Symbol, the Infinite, and Paradox:

Euclidean Essentials

Don Thompson

Dean of Academic Programs

Pepperdine University

Werner Heisenberg observed "It is probably true quite generally that in the history of human thinking the most fruitful developments frequently take place at those points where two different lines of thought meet." (Heisenberg 1958) In the spirit of such convergence, this paper will investigate the convergence of two different lines of thought lying at the nexus between mathematics and natural philosophy.

Euclid’s Elements is a foundational treatise in mathematics providing a first level description and axiomatic practice in the essentials of symbol, the infinite, and paradox. Furthermore, these essentials transcend their mathematical framework and find life as echo-structures resonating in other great writings within natural philosophy. In particular, this paper will discuss ways in which the writings of Plato, Aquinas, and Kierkegaard, when viewed in juxtaposition to Euclid, provide fruitful developments.

Euclid begins his treatise on geometry with twenty-three definitions, five common notions, and five postulates, following which he constructs some four hundred sixty five propositions. Thus, he builds a massive intellectual edifice from a mere mustard seed. Euclid creates life in vacuo, without even so much as a test tube. How and where would one begin such an undertaking?

Euclid’s first definition is both intriguing and problematic: "A point is that which has no part" (Euclid 1956). Here is an object which, by definition, contains no thing. It is an entity which is, of itself, empty. This can only mean that it exists conceptually. Thus, Euclid is telling us that his geometric system must rely, at the atomic level, on those objects which have no interior, no inside, no substance. A bold beginning, to be sure! Seen from a different vantage point, Euclid is telling us that his system will be purely symbolic, i.e. pointing to substance but not consisting therein. Thus, we are dealing at the base level with symbolic entities or representatives that point back to our world of parts and approximations. Though we may take issue with this slippery definition, how can we start elsewhere? How can we avoid the symbolic when attempting to capture mathematical precision or perfection itself? Furthermore, hasn’t Euclid given us the perfect definition of symbol? It does not of its own sense exist (i.e. it has no substance) but only conveys or transcends or points back to our imperfect models. Put another way, symbols are the hidden skeletons within the images we unconsciously sketch on paper or draw on chalkboards. Moreover, Euclid constructs trees and forests from these sinewy, skeletal forms, thereby capturing the essence of symbol without violating its purity, without being sucked into the imperfect, relativistic copies. Furthermore, he says a point is "that" as if to say, "I am pointing at it (the point) with my words and it is over there." Thus his very language is symbolic. His words are, themselves, pointers, their own poetic!

Plato enters. Plato understands. Plato points, similarly, through his language of forms, his language of symbols. Casual readers of Plato already know this obvious connection. Plato would have been disappointed to have us stop there. We need to look at his words regarding unity: "The soul … would be compelled to summon up thought and inquire into the true nature of unity. Hence the study of unity will be among those studies that guide and turn the soul to the contemplation of reality. ... We should persuade those who are to perform high functions in the city to undertake calculation, but not as amateurs. They should persist in their studies until they reach the level of pure thought, where they will be able to contemplate the very nature of number" (Plato 1985). Thus, Plato engages us in pure thought through contemplation of the very nature of number, the essence, the part which has no part. We are to contemplate the center, the symbol itself. Euclid and Plato are on the same wavelength – or at least pointing to the same sinusoid.

In addition to exposing and engaging the reader in the symbolic, Euclid also gives us an indirect look at the infinite: "It is possible to bisect a given finite straight line" (Euclid 1956). Implicitly, we can continue bisecting any finite straight line indefinitely. Thus, there are infinitely many points on any finite straight line. Furthermore, Euclid proves this possibility by a "compass—unmarked straightedge—stylus" construction. He does not rely solely on words, but on their completed action. In so doing, Euclid is showing us that the incremental, elemental, essential parts of the infinite consist of symbols in action. One does not contemplate the infinite without an accompanying activity, for infinity requires action. Thus, Euclid has hit the essence and the essence of the essence regarding the infinite. Infinity moves beyond any barrier we may erect. It takes a stand, then moves on, unwilling to be bound or left static. Glimmering and twinkling for a moment, infinity allow us to glimpse it, draw it briefly into our lungs, then it moves on to its next level. Infinity is the shark imperative – to understand, one has to keep moving.

Thomas Aquinas uses infinity to form several proofs of the existence of God. By infinite regression as manifested in infinite chaining, he proves that God exists, relying on "argument from motion", "efficient cause", "possibility … necessity", "gradation", and "order in the universe" (Aquinas 1988). At root, all of his arguments rely on the possibility of the infinite, the consistency of infinity, and on infinite processes. Infinity does not break down or careen out of control. It is manageable, performing and moving with predictability: "Everything that moves must be moved by something else" (Aquinas 1988). Without movement, we most certainly metastasize, freeze, and die. Indeed, death is the ultimate sensate absence of God, being infinite inactivity. Ironically, God built into the system the possibility and guarantee of the infinite. His very existence contains the possibility of our describing and discovering his very nature, as if He left the proverbial key under the front door mat. God, the perpetual motion machine reveals how he works, and both Euclid and Aquinas have caught him in the act.

As point and line intersect, Euclid presents us with a thorny paradox: "The extremities of a line are points" and "A line is breadthless length" (Euclid 1956). Apparently a line is made up of points and nothing else. It is made up of things having no part. We are led to the paradox of existence within non-existence, infinite and infinitesimal, being and nothingness. Euclid’s bisection proposition brings us to this paradoxical nexus, since, by continually, infinitely bisecting a finite line segment, we discover that there are infinitely many points between the original endpoints, and these aren’t even all the points that make up the line segment. The Eleatic philosopher Zeno took matters even further: "If a straight line segment is infinitely divisible, then motion is impossible, for in order to traverse the line segment it is necessary first reach the midpoint, and to do this one must first reach the one-quarter point, and to do this one must first reach the one-eighth point, and so on, ad infinitum. It follows that the motion can never even begin" (Eves 1969). Therefore, we are intellectually transfixed, unable to move; a far cry from Aquinas’ infinite chain of movers!

Finally, Kierkegaard confronts paradox in his own courageous way: "I shall rely on the strength of the absurd, on the strength of the fact that for God all things are possible" and "it takes a paradoxical and humble courage then to grasp the whole on the strength of the absurd, and that courage is the courage of faith" (Kierkegaard 1985). Rather than retreat from or remain transfixed by paradox, Kierkegaard encourages us to boldly embrace it, holding on to the tension of its absurd conflict. So, we are to allow its mysterious nature to grasp us and we, in turn , attempt to hold it within our grasp.

Thus, mathematical essentials intertwine with other realities. They inform and define. They connect and construct, transcend and transport, relying on symbolic unity, the infinite, and the paradoxical.


Literature Cited

Aquinas, Thomas. On Politics and Ethics. Translated and edited by Paul E. Sigmund. New York: W.W. Norton, 1988.

Euclid. The Elements. Translated by Sir Thomas L. Heath. 3 Volumes. New York: Dover, 1956.

Eves, Howard. 1969. An Introduction to the History of Mathematics. New York: Holt, Rinehart and Winston.

Heisenberg, Werner. Physics and Philosophy : The Revolution in Modern Science. New York: HarperCollins, 1958.

Kierkegaard, Soren. Fear and Trembling. Translated by Alastair Hannay. London: Penguin, 1985.

Plato. The Republic. Translated by Richard W. Sterling and William C. Scott. New York: W.W. Norton, 1985.