," Let There be Geometry"
I’m writing this piece in the wake of having speculated on the collapse of systems, objects and logos of extreme power and centrality to our species, that governed and defined meaning to human civilizations. We explored the possibility that language has taken the place of iconotypes and archetypes in constructing meaning through differance and supplementarity and then wondered if this had happened in the sciences. We observed the melting down of paradigmatic grand assumptions about the world in the culture of science yielding to a certain kind of chaos of discovery facilitated mainly by machinistic intelligence. It is now time to bring the last horizon, the spiritual pole in the cathedral of ideality; mathematics into question. It is a culture that, for most of the history of its practitioners, from Plato onwards, believed it was a process of discovery, forming the tools to view the eidos; ideal objects. Indeed, many schools of mathematical philosophy have rejected the notion of mathematical objects that numbers and geometrical shapes pre-existing human consciousness in an a priori form. Jan Brouwer [1],Leopold Kronecker[2], Bertrand Russell[3] and others have attacked the issue in various ways showing how mathematical truths are constructions of the mind and not eternal, unchanging, non-causal and non-spatiotemporal. In this blog, we shall attempt to look at this issue through phenomenological investigations of the ideal mathematical objects by Edmund Husserl in the ‘Origin of Geometry’ (OG) and Derrida’s critique of his essay.
There are many political consequences of these assumptions about mathematical objects. The Greeks, for example, believed that 1 was a unit of arbitrary length and didn’t believe there were true numbers. Hippasus[4], a disciple of Pythagoras, is said to have discovered irrational numbers; the and the incommensurability of a square’s diagonals with their edges. This feat caused such a fundamental reversal in Greek philosophy of ideality that Hippassus was drowned for his heretical ideas and disturbing the Pythagorean order of the world. To understand the scale of the idea being questioned, let us not imagine, but consider the idea of infinity or to invoke Euclid, the floor upon which you sit, stand or sleep extending infinitely in all directions. According to Husserl, to truly consider this as we are all capable of doing outside of imaginative and sensory experience, it would require a transcendental consciousness[5]. To imagine the ideal object of infinity, one would have to follow the Husserlian steps of “ruckfrage” or reactivation. There is the original object in its ideal and objective form, then the primordial sense of the object, followed by sedimentation (erosion of absolute form, linguistic and cultural pollution) of this original sense and the subsequent attempts to reactivate the original form by working through language to reach the horizon. Historicity, in the phenomenological sense, is the constituting of an intended object and in the case of geometric objects; this process would be its historicity.
This operation of consciousness, turning perception into an impression happens in what Husserl terms the living-present (“the consciousness of succession that makes possible the apprehension of a succession of consciousness”). There is the primal impression of an object, the retention and the protention. Think of a ball being thrown towards you. The mind perceives temporality differently from the register of the moment. We perform retention, in that we retain the previous points of the ball beyond the sense’s perception of the moment to understand the present place of the ball through its trajectory. At the same time we also perform protention in that we anticipate the next location of the ball in the trajectory. This protracted consciousness of temporality is necessary for any intentional historicity of geometric objects. The question then arises as to whether the ideal objects exist as the transcendental signifier outside the influence of language and hence history or if it is merely a guiding force. Derrida compares this dilemma with the fate of another transcendental signifier:
“The fragments which mention God are marked with the same apparent ambiguity. God is no longer invoked, as for example in Ideas I, only as the exemplary model and limit of all consciousness of impossibility in the proof of an eidetic truth…God is no longer designated as the transcendent principle…of every universal factual teleology , either of Nature or the spirit, i.e., of history. Divine consciousness, which reveals the intangibility of constituted essences is a fictional content and the directing Telos for the real universe. As such it is a factuality. The reduction of God as factual being and factual consciousness sets free the signification of transcendental divinity, such as it appears in the last writings. The ambiguity we announced a moment ago concerns precisely the relation of the transcendental Absolute as divinity and the transcendental Absolute as historical subjectivity. In its transcendental sense, God is sometimes designated as the one toward which "I am on the way" and "who speaks in us," at other times as what "is nothing other than the Pole.” At times the Logos expresses itself through a transcendental history, at other times it is only the absolute polar authenticity of transcendental historicity itself.”
If we think the former, that the logos in mathematics expresses itself through history or that it is the product of a transcendental history, then it would be derived from experience. The infinity, in other words, would unfold itself in a historical discursiveness. If we were to think the latter, that the logos is the impossible limit, then the concepts borrowed from the idea of infinity, for example, can only be metaphors that don’t affect the original purity. To settle these contending views, one would have to risk the bewilderment that Plato warns of, and go into the light.
What is the origin of the ideal objectivity of these objects that differentiate them from other entities in the lebenswelt, the life world? Are they historical or ahistorical? Husserl seems to contend with this question of objectivity by saying the object at its origin must already have objectivity for it to be recognized by the inventor. Otherwise, (we know this now after the fact) her invention was only ever hers. Derrida says, “the sense of the constituting act can only be deciphered in the web of the constituted object”. Therefore, only after the object has been constituted can we make statements about its primordial sense of objectivity and recognizability so in a way, intentionality has been forced by the objectivity (we must remember the omnitemporal nature of the living-present and not be limited by our impulse to causation to consider this proposition).
