REFLECTIONS ON PHILOSOPHY The Philosophy of Mathematics Swami Sarvottamananda ‘Ah! then yours wasn’t a really good school,’ said of paper away. At the end of the day, he went home the Mock Turtle in a tone of great relief. ‘Now (to the old Cambridge address, of course). When at OURS they had at the end of the bill, “French, he got there he realized that they had moved, that music, AND WASHING—extra.”’ he had no idea where they had moved to, and that ‘You couldn’t have wanted it much,’ said Alice; the piece of paper with the address was long gone. ‘living at the bottom of the sea.’ ‘I couldn’t aord to learn it,’ said the Mock Tur- Fortunately inspiration struck. ere was a young tle with a sigh. ‘I only took the regular course.’ girl on the street and he conceived the idea of ask- ‘What was that?’ inquired Alice. ing her where he had moved to, saying, ‘Excuse me, ‘Reeling and Writhing, of course, to begin with,’ perhaps you know me. I’m Norbert Weiner and the Mock Turtle replied; ‘and then the dierent we’ve just moved. Would you know where we’ve branches of Arithmetic—Ambition, Distraction, moved to?’ To this the young girl replied, ‘Yes Dad- Uglication, and Derision.’ dy, Mummy thought you would forget!’ —Alice’s Adventures in Wonderland e world of mathematics is beautifully reect- following story is told about the reputed ed in Alice’s Adventures in Wonderland—a world mathematician Norbert Weiner: When they of ideas, where absurdity is a natural occurrence. Tmoved from Cambridge to Newton, his wife, Mathematics takes us to a world of ideas away from knowing that he would be absolutely useless on the the ordinary, so much so that the archetypal mathe- move, packed him o to while she directed the matician is typied by the absent-minded professor. move. Since she was certain that he would forget In fact, the world of mathematics is an imaginary that they had moved and where they had moved to, world, a creation of brilliant minds who live and she wrote down the new address on a piece of paper thrive in it. Mathematics, and the mathematicians and gave it to him. Naturally, in the course of the who live in its abstract world, alike create a feel- day, he had an insight into a problem that he had ing of unworldliness in the common mind. Math- been pondering over. He reached into his pocket, ematics is itself abstract; more so is the philosophy found a piece of paper on which he furiously scrib- of mathematics—the subject of the present article. bled some notes, thought the matter over, decided there was a fallacy in his idea, and threw the piece Why Study the Philosophy of Mathematics? ‘Would you know Before we enter the subject, we must answer some where we’ve moved to?’ questions: What is the utility of studying the phi- losophy of mathematics? And what specically is the utility in the context of a journal dedicated to Vedanta? e word philosophy is derived from the Greek philo-sophia, ‘love of wisdom’. us, in essence, phi- losophy as a subject tries to supplement our knowl- edge by nding out what is knowable and what is CHANDRA PB September 2007 e Philosophy of Mathematics 19 not; just as, in essence, logic as a subject deals with Contrary to common belief, the real impor- what is provable and what is not, ethics with what tance of mathematics does not rest in the fantastic is right and what is wrong, aesthetics with what is theorems discovered; it is in the way mathematics beautiful and what is ugly, and religion with what is done—the mathematical process or methodol- is good and what is evil. Vedanta deals with what ogy. It is this that is the matter of our careful scru- is real and what is unreal, and asserts satyam-shiv- tiny. Physics has its own methodology too, which am-sundaram as a triune entity—that which is real is of equal importance. ough it may not appear is also good and beautiful. So if we view Vedanta obvious, both streams stress equally their respec- from this angle, then it is religion, ethics, aesthetics, tive methodologies more than the laws, theories, and philosophy—all rolled into one. and hypotheses—that is, the content of physics e philosophy of mathematics deals with meta- or mathematics—that they discover or propound. physical questions related to mathematics. It dis- at is one of the chief reasons why there is no crisis cusses the fundamental assumptions of mathemat- in scientic circles when one scientic theory fails ics, enquires about the nature of mathematical en- and another takes its place. tities and structures, and studies the philosophical Contrast this with the philosophies of old, par- implications of these assumptions and structures. ticularly those which were not based on the rm ough many practising mathematicians do not foundation of logic. ere the methodologies, the think that philosophical issues are of particular