## The Mathematical Projection of Nature

#### Knowledge Questions in Mathematics (Scope)

• Why is mathematics so important in other areas of knowledge, particularly the natural sciences?

• How have technological innovations, such as developments in computing,

affected the scope and nature of mathematics as an area of knowledge?

• Is absolute certainty attainable in mathematics?

• Is there a distinction between truth and certainty in mathematics?

• Should mathematics be defined as a language?

• Is mathematics better defined by its subject matter or its method?

• Does mathematics only yield knowledge about the real world when it is

combined with other areas of knowledge?

• Is there a hierarchy of areas of knowledge in terms of their usefulness in solving

problems?

**What is Mathematics?**

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 bring 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 r*epresentation* 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

Modern: 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

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.