio. There is a connection between reason and language here. The ground of the truth of judgement is represented as ratio. The first principle for Leibniz is the fundamental principle of rendering reasons.

With regard to the second question “how come reasons must be brought forward whatever reasons”, Leibniz says that reason is ratio, that is, “an account”. If an account is not given, a judgement remains without justification. It lacks evidence of its correctness. The judgement itself is not truth. Judgement only becomes truth when the reason for the connection is specified and accounted for, when the ratio, that is, an account, is given. Such a giving of an account is in need of a site where the account can be delivered and rendered. This site may be as formal as an experiment or an essay or an exhibition, or it can be as informal as a statement made over coffee and donuts. The rendering of reasons is because reason is ratio, an account. If it is not given, the judgment remains without justification. It lacks the evidence, the support or the ground, for its correctness. It remains “subjective”.

In answer to the third question: to whom or to what must reasons be rendered, the answer is to human beings who determine objects as objects by way of a representation that judges. “Representation” is representare: to make something present to humans, to present something, to bring something to a presence, to bring it forward. The “account” is that which brings forward into presence.

Since Descartes, and followed by Leibniz and all modern thinking, humans are experienced as an “I” (an ego, a self) that relates to the world such that it renders this world to itself in the form of connections correctly established between its representations, its ideas and images—its judgements—and this “I” sets itself over and against this world as to an object. Judgements and statements are correct, that means true, only if the reason for the connection of the subject to its predicate is rendered, given back to the representing “I”. A reason is this sort of reason only if it is a ratio or an account that is given about something that is in front of a person as a judging “I”, and is given to this “I”. An account is an account only if it is handed over. This handing over of reasons can be experienced in the human cognition in the form of works of art either as performances, paintings or language, as discoveries in the sciences through experiment or observation, or the personal experiences that one grasps and possesses through one’s own cognition. A reason is a reason to be rendered. When the reason for the connection of representations has been directed back and expressly rendered to the “I”, what is represented first comes to a stand so that it is securely established as an object, that is, as an object for a representing subject.

But a rendered reason only effects such a bringing-to-a-stand of objects when it gives in a sufficient way an account that is adequate for the secure establishing of objects. The reason rendered must be a ratio sufficiens or a “sufficient reason”. This is the principle behind all assessments in the IB Diploma and in all human cognition in general. It is the ‘mindset’ that demands “results” which in themselves satisfy the principle of sufficient reason. Doing well or not doing well in your assessments is whether or not you have sufficiently rendered the reasons in securely establishing the object about which you are making assertions whether it be in mathematical equations or in writing the TOK essay.

Leibniz says: “Nothing exists for which the sufficient reason for its existence cannot be rendered.” The reason that demands its being rendered in every judgement about an object at the same time demands that, as a reason, it suffices—which means that it be completely satisfactory as an account. Of and for what? So that in every way and for everyone it can bring an object to stand in the entirety of its stance. The completeness of the reasons to be rendered—perfectio—is what guarantees that something is firmly established—secured in its place—as an object for human cognition. Only the completeness of the account, perfection, vouches for the fact that every cognition everywhere and at all times can include and count on the object and reckon with it. It is the principle of reason that gives security to the woman in Moscow, Idaho and the man in Moscow, Russia that their proceedings in their experiments or their mathematical propositions are correct. “Nothing is without reason”. The principle now says that every thing counts as existing when and only when it has been securely established as a calculable object for cognition. It is from this reckoning and calculability that we have “subjective” and “objective” statements regarding the things that are. “Subjective” statements are denigrated because they lack “reality”, they lack “objectivity”, they lack sufficient reasons in their rendering.

It is not accidental that what is called the “theory of aesthetics” and the term “aesthetics” itself as the determination of works of art appears coincidentally with the announcement of the principle of reason. “Aesthetics” and its various theories rule in the domain or AOK the Arts. Sufficient reasons must be rendered for our “experiences” of a work of art and in our determinations of what a work of art is. The work of art must be experienced as “object” otherwise our responses to it are “subjective”. The Greeks, for example, never had any theories of aesthetics. They did not view or experience their art in the manner we are asked to experience it.

This distinction between “subjective” and “objective” statements is what Leibniz determined as the “grandness” of the principle of reason. In the thinking of Leibniz, the Principle (here capitalized because it means the “first” or primary, the arche, or the axiom) decrees what may count as an object of cognition, or more generally, as a being/thing. What Leibniz is saying here is that human cognition is governed by the principle of reason and is under its power. Cognition becomes Rational and governed by Reason. For over 2000 years, ratio has meant not only an “account” in the sense of that which stands to account for something else, but also ratio means to “account for” in the sense of “vindicating”, of confirming something as being in the right, of correctly figuring something out and securing something through such reckoning or “accounting” so that it may be “counted on”. Reckoning is the way humans take up something, deal with it, and take it on; how, in general, human beings perceive something. Ratio is a manner of perceiving, which means, it is Reason. It is the determining power of our “mindset” which is sometimes called “world-view” in these writings.

Rational cognition follows the principle of reason. Reason first fully develops its essence (what it is) as Reason through the principle of reason. The principle of reason is the fundamental principle of Rational cognition in the sense of a reckoning (an accounting) that securely establishes something. One speaks of rational grounds, of evidence. Leibniz’s articulation of the principle of reason brings to fruition what we call “modernity”. The principle of reason comes to determine all cognition and behaviour, in other words, our “personal knowledge”. Since Leibniz’s articulation, the principle of reason has embedded itself in our human being and determines the manner in which we, as human beings, are moving forward into the future. But we are not fully aware of how the principle of reason operates in our day-to-day activities.

How do we hear this claim of the principle of reason in the determination of our “mindset”, how we understand our “experiences” in our day to day activities? The manner in which the claim of the principle of reason is most heard is in the distinction between “subjective” and “objective” mentioned earlier. Today, we measure what is “great” and what is “grand” only where the principle of reason is authoritative. We see the evidence of the principle of reason in our technology as it drives forward the bringing of its contrivances and products to an all-encompassing greatest possible perfection. Perfection consists in the completeness of the calculably secure establishing of objects, in the completeness of reckoning with them, and with the securing of the calculability of possibilities for reckoning. Our contrivances and products (computers and hand phones, for instance) are not merely instruments, equipment and tools like hammers and pens. The contrivances and products of technology rest on the understanding of the world about us that has become secure in its calculability. This calculability arranges the objects about us so that they are secure and at our disposal; the things are turned into “data”, “information”. It is this securing of the disposability of the objects about us which brings algebraic calculation to its height as the determination of what is considered knowledge in our age. This knowledge comes about through the applications of the methodologies in the various AOKs which follow the principle of reason.

The striving for perfection in our technology is an echo of the demand for perfectio which means here the completeness of a foundation. It asks and answers the two fundamental questions: “why” and “because”. The “how” questions are secondary responses to the fundamental questions. The principle of reason is a striving which demands the rendering of sufficient reasons for all that is. Perfection is based on the thoroughgoing calculability of objects. The calculability of objects presupposes the validity of the principle of reason. The authority of the principle of reason determines the essence of the modern, technological age and it empowers the modern age.

What role does human freedom play in this ceaseless technological striving for perfection? In our personal knowledge and how we experience our lives, we must come to terms with the distinction between calculative thinking and reflective thinking. We may begin our reflection on why this age is called the “Information Age” and “The Atomic Age” in order to illuminate the differences in the forms and ways of being-in-the-world in which human beings are captured and enslaved by the principle of reason. We shall attempt to determine the distinction between the calculative thinking which the principle of reason prescribes and reflective thinking.

The Principle of Reason and Information

How does the principle of reason operate within the “information age”? “Information” is sometimes called knowledge by students in their essays and exhibitions. Information is the bringing of what is encountered to a stand in the “form” in which it can “in-form” (in + form + ation). It is the principle of reason that is the suffix –ation, from the Greek aitia, or “that which is responsible for” the “form” so that it may “inform”. Data are those things that must be placed within a form so that they can become things that can “inform” or be rendered. The rendering of data as information requires the principle of sufficient reason to organize and classify the data so that it can “be” as object and as something calculable.

To “inform” is to render an account, to pass on what has been brought to a stand in human cognition as representational thinking. We require that this rendering or “giving an account” be as quick, comprehensive and efficient in bringing about results in the most efficient manner possible in order to assist us in securing our necessities, requirements, and satisfactions. We speak of this rendering of accounts as “empowerment”.  So it is that in our age the representation of language as an instrument of information has come to dominance and shows itself in our attempts to create machines with artificial intelligence and ever bigger, greater, more efficient computing frameworks with capacities for ever larger calculations. These attempts are based on our understanding of “intelligence” as information and contribute to the organizing within the framework that the principle of reason as the technological has established for itself.

In order to be passed on, what is encountered must be “trans-formed” into data so that it can be manipulated and controlled. As said above, the suffix “a-tion” comes from the Greek aition which was interpreted and translated as “cause” by the Romans. In this trans-formation of what is encountered into what is called in-formation, into data, what is encountered ceases to be an “object” for us and only retains its validity, its reality, as long as it retains its sense as data. As data, it ceases to be an independently standing object. The principle of reason requires that all that is encountered is understood as data. Until it is so understood, the thing encountered does not have a “reality” for us; it is not a “fact”.

Why this need for everything we encounter to be rendered as “information”? Because in its rendering as “information”, the principle of sufficient reason can hold sway. What is the consequence of seeing and hearing language and speaking as information? Because of this hearing and speaking, the possibility of a thoughtful conversation with a tradition that is considered to be our shared knowledge, a shared knowledge that could invigorate and nurture us, is lacking. Because language has been consigned to information, reflective thinking is pushed aside and is considered as something useless and superfluous. It is to the IB’s credit that it wishes to have TOK at the core of the Diploma program so that whatever embers might lie within our thinking that are the remnants of reflective thinking may still be able to catch fire and flame out as something other than calculative thinking.

What is the relation of the principle of reason to our personal knowledge and what we have come to call empowerment? It is the power of the principle of reason that “empowers” what we think personal knowledge is. The principle of reason governs all modern thought and action in the sense that it makes all modern thought and its consequences possible. It is the principle of reason that “empowers” the modern age to be what it is. At the same time, the principle of reason “overpowers” all thought and action making it difficult, if not impossible, to think and act except in the manner prescribed by the principle of reason. Our enchainment to the principle of reason requires that we “hear” what is being said in it and, at the same time, how the “mighty” principle” (in Leibniz’ word) has come to determine what is understood here as “technology” and its “empowerment” of human beings in the modern age. This attentive “hearing” requires that we begin to listen to what we hear which we have previously been inattentive to in the principle of reason; and this hearing and seeing requires a responsiveness, responsibility on our part to what is and what we are, and what we conceive ourselves to be, as human beings.

 

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OT 1: Knowledge and Technology: Ethics

Digital data hex code symbolsInquiry question: How does technology, when viewed as merely instruments and tools that are used in assisting human beings to achieve their ends or in other human activities,, obfuscate the ethical issues that arise from within it?

As anyone who is involved in the top level of the informational technological sciences can tell us, it is impossible to work in the field without engaging in social engineering or cybernetics. One of the aspects of cybernetics is the creation of “communities” within which human beings feel the “freedom” to create themselves, to be “empowered”, but with the corollary consequence of their dehumanization through lack of empathy and humanity, or their sense of “otherness” and “owingness”.

The instrumental view of technology sees technology as a tool like any other and that it can be used for good or ill.  As we have gone along our path to thinking of technology, we have seen that technology is more of a “fate”; it is a mode or way of being-in-the-world that has arisen from particular historical conditions (Western European sciences) and social circumstances (historical contexts). The view of information technology examined here arrives from the view of reason and nature that came from these mastering sciences. Such a view cuts human beings off from any notion of a transcendent good (the Sun in Plato’s allegory of the Cave) and from any notion of a transcendent justice (a standard of justice other than that of our own making). The ethical implications should be made clear from this understanding of what allowed the technological to become possible. The essence of the technological is not left behind when its results are brought into being. It is these technological products and activities (techniques) that we view as what technology itself is, but this view is insufficient.

The situation in which we find ourselves currently seems obvious: we are faced with calamities concerning climate, the environment, population, resources, and pollution if we continue to pursue the policies that we have pursued over the last few centuries. The attempts to deal with these interlocking emergencies will require a vast array of skills and knowledge; and that is what most of you are being educated towards. Technological mastery will need to be used to solve the problems that technology has created. The focus of this mastery will be in the human sciences with efforts to change human behaviour. As the German philosopher Martin Heidegger has pointed out, the governing and determining science of the future is inevitably going to be cybernetics.

The realization of the cybernetic future will find its place most securely in the medical profession, particularly the biomedical field. We here in Singapore see a realization of this through the Singapore government’s focus on bio-medicinal research as one of its core industries of the future. What has been called “late stage capitalism” increasingly attempts to establish itself as “the mental health state” with the necessary array of dependent arts and sciences. The practical wisdom of politics was called by Plato “the royal techne”—that art which is higher than all particular arts because it is called to put the other arts in a proper order of least important to most important. It established a hierarchy. We have noticed in the TOK that the hierarchy established is “our self” as knower in the centre along with “ourselves” or “a community of knowers”. Our living in communities is “politics”, both in the ancient and modern sense. So what had been called “politics” by the ancients has been replaced by “social psychology” for the moderns. This “social psychology” is “cybernetics”—the mastery and manipulation of humans by other humans through various machinations. From this perspective, we get our terms “human resources” or “human capital” which appear benign in themselves but really are not.

In most of the TOK discussions that occur (and will occur), the difficult choices which will be necessary in the future are discussed within the assumptions of the ‘values’ and ‘ideals’ which shall direct our creating of history.  If we are to deal with the future “humanely” (that is, in a “human” fashion), our acts of ‘free’ mastery in creating history must be decided within the light of certain ‘ideals’ so that we can preserve certain human ‘values’ and see that ‘quality of life’ and quantity (economic prosperity) is safeguarded and extended. Clearly, the problem of dealing with these future crises involves great possibilities of tyranny, and we must be careful that in meeting these decisions we maintain the ‘values’ of free government.

In our TOK discussions, the way we put the questions/themes that relate to the tasks of the future, the future of our students (your futures) as the leaders of that future, involves the use of concepts such as ‘values’, ‘ideals’, ‘persons’ or ‘our creating of history’. The use of these concepts obscures the fact that these very concepts have come forth from within the ‘technological world-view’ to give us an image of ourselves from within that within. These terms are used “unthinkingly” from within this “world-view” and do not allow us to gain the openness necessary to be able to discuss the questions in any meaningful way.

The task in TOK is thus a negative one: to allow the concepts to come to light in their essence so that we may be free for something positive beyond them.

To do this we will look at “information technology as a fate” or a destining of human beings. This discussion arises from our radio show from last year, in which our two guests, experts in information technologies, both held the instrumental view of technology: that the information technology does not impose on us the ways that they should be used. They believed that human beings have the command and choice to determine whether information technologies will be used for good or ill.

The use of the word “should” implies a choice. The statements made by these men came from their intimate knowledge of information technology. But such a statement transcends that intimacy in the sense that the statement is more than a description of any given information technology or what is technically common to them as machines; the question goes beyond hardware and software. Because our guests wished to make statements about the possible good or evil purposes for which information technologies can be used, they expressed what information technologies are in a way which is more than a technical description. According to our guests, they are instruments made by human skill for the purpose of achieving certain human goals. They are “neutral” instruments in the sense that the morality or ethics of the goals for which they are or can be used is determined outside of them.

In expressing the instrumental view of technology, we can see that information technologies are obviously instruments because their capacities have been built into them by human beings; and it is human beings who must set up the operating of those capacities for the purposes that they have determined. All instruments can potentially be used for wicked purposes and the more complex the instrument, the more complex the possible evils. But if we apprehend information technologies for what they are, as neutral instruments, (according to these gentlemen) we are better able to determine rationally their potential dangers. That is clearly the first step in coping with these dangers. We can see that these dangers lie in the potential decisions human beings make about how to use information technologies, and not to the inherent capacities of the machines themselves.

This view is the instrumental view of most of us regarding technology and it is so strongly given to us that it seems common sense itself. It is the box. We are given an historical situation which includes certain objective technological facts. It is up to us as human beings in our freedom to meet that situation and to shape it with our ‘values’ and ‘ideals’, to put our IB Learner Profile into action and to act ethically .

Despite the decency and common sense of the statement “Information technologies do not impose on us the ways they should be used”, when we try to think about what is being said in the statement, in our thinking it becomes clear that information technologies are not being allowed to appear before us for what they are. They remain in the “shadows” for us.

The “not” or negation in the statement “information technology does not impose” concerns information technology’s capacities or capabilities, not its existence. Yet, clearly, information technologies are more than their capacities or capabilities. They are put together from a variety of materials, beautifully fashioned by a vast apparatus of fashioners. Their existence has required generations of sustained efforts by chemists, metallurgists, and workers in mines and factories. They require a highly developed electronics industry and the physics that lies behind that industry in the history of science and technique and their reciprocal relations. They have required that human beings wanted to understand nature, and thought the best way to do so was by putting it to the question as object so that it could reveal itself. They have required the discovery of modern algebra and the development of complex institutions for developing and applying that algebra. Nor should this be seen as a one-sided relationship in which the institutions necessary to the development of the machines were left unchanged by the discovery of algebra (here I am speaking of the universities and the more recent colleges of applied arts and technology).

To understand our educational system is to know that the desire for these machines shapes our institutions at their heart in our curriculum, in what the young (you) are encouraged to know and do (any view of the universal student choices in Group subjects in the IB Diploma indicates this). The information technology’s existence has required that the clever of our society be trained within the massive assumptions about knowing and being and making which have made algebra actual. Learning within such assumptions is not directed towards a “leading out” (educare + ation = that which is responsible for the “leading out” i.e. “education”) but towards an “organizing within”. This means and entails that those who rule any modern society will take the purposes of ruling increasingly to be congruent with this account of knowing. The requirements for the existence of information technologies is but part of the total historical situation (the word ‘fate’ or ‘destiny’ might be too ambiguous to be used here) which is given to us as modern human beings. The conditions of that historical situation are never to be conceived as static determinants (as something which cannot be changed), but as a dynamic interrelation of tightening determinations (the “box” gets smaller in terms of choices).

Information technologies are, obviously, within modern common sense, instruments, and instruments are always things which are made to be at human disposal. However, when the capacities or capabilities of these machines are abstracted from their historical existence, and when their capacities or capabilities are morally neutralized in the negative ‘do not impose’, we shut ourselves off from what ‘instrumentality’ has come to mean and how it has changed its meaning in the modern world.

Information technologies are one kind of technology. But “technology” is a very recently arrived word. Two Greek words, techne and logos are brought together in a combination that would have been unthinkable until recently. The word ‘technology’ is not to be found anywhere in the Greek lexicon. The new word ‘technology’ is able to stand and perdure because it brings forth to us the new situation: a quite novel dependence of science upon art and a quite novel dependence of art upon science—in fact, a quite novel reciprocal relation between ‘knowing’ and ‘making’. Look at the Mac Book Pros, hand phones and tablets in front of us and one can see the flowering of this reciprocal relationship. One can see here how aesthetics meets physics, how the “knowing” and the “making” come together.

This novel relationship of making and knowing stands at the heart of the modern era (by the “modern era” I mean since Newton’s science). The simple characterization of  information technologies as neutral instruments makes it sound as if instruments are now what instruments have always been and so hides from us what is completely novel, unique and new about modern instrumentality. This gulf in our understanding was made explicit by our guests’ use of the discovery of fire as an example of technology’s neutrality. In comparing the discovery of fire to the making of information technologies, our guests hid from us (not in any malevolent way) what we have to understand if we are to understand technology, as if the instrumentality of modern technologies could be morally neutral.  This account of information technology as neutral rises up in the statement, in opposition to that neutrality, an account of human freedom which is just as novel as our new instruments.

Human freedom is conceived in the strong sense of human beings as autonomous—the makers of our own laws and our own selves. This is also a quite new conception. It is first thought systematically in the writings of the German philosopher Immanuel Kant. It is also a conception without which the coming-to-be of our modern civilization would not and could not have been. But it is a conception the truth of which needs to be thought because it was not considered true by wise men of many civilizations before our own. The statement “information technology does not impose” holds a view of the world with neutral instruments on one side and human autonomy on the other. But it is just this view that needs to be thought if we are concerned with understanding the essence of technology and of understanding the essence of modern instrumentality, and if we are to see these as being a ‘fate’ or ‘destiny’.

How widely are we being asked to take the word ‘ways’ in the assertion that information technology does not impose the ways? Even if the purposes for which the tools or gadgets of information technology’s capabilities should be used are determined outside of itself, are not its inherent capabilities determinative of the ‘ways’ it can be used? We use information technologies to record students’ skills and ‘behaviours’. We use the data to control or assist teacher training in our PYP, MYP, and DP programs. The facts of our day-to-day instruction are abstracted so that they may be classified. Where classification rules, identities and differences can only appear in its terms (results as data). Classification is used by us both in our desire to know but also because of the convenience of organization. As our institutions of education grow larger, this ‘convenience of organization’ will come to dominate and will eliminate the heterogeneity of what those institutions were in the past: uni-versities become multi-versities. The point being made here is simply that the statement about information technologies tends to hide the fact that their very capabilities entail that the ways they can be used are never neutral. They can only be used in homogenizing ways. And the question about the goodness of homogenization or decentralization is excluded from thinking about the essence of technology.

A clearer example might be in using the automobile: “the automobile does not impose the ways it will be used”. All of us have experienced the inconvenience in this part of the world of societies in which the automobile has not, as of yet, come to dominate. Societies where automobiles dominate tend to be much the same as each other and we find these societies much more efficient and convenient for ourselves.  Yet, we cannot represent the automobile to ourselves as a ‘neutral instrument’. Here in Singapore, 20% of land use is given over to the infrastructure required for the automobile. But also, if we represent the automobile as a neutral instrument, we have abstracted the productive functions of Honda, Toyota and General Motors or Standard Oil and the other major oil conglomerates from their political and social functions, just as their public relations people would want. Moreover, we would have abstracted the automobile from the relations between such corporations and the public and private corporations of other countries. After all, to any sane person, the Iraq War was over oil; and the subsequent loss of lives, according to the British Medical Journal, The Lancet, was one and a half million Iraqi citizens, a number significantly higher than that given by the members of “the coalition of the willing”. When one thinks of ‘values’ and ‘ideals’ from within technology, one cannot ignore the continued homogenization of the central corporations in our everyday lives and the tremendous growth in their power over our lives, including the ability of driving us into wars.

Aristotle has pointed out that human beings are the ‘religious animal’, and the religion for most human beings who have lost any kind of transcendental faith in a god is the ‘belief in progress’. This belief can be described as the good progress of the race in the direction of the universal society of free and equal human beings, that is, towards the universal and homogeneous state. It is captured in the phrase “the ascent of man”. The followers of this religion of progress assert that the technology, which comes out of the account of reason given in the modern European sciences, is the necessary and good means to that end. That account of reason assumes that there is something which we call ‘history’ over against nature, and that it is in that ‘history’ that human beings have acquired their rationality. In the thought of the French philosopher Rousseau about the origins of human beings, the concept of reason as historical makes its extraordinary public arrival. Darwin’s Origin of Species is not possible without, first, the thought of Rousseau. Technology and The Human Sciences Pt 2: Rousseau, Kant, Hegel, Marx

The German philosopher Heidegger has said that capitalism and communism are simply predicates of the subject technology: the Presidents of the USA and China float down the same river (technology) in different boats (political ends). To put this in the context of our discussion, the same apprehension of what it is to be ‘reasonable’ leads human beings to build information technologies and to conceive of the universal, homogeneous society as the highest political goal. The ‘ways’ such machines can be used must be at one with certain conceptions of political purposes, because the same kind of ‘reasoning’ made the machines and formulated the purposes or the ends. To put the matter extremely simply: the modern ‘physical’ sciences and the modern ‘human sciences’ have developed in mutual interpenetration, and we can only begin to understand that mutual interpenetration in terms of some common source from which both sciences found their grounding. This common source is technology understood as a way of knowing the world and as a way of being-in-the-world.

To think ‘reasonably’ about the modern account of reason is of such difficulty because that account has structured our very thinking over the last centuries. Because we are trying to understand reason in the very form of how we understand reason is what makes it so difficult; that is, we are trying to use reason to grasp the essence of reason. The very idea that ‘reason’ is that reason which allows us to conquer objective human and non-human nature controls our thinking about everything; in other writings we have called it the principle of reason. This principle of reason is the box that we are required to try to somehow to think out of.

The root of modern history lies in our experience of ‘reason’ or the principle of reason and the interpenetration of the human and non-human sciences that grew from that root, or what has come to be called “the reduction thesis”. It is an occurrence that has not yet been understood, and it is an event that must come to be thought here in TOK. The statement ‘information technologies do not impose on us the ways they should be used’ hides that interpenetration. To repeat: the instrumental understanding of technology simply presents us with neutral instruments that we in our freedom can shape to our ‘values’ and ‘ideals’. But the very concepts of ‘values’ and ‘ideals’ come from the same form of reasoning that built the information technologies. ‘Information technology’ and ‘values’ both come from that stance which summoned the world before it to show its reasons and bestowed ‘values’ on that world. Those ‘values’ are supposed to be the creations of human beings and have, linguistically, taken the place of the traditional concept of ‘good’ which was not created but recognized. Information technologies do not present us with neutral means for building any kind of society. All their alternative ‘ways’ lead towards the universal, homogeneous state. Our use of them is exercised within that mysterious modern participation in what we call ‘reason’, and it is this participation that is most difficult to think in its origins.

TOK Question: Should we hold people responsible for the applications of technologies they develop/create?

The strongest ambiguity in the statement ‘ information technologies do not impose on us the ways they should be used’ is presented to us as if human beings ‘should’ use these machines for some purposes and not for others. But what does the word ‘should’ mean in advanced technological societies? Is not the essence of our difficulty contained in that this ‘shouldness’, as it was once understood and affirmed, can no longer hold us in its claiming?

‘Should’ was originally the past tense of ‘shall’. It is still sometimes used in a conditional sense to express greater uncertainty about the future than the word ‘shall’: (‘I shall get a raise this year’ is more certain than ‘I should get a raise this year’.) ‘Should’ has gradually taken over the sense of ‘owing’ from ‘shall’. (In its origins ‘owing’ was given in the word ‘shall’ when used as a transitive verb. See the concepts of ‘indebtedness’ and ‘responsibility’ in the discussion of technology in the unit blog on technology as a way of knowing.) In the sentence ‘information technology does not impose on us the ways it should be used’, we are speaking about human actions that express ‘owing’. If we change the statement to a positive form “information technology does impose on us the ways it should be used’, the debt would probably be understood as from human beings  to the machine. We can say of a good car that we ‘owe’ it to the car to lubricate it properly and maintain it properly if we want the car to do what it is fitted for—which is, in the traditional usage, its good—then we must look after it. But the ‘should’ in the statement about information technology is clearly not being used about what is owed from human beings to the machine. What is, then, the nature of the debt spoken? To what or to whom do we human beings owe it? Is the debt conditional? For examples, if human beings ‘should’ use information technologies only in ways that are compatible with constitutional government and never as instruments of tyranny, to what or to whom is this required support of constitutional government owed? To ourselves? To other human beings? To evolution? To nature? To history? To reasonableness? To God?

To characterize the great change that has taken place among those who consider themselves to be ‘modern people’, ‘goodness’ is apprehended in a much different way from previous societies. ‘Goodness’ is now apprehended in a way which excludes from it all sense of ‘owingness’ or ‘indebtedness’. What was the traditional Western view of ‘goodness’ is that which meets us with an excluding claim and persuades us that in obedience to that claim we will find what we are fitted for as human beings i.e. justice. Macbeth, for example, knows that he should not kill Duncan. The modern view of ‘goodness’ is that which is advantageous to our creating ‘richness of life’ or ‘quality of life’ i.e. it is exactly the choice Macbeth does make in choosing to kill Duncan because by doing so he believes he will increase his richness and quality of life.

What is true of the modern conception of goodness (which appears in advanced technological societies and distinguishes them from the older conceptions of goodness and the societies realized within those conceptions) is that the modern conception of goodness does not include the assertion of an ‘owed’ claim which is intrinsic to our desiring. ‘Owing’ is always provisory on what we desire to create. Our discussion of Aristotle’s conception of causality in our attempts to understand the essence of technology are relevant here. OT 1: Knowledge and Technology

Obviously, we come upon the claims of others and our creating may be limited particularly by the state because of what is currently permitted to be done to others. However, such claims whether within states or internationally, are seen as contractual, that is, provisional. This exclusion of non-provisional owing from our interpretation of desire means that what is summoned up by the word ‘should’ is no longer what was summoned up among our ancestors.  It always includes an ‘if’. The arrival in the world of this changed interpretation of goodness is interrelated to the arrival of technological civilization. The liberation of human desiring from any supposed excluding claim, so that it is believed that we freely create ‘values’, is a face of the same liberation in which human beings overcame chance by technology—the liberty to make happen what we want to make happen; to change the world through mastery.

The statement ‘information technology does not impose on us the ways it should be used’ asserts the very essence of the modern view (the human ability to freely determine what happens) and then puts that freedom in the service of the very ‘should’ that the same modern apprehension has denied. This is only possible with the conception of technology as instrument. The resolute mastery to which we are summoned in ‘does not impose’ is the very source of difficulty in apprehending goodness as ‘should’. Therefore, the ‘should’ has only a masquerading resonance when it is asked to provide moral content to the actions we are summoned to concerning information technologies. It is a word carried over from the past to be used in a present that is only ours because the assumptions of that past were criticized out of existence. The statement therefore cushions us from the full impact of the uniqueness it asks us to consider. It makes us forgetful against wondering and questioning about the disappearance of ‘should’ in its ancient resonance, and what this disappearance might portend for the future.

The commonality of statements in our modern world and in our education such as ‘information technology does not impose on us the ways it should be used’ are needed to buttress our morality in our daily decisions. The more it becomes possible to conceive that we might not be able to control the immensity of the technological apparatus and the constant emergencies it presents us with, the more intense become the calls for moral ‘values’ and ‘ideals’ as is demonstrated in the Guide for May 2022. Technological society is presented to us as a neutral means, something outside ourselves, and human beings are presented as in touch with some constant or permanence, from out of which they are called upon to deal with the new external crises. But obviously, all that is given us in the technological sciences denies that constancy or permanence, that standard, that eternality. What happens is that constancy is appealed to in practical life and denied in intellectual life. The language of the ‘eternal’ or ‘standards’ that we do not measure but by which we are measured is removed from all serious public realms. The residual and unresonant constant appealed to in the statement about information technology is ‘should’, but the intellectual life that allowed the coming into being of that information technology has also made that ‘should’ unthinkable.

When we speak of ‘values’ and ‘ideals’ in education as a way of approaching technological situations, we must realize that ‘values’, ‘ideals’, ‘persons’, and ‘the creating of history’ are at the very heart of what technological civilization is and are a language that has developed from out of this technological civilization.

Ontology refers to our way of being in the world. Every scientific discovery or application emerges from an ontology which so engrosses us that it can be called our Western destiny. Technology is not something over against ontology; it is the ontology and metaphysic of the age. It is for us an almost inescapable destiny. The question is: what is the ontology which is declared in technology since technological civilization enfolds us as our destiny?

Coming to meet us out of the very substance of our past, that destiny has now become not only our own but that of the species as a whole. Moreover, this destiny is not alone concerned with such obvious problems that we can blow ourselves up or can cure diabetes or have widespread freedom from labour or watch our distant wars on television or other media devices. It is a destiny that presents us with what we think of the whole, with what we think is good, with what we think the good is, with how we conceive insanity and madness, beauty and ugliness. It is a destiny which enfolds us in our most immediate experiences: what we perceive when we encounter a bird or a tree, a child, or a road. This destiny is not one in which we can pick and choose: it is a package deal. As the Greeks said, “the future comes to meet us from behind”.

[1] Martin Heidegger in 1935 defined the political movement of National Socialism in Nazi Germany as “the meeting of modern man with a global technology”. Today, we define this coming together of man and technology as ‘globalization’. Having an opportunity to change this definition of National Socialism in 1953 with the publication of An Introduction to Metaphysics, Heidegger chose not to do so.

AOK: Mathematics

History of Mathematics: Its relation to CT 1


“The book of nature is written in the language of mathematics”. –Galileo

To be is to be the value of a bound variable.” —Willard Van Orman Quine

However, I maintain that in any particular doctrine of nature only so much genuine science can be found as there is mathematics to be found in it”. — Immanuel Kant, Preface to “Metaphysical Beginning Principles of Natural Science”

Questions: Is absolute certainty attainable in mathematics? Is there a distinction between truth and certainty in mathematics? Should mathematics be defined as a language? What does it mean to say that mathematics is an axiomatic system? How is an axiomatic system of knowledge different from, or similar to, other systems of knowledge?

 

Science as “the theory of the real”, the “seeing of the real”, is the will of this science to ground itself in the axiomatic knowledge of absolutely certain propositions; it is Descartes’ cogito ergo sum, “I think, therefore I am” . An axiom is a statement that is taken to be true, and serves as a premise or starting point for further reasoning and arguments. The word comes from the Greek axíōma: ‘that which is thought worthy or fit in itself’ or ‘that which commends itself as evident’. This “fittedness” and “self-evidentness” relates to the correspondence theory of truth, but it has its roots in the more primal Greek understanding of truth as aletheia, that which is “unconcealed” or “that which is revealed”. The axiomatic ground-plan or blueprint for all things allows the things to become accessible, to be able to be known, by establishing a relation between ourselves to them. But today, the relation of the knower to what is known is only of the kind of calculable thinking that conforms to this plan which is established beforehand and projected onto the things that are. Initially, this relation to things was called logos by the Greeks. The word initially meant “speech” or “communication”, but today it means “reason”, “logic” and is sometimes referred to as “theorems”.

If we use an analogy, we see the things as “data” or “variables” that are much like the pixels on a computer screen that require a “system”, a blueprint, a framework so that the pixels/data/variables can be defined and bound, and in this defining and binding the things are made accessible so that they can conform to something that can be known, some thing that we bring with us beforehand which will allow them to be “seen” i.e. the body of the bodily, the plant-like of a plant, the animal-like of the animal, the thingness of a thing, the utility of a tool, and so on. The blueprint or mathematical projection allows the “data” to become “objective”; the data are not objective until they are placed within the system or framework. If they cannot conform to the blueprint, the framework, the system, to this manner of knowing, then we consider them “subjective” and they somehow have less “reality”; they are not a “fact” because they are less “calculable”. One sees the effect of this framing in our language and the texting that is now a popular mode of discourse for us. Grave consequences are the result of the thinking that is bound by, and bound to, the “mathematical projection”.

The mathematical and numbers are obviously connected, but what is it that makes “numbers” primarily mathematical? The mathematics and its use of number and symbol that we study in Group 5 is a response to but does not ground our will to axiomatic knowledge i.e. the knowledge that comes from the axioms and the first principles that follow from those axioms. Modern mathematics, modern natural science and modern metaphysics all sprang from the same root that is the mathematical projection in the widest sense. It is within the “mathematical projection” that we receive our answers to the questions of “what is knowing?” and “what can be known?” i.e. to those chief concerns of our “Core Theme”.

The change from ancient and medieval science to modern science required not only a change in our conceptions of what things are but in the “mathematics” necessary to realize this change, our “grasping” and “holding”, our “binding” of what the things are, what we ourselves bring to the things. The change is one from “bodies” to “mass”, “places” to “position”, “motion” to “inertia”, “tendencies” to “force”. “Things” become aggregates of calculable mass located on the grid of space-time, at the necessity of forces which are partly discernible and with various predictable jumps across the grid that we recognize as outcomes, values or results. When new discoveries in any area of knowledge require a change in design (what is sometimes called a “paradigm shift”, but are not, truly, paradigm shifts), the grid itself remains metaphysically imposed on the things. This grid, this mathematical projection, is at the mysterious heart of what is understood as technology in these writings.

Modern Natural Science (physics, chemistry, biology) is dependent on mathematical physics. Modern Natural Science views the world through the lens of what is known as the “Reduction Thesis”: that there is a correspondence between science and the world, and that this correspondence can be demonstrated within the correspondence theory of truth using the principle of reason, the principle of non-contradiction, the principle of causality, and the principle of sufficient reason. Science is the theory of the real. The world, in ascending order of complexity, is composed of elementary particles (states of energy), higher, more complex, structures such as those observed by chemistry, yet more complex ones such as organisms that are observed in biology, and, lastly, human beings and their institutions (the Human Sciences). In a similar fashion, the sciences can be rank-ordered in a corresponding way with mathematical physics at one end and, at the other, the sciences concerned with the human: sociology, psychology, political science, among others which require more than simple mathematical results.

The status of mathematical physics (where algebraic calculation becomes authoritative for what is called knowledge) turns on its ability to give us an account of the essential character of the world (essence = its whatness), rather than merely describing some of its accidents (an “accident” is a “non-essential” category for what a thing is. You have brown eyes and I have blue eyes but these are “accidents” and have no impact on our both being, essentially, human beings). Can mathematical physics make such a claim i.e. does mathematical physics describe or give an account of what and how the world really is? its essence?

 Ancient and Modern Representation of Number:

“Representation”, through the correspondence theory of truth, includes the conceptual tools which inform a world-view, or, to mix ancient and modern analogies, “representation” refers to the horizons, the limits defining this or that Cave, city, nomos (convention), civilization, or age. These definitions or horizons are the ‘paradigms’, ‘the stamp’ of what is considered to be knowledge in those Caves and determines what will be education in them. In the narrower sense, representation refers to the operations of the mind as it deals with concepts as well as its reflections on those operations, such as what we are trying to do here in TOK. We will examine the narrower sense here. We will note that the notion of a “concept” has been completely taken up in modern representation through imagination and reason, and these bring about the “knowing” and “making” that is the essence of technology. We shall try to do this with a reflection on the nature of number.

The Greek concept of number has a meaning which, when considered by First Philosophy (metaphysics), yields an ontology (the knowledge of ‘being-in-the-world’ and the beings in it) of one sort. The modern concept of number, on the other hand, while remaining initially faithful to this Greek meaning, yields an ontology or a way of being-in-the-world of a very different sort.

For the Greeks (and the tradition subsequent to them) number, the Greek arithmos, refers, always, to a “definite number of definite things”. Five or cinq or penta can refer to either five apples or five people or five pixels, but it must refer to a definite number of definite things. Alexander, one of the Aristotelian commentators, said: “Every number is of some thing”; the Pythagoreans said “The things are numbers”.  As for counting per se, it refers to things or objects of a different sort, namely monads or units, that is, to objects whose sole feature is unity, being a “one”. For example, it would be as unthinkable for an ancient mathematician such as Diophantus to assume that an “irrational ratio” such as pi, which is not divisible by one, is a number as it is for us moderns to divide a number by zero. (The neologism, “irrational ratio”, only means a ratio which yields, in our terminology, an irrational number.)

Similar considerations hold for geometry. A triangle drawn in sand or on a whiteboard, which is an “image” of the object of the geometer’s representation, refers to an individual object, for example, to a triangle per se, if the representation concerns the features of triangles in general. For the Greeks, the objects of counting or of geometry are, if considered by the arithmetical or geometrical arts, in principle, incorporeal, without body. Hence a question arises as to their mode of existence.

Plato’s and Aristotle’s answers (whatever the differences between them, they are agreed on this) are that to account for what it means to say that there are pure monads or pure triangles must begin from the common ground which has been condescendingly called “naive realism” by the moderns. For Plato, pure monads point to the existence of the Ideas, mind-independent objects of cognition, universals; for Aristotle, monads are to be accounted for on the basis of his answer to the question “What exists?”, namely mind-independent particulars, like Socrates, and their predicates, that is, by reference to substances (subjectum, objects) and their accidents. An accident, in philosophy, is an attribute that may or may not belong to a subject, without affecting its essence. Aristotle made a distinction between the essential and accidental properties of a thing.

A few words on “intentionality” are needed here and to distinguish between first-order intentionality and second-order intentionality. We say that computers can be said to know things because their memories contain information; however, they do not know that they know these things in that we have no evidence that they can reflect on the state of their knowledge. Those computers which are able to reproduce haikus will not do so unless prompted, and so we can really question whether or not they have “knowledge” of what it is that we think they are capable of doing i.e. constructing haikus. They do not have “intelligence”, per se.

Much of human behaviour can be understood in a similar manner: we carry out actions without really knowing what the actions are or what the actions intend. Intentionality is the term that is used to refer to the state of having a state of mind (knowing, believing, thinking, wanting, intending, etc) and these states may only be found in animate things. In these writings these states are referred to as Being or ontology. Awareness of the thought of Being is the purpose of this TOK course and this may be called a “second-order” intention. So first-order intentionality refers to the mind directed towards those beings or things which are nearby, ready-to-hand. They are the concepts that we use to understand the non-mental or material things. Second-order intentions deal with abstract, mental constructs. Much discussion of this is to be found in Medieval philosophy in their attempts to understand Aristotle.

“First intention” is a designation for predications such as: ‘Socrates is a man’, ‘Socrates is an animal’, ‘Socrates is pale’. It not only serves as a designation for such statements or assertions about a thing, but it also characterizes their ontological reference or the ‘thing’ to which they refer i.e. to the being of what the thing is. Each of the predications listed above (man, animal, pale) has as an object of reference, a “first intention”; in Aristotelian terms a substance, in the Latin subjectum e.g., Socrates. It carries with it a “pointing towards”. (In this explanation, it is important to note language as “signs” in the word “de-sign-ation”. It is also important to note how our “reasoning” is based on the grammar/language of our sentences in English due to its roots in ancient Greek and Latin.) “First intentions” refer to our “first order” of questioning i.e. asking about the categories or characteristics of the things, their descriptions. We may say that the questioning about these characteristics is “first order” since they look at our assertions about the character of the the things and not about the thing’s “essence”. They are of the “first order” because they arise from our initial perceptions of the thing.

According to the Greeks number refers directly, without mediation, to individual objects, to things, whether apples or monads. It is, in the language of the Schools (the medieval Scholastics), a “first intention”. Number, thus, is a concept which refers to mind-independent objects. In order to understand the modern concept of number, it is useful to say a few words about the distinction between first and second intentions and show how these have come to be related to our understanding of “first order” and “second order” questioning.

With reference to representational thinking as understood by the ancients, not only is abstractness misapplied in this case of a ‘subject’ and its ‘predicates’, but the modern concept of number stands between us and an appreciation of why this is so. The Greek concept of number, arithmos, as stated in, say, penta, is a first intention i.e. it refers to mind-independent entities, whether it is apples or monads (things, units). The modern concept of number as “symbol generating abstraction” results from the identification, with respect to number, of the first and second intentions: both the mind-independent objects and the inquiring mind and its concepts are combined. It is what we have been calling the mathematical projection here. In order to make sense of the notion of a “symbol-generating abstraction”, we need to go to the modern concept of number.

Symbolic mathematics, as in post-Cartesian algebra, is not merely a more general or more abstract form of mathematical presentation. It involves a wholly new understanding of abstraction which becomes a wholly new understanding of what it means for the mind to have access to general concepts i.e., second intentions, as well as implying a wholly new understanding of the nature and the mode of existence of general concepts, and thus, a wholly new determination of what things are through a wholly new manner of questioning. This new ‘representation’ allows symbolic mathematics to become the most important achievement of modern natural science. Let us look at how this came about.

Viete and Descartes and the New Understanding of the Workings of the Mind:

Viete

 In order to display where Viete departs from the ancient mode of representation, we need to focus on the use of letter signs and Viete’s introduction of letter signs into mathematics in the West. We think that a letter sign is a mere notational convenience (a symbol in the ordinary sense of the word in our day) whose function is to allow for a greater generality of reference to the things it refers to. But this use of symbols, as the character of “symbol generating abstraction”, entails a wholly new mode of ontology or being-in-the-world and the representation of things of the world.

Every number refers to a definite multitude of things, not only for ancient mathematicians but also for Viete. The letter sign, say, ‘a,’ refers to the general character of being a number; however, it does not refer to a thing or a multitude of things. Its reference is to a concept taken in a certain manner, that is, to the concept’s and the number’s indeterminate content, its variableness. In the language of the Scholastics, the letter sign designates a “second intention”; it refers to a concept, a product of the mind. But what is of critical importance:  it does not refer to the concept of number per se but rather to its ‘what it is’, to “the general character of being a number”. The letter sign, ‘a‘, in other words, refers to a “conceptual content”, mere multiplicity for example which, as a matter of course, is identified with the concept.  This matter-of-course, implicit, identification is the first step in the process of “symbol generating abstraction”. This step, which is entailed by Viete’s procedures and not merely by Viete’s reflections on his procedures, makes possible modern symbolic mathematics. In other words, at the outset, at the hands of its “onlie begetter” Viete, the modern concept of number suggests a radical contrast with ancient modes of representation.

KleingFor Plato and Aristotle logos, discursive speech/ language, is human beings’ shared access to the “content” of a concept, what was known as “dialectic”. It is through language, and as language, that mathematical objects are accessible to the Greeks.  Not so for modern representation. The letter sign refers and gives us access to “the general character of being a number”, mere multiplicity (arithmos) (although it was left to Descartes to work out the implications of this mode of representation. More will be said on Descartes below.) In addition, the letter sign indirectly, through rules, operational usages, and syntactical distinctions of an algebraic sort, also refers to things, for example, five units. This leads directly to the decisive and culminating step of “symbol generating abstraction” as it emerges out of Viete’s procedures. It occurs when the letter sign is treated as independent; that is, when the letter sign, because of its indirect reference to things or units, is accorded the status of a “first intention” but, and this is critical, all the while remaining identified with the general character of a number, i.e. a “second intention”. Jacob Klein in Greek Mathematical Thought and the Origin of Algebra sums up this momentous achievement: a potential object of cognition, the content of the concept of number, is made into an actual object of cognition, the object of a “first intention”. From now on, number is both independent of human cognition (not a product of the imagination or mind) i.e. objective, and also without reference to the world or any other mind-independent entity, which, from the point of view of the tradition (if not common sense) is paradoxical.

What all of this means, according to Klein, is that “the one immense difficulty within ancient ontology, namely to determine the relation between the ‘being’ of the object itself and the ‘being’ of the object in thought is . . . accorded a ‘matter-of-course’ solution . . . whose significance . . . (is) . . . simply-by passed”. We can see now how the Quine statement beginning this writing (“To be is to be the value of a bound variable”) relates to this arrival of algebraic calculation. The mode of existence of the letter sign (in its operational context) is symbolic.

Let us try to grasp Klein’s suggestion about what symbolic abstraction means by contrasting it with the Platonic and Aristotelian accounts of mathematical objects. For Plato the correlate of all thought which claims to be knowledge is the mind-independent form, the “outward appearance” (eidos) and the idea (idea) or, in the case of number, the monad, the “unique”, singular one; none of these are the ontological correlates of the symbolic, modern grasp of mathematics. For Aristotle the object of the arithmetical art results from abstraction, but abstraction understood in a precisely defined manner. The abstraction of Aristotle is diaeresis  where attention is paid to the predicates of things rather than the whole of a thing and the predicate is subtracted from the whole so that individual attention may be given to it. The subtracted thing has real existence outside of the mind.

The mode of existence of what the letter sign refers to in modern mathematics is not abstract in this Aristotelian sense, but is symbolic; it is more general. In the modern sense, both the symbol and what it refers to are not only unique, arising out of the new understanding of number implied by the algebraic art of Viete, they are, as well, logical correlates of one another, symmetrically and transitively implying each other i.e. such that, if a relation applies between successive members of a sequence, it must also apply between any two members taken in order. For instance, if A is larger than B, and B is larger than C, then A is larger than C.. That is, symbol in “symbol generating abstraction” is not a place marker which refers to some thing, as in the ordinary sense of symbol of our day such as a stop sign; rather it is the logical, conceptual, and thus quasi-ontological correlate of what it refers to, namely the “conceptual content” of the concept of number i.e. multiplicity. From this will follow (Newton) that all ‘things’ become ‘uniform’ masses located in ‘uniform’ spaces. The philosopher Kant will ground this viewing in his Critique of Pure Reason.

But at the same time, while bound to the ancient concept, the modern version is, paradoxically, less general. “Abstraction” in the non-Aristotelian sense, the label for symbolic modes of thought, can be grasped in at least two ways. First, it presents itself as a term of distinction as in the pair abstract/concrete. Whereas the concrete stands before us in its presence or can be presented through or by an image, the “abstract” cannot. Alternatively, “abstract” in the modern interpretation can also be illustrated by an ascending order of generality: Socrates, man, animal, species, genus. The scope of the denotation, or the extension, increases as abstractness increases, and, once again, the more general is also the less imaginable. But this is precisely what symbolic abstraction is not. The mathematical symbol ‘a‘ in context has no greater extension than the ancient number, say, penta. Rather, the symbol is a “way” or, in the modern interpretation of method which Descartes inaugurates, a step in a “method” of grasping the general through a particular (links to inductive reasoning and the scientific method may be made here as well as to the Greek understanding of dianoia).  It is a way of imagining the unimaginable, namely the content of a “second intention”, which is at the same time through procedural rules, taken up as a “first intention”, i.e., something which represents a concrete ‘this one’. One consequence of this reinterpretation of the concept of arithmos is that the “ontological” science of the ancients is replaced by a symbolic procedure whose ontological presuppositions are left unclarified” (Klein, Greek Mathematical Thought, p. 184). What are the things which are represented here?

Descartes
Rene Descartes

Descartes’ suggestion that the mind has such a power answers to the requirements of Viete’s supposition that the letter sign of algebraic notation can refer meaningfully to the “conceptual content” of number. The “new possibility of understanding” required is, if Descartes is correct, none other than a faculty of intellectual “intuition” (which we commonly call imagination). But this faculty of intellectual intuition is not understood in terms of the Kantian faculty of intellectual intuition. The Cartesian version, implied by Descartes’ account of the mind’s capacity to reflect on its knowing, unlike its Kantian counterpart, is not informed by an object outside of the mind. (Of course, since for Kant the human intellect cannot intuit objects outside the mind in the absence of sensation, there is no innate human faculty of “intellectual intuition”. It is, for Kant, a faculty that is impossible and illustrates a limitation on human knowing.)

Moreover, this power of intuition has “no relation at all to the world . . . and the things in the world” (Klein, p. 202). In other words, it is not to be characterized so much as either incorporeal or dealing with the incorporeal but, rather, as unrelated to both the corporeal and the incorporeal, and so perhaps is an intermediate between the “mind the core of traditional interpretations of Descartes. In the simplest terms, the objects of mathematical thought are given to the mind by its own activity, or, mathematics is metaphysically neutral; it says nothing about the being of a world outside of the mind’s own activities; it stresses subjectivity and subjectiveness.” The consequences of such thinking are immense and have been immense.

Nonetheless, this unrelatedness of mathematics and world does not mean that mathematical thought is like Aristotle’s Prime Mover merely dealing with itself alone. It requires, according to Descartes, the aid of the imagination. The mind must “make use of the imagination” by representing “indeterminate manyness” through symbolic means” (Klein, p. 201). A shift in ontology, the passage from the determinateness of arithmos and its reference to the world, even if it is to the world of the Forms of Plato, to a symbolic mode of reference becomes absorbed by what appears to be a mere notational convenience, its means of representation, i.e., letter signs, coordinate axes, superscripts, etc., thus preparing the way for an understanding of method as independent of metaphysics, or of the “onto-language” of the schools of our day. The conceptual shift from methodos (the ancient “way” particular to, appropriate to, and shaped in each case by its heterogeneous objects) to the modern concept of a “universal method” (universally applicable to homogeneous objects, uniform masses in uniform space) is thus laid down. Through this, the way is prepared for a science of politics (and all human sciences) whose methodology is “scientific” and to their reference within these sciences of human beings as objects and ‘masses’.

The interpretation of Viete’s symbolic art by Descartes as a process of abstraction by the intellect, and of the representation of that which is abstracted for and by the imagination is, then, “symbol generating abstraction” as a fully developed mode of representation (Klein, pp. 202, 208; cp. pp. 175, 192). Consider two results of this intellectual revolution.

1. In order to account for the mind’s ability to grasp concepts unrelated to the world, Descartes introduces a separate mode of knowing which knows the extendedness of extension or the mere multiplicity of number without reference to objects universal or particular outside of the mind. This not only allows, but logically implies, a metaphysically neutral understanding of mathematics.  A mathematician in Moscow, Idaho, and one in Moscow, Russia, are dealing with the same objects although no reference to the world, generic or ontological, needs to be imputed.

2. “Symbol generating abstraction” yields an amazingly rich and varied “realm” (to use Leibniz’s sly terminology) of divisions and subdivisions of one and the same discipline, mathematics. For confirmation, one need only glance at the course offerings of a major university calendar under the heading “Mathematics”. Yet the source of this “realm” is at once unrelated to the world and deals with the “essence” of the world through mathematical physics in its essentialist mode. This is possible because the imagination is Janus-like. It is the medium for symbol generating and also a bridge to the world, since the world and the imagination share the same “nature” i.e., corporeality or, what comes to the same thing, the “real nature” of corporeality, extension.

Viete for one, as well as Fermat, simplified their achievements. They understood the “complex conceptual process” of symbol generating abstraction as merely a higher order of “generalization” thereby setting the stage for what has come to be habitual for modern consciousness, the passing over of the theoretical and exceptional, so that, in Klein’s phrase, it is simply “by-passed” or overlooked (Klein, p. 92). (All this is an inversion of Heidegger’s insistence that the passing over of the ‘proximal’ and ‘everyday’ must be overcome to appropriate Being in our day.) But this blindness to its own achievements, from which the modern science of nature suffers, is a condition of its success. Only if the symbol is understood in this way merely as a higher level of generality can its relation to the world be taken for granted and its dependence on intuition be “by-passed”. Only if symbol is understood as abstract in modern opinion’s meaning of the word would it have been possible to arrive at the bold new structure of modern mathematical physics on the foundations of the old.

It is important to grasp the conditions of the success of the modern concept of number. One of these is that modern mathematics is metaphysically neutral. This means, first of all, that modern mathematics does not entail, of itself, or presuppose of itself, metaphysical theses concerning what exists or what is the meaning of Being. For a contrast, one need only follow Klein’s patient exegesis of Diophantus’ Arithmetic; there, object, mode of presentation, scope of proof, and rigor of procedure are intermingled with metaphysics (Klein, pp. 126-49). Klein shows that “Aristotle’s theory … of mathematical concepts . . . was assimilated… by Diophantus and Pappus. Secondly, and more conclusively, the proofs and content of modern mathematical arguments need not be considered in conjunction with the metaphysical orientation of the mathematician presenting the argument, and so, whereas the pre-modern world could distinguish between Platonic and, say, Epicurean physics, no analogous distinction is viable in the modern world. There is yet a third way in which modern symbolic mathematics is metaphysically neutral and this in the strongest sense. It is neutral because it is all consistent with all metaphysical doctrines, nominalist or realist, relativist or objectivist. Whatever the metaphysics, to date, there is an interpretation of modern mathematics which leaves it unscarred. This is not the case for the ancient conception. For example, Euclid’s division of the theory of proportions into one for multitudes and another for magnitudes is rooted in the nature of things, in an “ontological commitment” to the difference between the two. Only after the metaphysical neutrality of the modern conception is taken for granted and bypassed, is it possible to do away with Euclid’s division as a matter of notational convenience.-

None of this holds true for mathematical physics in its authoritative mode, as arbiter of what there is (and what can, therefore, be claimed to be knowledge),  in the version it must assume to serve as a ground for the acceptance of the victory of the Moderns over the Ancients at the level of First Principles (metaphysics). Mathematical physics does make in this mode metaphysical claims. It is not metaphysically neutral. Elementary particles are, for example, if mathematical physics is arbiter of what there is. But are they? One can see a corollary application of this thinking in the “objectlessness” of modern art. 

Take, to begin with, the most influential version of ontology for those who accept the Reduction Thesis, that is, Willard Van Orman Quine’s famous dictum that “to be means to be the value of a bound variable.” Drawn as the dictum is in order to make metaphysics safe for physics, does it entail the existence of, say, elementary particles? All we know is that if we claim that particles are, that is, are in reality and not merely operationally defined then our claim will fit this semantic model. Conversely, sets, aggregates, mathematical infinities also qualify as “existents” in this semantic sense, but they cannot give us any knowledge of the world, since we need not impute to them any reference to a world outside the mind when we deal with them as pure objects of mathematics. In other words, as long as, in Cartesian terms, the identification of the real nature of body as extendedness with the objects of mathematical thought remains unproven and is merely, in effect, asserted, Sir Arthur Eddington’s hope that mathematical physics gives us an essentialist account of the world will remain just that, a hope.

All of the above means that Klein’s book is a key to understanding modernity’s most profound opinion about the nature of Being, of bringing to light the very character of these modern opinions in a manner which discloses not only their historical genesis but lays open to inspection why they are not only opinions but also conventions. Thus his book Greek Mathematical Thought and the Origin of Algebra is a key to renewing that most daunting of human tasks, liberating us from the confines of our Cave.

CT 1: Knowledge and the Knower

“I know because…”/ “We know because…”

Inquiry Questions:  1) Why (how) has algebraic calculation come be the paradigm of knowledge for our age and how does this affect my thinking? 2) How is learning a “giving to one’s self what one already has” and how does this determine how I understand what knowledge is? How do we acquire knowledge? What constitutes a “good reason” for us to accept a claim? Are intuition, evidence, reasoning, consensus and authority all equally convincing methods of justification? Does knowledge always require some kind of rational basis? How do our expectations and assumptions have an impact on how we perceive things? What are the advantages and disadvantages of requiring that all knowledge is verified by a group?

The priority of understanding who we as the “knowers” is obvious since knowers are either individuals or communities. What is this “knowledge” that they and I “know”? And how do they and I know it? In the Renaissance, human beings were given priority by being placed in the centre of the world as it was given, and from this, “humanism”, in any of its various guises, came to dominate thinking in our age. Our study of TOK remains within this “humanistic” legacy that we have inherited.

In arriving at how we know things, we have come to view reason as an instrument because our encounters with the world we live in are encounters with chaos, a chaos that must be controlled through the dominating application of the knowledge which we have received from the reasoning of our sciences. When we say “I know be-cause…” and “we know be-cause…”, we are stating that the principle of causality dominates how we understand the being of what something is and how it is i.e. “be”, the being of what is, “cause” the reason for the being of what is. The principle of causality is but one aspect of the principle of reason: nihil est sine ratione “nothing is without (a) reason“.

Thinking in our age is an empowering and overpowering activity and we shall try to see how it might be possible to have a receptive kind of thinking and what this might entail. It is this priority, to think about thinking, which spurs the inquiry into who the knowers are and what the things that they know are, how the knowers establish the horizons of things in their definitions and classifications of those things, their possibilities. The inquiry into the “how” of definitions and classifications is a search for an understanding of the “key concepts” that are used in TOK and in learning today. We may learn of these by learning of their origins. We are driven at the same time to find answers to the questions of who and what we are as human beings since this knowing determines how we understand ourselves and this understanding determines our actions, our ethics, in the world we live in. It is the old Delphic command of “Know Thyself” which is both a command to know who we are as an individual and, as an individual, to take up the journey that is the search for knowledge.

Historical Background: The Key Concepts

We return to the ancient Greeks to understand the essential beginnings of our principles and “key concepts” and how we know something. This return is required to understand the thinking that has come to be the historical knowledge of the West, the historical background if you like, but it is also the map we use in our search for knowledge. This thinking begins with the assertion about what some thing is and its key is to be found in logic. The human being is understood as the animale rationale, the “rational animal” that uses “logos” to understand the world around it.

As a proposition, a “position” put forward, a “stand” through which we hope to “understand” the simple assertion is a saying, a logos (we cannot, ultimately, separate the ways of knowing of reason and language) in which the “how” and the “what” of something is is said or asserted about something e.g. “The book is green”.  Here ”green” is said of the book.  That of which it is said (“the book”) is what underlies; it is the subject. Therefore, in the attribution of “greenness” something is said from above down to what lies underneath. In the Greek language, kata means “from above down to something below”. “Greenness” and all color is a category or an attribute of some thing. The some thing itself is the subject in Greek not what we understand as “object”.

Much can be said “down to a thing”, about it. “The book is green”. “The book is thicker than the one beside it”. “The book is big”. “The book is on the desk”. “The book is a new IB Higher Level Physics textbook”. It is the categories which determine the “thingness” of some thing. The statements that we make about the categories of the things are assertions

Using these assertions as guides, we can follow how some thing is determined at any given time to be a thing. Now, we do not pay attention to this particular thing in the example, the Physics textbook, but to that which in every such assertion of this sort characterizes every thing of this kind in general. “Green” says in a certain respect, namely in respect of color, how the thing is constituted.  A trait or quality is attributed to the thing. In the attribution, “big” becomes size, extension (quantity). With the attribution “thicker than”, there is asserted what the book is in relation to another book; “on the desk”: the place; “new”: the time in which the book came into appearance. This representation of the thing is called the correspondence theory of truth i.e. what is spoken about the thing corresponds to what we believe the reality of the thing is. This “truth” lies in the correspondence of the categories: that the statements made about the thing are true. The statements bring the thing to presence and illuminate the thing so that we may have certainty about what is being spoken about. 

Quality, extension, relation, place are determinations that are said in general of the book but also about any thing (the categories are universals). These determinations name the characteristics of the things and how they exhibit (show) themselves to us if we address them in the assertion and talk about them; they are the perspectives from which we view the things. Insofar as these determinations are always said down to a thing, the thing itself is already co-asserted as already present.

What is said or asserted about the thing, the subject, is called by the Greeks katagoria, which we understand in English as “categories”. What is attributed to the thing is then nothing other than the being characterized (green), being extended (big), being in relation to (next to), being there (on the desk), and the being “now” of the book as something that is. In the categories, the most general determinations of the being of some thing that is are said and we have provided a description of the thing. When we talk about “the things known” we mean the being of the things as some thing that is; the being of the thing has presence in itself and is something that is shared. Those determinations, which constitute the being of some thing that is i.e. of the things themselves, have received their name from assertions about them. The assertions define the limits and horizons of the thing so that it can be known to be what it is: “How do I know x? How do we know y?” From these assertions about the thing, we are able to classify it as some thing; it is a “this” and “not this”.

In naming the being of things as modes of assertedness lies a unique interpretation of the being of some thing, of who and what we are as human beings, and what the things about us are. In Western thinking, the determinations of being and beings are called “categories”: the structure of some thing (what some thing is) is connected with the structure of the assertion (corresponds) about it. It is here that what is called Western metaphysics begins and this beginning is to be found in the principle of reason which we have determined is to be found in logic. These beginnings are to be found in a text called Aristotle’s Physics.

Aristotle_Altemps_Inv8575The knowledge embedded in an assertion is true insofar as it conforms to its object. Truth is the “correctness” of the correspondence. In Medieval times, this correctness was called “adequation”, “assimilation”, or “correspondence”. These conventions or key concepts belong to Aristotle. Aristotle conceives of truth in the logos (assertion) as “assimilation”. The representation, the idea in the mind, is assimilated to what is to be grasped. The representational assertion about the book being on the table, or representation in general, pertains to the “psyche” or “soul”, something “spiritual” according to Aristotle; we, of course would say “the mind” or “the brain”. Latin interpretations of Aristotle arrived at the definition of human being as the animal rationale. What does this definition of human being imply?

OT 2: Knowledge and Language: Logos–Ratio—Reason

The assertion about the thing is a kind of legein—“addressing something as something” for the Greeks. This implies that something is taken or grasped as something. Considering and expressing something as something in Latin is called reor, ratio. Therefore, ratio becomes the translation of logos. The simple asserting simultaneously gives the basic form in which we articulate meaning and think something about things. This basic form of thinking, and thus of thought, is the guideline (principle) for the determination of the “thingness of the things”. The categories or universals determine, in general, the being of what is. To ask about the being of what is, what and how what is is at all, is called prima philosophia or “first philosophy”. We come to understand this word as what we mean by metaphysics.

Thought as simple assertion, logos, ratio is the “guideline” (principle) for the determination of the being of what is i.e. “the things known”. “Guideline” (principle) means that the modes of asserting direct the view (cognition) in the determination of the presence of something i.e. of the being of what something is (this is called hypothesis which combines the prefix hypo meaning “underneath” or “below” with thesis meaning “assertion” about what some thing is that needs to be proved or supported). This thinking has brought about our relation to all that is as “object”, and the object must respond to the manner of the questions which are imposed on it in order to be considered a “being”.

Logos and ratio are translated into English as “reason” i.e. logic and rational. Human being is determined as the “rational animal”. There is, thus, a connection between the things that are known, the what and how they are as known, and the what and how of human beings as knowers, and reason. The history of Western philosophy is a long discussion about this connection.

The Modern Mathematical Science of Nature and Reason:

The rise of modern natural science became decisive for the definition of what something is and, at the same time, what we as human beings are. That this should be the case required a transformation of human beings in their relationship to the things that are (this transformation is what we call ontology, the science of being and beings). How this transformation came to be requires that we get a clear picture of the character of modern natural science. To do so, we will avoid specific or special questions and deal with the general. Three modes are involved: the thing, our stance toward the thing (here referred to as ontology), and human being. How do our “key concepts” devolve from this?

The transformation of science basically took place through centuries of discussion about fundamental concepts and principles of thought i.e. the basic approach to things and toward how what is is at all. The paradigm shifts which Thomas Kuhn speaks of in The Structure of Scientific Revolutions are related to the twofold foundation of science: 1) experiment (or experience) i.e. the direction or method and the mode of mastering and using what is; 2) metaphysics i.e. the pro-jection of the fundamental knowledge of being, out of which what is knowledge develops. Experience (experiment) and the pro-jection of being (key concepts) are reciprocally related to one another and always meet in a basic feature of attitude or disposition (stance/ontology; ethics) towards what knowledge is. What this stance or stand may be is a product of the historical situation, or so we understand it. Is it possible to find a “stand” beyond the historical situation (or what for the Greeks was called “nature”/physis)? Are ethics, human actions, historical and therefore subject to change or are they arrived at through universal principles and therefore permanent?

Galileo
Galileo

It is sometimes said that modern science starts from “facts” while medieval science started from general speculative propositions and concepts. This is true in a certain way. But it is equally true that the ancients and medieval scientists also observed the facts and modern science also works with universal propositions and concepts. His contemporaries criticized Galileo, one of the founders of modern science, in much the same manner. The contrast between ancient and modern science is not “there concepts and principles and here facts”; both deal with them. It is the way the facts are conceived and interpreted and how the basic concepts are established that is decisive.

The scientists of the 16th and 17th centuries understood that there are no mere “facts”: a “fact” is only what it is in light of the fundamental conception (the Principle of Reason) and how far that conception reaches.  Please understand that we are not talking about the absurd notion of “alternative facts” here. Science has always attempted to get beyond sophistry in its search for the truth, and the interpretations of science are not to be confused with, or placed on, the same level as has been asserted by the “alternative facts” followers. Our current experience of the Covid-19 pandemic should give us ample evidence of this where the prevalence of an “anti-science” perspective has resulted in thousands upon thousands of deaths and millions of infections.

Positivism, which relies on sensory perception, thinks that it can sufficiently manage with “facts” and “new facts” while the concepts are merely expedients which one somehow needs but should not get too involved with since that would be philosophy or metaphysics. Such a view may, perhaps, be the reason that positivist scientists are only (and have only been) capable of average and subsequent work as compared to those who change or “revolutionize” science, such as Einstein and Heisenberg. The positivist view remains present in the current TOK course: we are to utilize and question the “key concepts” already given to us and apply them to “real life situations” without too much concern for “truth” or “philosophy”. Those who shift the “paradigms”, in Kuhn’s words, the Einsteins, Bohrs and Heisenbergs, the founders of modern nuclear physics, were first philosophers and created new ways of posing questions and in holding out in the questioning of what is questionable. Their science was a product of their means of questioning and of their imaginations in their search for the language (mathematics) in which to express their thinking and their findings. They had to literally think “outside of the box” or frame that the principle of reason and causality constructs in order to arrive at the truths of their propositions. The principles of reason and causality as the grounds for establishing the nature of things have become highly questionable under their scrutiny and interrogation. New questions regarding the nature of knowledge have arisen.

It is sometimes said that the difference between the old and new science is that modern science “experiments” and “experimentally” proves its cognitions (sense perceptions).  But the experiment, the test, to get information concerning the behaviour of things through a definite ordering and arrangement of things and events was also familiar in ancient times and in the medieval period. It is not the experiment as such in the wider sense of testing through observation, but the manner of the setting up of the test and the intent with which it is undertaken and in which it is grounded that is decisive. The scientific method is connected with a kind of conceptual determination of the facts and the way of applying concepts i.e. with the kind of hypothesis about things. It is primarily a way of viewing. For the Greeks “viewing” was called theoria, the root of our word “theory”. Science is the theory of the real or the way we have of looking at the real.

Besides the two characteristics noted: 1. Science of facts; 2. Experimental research, there is the third, and that is that modern science is a calculating and measuring investigation based on a synthesis of the categories that were spoken of earlier. But this is also true of ancient and medieval science which worked with measurement and number. Again, it is a question of how and in what sense calculating and measuring were applied and carried out, and what importance they have for the determination of the being of the objects themselves.

With these three characteristics of modern science, that it is a factual, experimental, measuring science, we are still missing its fundamental characteristic which determines the basic movement of science itself.  This characteristic is the manner of the working with the things and the metaphysical projection of the “thingness of the things”. This fundamental feature is that modern science is mathematical.

What do “mathematics” and the “mathematical” mean here? Mathematics, the Group 5 subject area and one of our Areas of Knowledge, is itself only a particular formation of the “mathematical”. So, what is the “mathematical”?

Learning/Knowing as Practice: Techne as Knowledge

Learning is a “grasping” and “a making one’s own” (“appropriating”, we take something into ourselves). We have the wonderful phrase in English “I get it” when we feel we have learned something. But not every “getting” or taking is a learning. We can get or take a seashell and make it part of a collection. In a recipe, it says “take two spoonfuls of sugar” i.e. use. “To take” means to take possession of a thing and have some disposal over it. Now, what kind of taking is learning? Mathemata—things insofar as we learn them. But strictly speaking, we cannot learn a “thing”; we can only learn of its use. Learning is therefore a way of taking and making one’s own in which the use of the thing is made “one’s own”. Such making one’s own occurs in the using itself. We call it practicing. But practicing is only a kind of learning. Not every learning is a practicing. What is the essential aspect of learning in the sense of mathesis? Why is learning a taking? What kinds of things are taken, and how are they taken?

Let us consider again practicing as a kind of learning. In practicing we take the use of the computer i.e. we take how to handle it (the keyboard; the software) into our possession. We master the way to handle its various commands in order for it to do what we intend. This means that our way of handling the computer is focused upon what the computer itself demands; “computer” does not mean just this individual computer of a particular serial number. We become familiar with the thing; learning is always “a becoming familiar with”. Learning has different directions: learning to use and learning to become familiar. Becoming familiar also has different levels. We become familiar with one particular model of the computer, but also with all computers in general be they PCs or Macs. With practice, which is learning its use, the “becoming familiar” involved in it remains within a certain limit. There is “more” to become familiar with about the computer, the thing i.e. programming, web design, the raw materials needed to make the computer, and so on. But to use the computer, we do not need to know all these things. How the computer works belongs to the thing. When a computer we are practicing to use must be produced, in order to provide and produce it so that it can be at our disposal, the producer of the computer must have become familiar beforehand with how the thing works and how the thing is supposed to work. With respect to the computer, there is still a more basic familiarity, whatever must be learned before, so that there can be such models and their corresponding parts and software at all; this is a familiarity with what belongs to a computer at all and what a computer is and what it is supposed to do.

This familiarity with the computer must be known in advance, and must be learned and must be teachable. This becoming familiar is what makes it possible to produce the computer; and the computer produced, in turn, makes its practice and use possible. What we learn by practice is only a limited part of what can be learned of the thing. We do not first learn what a computer is when we become familiar with a PC or a Mac. We already know that in advance and we must know it; otherwise, we could not perceive the computer as such at all, nor whether it is a Mac or PC and these names would make no sense to us. We might make the mistake of seeing a media pad as a cutting board. Because we know in advance what a computer or a tablet is, and only in this way, does what we see laid out before us become visible to us as what it is.

Of course, we know what a computer is only in a general and indefinite way. When we come to know the computer in a special and determined way, we come to know something which we really already know. It is this “taking cognizance” (grasping, appropriating, “getting it”, cognition) that is the genuine essence of learning, the mathesis. The mathemata are the things insofar as we take cognizance of them as what we already know them to be in advance: the body as the bodily, the plant-like of the plant, the animal-like of the animal, the thingness of the thing, and so on. This genuine learning is therefore an extremely peculiar taking, a taking where the taken (what is learned) is something that one actually already has. It is from this that the AOKs are determined and it is the ground of the methodology used in the AOKs.

Teaching, in whatever mode we may feel is most “useful”, corresponds to this learning. Teaching is a giving, an offering; but what is offered in the teaching is not the learnable, for the student is merely instructed to take for himself what he already has. If the student only takes over something which is offered (rote learning) he does not learn. The student comes to learn only when they experience what they take as something they themselves already have. True learning only occurs where the taking of what one already has is a self-giving and is experienced in this way. Today we call this “empowerment”. Teaching does not mean anything else than letting the others learn i.e. to bring the others to learning, to facilitate the learning. Learning is more difficult than teaching; only the one who can truly learn can truly teach. The genuine teacher differs from the student only in that he or she can learn better and that the teacher more genuinely wants to learn (the necessity for “passion” in teaching). In all genuine teaching, it is the teacher who learns the most.

The most difficult learning is to come to know all the way what we already know. In TOK we continually ask with a mind to their usefulness, the same obviously useless questions of what a thing is, what technology is, what tools (instruments) are, what a human being is, what a work of art is, what the state and what the world are. This is disorientating and disruptive for students: they want their learning to be useful and such use is usually directed towards the future, but their desire for “results” is already pre-determined by the system that is already in existence and has been in existence for a long period of time.

The mathemata, the mathematical, is that “about” things which we already know. We do not first “get it” out of things, but in a certain way we bring it already with us. From this we can understand why number is something mathematical. We see three chairs and say that there are three. What the “three” is the three chairs do not tell us, nor three apples, nor three cats, nor any other three things. Moreover, we can count three things only if we already know “three”. In grasping the number three, as such, we explicitly recognize something which, in some way, we already have.

This recognition is genuine learning; it is a “taking cognizance” of something. The number is something in the proper sense “learnable” i.e. something mathematical. Things do not help us to grasp “three” i.e. its “threeness”. What is a “three”? It is the number in the natural series of numbers that stands in the third place. In “third”? It is only the third number because it is a three. And “place”—where do places come from? “Three” is not the third number but the first number. “One” really isn’t the first number. For instance, we have before us a book, a desk. This one and, in addition, another one. When we take both together we say “both of these”, the book and the desk. Only when we add a whiteboard marker to the book and desk do we say “all”. Now we take them as a sum i.e. a whole of so and so many. Only when we perceive it from the third is the book a one, and the desk a second, so that one and two arise, and “and” becomes “plus”, and there arises the possibility of places and series. What we now “take cognizance” of is not created from any of the things. We take what we ourselves somehow already have. What must be understood as mathematical is what we can learn in this way.

We “take cognizance” of all this and learn it without regard for the things. Numbers are the most familiar form of the mathematical because, in our usual dealing with things, when we calculate or count, numbers are the closest to that which we recognize in things without creating it from them. For this reason, numbers are the most familiar form of the mathematical. In this way, this most familiar mathematical becomes mathematics.

In TOK, when we speak of “knowledge and the knowers”, mathesis is the manner of learning and the process itself while the mathemata is what can be learned in the way indicated i.e. what can be learned about the things without taking it from the things themselves. The mathematical is that evident aspect of things within which we are always already moving and according to which we experience them as things at all, and as such and such things. The mathematical is the fundamental position we take toward things by which we take up things as already given to us, and as they should be given. Therefore, the mathematical is the fundamental presupposition of the knowledge of things.

Plato-raphaelPlato is noted in the 6th century A.D. Neo-Platonist philosopher Elias Philosophus’ Commentary on Aristotle’s Categories to have put over the entrance to his Academy: “Let no one who has not grasped geometry enter here!” For Plato, the mathematical was geometry (not only one subject, but the foundation of all knowing). Those who enter the Academy must first grasp that the fundamental condition for the proper possibility of knowing is the knowledge of the fundamental presuppositions of all knowledge and the position (stand; the ethical) we take based on such knowledge. This type of knowledge is to be distinguished from opinion. Plato also states: “The god is forever the geometer”. By this he means “the god” is forever present in the learnable and the knowable.

Summation:

Reason as the principle of reason and as a way of knowing is related to the mathematical. Our maintaining that the basic character of modern science is the mathematical brought about this “short” reflection on the essence of the mathematical. After what has been said, this cannot mean simply that modern science employs mathematics. But how does the principle of reason as a way of knowing and the mathematical come to be algebraic calculation? What happens to “nature” and “the world” once “knowledge as calculation” comes to the fore? How this unfolding came about and how mathematics unfolds its essence in the modern sciences needs to be examined in the next section. The discussion on Mathematics as an Area of Knowledge also attempts to reveal the essence of the mathematical. In these reflections we are trying to illuminate the mystery that is technology and arrive at a greater knowledge of who we are as human beings.

Descartes’ “Cogito ergo Sum”: The Subject/Object Distinction

Descartes
Rene Descartes

Modern philosophy is usually considered to have begun with Descartes (1596-1650) who lived a generation after Galileo. It is no historical accident that the philosophical formation of the mathematical foundation of the modern stance/stand in Being is primarily achieved in France, England and Holland.

During the Middle Ages philosophy stood under the exclusive domination of theology and gradually degenerated into a mere analysis of key concepts and elucidations of traditional propositions and opinions, an approach similar to what is taken in TOK currently. Descartes appeared and began by doubting everything, but this doubt ran into something which could no longer be doubted, for inasmuch as the skeptic doubts, he cannot doubt that he, the skeptic, is present and must be present in order to doubt at all. As I doubt I must admit that “I am”. The “I” is indubitable. As the doubter, Descartes forced human beings to doubt in this way; he led them to think of themselves, of their “I”. Human subjectivity came to be declared the centre of thought. From here originated the “I”-viewpoint of modern times and its subjectivism, and also the grounding of what we call “humanism”. Concurrently, the world came to be viewed as “object” and the things of the world understood as objects, ob-jacio “the thrown against”. What is “thrown against” the world when it is understood as “object”?

Philosophy was brought to the insight that doubting must stand at the beginning of philosophy: reflection upon knowledge itself and its possibility. This is in contrast with the Greeks where “trust” stands at the beginning of philosophy and “doubting” led one to see why that “trust” was an appropriate response to the things that are. With Descartes, a theory of knowledge had to be erected before a theory of the world i.e. a “map” of the mind and its seeking had to be created before the world could be “discovered”. Descartes’ stand required ‘certainty‘ and ‘correctness‘ regarding the world and its being and these were to be derived through theory. (Our course is called Theory of Knowledge. Its description in the TOK and its contents illustrate that it is conceived as a “modern” product. The Greeks, for example, did not have “theories of knowledge”.) From Descartes on, epistemology is the foundation of philosophy (TOK is really a course in epistemology), and this is what distinguishes modern from medieval philosophy. Much of the modern translations and interpretations of Plato and Aristotle are attempts to make them epistemologists.

The main work of Descartes is called Meditations on First Philosophy (1641). This is the first philosophy of Aristotle, prima philosophia, the question concerning the being of what is in the form of the question concerning the thingness of things. Meditations on First Philosophy—nothing about “theory of knowledge”. The sentence in its assertion (subject + predicate) or proposition constitutes the guide for the question about the being of what is (for the categories, what is spoken down to something).  (The connection between Christianity and Greek metaphysics that prioritized certainty and which made the development and the acceptance of the mathematical possible (the certainty of Christian salvation), the security of the individual as such—will not be considered here, though these are the roots of what is called “humanism” and why we as human beings have a special place in the TOK course design.)

In the Middle Ages, the doctrine of Aristotle was taken over in a very special way. In later Scholasticism, through the Spanish philosophical schools, especially through the Jesuit Suarez, the “medieval” Aristotle went through an extended interpretation. Descartes received his philosophical education from the Jesuits. The title of his main work expresses both his argument with this tradition and his motivation to take up anew the question of the being of what is, the thingness of things, and “substance”.

For about a century following Galileo, mathematics had already been emerging more and more as the foundation of thought and was pressing toward clarity. Algebra was becoming the language in which the mathematical spoke. The world-view was changing and needed “grounding”.

“The mathematical” wills to ground itself in the sense of its own inner requirements which are based on the principle of reason. It expressly intends to make explicit that it is the standard of all thought and to establish the rules that require that it be so. Descartes participates in this reflection upon the fundamental meaning of the mathematical (that which can be learned and that which can be taught). Because this reflection concerned the totality of what is and the knowledge of it, this had to become a reflection on metaphysics—a meditation on first philosophy. This need for a foundation of mathematics (the mathematical) and of a reflection on metaphysics characterizes his fundamental philosophical position. We can see this outlined in his Rules for the Direction of the Mind. 

“Rules”: basic and guiding propositions in which mathematics submits itself to its own essence (axioms); “for the Direction of the Mind”: laying the foundation of the mathematical in order that it, as a whole, becomes the measure or standard of the inquiring mind, the compass which provides the direction for the mind in its questioning. By announcing the mathematical as subject to rules as well as the “freedom” of the determination of the mind, the basic mathematical-metaphysical character is already expressed in the title. By way of reflection upon the essence of mathematics, Descartes grasps the idea of a “universal science” (scientia or knowledge), to which everything must be directed and ordered as the one authoritative science. Descartes expressly states that it is not a question of “vulgar mathematics” (common calculation or what we know as “arithmetic”) but of “universal science”. We will only look at three of the twenty-one rules, namely, the third, fourth and the fifth. Out of these, the basic character of modern thought leaps before our eyes.

 Rule Three: 3. “As regards any subject we propose to investigate, we must inquire not what other people have thought, or what we ourselves conjecture, but what we can clearly and manifestly perceive by intuition or deduce with certainty. For there is no other way of acquiring knowledge.” (See both the Coherence theory of truth and the correspondence theory of truth as well as the principle of reason). This is what we have come to call “Scope” and “Perspectives” in our latest TOK guide.

Rule Four: 4. “There is need of a method for finding out the truth.” This rule does not mean that a science must also have its “method” but it wants to say that the procedure i.e. how in general we are to pursue (proceed) to the things, our path to the things decides in advance what truth we shall seek out in the things. Method or the methodology is not one piece of equipment of science among others but the primary component out of which is first determined what can become an object (objectified) for the science and how it becomes an object. This entails all areas of knowledge for “method” is what determines what can be called “knowledge” in all areas of knowledge. In our latest guide, this is referred to as “Methods and Tools”; and while the plural indicates a variety of methods and tools, they are all fundamentally grounded in the axiomatic nature of mathematics.

A note on the distinction between abstract and concrete is required here. One reaches the abstract when one “skips over” or abstracts from some features implied in the “concrete” or “the real”. When in speaking of a tree, for instance, one abstracts everything which is not a tree (the earth, air, the sun) and one is speaking of an abstraction that does not exist in reality, for the tree can only exist if there is earth, air, and sun etc. Hence, all the particular sciences deal, in varying degrees, with abstractions and must do so if they are to be “mathematical”. The “isolated particular” is by definition “abstract”. The journey of the mind is an attempt to rise to the “general ideas” which are the “concrete”.

Rule Five: 5. “Method consists entirely in the order and disposition of the objects towards which our mental vision must be directed if we would find out any truth. We shall comply with it exactly if we reduce involved and obscure propositions step by step to those that are simpler, and then starting with the intuitive apprehension of all those that are absolutely simple, attempt to ascend to the knowledge of all others by precisely similar steps.” That “method” is the “ordering and the gathering” or disposition of the objects which are under investigation is the mathematical pro-jection that is part of the essence of technology.

From these three rules we must now determine the relationship of the mathematical (that which can be taught and that which can be learned) with traditional “first philosophy” (metaphysics) and how modern philosophy came to be determined and so, too, to understand the reason why algebraic calculation has come to be what is called “knowledge” today.

To the essence of the mathematical as a “projection” (a “throwing forward” or a “throwing toward”) belongs the axiomatical, the arche or the beginning of the basic principles or concepts upon which everything further is based in a “coherent”, insightful order. If mathematics, in the sense of a universal learning, is to ground and form the whole of knowledge, then it requires the formulation of special axioms.

These axioms must: (1) be absolutely first in order, intuitively evident in and of themselves, i.e. absolutely certain. This certainty participates in deciding their truth. (2) The highest axioms, as mathematical, must establish in advance, concerning the whole of what is, what is in being and what being means, from where and how the thingness of things is to be determined. According to the tradition, this happens along the guidelines of the proposition. But up till now, the proposition, “the position that is thrown forward”, had been taken only as what offered itself, as it were, of itself. The simple proposition about the simply present things contains and retains what the things are. Like the things, the proposition is the framework of the things and for the things. 

However, there can be no pre-given things for a basically mathematical position. The proposition cannot be an arbitrary one. The proposition must itself be “grounded”. It must be a basic principle—the basic principle absolutely. One must find the basic principle of all “positing”/”projecting” i.e. a proposition in which that about which it says something, the subjectum is not just taken from somewhere else. That underlying subject must emerge for itself in this original proposition and be established. Only in this way is the subjectum an “absolute ground” purely posited from the proposition as such, a basis and, as such, an “absolute ground” that is unshakable and absolutely certain. Cogito, ergo sum. Because the mathematical now sets itself up as the principle of all knowledge through the principle of reason, all knowledge up to now must necessarily be put into question, regardless of whether it is tenable or not.

Descartes does not doubt because he is a skeptic; he must doubt because he posits the mathematical as the absolute ground and seeks for all knowledge a foundation that will be in accord with it. It is a question of finding not only a fundamental law for the realm of nature, but finding the very first and highest basic principle for the being of what is in general. This absolutely mathematical principle cannot have anything in front of it and cannot allow what might be given to it beforehand. If anything is given at all, it is only the proposition in general as such i.e. as a thinking that asserts. The positing, the proposition, only has itself as that which can be posited. Only where thinking thinks itself, is it absolutely mathematical i.e. a “taking cognizance” of that which we already have. Insofar as thinking and positing directs itself toward itself, it finds the following: whatever and in whatever sense anything may be asserted, this asserting and thinking is always an “I think”. Thinking is always an “I think”, ego cogito. Therein lies: “I am”, sum. Cogito, sum—this is the highest certainty lying immediately in the proposition as such. In “I posit”/”I assert”, the “I” as positer is co- and pre-posited as that which is already present as what is. The being of what is is determined out of the “I am” as the certainty of the positing.

The formula which Descartes’ proposition sometimes has (“Cogito ergo sum”) gives the common misunderstanding that there is an inference here. Descartes emphasized that no inference is present. The sum is not a consequence of the thinking, but vice versa: it is the ground of the thinking. In the essence of positing lies the proposition: I posit. That is a proposition which does not depend upon something given beforehand, but only gives to itself what lies within it. In it lies: “I posit”. I am the one who posits and thinks. This proposition is peculiar since it first posits that about which it makes an assertion, the subjectum. What it posits in Descartes’ case is the “I”.  The “I” is the subjectum of the very first principle. The “I” is therefore a special something which “underlies” (subjectum) the subjectum of the positing as such. Here one sees Aristotle turned upside down.

Since Descartes’ time, the “I” has been called the “subject”. The character of the ego as what is especially already present before one remains unnoticed. Instead, the subjectivity of the subject is determined by the “I-ness” of the “I think”. That the “I” comes to be defined as that which is already present for representation (the determination of what is “objective” in today’s sense) is not because of an “I-viewpoint” or perspective, or any subjectivist doubt, but because of the essential predominance and the definitely directed radicalization of the mathematical and the axiomatic.

This “I” which has been raised to be a special “subject” on the basis of the mathematical, is, in its meaning, nothing “subjective” at all, in the sense of an incidental quality of just this particular human being. This “subject” designated in the “I think”, this I, is subjectivistic only when its essence is no longer understood i.e. is not looked at from its origin considered in terms of its mode of being: “I am this thinking…”

Until Descartes, everything present-at-hand for itself was a “subject”; but now the “I” becomes the special subject, that with regard to which all the remaining things first determine themselves for what they are as such. Because—mathematically—they first receive their thingness only through their founding relation to the highest principle and its “subject” (the “I”), they are essentially such as stand as something else in relation to the “subject”, something which lies over against it as objectum. The things themselves become “objects”, the “over against”.

The word objectum goes through a corresponding change of meaning. Up to Descartes, the word objectum denoted what was thrown up opposite as one’s mere imagining: I imagine a golden mountain. This representation—an objectum in the language of the Middle Ages—is according to the usage of language today, merely something “subjective”; for a golden mountain doesn’t exist “objectively” in the new meaning.

The reversal of the meanings of the words subjectum and objectum from Aristotle’s understanding of these concepts is simply not a casual change of usage; it indicates a radical change in human beings’ orientation to what is i.e. the enlightenment of the being-of-what-is on the basis of the predominance of the mathematical. To say that human being is “enlightened” means that it is enlightened in itself as “being-in-the-world” but not through any other entity, so that it is itself this enlightenment. This enlightenment is the principle of reason’s unfolding in the essence of the mathematical. What is present-at-hand but hidden in the dark becomes accessible only for an entity enlightened in this way. With Descartes begins the era called the Age of Enlightenment.

Reason as the Highest Ground: The Principle of the “I”: The Principle of Contradiction:

After Descartes, the I as “I think” is the ground upon which all certainty and truth becomes based. But thought, assertion, logos is, at the same time, the guideline for the determination of the being of some thing through the categories. These are found in the “I think”, in the viewing of the “I”. Because of the fundamental significance of the foundation of all knowledge in the “I”, the “I” becomes the essential definition of a human being. With this emphasis on the “I” i.e. with the “I think”, the determination of the rational and of reason takes priority—for thinking is the fundamental act of reason. Up to Descartes, and later, human beings had been apprehended as the animal rationale as a rational living being. With the “cogito—sum” reason becomes explicitly posited according to its own demand as the first ground of all knowledge and the guideline for the determination of the things. The philosopher Kant will later assert: “the mind makes the object”. 

Already in Aristotle, the assertion, the logos, was the guideline (axiom) for the determination of the categories i.e. the being of what is, the “how” of what is. However, the centre of this guideline (axiom)—human reason, reason in general—was not characterized as the subjectivity of the subject. With Descartes, reason has been set as the “I think” and becomes the “highest principle” as the guideline (axiom) for all determinations of being and of what things are. The highest principle is the “I” principle: cogito—sum. It is the ground axiom of all knowledge; but it is not the fundamental (ground) axiom, simply for this one reason, that in this I-principle itself there is included and posited yet another one, and therefore with every proposition. When we say “cogito—sum”, we express what lies in the ego (subjectum), the subject. If the assertion is to be an assertion, it must always posit what lies in the subjectum. What is posited and spoken of in the predicate cannot speak against the subjectum. The assertion must always be such that it avoids the “saying that is a speaking against”, the contradiction: the principle of contradiction.

Since the mathematical as the axiomatic project posits itself as the authoritative principle of knowledge, the positing is established as “the thinking”, as the “I think”, the “I-principle”. “I think” signifies that I avoid contradiction and follow the principle of contradiction. This is why the position of “alternative facts” is not tenable: it posits contradictions i.e. it is a form of “madness” because it is not “rational”.

The “I-principle” and the principle of contradiction spring from the nature of thinking itself, and in such a way that one looks only to the essence (what something is) of the “I think” and what lies in it and in it alone. The “I think” is reason, and the is its fundamental act (“I am”); what is drawn solely from the “I think” is gained solely out of reason itself. Reason so understood is purely itself, pure reason (and, thus, we later have Kant’s Critique of Pure Reason).

Descartes’ principles, which agree with the fundamental “mathematical” feature of thinking, spring solely from reason, and become the principles of knowledge proper i.e. metaphysics, the determination of the being of what is. The principles of “mere reason” become the axioms of pure reason. Pure reason, logos so understood, the proposition in this form (the assertion) becomes the axiom and standard of metaphysics i.e. the court of appeal for the determination of the being of what is, the thingness of things. The question about what something is is now anchored in pure reason i.e. the mathematical unfolding of its principles through the principle of reason, nihil est sine ratione: “Nothing is without reason. What is “subjective” is that which is confined to the individual alone and is not provided with sufficient reasons for its being. 

In Kant’s Critique of Pure Reason lies the logos of Aristotle, and in the “pure” a certain special formation of the “mathematical”.

Summary of Knowers and what is Known:

In following the history of the question of the thing, we noticed that it was characterized by the mutual relation of the thing and the assertion (logos), the axiom along which the universal determination of what something is is established. The assertion, the proposition was viewed in a “mathematical” way as principle; and the principle sets forth the principles that lie in the essence of thinking (reason), of the proposition as such i.e. the I-principle and the principle of contradiction. With Leibniz there is added the principle of sufficient reason, which is also already co-posited in the essence of a proposition as a principle. These propositions originate purely out of mere reason, without the help of a relation to something previously given before one. They are thinking’s giving to itself that which thinking in its essence already has in itself. It is the essence of our knowledge questions.

For Descartes, the fundamental axioms i.e. the absolute axioms are the I-Principle, the principle of contradiction, and the principle of sufficient reason. The whole of our understanding of what something is is to be based on them, and that which we call “cognition” (sensory perception, awareness) is also to be based on them. This means that we must address what is as a whole and our questioning of the particulars is already determined by our understanding of what that whole is. 

In our writing on knowers and the things known, we attempted to describe the turn from earlier knowledge of nature to modern thought. We limited ourselves to a part of what is as a whole. We also did not discuss how this limited part (nature) belongs into the whole of what is.

Since the ascendancy of Christianity in the West (not only in the medieval period but also in the modern), nature and the universe were considered as created. In Christianity, a hierarchy of what is as a whole is established. What is most real and the highest is the creative source of all that is, the one personal God as spirit and creator. All of what is that is not godlike is the created. Among all that is created, humanity is distinctive, and this is because the eternal salvation of humanity is at stake and in question. God as creator, the world as created, humanity and our eternal salvation, these are the three domains defined by Christian thought within what is as a whole.

In Western thought, the questions of the “what is” kinds are called “metaphysics”: what is as a whole, what something within the whole is, why it is as it is. The West has been concerned with God (theology), the world (cosmology) and humanity’s salvation (psychology). In agreement with the character of modern thought as mathematical, Christian metaphysics, too, is formed out of the principles of pure reason, the ratio. Thus the metaphysics of God becomes a “rational theology”, the doctrine of the world becomes a “rational cosmology” and the doctrine of humanity becomes a “rational psychology”.

Christianity’s impact on modern metaphysics can be arranged in this way: (1) the Christian conception of things as “created”; and (2) the basic mathematical character of the things. The first indicates the content of metaphysics; the second its form. This structure as determined by Christianity forms not only the content of what is treated in thought, but also determines the form, the “how” it is treated. Insofar as God as the creator is the cause and the ground for all that is, the how, the way of asking the questions, is orientated in advance toward this principle. Vice-versa, the mathematical is not only a form clamped over this Christian content, but it itself belongs to the content. Insofar as the I-principle, the “I think” becomes the leading principle, the “I” and consequently, human beings, reach a unique position within the questioning about what is. The “I” designates not only one area among others, but just that one to which all metaphysical propositions (“what is” questions) are traced back and from which they stem. Metaphysical thought moves in the variously defined domains of subjectivity (dispositions, attitudes, metacognition). After Descartes, Kant will say “All questions of metaphysics i.e. those of the designated disciplines (our AOKs) can be traced back to the question: What is man? (i.e. who or what is the knower?)”. In the priority of this question is concealed the priority of the method outlined in Descartes’ Rules for the Direction of the Mind.

If we use the distinction of form and content to characterize modern metaphysics (such as in done in empiricism), then we must say that the mathematical belongs as much to the content of this metaphysics as the Christian belongs to its form.

The essence and the possibility of this “what is” must be determined in each case rationally, out of pure reason i.e. from concepts gained in pure thought. If what is and how it is must be decided in thinking and purely from thought, then before the definitions of what is as God, the world, and humanity there must be a prior guiding concept of what is as such. Especially where this thinking conceives itself mathematically and grounds itself mathematically, the projection of what is as such must be made the foundation (axiom) of everything. Thus, the inquiry that asks about what is in general must precede the inquiry into the particulars of the areas of knowledge.

But because metaphysics has now become the “mathematical” (what can be learned and what can be taught), the general cannot remain what is only suspended above the particular, but the particular must be derived from the general as the axiomatic according to the principles (“the mind makes the object”). This signifies that in the general of what can be learned and what can be taught what belongs to what is as such, what determines and enframes the thingness of the things as such must be determined in principle according to axioms, especially according to the first axiom, according to the frame of positing and thinking as such. What is a thing must be decided in advance from the highest principle of all principles and propositions, i.e. from the principle of pure reason, before one can reasonably deal with the divine, worldly and human.