“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.
The 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?
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 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.
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
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.