Theorem 2.0

The Oulipo did much to thematize the connections between literature and mathematics, and the Oulipian Jacques Roubaud describes how the method of constructing mathematical texts can be used in poetic composition: “A constraint is an axiom of a text. Writing under Oulipian constraint is the literary equivalent of the drafting of a mathematical text, which may be formalized according to the axiomatic method” (Roubaud 89). The axiomatic method in mathematics and formal logic establishes axioms that consist of a set of symbols, a set of valid combinations of the symbols, and rules to transform valid combinations of symbols to other valid combinations of symbols. A combination of symbols is a theorem. A theorem is proven true if you can start from an axiom and apply a series of transformation rules to arrive at the theorem.

The rigorous theory of the axiomatic method took place as a development in the ongoing search for the foundations of mathematics that arose after the 19th century crisis. Until the 19th century geometry had been the most secure and certain knowledge of which philosophy could boast. The theory of geometry had been handed down unchallenged for thousands of years through Euclid's Elements. In the 19th century one of Euclid's postulates gave way. The parallel postulate states that "Through a given point, only one line can be drawn parallel to a given line." In the first quarter of the century three different mathematicians (Gauss, Bolyai, Lobachevsky) created geometries that denied the fifth postulate, and became convinced that the non-Euclidean geometries did not lead to contradiction. This knowledge was scandalous, and Gauss hesitated to publish his results. Some extremely important and useful results required the parallel postulate, including the Pythagorean theorem. The scandal waited in the wings until Beltrami proved in 1868 that Lobachevsky's non-Euclidean geometry was as consistent (ie. free from contradiction) as Euclidean geometry. Mathematics until that point had Geometry, now it had geometries. Mathematical research in the 19th century shook the two thousand year old geometry at its roots, and the crisis arose with the loss of confidence in the oldest and surest form of mathematics.

Due to the privileged place accorded to geometry in philosophy, the crisis was more than mathematical, it shook the entire epistemology of Western thought:

The situation was intolerable because geometry had served, from the time of Plato, as the supreme exemplar of the possibility of certainty in human knowledge. Spinoza and Descartes followed the 'more geometrico' in establishing the existence of God, as Newton followed it in establishing his laws of motion and gravitation. The loss of certainty in geometry was philosophically intolerable, because it implied the loss of all certainty in human knowledge. (Hersh 15)

The crisis inaugurated the search for the Foundations of Mathematics, which aspired to provide a single theory from which all of mathematics could be derived. The first comprehensive attempt at providing foundations came by way of Gottlob Frege's logicism. If geometry could no longer be counted on to ground all of mathematics, then perhaps logic could do so. The sure footing for mathematics would be found in the logic of sets. Unfortunately for Frege, just as he was about to publish the final volume of his series of works on the foundations, Bertrand Russell showed how Frege's theory contained a paradox, known commonly as Russell's paradox or the Barbers paradox.

A man of Seville is shaved by the Barber of Seville if and only if the man does not shave himself.
Does the Barber of Seville shave himself? (Russell, ctd. Bunch 86)

If the Barber of Seville shaves himself, then he must be a man who does not shave himself, which is a contradiction. If the Barber of Seville does not shave himself, then he must be one of the men who is shaved by the Barber of Seville: another contradiction. Russell translated this paradox into Frege's formal system at its roots, and thus dismantled it, since any logical system that permits contradictions to be derived from its axioms can be used to prove contradictory statements like "1=2". Mathematics clearly cannot admit such statements. Frege went through with the publication of his foundations, but wrote in a postscript to the final volume: "Just as the building was completed, the foundation collapsed" (Frege).

Bertrand Russell and Alfred North Whitehead attempted to fill the cracks in Frege's foundations by reformulating the concept of sets to exclude the Barber paradox, which led to the publication of their magnum opus Principia Mathematica. The title echoed the single most important scientific publication to that date, Newton's Philosophiae Naturalis Principia Mathematica, and its aims were as ambitious in mathematics as Newton's were in physics, to revolutionize and ground their respective sciences. Although noone found any paradoxes in the Principia, most mathematicians considered the machinery used to eliminate the Barber paradox to be unjustifiable and obscure, involving constructions that strayed far from logic and any certainty it provided.

David Hilbert responded to the problem by shifting the focus to mathematical methodology and began the study of the axiomatic method itself, which came to be known as metamathematics. He advocated a formalism that reduced mathematical activity to manipulating a set of symbols according to rules stated as axioms. Any mathematical theorem was then either an axiom, or it was deducible from the axioms by following a series of valid applications of the transformation rules. In this way checking the validity of a theorem's proof would be equivalent to checking that a series of mechanical manipulations of symbols followed the rules. By reducing mathematics to the study of formal systems, Hilbert sought the same end as Leibniz with his Universal Characteristic: "For then reasoning and calculating would be the same thing" (Leibniz, ctd Derrida 78). Hilbert believed that through this method mathematics could be shown to be complete (every theorem could be proven true or false), and consistent (no theorem can be both true and false, ie. no paradoxes). The effort undertaken to do so is known as Hilbert's program. Unfortunately for Hilbert, Kurt Godel imported a version of Epimenedes' Liar Paradox into Russell & Whitehead's Principia Mathematica to show that mathematics could not be proven both consistent and complete.

Epimenedes the Cretan said: "All Cretans are liars." Is Epimenedes a liar? If Epimenedes the Cretan is telling the truth, then all Cretans are not liars, and Epimenedes is lying, which is a contradiction. If Epimenedes is lying, then not all Cretans are liars, so the statement he made is true. A better formulation of the paradox was given by Eubulides: "This statement is false". Godel brought Epimenedes to bear on Principia Mathematica by translating the Liar paradox into a theorem to say "This theorem is not provable."

