Talk:The philosophy of mathematics
The Philosophy of Mathematics Swami Sarvottamananda (September 2007 ( 512)
‘Ah! then yours wasn’t a really good school,’ said the Mock Turtle in a tone of great relief. ‘Now at OURS they had at the end of the bill, “French,
music, AND WASHING—extra.”’ ‘You couldn’t have wanted it much,’ said Alice; ‘living at the bottom of the sea.’ ‘I couldn’t aford to learn it,’ said the Mock Tur-tle with a sigh. ‘I only took the regular course.’‘What was that?’ inquired Alice. ‘Reeling and Writhing, of course, to begin with,’ the Mock Turtle replied; ‘and then the diferent branches of Arithmetic—Ambition, Distraction, Uglifcation, and Derision.’
—Alice’s Adventures in Wonderland
The following story is told about the reputed mathematician Norbert Weiner: When they moved from Cambridge to Newton, his wife, knowing that he would be absolutely useless on the move, packed him of to MIT while she directed the move. Since she was certain that he would forget that they had moved and where they had moved to, she wrote down the new address on a piece of paper and gave it to him. