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REFLECTIONS ON 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 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 . 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 ? e word philosophy is derived from the Greek philo-sophia, ‘love of wisdom’. us, in , 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, as a subject deals with Contrary to common , the real impor- what is provable and what is not, with what tance of mathematics does not rest in the fantastic is right and what is wrong, 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 why there is no crisis cusses the fundamental assumptions of mathemat- in scientic circles when one scientic theory fails ics, enquires about the of mathematical en- and another takes its place. tities and structures, and studies the philosophical Contrast this with the 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 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 — 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 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 . 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 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 , PB September 2007  Prabuddha 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 De ning 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 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? () nation given by him as well as • Are theorems true? () similar arguments • Do mathematical theorems constitute knowl- advanced by other edge? ()

CHANDRA influential peo- • What makes mathematics correspond to experi- ple, traditionally, ence? () 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 do mathematics? (Phi- deals with the philosophical issues in mathemat- losophy of Mind) ics, something that is to be done by . • 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. ism, empiricism, intuitionism, and constructivism us, we shall only try to look at answers to ab- when studying the philosophy of mathematics. But stract questions related to mathematics—the form, we shall try to steer clear of these murky depths language, and content of mathematics; the nature here. of mathematical concepts; and the truth and reality of mathematical discoveries and inventions. Philos- Nature of Mathematics ophy of mathematics, hence, is truly the metaphys- Mathematics is a formal and not empirical science. ics of mathematics—meta-mathematics, the higher What is a formal science? A formal science endeav- of mathematics. ‘Normal mathematics’, ours to extract the form from a given piece of de- on the other hand, deals with the relatively mun- ductive argument and to verify the logic on the dane, the concrete, the useful, and the visible. basis of the of form, rather than directly to interpret the content at every step. us, a favour- The Subject Matter ite technique to prove the fallacy of an argument Let me clarify a misconception. We are apt to think is to substitute hypothetical axioms in its form so  PB September 2007

22 Prabuddha Bharata equal to and subtracted  respectively so that   true. ey may not be true in an alternative world   could be read as  equals  subtracted from . or in an alternative physical system. Again,  could also be interpreted as greater than or For example, take the speed of light. Physicists equal to instead of equal to, in which case the above tell us that the speed of light is a constant, nearly would read  plus  is greater than or , km/sec. Now why should the speed of equal to . In each of these cases we have a - light be this value? Can it not be a dierent value? able interpretation of the axioms, though the dif- Would the physical world appear dierent if the ferent interpretations of the operators  and  are speed of light were dierent? When we say that not mutually compatible. us we see that the same ‘e speed of light is nearly , km/sec’ is not model can be interpreted in three dierent ways. a necessarily true statement, then we mean that we I always like comparing this triad of the math- can postulate, without fear of any technical objec- ematical model, the objective world, and our inter- tions, another universe where the speed of light pretation to that of śabda, , and jñāna—word, is dierent, say, , km/sec. Of course, that object, and meaning. e mathematical model of world would be unlike ours and is not known to the world is equivalent to śabda, the world to artha, exist, but this line of thinking gives us a hint that and the interpretation to jñāna. is is the way in there is no a priori reason for physical constants to which mathematical concepts relate to the objects have the immutable values that characterize them— of through an interpretation of events however real they may be for us. In fact, Vedanta that is entirely a product of our thinking. boldly proclaimed a long time ago that the physi- cal universe does not have any a priori reason for its Mathematics and Physics , and Buddhist thought has also followed Let us now compare the theories of mathematics this great tradition. and physics. What we rst notice is that mathemat- Here it may be of interest to draw a comparison ical are necessary truths, that is to say, truths with Nyaya, the traditional Indian system of logic. deducible from axioms, and true in each and every Nyaya is an empirical philosophy and is fully im- alternative system (or universe) where the axioms bued with realism. erefore, in its traditional ve- hold. In other words, mathematical truths are true step (pañcāvayava anumāna), it is man- by denition and not incidentally. , datory to cite a real-life example (dṛṣṭānta) while the celebrated German , called them a drawing an from given . is step priori truths. Empirical truths, on the other hand, is much like deducing a specic instance from a gen- are a posteriori truths, only incidentally true. All eral principle. And because of this thoroughly real- physical facts are, surprisingly, only incidentally istic approach, postulating a hypothetical universe within Nyaya discourse is virtually impossible, be- cause that would lack real-world examples. In the mathematical domain, on the other hand, every en- tity is hypothetical, and entities get connected to the real world only through the interpretations applied to them. So we can postulate a hypothesis anytime and anywhere. ough Nyaya too, as a system of formal logic, has its own hypothetical concepts, its grounding in the real world restricts its conceptual Abstract thinking: Is doing mathematics exibility. Hence, Nyaya as a logical system is able an exclusively to deduce only a subset of the truths which math- human trait? ematical logic is able to derive. (To be concluded)

 GOPI PB September 2007