If the assertion "This theorem is not provable" is false, then there are theorems that are false and provable (mathematics is inconsistent). If the assertion is true, then there are theorems that are true and not provable (mathematics is incomplete). Faced with the choice, mathematicians chose incompleteness. To choose inconsistency, where a theorem can be proven both true and false, would be to countenance proofs that "1=2", and this way lies madness. The consequence of the incompleteness of mathematics is that there are undecidable propositions: theorems that are true, but cannot be proven from the axioms.

These paradoxes involve an intrusion of one order of signification into another, where an element considered on the outside is smuggled to the inside, forming a vicious circle due to self-referentiality. Paradoxes are turned into fallacies by denying the premises from which they issue; this death of the paradox is necessary, for each time a paradox is reached in mathematics, we glimpse the death of the system. Mathematics retreats from the death of reason, for with it goes all truth. Opposed to mathematics, poetics seeks out and revels in the death of language, in the dissolution of the reading subject in a paradox of signification.

In The Pleasure of the Text, Roland Barthes theorizes the reader who seeks the dissolution of his subjectivity in the reading act by opposing the texts of a comfortable pleasure with the texts of a rapturous jouissance:

Texts of pleasure: the text that contents, fills, grants euphoria; the text that comes from culture and does not break with it, is linked to a comfortable practice of reading. Text of jouissance: the text that imposes a state of loss, the text that discomforts [...], unsettles the reader's historical, cultural, psychological assumptions, the consistency of his tastes, values, memories, brings to a crisis his relation with language. Now the subject who keeps the two texts in his field and in his hands the reins of pleasure and jouissance is an anachronic subject [...]: he enjoys the consistency of his selfhood (that is his pleasure) and seeks its loss (that is his jouissance). (14)

While reading a text of pleasure, the reader participates in creating meaning and narrative, which constitutes their subjectivity within a discourse. When the materiality of the text insists upon itself through various methods of linguistic excess, the illusion of discourse dissolves and we lose meaning as a settled given. When we are forced out of participating in a discourse, we become the text's voyeur, similar to the effect that occurs in reading when I say to you that the word at the end of this sentence is meaningless. Mathematics seeks out its paradoxes to eliminate the loss they incur, while poetics seeks out its paradoxes to revel in the loss they produce:

Paradoxes are recreational only when they are considered as initiatives of thought. They are not recreational when they are considered as "the Passion of thought," or as discovering what can only be thought, what can only be spoken, despite the fact that it is both ineffable and unthinkable-a mental Void (Deleueze 75).

For Bertrand Russell, paradoxes are the "Passion of thought":

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new field of mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering [by arriving at paradoxes], and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some twenty years of very arduous toil [which encompassed the arrival of Godel's results], I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge more indubitable (Russell).

For Roland Barthes, paradoxes are recreational initiatives of thought:

Imagine someone [...] who abolishes within himself all barriers, all classes, all exclusions, not by syncretism but by simple discard of that old specter: logical contradiction; who mixes every language, even those said to be incompatible; who silently accepts every charge of illogicality, of incongruity; who remains passive in the face of Socratic irony (leading the interlocuter to the supreme disgrace: self-contradiction)[...]. Such a man would the mockery of our society: court, school, asylum, polite conversation would cast him out: who endures contradiction without shame? Now this anti-hero exists: he is the reader of the text at the moment he reaches jouissance. Thus the Biblical myth is reversed, the confusion of languages is no longer a punishment: the text of jouissance is a sanctioned Babel. (3)

The Oulipian formulation of Hilbert's program for literature has clear and distinct limits, for the analogy of drafting a mathematical proof according to the axiomatic method and a poetic text according to constraint can only be carried so far:

[E]ven if the 'axioms' of an Oulipian constraint may be established with sufficient precision (as in the case of the lipogram), what will play the rather primordial role of deduction in mathematics? What is an Oulipian demonstration? One may think that a text composed according to a given constraint (or several constraints) will be the equivalent of a [proof of] a theorem. It is a fairly interesting hypothesis. It is nonetheless true that the foreseeable means of passage from the statement of the constraint to its 'consequences', the texts, remains in a profound metaphorical vagueness (Roubaud 89).

Mathematical proofs are teleological, whereas poetic texts are not. The profound metaphorical vagueness separates mathematics and poetry at the site of their paradoxes. Hence, by inspection

Works Cited

Barthes, Roland. The Pleasure of the Text. tr. Richard Miller. New York: Hill and Wang, 1975.

Bunch, Bryan. Mathematical Fallacies and Paradoxes. New York: Dover, 1982.

Deleuze, Gilles. The Logic of Sense. tr. Mark Lester w. Charles Stivale. ed. Constantine Boundas. New York: Columbia University Press, 1990.

Derrida, Jacques. Of Grammatology. tr. Gayatri Chakravorty Spivak. Baltimore: The Johns Hopkins University Press, 1998.

Frege, Gottlob. Foundations of Arithmetic.

Hersh, Reuben. "Some Proposals for Reviving the Philosophy of Mathematics." New Directions in the Philosophy of Mathematics. Princeton: Princeton University Press, 1998. 9-28.

Roubaud, Jacques. "Mathematics in the Method of Raymond Queneau". Oulipo: A Primer of Potential Literature. Normal: Dalkey Archive Press, 1998.

Russell, Bertrand. Portraits from Memory.