Thinking Outside of the Euclidean Box
by Brett Holverstott
"Let no one come to our school, who has not first learned the Elements."
This well-known phrase was inscribed on the doors of the philosophers of Ancient Greece, including Plato's Academy. Euclid's Elements appeared about 300 BC, and until the early 20th Century-over 2200 years later-it served as the standard textbook for elementary geometry. It is the most successful textbook in the history of mankind-precisely because it has been the bane of philosophers and mathematicians for a millennia.
Euclid was a mathematician at the University in Alexandria. Although many texts in geometry preceded him (all called the Elements), Euclid's innovation was to order and perfect a comprehensive set of 465 theorems, resting upon the fewest necessary assumptions. Before he embarks on his theorems, he gives 23 Definitions including "point" and "line," 5 simple rules of logic called Common Notations, and 5 basic assumptions called Postulates.
The 5th Postulate, called the "Parallel Postulate," is a complex expression (with many geometrical equivalents) which basically states that two parallel lines will not get closer or farther away from each other, if extended to infinity. This was self-evident to all geometers before the 19th century, and they could not conceive of a geometry where this was false. So they tried to prove it as a theorem from the remaining postulates.
Poseidonios tried in the mid 2nd-century BC, and was succeeded by Archimedes, Playfair, Ptolemy, Proclus, Aghanis, Simplicius, Al-Jawhari, Qurra, Al-Nayrizi, Grigoryan, Nasiraddin, Wallis, Sina, al-Haytham, Khayyam, al Salar, al-Tusi, Cataldi, Giordano, Lambert, Bertrand, Legendre, Gur'ev, and countless others, a millennia-long line of the world's greatest mathematical minds leaving only a wake of failed proofs. The wreckage now makes up a repertoire of fallacies in mathematical logic. The most frequent error was to beg the question, by assuming a geometric or philosophical equivalent to the 5th Postulate. Aristotle, who died shortly before the time of Euclid, prophetically wrote:
...whenever a man tries to prove by means of itself what is not known by means of itself, then he begs the point at issue... This is what happens with those who think they describe parallel lines, for they unconsciously assume things which it is not possible to demonstrate if parallels do not exist.
[Prior Analytics, 65a]
Imre Toth uses Aristotle to show that he and his contemporaries had a more logical grasp of the Problem of Parallels then the mathematicians from Euclid onward. He suggests that Aristotle and his contemporaries pursued a line of logic unmatched until Saccheri in 1733. Saccheri had started with the opposite of the Parallel Postulate and tried to use reductio ad absurdum to show a contradiction-but none was reached. Unable to go further, he hastily concludes that the results are "repugnant to the nature of a straight line." Aristotle too, could not go further.
In the early 19th Century, the philosophic work of Immanuel Kant (1724-1804) was gaining favor in the mathematical community. Kant held that there is a difference between knowledge independent of (a priori) and reliant upon (a posteriori) experience. In order to defend the validity of geometric knowledge in this way, Kant proposed:
...mental space renders possible the physical space... space is not at all a quality of things in themselves, but a form of our sensuous faculty of representation... the space of the geometer is exactly the form of sensuous intuition which we find a priori in us...
[Prolegomena to Any Future Metaphysics, Remark 1]
He explains that the basis of geometry is intuitive, built into our process of thought. Armed with this logic, the mathematical community assigned Euclidian Space to be an a priori principle without need of proof, and without possible exception. Those who still labored on the issue were considered cranks.
Unaffected by Kant's doctrine, the "crank" to discover Non-Euclidean geometry was Carl Friedrich Gauss, who did so by 1813.
Gauss knew he held the solution to the epic question. He had heroically reasoned beyond a millennia of predecessors, and had described a consistent system of geometry where parallel lines are not equidistant. But meeting only disinterest with whom he spoke, and in fear of being misunderstood by the weak-minded community, he did not make his discovery public. In a private letter dating 8 November 1824, Gauss sketches his new geometry and writes,
We know, despite the say-nothing word-wisdom of the metaphysicians, too little, or too nearly nothing at all, about the true nature of space, to consider as absolutely impossible that which appears to us unnatural.
The discovery was later made by three others, independently. Gauss would never need to publish his findings. His notes on the subject halt in 1932, the day he received a copy of Johann Bolyai's famed Appendix, describing the geometry in detail. "My intention was... not to allow it to become known during my lifetime," Gauss wrote, and was happy to be spared the effort of preserving his notes for posterity.
The influence of Gauss and his followers formulated the mathematical theory Einstein would use, early in the next century, to describe the curvature of space. Einstein showed that in the presence of a gravitational field, all matter and energy moving through space follows curved paths describing geodesics-forms of consistent non-Euclidean geometry. Thus, the Non-Euclidean geometry of the real world is now a fact. Kant's Doctrine of Space, like the other failed proofs, should be thrown in the junk heap-and the fallacy appropriately identified.
An alternative theory of the origin of knowledge, proposed in 1966, suggests:
...concept-formation and applied mathematics have a similar task, just as philosophical epistemology and theoretical mathematics have a similar goal: ...of bringing the universe within the range of man's knowledge, by identifying relationships to perceptual data.
[Intro to Objectivist Epistemology, pg. 14]
In opposition to Kant's distinction between the a priori and a posteriori, Ayn Rand unifies mathematics, science, and philosophy under a common process of thought-of abstraction from perceptual experience. She explains mathematics as special case "that defines the entities it deals with very simply" which, after the base is established, can proceed without need of any further definition [IOE pg. 201].
But systems of mathematics are (more often than not) prompted by man's interaction with realty, even after the fundamental concepts of mathematics, such as the arithmetical series, have been established. Rand mentions "A vast part of higher mathematics, from geometry on up, is devoted to the task of discovering methods by which various shapes can be measured" [IOE pg. 14] Under this theory, mathematics remains valid though it derived from the empirical concept "shape," and prompted by the various shapes man strives to measure.
But this is not to say Kant's distinctions are not useful concepts. In the sciences, it is often useful to take knowledge you have and apply it to new problems-to form hypothesis based on principles taken elsewhere. I would define a priori as: knowledge already formed, which is being applying to a new system without confirming the unique subtleties in the system. Thus a priori takes on a useful significance in the act of problem solving. Indeed, this definition is already in use in the natural sciences. But contrary to Kant's definition, this implies that a priori knowledge is inherently uncertain, and is only capable of being confirmed by the a posteriori, the "real" world.
I have attempted to show that there exists a historical relationship between philosophy and mathematics, which has influenced the progress of mankind in mathematics, and thus in the sciences. The problem of parallels, first introduced in Ancient Greece, is a case in point. Kant's distinction between the a priori and a posteriori was an attempt to prove the absolute certainty of mathematics by showing how it required no outside experience-how it was free of the errors of observation and the process of induction. But as mathematics moves forward, we cannot escape the need to treat mathematics like a natural science, to question our assumptions and discover new laws of nature. The reality-oriented epistemology of Rand may hold the key to formulating a new philosophical foundation for the "science" of mathematics.
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