The means by which we come to sense this ideal objectivity is then language. In bound, real world objects, there is the word’s ideal objectivity which can only remain objective in a factual sense. The word “lion” is factually in the English language but sensually doesn’t have such a unified signification. Similarly, the objectivity and unity of sense that Husserl calls “expression…to the intentional content” can be grounded and fixed by the identity of the object. However, the object itself, (the lion) is neither the expression nor the sense. Every person perceives the sensible lion differently; there is neither an absolute nor universal understanding of the lion so it is bound by empirical subjectivity. When it comes to geometrical objects, however, the ideality is not located in the expression or the intended content, but in the object itself. L2+W2= H2 in a right angled triangle, or better known as the Pythagorean Theorem is unbound by language for its expression and remains true across all cultures. Here, the object and sense of the object are distinguished at the level of the very possibility of language in the object.
Derrida says, at this point, that there seems to be a paradox. Ideal objectivity of the geometrical object is revealed to us only through language, something that they purport to transcend in order to remain universally true. According to Husserl, the geometrical object doesn’t live in a “topos ouranios” (heavenly plane), so it comes as an Idea in the inventors mind. The living present’s internal dialectic of retention and protention allows for the object’s intrapersonal communication. It is useful to imagine that without the immersion into language, and hence into history, the idea would remain imprisoned in the inventor’s mind. It is the “one and the same world” that we live in and our consciousness of this fact that allows for the universal language of mathematics, making the vehicle for objectivity transcendental subjectivity.
The transcendental historicity or the intentional constitution of the ideal object occurs through the living present in the inventors mind and through writing in the community. Therefore, writing “sanctions and completes the existence of pure transcendental historicity” as it gives the sense of non-spatiotemporal objects the ability to become non-spatiotemporal pushing humans into a transcendental community.
Constituted object ---> objectivity---> constituting act---> inventors mind---> idea---> intrapersonal language---> interpersonal writing---> community
Now we come to the essential question posed earlier; the origin of the ideality. It could be that geometrical ideality is based on the morphological idealities of imagination and sense, that we all imagine and sense the same way. We have already discussed that Husserl rejects the notion of platonic idealism. There is a middle ground that Husserl locates in the pre-scientific life-world which is the non-imaginative and non-sensible realm of pure thought. We are forced to examine the Idea in the living-present again. The idea exists in a unity of past (primal impression), present (retention) and future (protention) and it is not constituted but merely thought and experienced in time. The difference is between imagining the morphological roundness and making a “higher, absolutely objective, exact and non-sensible ideality occur- the circle.” This is Kant’s notion of idea, an ideation or an intuition of essence where facts and descriptions are forgotten and an ideal object is created. Basically, we work around the axioms and postulates of geometry to its original objects. Therefore, Derrida speculates “must we not say that geometry is on the way toward its origin, instead of proceeding from it…that pure ideality is announced in bound ideality” I think here Husserl would say this leap draws its support or appeal from the sensible ideality, for example, “roundness”. Therefore, the Idea allows for ideality’s origin eventually and it is eternal yet historical because eternity is just a mode of historicity; of constituting an intentional object.
Being, or living in the tradition, Derrida says, “is silently shown under the negativity of the apeiron”. The infinity, the absolute origin to come is always deferred for anything to appear in the living present. He says, “ and thought’s pure certainty would be transcendental, since it can look forward to the already announced Telos only by advancing on (or being in advance of) the Origin that indefinitely reserves itself…to learn that Thought would always be to come”. John P. Leavey says, “an origin, an absolute Origin, must be a differant Origin- the never-yet-always-already-there as the “beyond” or “before” that makes all sense possible.”
Once within the realm of differance, through language at the origin of ideality and through the anteriority/ posteriority of the absolute origin, the pure mathematical object loses its negative theological status. In fact, when asked by an audience member at one of his lectures that if “differance is the source of everything and one cannot know it: it is God of negative theology”, Derrida replied “it is and it is not.” As opposed to the negative theological place of mathematical objects as something supereminent and yet concealed and ineffable, differance is always deferred from entering the present. With the Origins of Geometry so precariously placed at the poles and hence the limits of historicity, they are “but a quasi-transcendental anteriority unlike the supereminent, transcendental ulteriority” of God as John Caputo puts it. This is the essential philosophical difference which then leads to the political decentering from the first cause or transcendental signified yet not quite to the realm of other discarded signifiers.
The shared nature of the same world allows for universally acceptable truths even within this imagination of mathematical differance. According to Husserl, science is the exemplary, unique and archetypal culture precisely because it doesn’t produce culturally-specific truths and aims at a completed state or telos as every culture does. The labor within this culture that has decentered its mathematical transcendental logos can be imagined in wildly different ways. It can be thought of in a constructivist school that only admits mathematical entities which can be explicitly constructed to mathematical discourse. It can be thought of in an intuitionist school of Jan Brouwer that believes that “there are no non-experienced mathematical truths”. This would engage machine labor in the mathematical enterprise in interesting ways as it has already done by using concepts of the Turing Machine to fill gaps within it. Automated Theorem Proving is the just the beginning and many more vehicles for ruckfrage will emerge. We will continue to explore the validity of constructivism, intuitionism and other attempts to bring mathematics into the world of the post, dominated by ghosts, machines and monsters. Until then, however, while seeking the phenomenological origin of geometry, we may have cast the diamond cubic crystal lines around the poles of its historicity, reining it within the vast web of language.
References:
[1] Franchella, Miriam . "L.E.J. Brouwer: Toward Intuitionistic Logic." ScienceDirect: n. pag. Web.
[2] Boniface, Jacqueline . "Leopold Kronecker’s conception of the foundations of mathematics." Philosophia Scientiæ: n. pag. Web.
[3] Principia Mathematica
By: Whitehead, Alfred North, and Bertrand Russell.
The University Press
[4] Metaphysics
By: Apostle, Hippocrates George.
Indiana University Press
1966
[5] Derrida, Jacques, and Edmund Husserl.
N. Hays ;
1978
