Student Journal

On Mathesis Universalis

“To read the unconscious of a text is to reconstruct a seemingly limitless number of meanings.”

- Iain Morland, “Reading’s Reason” 

Rene Descartes is christened the “father” of modernity on many scores and the term is aptly chosen. His methodology, with which modernity has supplanted his philosophy, served the patriarchal status quo of his time, and divorces science from poetic knowledge via a schism that echoes the Platonic misogyny found in Republic. Moreover, the Cartesian climate (not necessarily Descartes himself) has shifted the import of science from one of product to process, stripping poetics of ideological impact in light of its non-rational (immeasurable) characteristics. Following the example set out by Iain Morland in “Reading’s Reason,” I will show how the essential apparatus of Descartes’ mathesis universalis, alongside Cartesian dualism has produced a clear favoring of method over principle, and thus a relationship to technology (as Martin Heidegger informs us that technology is method) that carves out certain immortality for symbolic hierarchies and objectivity vs. more culturally symbolic and imaginative ways of knowing. While Heidegger and Morlanddescribe how we cannot simply “opt in” or “opt out” of the modern relationship to method/technology, I lean my argument not against our dependence on processes, but towards contemplation of alternative processes. Mathesis universalis, a systemized mathematical approach to learning and experimentation, was created as a universalizing mechanism for the objective sciences. As such the political structure of that which is determining the constitution of this universality must be examined; after all, those in position to universalize are those for whom the universal is already determined.

In the following, I endeavor to mine the different conceptions of mathesis or mathesis universalis, as it is understood by philosophers Rene Descartes, Edmond Husserl, Martin Heidegger, and Michel Foucault. This exploration gives a grounding of the project, begun by Descartes, of defining mathesis universalis in relation to our current episteme.  I argue that mathesis universalis, as established by Descartes and maintained today by the current relationship between technology and knowledge, excludes poetics from the epistemic catalogue via means which often uphold patriarchal discourse in the sciences. By utilizing Julia Kristeva’s methodologies, which account for cultural and poetic knowledge, an alternative to mathesis universalis becomes discernible. This alternative offers a non-exclusive universality inclined towards immersion and intuition, closing the gap between object and subject that Cartesian dualism requires. In this chapter I show the patriarchal, exclusory nature of modern, Cartesian scientific method as germane to modern science and modern ways of knowing.  

From Process to Product

Iain Morland’s essay “Reading’s Reason” explores the shift in ethical implications that reading (also writing) poetry has in our modern times. Immediately he addresses Adorno’s accusations of the barbarity of reading poetry after Auschwitz, but his examination of the act of reading has more implications than the justification of poetics. Adorno’s charge of barbarity carries with it the implication that poetics is simply a result; a thing, and has little value as necessary process to beings-as-such. Morland’s critique shows that poetics have a value beyond that which our current, mathematically-styled episteme typically allows for. Referencing Max Horkheimer’s bookThe Eclipse of ReasonMorland seeks to explain the how our modern intellectual climate that favors rationality over reason works (alaMarx) to maintain the status quo:

 Objective reason, Horkheimer suggests, has been eclipsed by its insipid, subjective counterpart, rationality. Rationality defines what is reasonable as what is useful; it does not seek truth as an end in itself. Reason has thus become merely a mental tool with which one can make effective plans in neocapitalist society. In short, rationality privileges means over ends. This subjective pastiche of reason, like poetry after Auschwitz, dumbly reiterates the status quo through devising ever more efficient procedures by which tasks may be carried out. It does not criticize the objective ends that are actually furthered by such procedures (Morland 86). 

Morland continues on to dissect the various political, social, and cultural values that can be ascribed to the process of reading, both before and after modernism. His examination introduces a materiality to the process of reading, granting the reader responsibilities for finding latent cultural content, and generating new meanings from it. He does not limit the scope of this responsibility to poetry or leisurely reading, rather, he insists that academic reading falls under the same auspices: “Academic reading similarly intervenes in the space of this interaction to produce patterns of meaning from the timeless cacophony of the text’s cultural unconscious. The text, mute and unchanging, appears to obey” (89). Although Morland grants the act of reading a kind of anthropomorphism that is unnecessary to explore the material value of the process in terms of cultural criticism and revolution, his findings are, however, convincing and provoking. He states:

Reading, in its reified and objectified position in rational culture, functions as a recreational opposite to serious activities. It serves to highlight the supposed usefulness of all that it is not: commerce, sport, information technology, engineering. While these “fields” are crudely drawn examples, they nonetheless together illuminate the peculiar cultural position of reading. That they broadly make sense as a collective example of what reading is “not” is a reflection of reading itself. Under the gaze of rationality, reading has become exaggeratedly unreasonable and derivative; it can make no claims for itself because it is popularly defined only through what it is not. In other words, it is precisely outopian, having no location of its own (91).

He determines via Horkheimer’s critique of reason and rationality, that reading has been reduced to a commodified, utopian activity with no value as rote-process. He then argues that reading can be re-established as a means to uncover the latent cultural content, and thereby provide new understandings of texts that allow for the disruption of the status quo. I re-read Descartes and this subsequent addendums to the Scientific Method in such a light, finding the latent content to be that of patriarchal positivism. Before I begin the investigation of mathesis universalis in this light, it is necessary to reinforce Morland and Horkheimer’s finding of process-orientation as symptomatic of modernity. 

Process as Technology and the Scientific Method

In his 1953 essay The Question Concerning Technology, Heidegger describes a two-fold definition of technology:

We ask the question concerning technology when we ask what it is. Everyone knows the two statements that answer our question. One says: Technology is a means to an end. The other says: Technology is a human activity. The two definitions of technology belong together. For to posit ends and procure and utilize the means to them is a human activity (Heidegger 312).

The process-oriented nature of technology is thus given. Heidegger later iterates the essence of technology in stating, “Technology is no mere means. Technology is a way of revealing” (318). As such, an examination of the latent cultural content of the basis of our technology, i.e. the Scientific Method, is due. In our time of such deep faith in the scientific method and its mathematical results, it is easy to forget that this method finds its basis in academic literature. This literature must be subject to the scrutiny of its cultural basis and the implications they could hold over our contemporary use of its findings. 

The teleological nature of the last sentence in Heidegger’s quote above gives us insight into how we might approach Descartes’ scientific method. If we are to understand the method as technology, that is, as process, we must uncover the cultural content that surrounds its literature. If Descartes created mathesis universalis, a process by which to accomplish the means of universalizing the findings of science, it can be seen as teleological in nature. Descartes desired a specific end, and thus created the means to accomplish it. He did so so successfully, it should be noted, that his technology was adopted as law, and irrevocably changed the very concept of technology itself. It is, after all, modern technology that is at stake in Heidegger’s essay. As he noted, modern technology 

moves us to ask the question concerning technology per se. It is said that modern technology is something incomparably different from all earlier technologies because it is based on modern physics as an exact science. Meanwhile we have come to understand more clearly that the reverse holds true as well; Modern physics, as experimental, is dependent upon technical apparatus and upon progress in the building of apparatus (319-20).

 Indeed the relationship between modern technology and the scientific method (that which posits physics as an exact science) is complex, and deserving of a clear synopsis and critique.

Descartes’ Rules for the Direction of the Mind and Discourse on the Method for Rightly Directing One’s Reason and Searching for Truth in the Sciences

For clarification, mathesis universalis and the scientific method are synonymous to Descartes. The two terms are used separately in the aforementioned works, but they carry the same connotations. Rules for the Direction of the Mind (Rules), unpublished during Descartes lifetime, pre-dates his Discourse on the Method for Rightly Directing One’s Reason and Searching for Truth in the Sciences (Method), the latter being the text in which he is best known for laying out his scientific methodology. It is in Rules, however, that Descartes crystalizes the need for mathesis universalis as germane to the unstructured nature of scientific research during his lifetime. This apparent need for structure begets some critique however, as it begs the question as to whom is generating the structure, and to what ends do they create such structure? What “given” norms are they supposing?  Cartesian dualism, the concept of the mind and body as separate entities, with the mind rightly having dominance over the puerile, self-serving body, is of key importance when considering such questions. With Descartes’ cogito, a re-awakening of Plato’s divide of the body and mind (TimaeusMain Sections I and III) came in the instance of the mind’s ability to doubt even its own body. Res cogitan(reason) is removed from res extensa(corporeal sensation). The body is the first of the exclusions that Descartes’ mathesis universalis will mandate. This initial schism warranted great criticism and in the next chapter will be discussed in greater detail as it is opposed by Julia Kristeva’s psychoanalytic investigation of language. What should be noted here is that an immersion of the mind within the sensations of the body as a way of knowing the world has been characterized as feminine in nature as opposed to the masculine, calculating nature of Cartesian dualism.  This characterization of femininity has its origins with in Book X of Plato’s Republic, in which emotions and, more specifically, the poetry that evokes them, are dubbed “womanish” and thus corruptive of the reason-driven, and thus stronger, male mind. Visceral emotions (grief, anger, laughter, sex, abjection, and other pains and pleasures) are chaotic, and beneath the goodness of reason and order. Descartes echoes Plato’s disdain for the un-orderly in this second of exclusions (emotions) from his mathesis universalis.

 Typically, Descartes’ demonstrative method is juxtaposed by that of more analytic scientists such as Isaac Newton, in that Descartes sought laws via a priori methods, rather than through observation and experimentation. This is however a gross over-generalization of Descartes methods as he in no way condemns knowledge via experience and experimentation, as can be seen throughout his works, although it is clear he requires that the subject proceed in an ordered manner, and within what he saw as the fixed nature of logic, which is to say discrimination of the complex in preference to the simple and self-evident [1] . It is not known whether or not Descartes was familiar with Occam’s razor (the simplest solution is usually correct), however, similar philosophy regarding the pursuit of knowledge via research can be found in the work of Thomas Aquinas and Aristotle. Nevertheless, Descartes cogito and his scientific method favor such economy of theory, giving us our first indication of preference to facility of process over the craftsmanship required of complexity.

 It is undoubtedly true that scientific discourse during Descartes’ time was chaotic. In his book, The Clockwork Univers: Isaac Newton, The Royal Society and The Birth of the Modern World,  Edward Dolnick paints a marvelous picture of the amalgam of superstitions and quack theories that passed for science, even after Descartes lifetime. Dolnick reviles Descartes himself as having crackpot theories, though credits him precision and veracity of methodology, stating, “Descartes was the ultimate skeptic … but he proposed a careful, scientific explanation for the well-known fact that if a person had been murdered and the killer later approached the victim’s body, the corpse ‘identified’ its killer by gushing blood”(Dolnick 54). Examples of this kind abound in French and English history, making it clear that causality, demonstrability, and the rationalization of theory were subjects of close debate. Descartes found the need for reining in scientific experimentation paramount, particularly if his own theories and experiments were to go untouched by the lunacy of his time.

 The litany of principles that comprises Rules serves not only to direct one’s thinking (indicating a primacy of the a priori), but also to direct one’s thinking about experience and experiments; it is a handbook for the ordering of one’s measurements (which can be gathered empirically or via thought experiment) insofar as those measurements are to be generated as facts. Returning to the criticism of Descartes’ scientific method as exclusory, Luce Irigaray makes a sharp assessment of the importance of measurement to mathesisuniversalis in her essay Wonder: Descartes, The Passions of the Soul, decrying Descartes for his dualism and non-immersive ways of gaining knowledge as patriarchal and phallocentric:

That which is great inspires esteem, magnanimity, even pride; that which is small inspires disdain, humility, even baseness. For Descartes, there is no magnanimity vis-à-vis smallness. This can be understood as an attempt to reduce smallness that has disappointed our wonder, or as the inability to admire the seed, that which is still being born, still becoming. Can this also be interpreted as the first determining of passion in terms of the quantitative? Yet sexual difference is not reducible to the quantitative even though it is traditionally measured by such standards: by more or less (Irigaray 108).

 In the last sentence, Irigaray references Plato’s philosophy of women as described in Book V of Republic, in which he describes women as similar in nature to men, but weaker in all ways. [2] A kind of exclusion from mathesis universalis occurs with such a quantitative mind-set, an idea that will be revisited later in this chapter with Heidegger’s essay, Modern Science, Metaphysics and Mathematics.

 Returning to Descartes’s Rules, knowledge, whether gained by thought or physical experimentation, should be demonstrative of generic laws that are present in generic, which is to say ideally simple, self-evident forms that are a priori. Thus Descartes calls for what scholar Fred Wilson refers to as the “law about laws.” He describes it as such,

This law about laws serves as an abstract generic theory, and it yields, in regard to any specific sort of situation falling under the genus, the conclusion that, for such a specific sort of situation, there is a law (this has been called a ‘Principle of Determinism’) and that this law will have a certain generic form and not any other sort of form (this has been called a ‘Principle of Limited Variety’) (Wilson 4).

 Descartes began his project of setting forth laws to govern logic, judgment, and the marshaling of research in order to ground such activities in light of his own theories of the thinking subject. The thinking subject as ever separate from its body (Cartesian dualism) must follow some sort of structure of activity, else it simply be allowed to fall into solipsism. With his famed cogito, Descartes comes very near to solipsism, a detriment when seeking absolute, external, essential truths, and his scientific method forces the mind into a structure of being that is non-immersive and, while allows for playful imagination, refutes imaginings that cannot fit the paradigm of “possible.” The scientific community and the subject required governing laws that would allow for proper conduct and dissemination of research, as he states succinctly in Rule IV, “There is need of a method for finding out the truth” (Descartes 5). He expounds on the anarchic state of research in his time, making clear the verity of Rule IV,

So Blind is the curiosity by which mortals are possessed, that they often conduct their minds along unexplored routes, having no reason to hope for success, but merely being willing to risk the experiment of finding whether the truth they seek lies there… This is the way in which most Chemists, many Geometricians, and Philosophers not a few prosecute their studies (5).

 When Descartes simultaneously declared the existence of the thinking subject and a need for laws about such thinking, he placed truth within the scope of measure, order, and meaning.  These three spheres are the essences of thinking, and, for Descartes, are the key to finding the essential truths of the inquiries of science. Drawing on the above named Principles of Determinism and Limited Variety, Descartes created Rules in kind, stating in Rule IV (See Appendix A) that there are laws to be found, and in Rules V and VI that these rules will be of a simple, generic, and fixed form. This last tenet of Rules, again recalls Plato. Timaeus, in which Plato proposes metaphysics based on similar ideas of fixedness, idealness, and purity of form to the exclusion of emotion and corporeality, is a clear predecessor to Rules and Method. [3] Thus the quest for mathesis universalis, a project Descartes, through his reliance on Platonism (ideal forms, in particular the arithmetic and geometric) begins with a solidly mathematical methodology. [4] With the development of the human sciences of psychology (i.e. Freud’s psychoanalysis theorizing the consciousness as subject to a unique personal history) and sociology (Marx and Weber’s theories that make posit consciousness as subject to a specific culture and system of power) and thwarting Platonic Ideals, all alongside Newton’s calculus, Eisntein’s Relativity and Bohr’s Quantum Theories, this project remains incomplete insofar as Descartes now rudimentary understanding of mathesis universalis does not account for the whole of understanding of human thinking, or the conditions of the a priori (to be later developed by Husserl, Heidegger and Foucault). To be sure, Descartes critique and methodology, while certainly applicable to all sciences, remain seated in his own needs as a mathematician. 

 It is the final of Descartes’ qualities of thinking, the fixed nature of logic and its forms, and the a priori, that philosopher Edmond Husserl first takes apprehension of throughout his body of work, beginning with Logical Investigations in which the nature of logic, that is, its fixed properties and source, is questioned. 

Husserl’s Logical Investigations—A Revision of Descartes

As mentioned above, the advent of the human sciences, transcendental idealism, and the various models of psychology all generated crises for Descartes’ conception of a mathesis universalis.  Where before the thinking subject (still a subject of God’s creation possessing an independent will, as described in the fourth discourse of Method) was still under the ironclad auspices of universal law, the “universal” nature of the subject was quickly coming under fire as a given truth. Even Universalist thinkers, such as Immanuel Kant, had to concede the sensus communus as pertaining to a society and not human kind as a whole. Thus Edmond Husserl took up the question of the universal nature of logic. In an echo of Descartes’ mission, Husserl sought to ground the conception of thinking (still confined to logic and reason) as the definition and source of logic had caused great schisms among the philosophers of his time. Unlike Descartes, Husserl worked within the phenomenological framework as he observed being as it appeared to the senses, rather than searching for fixed truths. It is also important to note that Husserl was working in direct criticism of mathematical psychologism—the idea that mathematical and logical concepts are grounded in psychological facts. That is to say, like Descartes, Husserl felt mathematical proofs to be transcendent of the psyche and based on truths outside the realm of subjectivity. Where they differ is in their approaches to observations about the conditions of logic, a subject that Descartes would have placed under the auspices of Rule II (See Appendix A), declaring it a subject beyond our mental capacity. It is not until Kant that our capacities stretch to thinking about our thinking. The work of the proceeding philosophers all owe a debt to Kantian subjectivity, as it has allowed for the vast meta-critiques of logic and mathesis that illuminate so much of epistemology and ontology.   

Husserl grounds his work in the ideas of intentionality and intentional content. The beginnings of these conceptions appear in Logical Investigations, wherein he first notes that logic has three different modes of direction: psychological, formal and metaphysical. In deciding that all of these forms have value (although Husserl clearly intends towards formal logic above all), and that one form will often move towards another, Husserl posits the question of logic in a more active, less Platonic fashion. Instead of logic being essential to mathematics (formal or pure logic) and thus to be applied to all forms of thinking and observation as Descartes would have done, Husserl maintains that all of the forms of logic can be placed within a context of intentionality. He states,

We must not overlook the fact that generality of meaning—whether this be generality of meaning-intention or generality of meaning-fulfillment—is something felt as immanent in each individual case where a general name is understood, and is applied to intuition in accordance with its sense, and that it must be the manner of consciousness (logic), the manner of our intention, that makes the difference (Husserl 90).

Husserl states that logic is directed towards some object (the intentional object) even if the object has no basis in reality, and possesses “intentional content.” Intentional content should not be understood as limited to being an act towards some object (as one intends to purchase an object or to study a topic) but also pertains to phenomenological observation and possession of meaning (one intends the furry, four-legged creature as a cat). While this theory of intentionality has been subject to much criticism, it has a direct link to mathesis universalis in that one can initially only intend towards certain objects as one might already possess them, as in objects of pure logic (mathematical formulae, for example)—a concept that is elaborated by Martin Heidegger in his essay Modern Science, Metaphysics and Mathematics. What Husserl has accomplished is giving an active, subjective quality to logic, disposing of Descartes’ passive and Platonic views of logic, which would in turn change the conditions of mathesis from a universal to a constructed nature. In the next section, as Heidegger works to uncover the historically situated nature of mathesis, he reflects on the nature of mathesis, adding yet another unfixed dimension to our understanding of logic.  

Heidegger’s Modern Science, Metaphysics and Mathematics

Throughout his career-long effort to situate the subject in terms of the ever-changing truth of the world, Heidegger placed Dasein at the center of his examinations of the various disciplines of human thinking. Predicating much of his work as a response to Cartesian subjectivity, as well as Husserl’s conceptions of time and intentionality, Heidegger defines Dasein as a thinking being, but a Being of the world and of its time, and thus the question became: how much of Dasein’s world is subjectively constructed, and historically constructed? Immersed in the culture of nuclear physics and the beginnings of quantum physics (Einstein and Bohr were certainly well known to Heidegger), and witnessing the exponential growth of technology, Heidegger took up the project of clarifying Dasein’srelationship to the bedrock of the era: mathematics.

In his 1935 essay, Modern Science, Metaphysics and Mathematics, Heidegger seeks to uncover our understanding of mathematics as an epistemic force. His work is of the first in philosophy to question modern science without resorting to nostalgia or calling for technological asceticism. He calls for an examination of what we understand as “Modern Science” as it opposes “Ancient Science” with the understanding that it is our relationship to technology that is to be reflected upon, not the option for or against technology, as Heidegger was well aware that humanity had moved beyond such choices. In the uncovering of the ancient understanding of science, the right-ness and naturalness of our modern relationship to science and technology is presented as non-universal and incomplete. The work opens the door for subsequent criticism of the Cartesian models of modern science and technology on the scores of the Kantian schema (universals belonging to particulars or vice versa). This first crack in the solidarity of Cartesian science is the historical beginning of the carving away of the layers of exclusion in a supposedly “universal” methodology; a necessary step in our understanding of Cartesian patriarchy as it pertains to the sciences.

Heidegger describes the difference between the two epochs in terms of their mathematical projection, the former being under the auspices of “facts” and the latter under the category of “concepts.” Both ancient and modern sciences develop with experimentation and the measurement of their results, but the experiments are measured with a kind of privilege to either facts or concepts—one is always given priority over the other. Through this work, Heidegger makes clear his opposition to the constraints of the axiomatic nature of modern science, and lobbies for a more conceptual, qualitative mathematical projection, as opposed to Descartes’ and Husserl’s belief that mathematical axioms are beyond the scope of subjective logic. For clarification, the term axiomatic has a dual meaning: (1) a self-evident truth that requires no proof; (2) a universally accepted truth or law. Through the course of this work, both definitions will be utilized, and, in fact, their relationship to each other as definitions will be questioned. At this point, it is sufficient to say that Martin Heidegger’s work largely concerns the former of the definitions in that he is searching for axioms developed via Dasein’s (the subject’s) mathesis or self-evident knowledge. It is worth noting that Heidegger has dropped the qualifier “universalis” in his use of mathesis, as the idea of any kind of fixed universal has been extinguished by his conceptions of Dasein as a being ever changing over time, and subject to its own Being-as-such.

He begins with what, in his view, is the fundamental separation between ancient and modern science—the philosophical disposition versus positivism. He writes,

They (sixteenth and seventeenth C. scientists) understood that there are no mere facts, but that a fact is only what it is in the light of the fundamental conception, and always depends upon how far that conception reaches (Heidegger 272).

 Heidegger makes it clear that the ancient philosophers, by working their experiments in regards to the over-arching concept rather than the objective study of facts, understood that their own preconceptions shaped their “mathematical projection,”

The manner of experimentation is presumably connected with the kind of conceptual determination of the facts and way of applying concepts, i.e., with the kind of preconception about things. Besides these two constantly cited characteristics of modern science, science of facts and experimental research, one also usually meets a third. This third affirms that modern science is a calculating and measuring investigation (272).

From this point Heidegger unpacks his own term of mathematical projection.  A decisive line is drawn by Heidegger between the mathematical and mathematics, concepts that in our current episteme are synonymously used. This split is initially made in the form of a criticism of Kant’s misuse of the term “mathematics.” [5]   Heidegger argues that what we understand as mathematics is truly only a manifestation of the mathematical in that it is dependent upon one’s preconceptions, as Husserl has demonstrated and as Michel Foucault will further develop. Heidegger makes this argument with an etymological focus on the term mathematical as it stems from the Greek, ta mathemata, which simultaneously means what can be learned and what can be taught. Heidegger, seeking a more-active, less-fixed sense of the word, connects the term mathemata with its inter-related disciplines of physica (things as they are themselves),poioumena (things as they are made by human hand), chremata (things as they are at our disposal), and pragmata (things as we have to do with them—the mode of use or consumption). [6]   In order to break out our preconceptions of what mathematics truly is, Heidegger not only disconnects mathematics from any specialized field, but goes so far as to disconnect numbers from mathematics and declares them to be mathematical on the premise that one must already know numbers to understand numbers. [7] This is the essence ofHediegger’s mathemata, “the things insofar as we take cognizance of them as what we already know them to be in advance, the body as bodily, the plant-like of the plant, the animal like of the animal, the thingness of the thing, and so on” (275). Mathemata is thus presented as the object of mathesis, or learning.

The distinction must be made here that Heidegger’s conception of mathesis is not Platonic in nature, as he gives no indication of the eternal fixedness of the mathemata. The ever-active state of learning what is to be learned is emphasized by the story of Socrates and the Sophist. When Socrates is chided by the Sophist for asking the same questions, Socrates retorts with “But you who are so extremely smart, you never say the same things about the same thing”(276). Aside from the Socratic discipline of examination, Heidegger chooses this story in order to stress that the mastery of mathemata is never a complete process, that as the thing to be learned changes through time and context, the mathesis is also changing as Dasein changes in its being.  It is critical to iterate that mathemata are taken in by taking cognizance of what we already have in regards to the mathemata, i.e. what is taken up is self-evident—recalling the first definition of axiom. This is the essence of the mathematical for Heidegger, that mathesis is a mastery over oneself as well as the mathemata, insofar as what is self-evident is what we already have to take-up. 

In order to divorce our vernacular understanding of mathematics from mathesis, Heidegger dispels the myth of numbers as constituting mathematics. Inevitably numbers are the first things associated with mathematics, as “…numbers are the closest to that which we recognize in things without deriving it from them” (277). As such numbers belong to the learnable, mathemata, and are subject to learning, mathesis, making them mathematical in nature and not vice-versa. The modern meaning of the “mathematical,” to Heidegger, is the active state of our moving towards mathemata insofar as we take them up as things already possessed by us. It is thus that Heidegger begins his description of Modern Science as mathematical in nature, rather than based on mathematics proper. One must note here that in his quest for more qualitative types of mathesis, Heidegger refers back to the idea first established by Descartes and refined by Husserl, that mathesis universalis is composed of measurement, order and meaning, but is proposing that we find measurement via means other than numerical, as the numerical is only a secondary characteristic of mathematics.

Heidegger shoulders Isaac Newton (and more subtly, Immanuel Kant) with the first burdens of modern thought yet does so without refuting his earlier laid principle that we learn what we already have. [8]   He describes Newton’s first principle of the Law of Motion as being existent, yet not fully-formed, in the works of Galileo and Democritus, thus making clear that modernity is not born of original thought, but that modernity comes from the mastery of what was historically already known. [9] He writes, “All great insights and discoveries are not only usually thought by several people at the same time, they must also be rethought in that unique effort to truly say the same thing about the same thing”(281). In this observation, he suggests a kind of a priori knowledge granted to Dasein, knowledge that is a part of Dasein’s being-in-the-world. The allusion to this precedes Foucault’s episteme, yet moves along the same horizons, as I discuss here elsewhere.

On this same line, Heidegger makes it clear that although Aristotle belongs to the Ancient Sciences, his viewpoint on the nature of things is indeed phenomenological, yet still based on principles rather than facts. Heidegger discusses how Aristotle’s conception of physical bodies in motion still relies on conceptions of the space in which it moves rather than of observation of the physical facts. The organization of space in Ancient Greece is key to understanding where the difference between Aristotle’s and Newton’s thinking begins. In Ancient Greece, space was stratified by the elements that inhabit it. The general cartography of how the elements occupied space was perceived as a four layer stratum of (top to bottom) fire, air, water, and then earth. While no doubt this layout of the world was conceived on observation, it was not codified by verification of facts, but was created by placing observations under the auspices of Aristotle’s preconceptions of the four elements. Heidegger writes:

Each body has its place according to its kind, and it strives toward that place. Around the earth is water, around this, the air and around this, fire—the four elements. A rock falls down to the earth, However, if a rock is thrown upward by a sling, this motion is essentially against the nature of the rock, para physin. All motions against nature are biai, violent. (284)   

Heidegger’s discussion of Aristotle’s elemental topography illustrates the make-up of the historical episteme of Aristotle’s time.  Prior to the Enlightenment, epistemes, which are foundations of knowledge and of the generation of knowledge that are unique to a culture and time, were formed under the auspices of concepts, and the facts observed were squarely organized in full support of such concepts. Aristotle’s four elements, while no longer a guiding force of knowledge, have left traces within our modern episteme, and thus must be uncovered in order to discern meaning of our own construction of reality. [10] Michel Foucault’s four-fold approach to generating signs (discussed in detail below) has vestigial traces of Aristotle, thus illustrating the historical nature of our understanding of the essences of understanding. Key to the elemental construction of space is the concept that beings are either in proper state of Being or they are moving towards their proper state of Being, ideas that have an ancestral relationship to the Modern, Newtonian conceptions of bodies in space.     

By way of demonstrating the shift from the tailoring of observed facts to over-arching concepts to the construction of concepts via observed facts, Heidegger discusses Newton’s simplification of Aristotle’s framework of the motion and placement of bodies. The shift towards modernity can be made clear via eight changes, all contained within Newton’s axiom:

Corpus omne perseverare in statu suo quiescendi vel movendi uniformiter in directum, nisi quatenus a viribus impressiscogitur statum illum mutare (a body at rest will remain at rest, and a body in motion will remain in motion with a constant velocity, unless acted upon by a force) (Newton 10).

The distinctions between Aristotle and Newton are as follows:

1.   Newton unifies bodies of all types (bodies in space or on earth, bodies of all make-up) with the phrase corpus omne (every body).
2.   Newton refers to all types of movement, whether biai or physin.
3.   Newton refers to all types of space, propriety to the type of Being is no longer a consideration.
4.   The essence of force is determined by the fundamental law of motion, rather than the reverse. 
5.   Motion is seen not as a change of a type of space, but a measurable change of position.
6.   The motion is no longer considered in respect to nature (there is no more violence against nature via a motion, there is simply motion).
7.   Nature is no longer the guiding principle of physics, but is instead is the mode in which the bodies find themselves moving in.
8.   Thereby the manner of questioning nature also changes and, in a certain respect, becomes the reverse (Heidegger 289).

It is with this essential reversal of the construction of motion that Heidegger illustrates the historic shift of mathesis made possible via the Cartesian revolution. Recalling that a condition of the thinking subject is that dogmatic principles are no longer the framework for facts, that it is facts that are the framework for laws, Heidegger has demonstrated that historic knowledge, while in the current time can be perceived as erroneous, still has value when understood in such a reversed fashion. When properly uncovered, the laws of historical mathesis are still relevant. To that same degree, the exclusions of the mathesis universalis are made clear when juxtaposed to historical models. Susan Bordo reminds us of the patriarchal nature of such exclusions, and the incompleteness of the modern model owing to such exclusions. She observes, “The specific consciousness we call scientific, Western and modern, is the long sharpened tool of the masculine mind that has discarded parts of its own substance, calling it ‘Eve,’ ‘female,’ and ‘inferior’” (Bordo 65).

In such a manner, Heidegger has described mathematics as nothing more than a manifestation of the mathematical—that is, that the field of mathematics and the axioms that compose mathematics, are predicated on historical knowledge (in this case about the uniformity of space, motion and bodies). He further describes the mathematical as:

1.   A project in which facts show themselves and already exist.
2.   A project in which things are taken as they are evaluated to be beforehand. These facts that are taken-for beforehand are the axioms of mathematics (axioo in Greek). Axioms are fundamental principles that are given beforehand.
3.   The axiomatic is anticipatory of the facts.
4.   The axiomatic provides the blueprint for the realm in which bodies exist. “Nature is now the realm of the uniform space-time context of motion, which is outlined in the axiomatic project and in which alone bodies can be bodies as a part of it and anchored in it” (Heidegger 292).
5.   The axiomatic, finally, determines the mode of taking-in and taking-for that governs the experiments and facts derived there-from. Things can only show themselves insofar as they can be seen within the axiomatic project.
6.   The axiomatic thus provides the system of measurement by which things may be sorted and compared. In Newton’s case, the uniformity of space, time, and bodies allowed for uniform measurements (numerical measurements).  

The new form of modern science did not arise because mathematics became an essential determinant, rather, that mathematics, and a particular kind of mathematics , could come into play and had to come into play is a consequence of the mathematical project (293). 

This last sentence iterates the fact of process-domination in modern science. The form of science has been made beholden to Cartesian methodology. Heidegger has shown how the axiomatic (the mathematical) has shaped modern thinking in regards to all of the sciences, a practice he casts doubt upon in such essays as The Question of Technology. The ethics of the various types of mathesis, while certainly a significant discussion, serves only to interrupt our understanding of the philosophical development of mathesis, so I will leave the moral implications of this topic to another section. Suffice it to say for the moment that process-domination serves the patriarchal status-quo in that it excludes the methods of knowing, learning and teaching of those without access to academic, meaning canonized, forms of mathesis, i.e. women, minorities, etc.  Returning now to the original pursuit of the significance of mathesisuniversalis, we explore what the modern mathematical mind means to Dasein.

Heidegger makes known his weariness of mathematical thinking in his essay “What is Metaphysics?” He criticizes Plato’s exaltation of the rational measurability of mathematics when he writes, “Mathematical knowledge is no more rigorous than philosophical-historical knowledge.  It merely has the character of ‘exactness’ which does not coincide with rigor” (Heidegger, 93). It with this in mind that we return to Heidegger’s essay on mathematics and the historical reasoning he gives for the turn of modern thought to the mathematical.

Heidegger credits Descartes with sowing the seeds of modernity and Newton with harvesting its first fruits. Descartes’ crystallization of the subject, Heidegger states, came along precisely during the time that the mathematical was, itself, solidifying in opposition to the theologically based metaphysics of the pre-Renaissance. These shifts are not coincidental. Heidegger argues that in the very core of our understanding of the subject lay the ultimate act of mathesis. Citing Descartes’ Rules, (referred to by its Latin title Regulae asdirectionem ingenii), Heidegger finds a reversal of subjectum and objectum, which he characterizes as the seat of our modern understanding of the subject and object. Of Descartes’ Rules, Heidegger [11] finds three that best illustrate the mathematical nature of Descartes’ construction of the subject:

·      Regulae III: “Concerning the objects before us we should pursue the questions, not what others have thought, nor what we ourselves conjecture, but what we can clearly and insightfully intuit, or deduce with steps of certainty, for in no other way is knowledge arrived at.”

·      Regulae IV: “Method is necessary for discovering the truth of nature.”

·      Regulae V: “Method consists entirely in the order and arrangement of that upon which the sharp vision of the mind must be directed in order to discover some truth. But we will follow such a method only if we lead complex and obscure propositions back step by step to the simpler ones and then try to ascend by the same steps from the insight of the very simplest propositions to the knowledge of all others.”

In just these three rules, one can see the mathematical couching of the subject as it seeks to understand the objects of its “thinking.” Heidegger elucidates that the “I” of Descartes is void of inference, but is instead an active subject, one in constant use of reason— “The sum is not a consequence of the thinking, but vice versa; it is the ground of thinking (Heidegger 302).” According to Heidegger, this miscalculation is precisely the mathematical at work. He writes:

The character of the ego, as what is especially already present before one, remains unnoticed. Instead, the subjectivity of the subject is dteremined by the “I-ness” [Icheit] of the “I think.” That the “I” comes to be defined as that which is already present (the objective in today’s sense) is not because of any I-viewpoint or any subjectivistic doubt, but because of the essential predominance and the definitely directed radicalization of the mathematical and the axiomatic” (303).

Mathesis, for Heidegger, begins and is founded, not upon the fixed axioms of already known mathematics, but upon the subject—the subject (Dasein) who is taking in what it already has. Dasien is subject to the propositions that stem from the mathematical nature of our modern thinking, which is fundamentally based on mathesis. First and foremost of the proposition is not “I think”, but “I posit.” The proposition is the subject of its own proposition; it is Dasein taking purely itself. Yet in Heidegger’s broader philosophy as laid out inBeing and Time, the subject is a product of its Being-as-such (Befindlichkeit), which widens the scope from that of the Being, to the Being-in-the-world, framing the question of mathesis in more sociological terms. What is the result of a society composed of Dasein as it takes in what it already possesses? Michel Foucault attempts to broaden mathesis in exactly this manner.

From Heidegger, we can trace the course of modern thought to the simultaneous crystallization of the subject and mathematical thinking with Descartes. But as philosophers of the Enlightenment and beyond began to work within this new episteme, the isolation of mathesis, that is of mathematical thinking, could not remain limited to the subject, or even to the intentionality of the subject as Husserl posited. Such a shift of paradigms not only begets a new paradigm, but consideration of previous paradigms as well and the reasons for their apparent demise. Descartes awoke not only the question of the subject “I,” but also the question of what “I” was before, and projections of what “I” will become in the future. It is here, I show next, that the question of aesthetics becomes key. When the constructed nature of past thinking is made clear, one cannot but help extrapolate upon how current thinking must also be constructed, and thus generate “future thought” upon what is flawed about the “now” construction of knowledge and how it will too, meet its ultimate shift to a new paradigm. But questions of this nature cannot be isolated to the ontology of Heidegger, as the scope of the individual, Dasein, is no longer centrally relevant when considering human beings as a society. Michel Foucault broadens the question of the thinking individual when he states, “Ultimately, the problem that presents itself is that of the relations between thought and culture: how is it that thought has a place in the space of the world, that it has an origin there, and that it never ceases, in this place or that, to begin anew?” (Foucault 50).  

Foucault’s Mathesis as in The Order of Things: An Archeology of the Human Sciences

In his book, The Order of Things: An Archeology of the Human Sciences, Michel Foucault mines the history of human sciences (psychology and sociology, in particular) in order to shed light on the nature of our knowledge of ourselves, and how we construct the knowledge of ourselves as societies. He makes the overall argument that it is not the measuring aspect of mathesis that is ultimately responsible for generating facts, but the process of ordering that bears the most weight when generating meaning to create facts. Thus far in this chapter we have discussed Rene Descartes’ initial call for mathesis universalis in Rules, which gives primacy to the mathematical processes of measurement, Edmond Husserl’s conception of intentionality and the nature of logic as varied in its source, thereby eliminating the universal quality of Descartes mathesis, and seen Martin Heidegger’s historical situating of mathesis as transcendental knowledge shaped by either an over-arching concept or by the arrangement of observed facts that generate laws, as well as his plea for more qualitative conceptions of mathesis. Foucault answers this plea in the favoring of the process of ordering (recalling that mathesis consists of measurement, order, and meaning) in mathesis, as well as providing historical evidence of ways of alternative ways of knowing and ordering. Foucault’s assembly of this theory as it serves to historically situate the contemporary episteme that the target of his criticism. Such an examination makes clear my assertion that the Cartesian scientific method, while striving for universality of the “objective” sciences, is in fact exclusive of immersive ways of knowing, and thus exclusive of those who practice un-canonized traditions, such as oral traditions of knowledge or meditative practices. The limitations put upon science by Descartes were designed to exclude the abject, the instinctual, and the experiential as viable means of knowing. I show that his doing so has created a rift between minority and majority in the various fields of science, one that continues even today. Such inequities force the path of epistemic development to serve the purpose of maintaining the patriarchal status quo, a status quo that has not always, historically, been the case. If an alternative is imaginable in history, as Foucault demonstrates next, then, surely, an alternative is imaginable for future societies. 

Foucault posits that each society, in various times, generates their epistemology under a specific grid of knowledge, known as anepisteme, as defined earlier in this chapter. This idea is similar to Thomas Kuhn’s conception of a paradigm which, he refers to as methodological conception (involving how things are done, reflecting the process driven society). Foucault’s episteme refers to an epistemological conception (concerning how things are known, reflecting a once-product oriented society). The episteme can be seen as the magnified concept of the axiom, when comparing Heidegger’s concept of mathesis, in the Heidegger’s axiom is a truth that is known among a society, while Foucault’s episteme is a pre-consciousness that is known throughout a culture. Foucault discusses how systems of knowledge are formed and then transformed over time. His analysis of the human sciences begins in the sixteenth century, as it was then with Descartes that the human sciences proper began, and it is that period’s episteme he first elucidates in order to understand the discontinuities that occurred with Cartesian thinking.  

Beginning his discussion, Foucault describes the epistemic foundation of sixteenth century as being one of similitudes. He posits that rather than the process of elimination and discrimination that has become the crux of modern mathesis via Descartes’ scientific method, the sixteenth century episteme consisted of finding resemblance and similitude. He discusses four key types of resemblance:convenientiaaemulatio, analogy, and sympathyConvenientia, “denotes the adjacency of places more strongly than it does similitude,” implies that objects adjacent to one another complement one another, a kind of connectivity that assures that all objects are in their space proper as their motions, syntax and even forms complement one another in space (18).  Aemulatio, is an idea similar toconvenientia, but is not limited to space. Foucault writes, “The relation of emulation enables things to imitate one another from one end of the universe to the other without connection or proximity” (19). Analogy, is the superimposition of either convenientia or aemulatioupon objects via the grid of knowledge that the subject possesses. Foucault describes how a sixteenth century and a nineteenth century analogy have special kind of validity for their understanding of the universe as it changes throughout the various epistemes. Sympathy, the freest form of resemblance, involves play and power. Foucault writes:

Sympathy is an instance of the Same so strong and so insistent that it will not rest content to be merely one of the forms of likeness; it has the dangerous power of assimilating, of rendering things, identical to one another, of mingling them, of causing their individuality to disappear—and thus of rendering them foreign to what they were before (29).

Because sympathy has the power to reduce the world to homogeneity, an idea that would run in direct contradiction to Descartes’, Husserl’s, and Heidegger’s conceptions of the subject, according to the Cartesian model, it must be paired with its antithesis, antipathy. It is thus that our first inklings of discrimination and elimination of possibility via communion or immersion is born. Foucault describes the necessity of such disconnection in the current Cartesian episteme, “Antipathy maintains the isolation of things and prevents their assimilation; it encloses every species within its impenetrable difference and its propensity to continue being what it is” (24).

So described, sympathy must be made nearly criminal if one is to uphold modern science. Iterating this point via feminist critique, SusanBordo describes sympathy as dangerous,

If the key terms of Cartesian hierarchy of epistemological values are clarity and distinctness—qualities that mark each object off from the other and from the knower—the key term in the alternative scheme of values might be designated as sympathy. Various writers have endeavored to articulate such a notion. Henri Bergson names it ‘intellectual sympathy’ and argues that the deepest understanding of that which is to be known comes not from analysis of parts but from ‘placing oneself within’ the full object and allowing it to speak (Bordo 64).

