MATHEMATICS AND PHILOSOPHY OF GEORG CANTOR

For starters, here is the very short (regrettably!) Cantor entry in my Webster’s Biographical Dictionary:

“Cantor, Georg. 1845-1919. German mathematician, born in St. Petersburg; professor, Halle (from 1872). Developed a theory of irrational numbers, an arithmetic of the infinite, and the theory of sets of points; he also introduced transfinite numbers.”

Georg Ferdinand Ludwig Philipp Cantor (1845-1919) was indeed a great German mathematician who developed all these things. He also introduced the above-mentioned transfinite numbers, that is, cardinal numbers or ordinal numbers which are larger than all finite numbers, yet do not necessarily qualify as absolutely infinite. The specific distinction of “absolute infinity,” also introduced by Cantor, identifies it with God, and thus his mathematics explicitly transcends into religious philosophy. In order to better understand what Dr. Cantor means by transfinite, as opposed to absolutely infinite, here is what he says about it in a very succinct and comprehensible fashion:

What I assert and believe to have demonstrated… is that following the finite there is a transfinite, that is an unbounded and ascending ladder of definite modes (compare this to Dèscartes’ distinctive pairs of definite-indefinite and finite-infinite!), which by their nature are not finite but infinite, but which, just like the finite, can be determined by well-defined and distinguishable numbers. (Today, many mathematicians do not care much for Cantor’s distinction, preferring the word infinite in all cases, but many still use the term transfinite which I believe is a very useful term, introducing an important nuance into the philosophical underpinnings of mathematics.)

Cantor was by no means the only mathematician who theologized and philosophized mathematics: the long history of the blending of mathematics with philosophy explicitly starts with Pythagoras, extending through Spinoza, Leibniz, and Kant into the twentieth century. Cantor himself identified his position as a mathematician-philosopher. It is therefore only natural for us to take him at his word, and rationalize his appearance in this section on philosophical grounds.

There are some very interesting facts about Cantor’s biography. He was born in Russia of a German father and Russian mother, and although the family moved to Germany when he was just eleven years old, Cantor retained a mystical spiritual affinity and nostalgia for Russia throughout all his life.

He is also known for erratic behavior that frequently cost him broken friendships. This fact is now ascribed to a mental illness, periodically driving him into a depression, and greatly complicating his professional and personal life. Bertrand Russell treats Cantor in his History of Western Philosophy in the chapter revealingly titled The Philosophy of Logical Analysis. But ascribing to Cantor his philosophical credentials, we do not even need Cantor’s or Russell’s word for it. Any mathematician and scientist whose work involves infinity, volens-nolens enters the domain of philosophy, and Cantor’s signature endeavor dwelled in infinity as in his permanent residence.

Cantor’s specific area of mathematics (incidentally, it is one of my favorite areas too!) is rather esoteric for a reader not well-versed in it, but, in describing it, Russell does a fairly good job of “Cantor-made-simple,” and for this reason his Cantor passage is well-worth quoting:

…(Georg Cantor) developed the theory of continuity and infinite number. Continuity until he defined it had been a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated. Cantor also overcame the long-standing logical puzzles about the infinite number. Take the series of the whole numbers from 1 onwards: this number must be an infinite number. But now comes the curious fact: The number of even numbers must be the same as the number of all whole numbers. Consider the two rows:

1, 2, 3, 4, 05, 06, ….

2, 4, 6, 8, 10, 12, ….

There is one entry in the lower row for every one in the top row; therefore the number of terms in these two rows must be the same, although the lower row consists of only half the terms in the top row. Leibniz, who noticed this, thought it a contradiction, and concluded that though there are infinite collections, there are no infinite numbers. Georg Cantor, on the contrary, boldly denied that it is a contradiction. He was right; this is only an oddity. Cantor defined an infinite collection as one which has parts containing as many terms as the whole collection contains. On this basis, he was able to build up a most interesting mathematical theory of infinite numbers, thereby taking into the realm of exact logic a whole region formerly given up to mysticism and confusion.

In the conclusion of this entry I would like to introduce several Cantor quotes which underscore his brand of bonding between mathematics and philosophy:

The actual infinite as distinguished from transfinite infinity arises in three contexts: first, when it is realized in the most complete form, in a fully independent otherworldly being, in Deo !!, where I call it the Absolute Infinite or simply the Absolute; second when it occurs in the contingent created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type. Here is a clear rationalization of Cantor’s invention of the term transfinite, to set mathematics apart from theology, and in this philosophical aspect, alongside others (such as logical, to set it distinctly apart), his invention is both perfectly valid, and also highly commendable on all grounds.

The essence of mathematics lies in its freedom. Again, the borderline between mathematics and philosophy is crossed by the introduction of the philosophical term freedom. The way I understand this statement, from my subjective perspective, of course, is that once we realize that mathematics is grounded in hypotheses, its progress lies in the expansion of the number of such founding hypotheses, turning the mathematician into an inventor, a creator, a thinker, a philosopher!

A set is a Many that allows itself to be thought of as One. This is perhaps the most spectacular and splendid example of a philosophical dictum, bringing to mind nothing less than PreSocratica Sempervirens!

In mathematics, the art of asking a question must be held of higher value than solving it. Yes!!! But why so restrictive--- to mathematics only? We are surely entitled to expand this judgment to all philosophical thinking (and, “needless to say,” because the reader knows it already, I have always subscribed to the expanded version of this statement). But, anyway, Georg Cantor has already amply established his philosophical credentials to us by now, and the reason for his mandatory inclusion into this section has now been explained and justified beyond any doubt.