rel- facts and theories, the lives and teachings of the evance to their activities, yet the fact remains that proponents, and, to a lesser extent, the mythologies these issues, like any other issue in life, do play an and cosmologies, were so intermingled, with no important role in shaping our understanding of clear cut demarcations between them, that systems reality as also in shaping the world of ideas. is is stood or fell as a whole. It was a favourite technique attested to by the fact that both the ongoing scien- of opposing schools of thought to point out a sin- tic revolution and the concomitant phenomenal gle fallacy or discrepancy somewhere in a gigan- rise of technology borrow heavily from the progress tic work: that was enough to invalidate the whole in mathematics—a dependence that can be seen philosophy. Seen in this light, the strange method throughout the evolution of civilization by the dis- of proving the supremacy of one’s philosophy that cerning mind. is oen seen in Indian philosophical dialectics— e importance of mathematics can be judged through intricate and abstruse arguments as well by the fact that it is used in every walk of life—and as ludicrously naïve squabbling—is not likely to this is no overstatement. It is invariably present surprise us. ere will be much to gain if we in- wherever we nd the touch of rational thought. It corporate the logic of mathematics and the meth- is the ubiquitous guide that shapes and reshapes odology of physics into our classical philosophies, our thoughts and helps us in understanding ideas and give up the esoteric dependence on classica- and entities, both abstract and concrete. Moreo- tion, enumeration, categorization, and obfuscation. ver, the foundations of mathematics are rock solid. We need both the ne edice of logic and the rm Never has a mathematical position needed retrac- foundation of methodology, because most of the tion. Even in physics, considered a glamorous eld Indian darshanas are not mere speculative philoso- in present-day society due to its numerous appli- phies but are also empirical—they have many ele- cations, one nds scientists backing out from po- ments of philosophical realism. Of course, the con- sitions they held some years earlier. But it is not tribution of the Indian philosophies in the realm of so in mathematics. Once a mathematical truth is mind and abstract thought is enormous. Equally discovered, it seems to remain a truth for eternity. important are the bold proclamations of the rishis Why is this so? about consciousness and transcendental realities, PB September 2007 Prabuddha Bharata which are beyond that when we talk about the philosophy of math- criticism. ematics we are dealing with all that is abstruse and complicated. Nothing can be further from the Dening truth. It is the simple facts and elementary theo- the Term rems of mathematics that pose the greatest di- In his Introduction to culty to philosophical understanding, by virtue of Mathematical Philosophy, Ber- their fundamental nature, a nature with essential trand Russell takes the ‘philosophy properties which we unknowingly take for grant- of mathematics’ and ‘mathemati- ed. To illustrate the point, we list here some of the cal philosophy’ to mean one and questions that philosophy of mathematics exam- the same thing. His argument is ines and the classical philosophical domains to that formal philosophy is mathe- which they belong: matics. And, because of the expla- • Are numbers real? (Ontology) nation given by him as well as • Are theorems true? (Rationalism) similar arguments • Do mathematical theorems constitute knowl- advanced by other edge? (Epistemology) CHANDRA influential peo- • What makes mathematics correspond to experi- ple, traditionally, ence? (Empiricism) works on mathematical philosophy also deal with • Is there any beauty in numbers, equations, or the- the philosophy of mathematics, and vice versa. But orems? (Aesthetics) a more commonsensical dierentiation between • Which mathematical results are astounding, el- these terms may be made thus: Mathematical phi- egant, or beautiful? (Aesthetics) losophy is essentially philosophy done mathemati- • Is doing mathematics good or bad, right or cally, hence falling within the purview of math- wrong? (Ethics) ematicians, whereas philosophy of mathematics • Can non-human beings do mathematics? (Phi- deals with the philosophical issues in mathemat- losophy of Mind) ics, something that is to be done by philosophers. • Can machines do mathematics? (Artificial Philosophy of mathematics, as we treat the subject Intelligence) in this article, is indeed philosophy taking a look It is customary to consider philosophical the- at mathematics, and therefore is not the same as ories like mathematical realism, logical positiv- mathematical philosophy.