It is not just to oppose Cartesian methodologies that Foucault and others have considered sympathy in the wake of exclusion. A meta-dialogue about technology and its repercussions on society, such as that in Hediegger’s essays, The Question Concerning Technologyand Modern Science, Metaphysics and Mathematics offers a reminder that we may not simply drop or opt out of the modern condition of science, rather that we have it in our power to imagine and enact beyond the modern episteme. It is clear Foucault has uncovered sympathy in answer to Heidegger’s call for qualitative-ness. What he is describing is not dissimilar to Heidegger’s conception of mathesis, as all the forms of resemblance are a taking-up of what is given, of what one already has, and applying it to the contemporary episteme. Foucault states, “Convenientiaaemulatio, analogy, and sympathy tell us how the world must fold in upon itself, duplicate itself, reflect itself or form a chain with itself so that things can resemble on another” (Foucault 26). The distinction is in the field of application. Heidegger is interested in the mathematical nature of our contemporary metaphysics, while Foucault turns his focus on the construction of our contemporary epistemology and how it relates to ideas of normalcy of the human being. As metaphysics informs epistemology, the two disciplines cannot but resemble one another. Foucault closes the circle of his resemblances with the concept of signatures. Resemblances, in order to be knowledge producing, must go beyond the surface of their likeness. Foucault explains: “Resemblance was the invisible form of that which, from the depths of the world, made things visible; but in order that this form may be brought out into light in its turn there must be a visible figure that will draw it out from its profound invisibility” (26).  Further in the text, Foucault clarifies the nature of the signature: “The signature and what it denotes are of exactly the same nature; it is merely that they obey a different law of distribution; the pattern from which they are cut is the same” (29).

As he describes the episteme of the sixteenth century, Foucault is emphasizing the unifying principle that was the aegis of language. Language was the system of understood signatures of similitudes prior to the Cartesian revolution. Put simply, what was seen resembled what was read. When the primacy of the written (read printed) word that came about in the seventeenth and eighteenth centuries with the advent of the printing press, a discontinuity between the seen and the read occurred. He writes:

Things and words were to be separated from one another. The eye was thenceforth destined to see and only to see, the ear to hear and only to hear. Discourse was still to have the task of speaking that which is, but it was no longer to be anything more than what it said (43). 

It is at this moment the political nature of Descartes’ universalization becomes most evident. Those who are without access to “what is said” are excluded from discourse. This excluded group would exclude the illiterate, the uneducated, and those who maintained un-canonized forms of history.

Returning to Descartes’ Rules, we find the exact moment of the abandonment of similitudes. In the subtext of Rule III, Descartes disavows similitude: “It is a frequent habit when we discover several resemblances between two things, to attribute both equally, even on points in which they are in reality different, that which we have recognized to be true of only one of them”  (Descartes 2).

From this point in history, the unifying quality of language via similitudes was lost, and the categorization and measurement of difference became the mode of language and meaning, and thus of knowledge. Of our contemporary times, Foucault discusses two people with whom the significance of similitude is not lost: the madman and the poet. The madman practices homosemanticism, the nonsensical gathering together of similitudes that can never reach an endpoint, and will eventually void the meaning generated by the signs. The poet practices allegory, seeking out similitudes in the signs that speak them. Both are on the fringes of society, according to Foucault: “Between them there has opened up a field of knowledge in which, because of an essential rupture in the Western world, what has become important is no longer resemblances but identities and differences” (Foucault 50).

The process by which we identify differences is comparison of two varieties—of measurement and of order. Measurement is comparison of the object to an original unit and thus dividing the object into multiplicities. Ordering is a comparison of simplistic elements that are shared between objects, and then an arrangement of complexity. The more complex an object, the more differences can be construed from its counterpart in the comparison. This is not to say that there is a base unit for comparison, but that instead a relative characteristic or function of the object is the determining grounds for finding difference. And it is thus that completes the revision of the episteme of Western Culture. According to Foucault, the moment Descartes gave primacy to mathematics (a system that requires no similitude) resemblance passed into the fringes of society (recall the poet and the mad man). He writes, “Resemblance, which had for long been the fundamental category of knowledge—both in form and content of what we know—became dissociated in an analysis based on terms of identity and difference” (54). The modification to the episteme can be summarized by the following four points:

1.   Comparative analysis is substituted for the organization of resemblances.

2.   A finite enumeration of comparisons is established.

3.   With the finite enumeration of comparisons, an absolute knowledge of identities and differences can be known.

4.   To know is to discriminate, and thus history and science will divorce from one another.

It is at this point that Foucault begins his critique of mathesis, criticizing Descartes’, Husserl’s and Heidegger’s insistence that mathesis be solely connected with measurement. He describes it as:  

[A] universal science of measurement and order. Under cover of empty and obscurely incantatory phrases ‘Cartesian influence’ or ‘Newtonian Model’ our historians of ideas are in the habit of confusing these three things and defining Classical rationalism as the tendency to make nature mechanical and calculable (56).  

Foucault makes clear the separate nature of measurement and order insofar as there are certain objects of life that cannot be measured. Objects, however, can always be ordered. He states:

The fundamental element of the Classical episteme is neither the success or failure of mechanism, nor the right to mathematize or the impossibility of mathematicizing nature, but rather a link with the mathesis which, until the end of the eighteenth century, remains constant and unaltered. The link has two essential characteristics. The first is that relations between things are indeed to be conceived in the form of order and measurement, but with this fundamental imbalance, that it is always possible to reduce problems of measurement to problems of order. So that the relation of all knowledge to the mathesis is posited as the possibility of establishing an ordered succession between things, even-non measurable ones (57).

Michel Foucault finds a particular tie to aesthetics and order as germane to mathesis. Through literature in his preface, Foucault develops an example of how the artist/poet acts as agent to the disruption of established order. It is found in the poet Borges’ listing of a bizarre taxonomy of animals found in a “Chinese encyclopaedia.” [12] In this list, Foucault identifies a disruption of what he calls the common tabula—the space (or table) in which one finds sensible order. He describes the disruption thusly: “The uneasiness that makes us laugh when we read Borges is certainly related to the profound distress of those whose language has been destroyed: less of what is ‘common’ to place and name. Atopia, aphasia” (XIX).

As discussed throughout this chapter, order is a key factor of mathesis—one that until Foucault has received little critical attention as compared to its counter-part of measurement. As Heidegger perceives, the nature of space was of the utmost concern for understanding the known universe. The question of thinking and communicating was taken as a universal and granted quality. As a result Foucault claims, “There would appear to be then …. a culture entirely devoted to the ordering of space, but one that does not distribute the multiplicity of existing things into any of the categories that make it possible for us to name, speak, and think (XIX).” Borges has found an instance in which the order of things, un-granted by mathematical measurement or established linguistics, has been disrupted and the constructed nature of order has become evident. Subsequently, an alternative to measured, ordered and linguistic understanding is found. An immersive kind of knowledge is conveyed. One can imagine the taxonomist encountering these criteria perhaps by memory, or simply throughout the course of their day—that is to say, organically, un-masculine-ly, the kinds of animals are gathered, rather than ordered, experienced rather than measured.

Conclusion

As with any argument, the risk of accusation to no benefit of improvement abounds. As I make it clear that the Cartesian scientific method upholds patriarchal discourse via its resemblance to Platonic sexism, it is not to blatantly accuse Descartes alone of patriarchy. Descartes was subject to his own episteme, one such that included inequities of the sexes. What should be made clear by this discourse is that there is historical proof that alternatives to our modern ways exist and they need not necessarily be in opposition. In fact the generation of a dichotomous relationship between ways of knowing is precisely what must be avoided. The next chapter will discuss how Julia Kristeva’s theory of the mathesis of ambiguity serves to improve upon mathesis universalis. The third chapter will offer a dialogical analysis of the two via the performative evaluation of Alan Sokal’s Hoax.

Works Cited

Aristotle. Poetics. Trans. Malcolm Heath.  London: Penguin, 1996.

---. Nichomachean Ethics, translated by Terence Irwin. New York: Hackett Publishing, 1999. 

Bordo, Susan editor Feminist Interpretations of Rene Descartes. University Park, PA:  Pennsylvannia University Press, 1999.

Descartes, Rene. “Rules for the Direction of the Mind.” Great Books of the Western World      edited by Robert Maynard Hutchins. Chicago: University of Chicago Press, 1952. PP ix-40.

Heidegger, Martin. Being and Time. Translated by Joan Stambaugh and Dennis J.Schmidt. New York: State University of New York Press, July 2010.

---. Basic Writings edited by David Farrell Krell. San Francisco: Harper, 1993.

---. Introduction to Metaphysics translated by Gregory Fried and Richard Polt. New Haven: Yale University Press, 2000.

Husserl, Edmond. Logical Investigations, Volume I. New York: Routledge, 2008.

Jaspers, Karl. Kant: Volume I of The Great Philosophers. New York: HBJ Publishers, 1962.

Kristeva, Julia. Powers of Horror: An Essay on Abjection. New York: Columbia University Press, 1980.

Newton, Isaac. Philosophiae Naturalis Principia Mathematica. Translated by I. Bernard Cohen. Berkely: University of California Press, 1999.

Plato. Republic. Translated with an Introduction by C. D. C. Reeve. Indianapolis:

       Hackett Pub. Co, 2004.

Shlain, Leonard. Art &Physics: Parallel Visions in Space, Time and Light. New York: Harper-Collins, 1991.

Steinberg, Leo. Velázquez' ‘Las Meninas. October, Vol. 19 (Winter, 1981), pp. 45-54.

Stempel, Daniel.  “Blake, Foucault, and the Classical Episteme.” PMLA, Vol. 96, No. 3 (May, 1981), pp. 388-407

Wilson, Fred. Rene Descartes: Scientific Method. University of Toronto: IEP. Last updated: February 9, 2008 | Originally published: February/9/2008.  http://www.iep.utm.edu/desc-sci/

Appendix A- Descartes’ “Rules for the Direction of the Mind, Rules I-XIII”

1.   The end of study should be to direct the mind towards the enunciation of sound and correct judgments on all matters that come before it.

2.   Only those objects should engage our attention, to the sure and indubitable knowledge of which our mental powers seem to be adequate.

3.   In the subjects we propose to investigate, our inquiries should be directed, not to what others have thought, nor to what we ourselves conjecture, but to what we can clearly and perspicuously behold and with certainty deduce; for knowledge is not won in any other way.

4.   There is need of method for finding out the truth.

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 step.

6.   In order to separate out what is quite simple from what is complex, and to arrange these matters methodically, we ought, in the case of every series in which we have deduced certain facts the one from the other, to notice which fact is simple, and to mark the interval, greater, less, or equal, which separates all others from this.

7.   If we wish our science to be complete, those matters which promote the end we have in view must one and all be scrutinized by a movement of thought which is continuous and nowhere interrupted; they must also be included in an enumeration which is both adequate and methodical.

8.   If in the matters to be examined we come to a step in the series of which our understanding is not sufficiently well able to have an intuitive cognition, we must stop short there. We must make no attempt to examine what follows; thus we shall spare ourselves superfluous labor.

9.   We ought to give the whole of our attention to the most insignificant and most easily mastered facts, and remain a long time in contemplation of them until we are accustomed to behold the truth clearly and distinctly.

10.        In order that it may acquire sagacity the mind should be exercised in pursuing just those inquiries of which the solution has already been found by others; and it ought to traverse in a systematic way even the most trifling of men’s inventions though those ought to be preferred in which order is explained or implied.

11.        If after we have recognized intuitively a number of simple truths, we wish to draw any inference from them, it is useful to run them over in a continuous and interrupted act of thought, to reflect upon their relations to one another, and to grasp together distinctly a number of these propositions so far as is possible at the same time. For this is a way of making our knowledge much more certain, and of greatly increasing the power of the mind.    

12.         Finally we ought to employ all the aids of understanding, imagination, sense and memory, first for the purpose of having a distinct intuition of simple propositions; partly also in order to compare the propositions to be proved with those we know already, so that we may be able to recognize their truth; partly also in order to discover the truths, which should be compared with each other so that nothing may be left lacking on which human industry may exercise itself.

13.        Once a “question” is perfectly understood, we must free it of every conception superfluous to its meaning, state in simplest terms, and, having recourse to an enumeration; split it up into the various sections beyond which analysis cannot go in minuteness.

 

 

 

 

 

 

[1] For example, the fifth discourse of “Optics describes experiments with ox eyeballs, from which knowledge about how light passes through a lens might be gathered (Olscamp 38-40).

[2] For example: “Then there is no way of life concerned with the management of the city that belongs to a woman because she’s a woman or to a man because he’s a man, but the various natures are distributed in the same way in both creatures. Women share by nature every way of life just as men do, but in all of them women are weaker than men” (Plato 456).

[3] From Main Section II: The Work of Necessity, of Timeaus in particular deals with similar concerns and practicalities as Descartes concerns himself with in Rulesand Method (38-64).

[4] Rule IV discusses in depth the “perfection” of these two forms in their simplicity and self-evident nature.

[5] The quote Heidegger uses is, “However, I maintain that in any particular doctrine of nature only so much genuine science can be found as there is mathematics to be found in it” (4).

[6] Husserl and Kant both would have been suspicious of the interconnectedness that Heidegger presents here, as both sought to preserve mathematics as pure logic in their respective fashions. Husserl states in Logical Investigations, “It was Kant who uttered the famous words on logic which we here make our own: ‘We do not augment, but rather subvert sciences, if we allow their boundaries to run together’” 13.

[7] Heidegger does this via the example of the numeration of the number three: “We can count three things only if we already know ‘three.’ In thus grasping the number three as such, we only expressly recognize something which, in some way, we already have. This recognition is genuine learning. The number is something in the proper sense learnable, a mathemata, i.e., something mathematical” (Heidegger 273).

[8] “When Kant speaks of ‘science,’ he means Newton’s physics” (279).

[9] Newton’s first Law is: “Every body continues in its state of rest, or uniform motion in a straight line, unless it is compelled to change that state by force impressed upon it” (Newton 21).

[10] Leonard Shlain, in his text Art and Physics: Parallel Visions in Space, Time and Light describes how Aristotle’s episteme has been morphed into the contemporary episteme, “In our present paradigm we still acknowledge four basic constructs of reality: space, time, energy, and matter. Space and time constitute the gridwork within which we conduct our lives, while inside their frame, energy, matter, and various combinations thereof create our world of appearance.  These four elemental constructs form a mandala of totality. All perceptions created in the dream room of our minds are constructed from these four building blocks” (Shlain 26).

[11] I have included Heidegger’s own translations here, alongside the ones in Appendix A for the purposes of veracity to Heidegger’s intention towards the text.

[12] The list is as follows: “(a) belonging to the Emperor, (b) embalmed, (c) tame, (d) sucking pigs, (e) sirens, (f) fabulous, (g) stray dogs, (h) included in the presentation, (i) frenzied, (j) innumerable, (k) drawn with a very fine camelhair brush, (l) et cetera, (m) having just broken the water pitcher, (n) that from a long way off look like flies,” (Foucault xv).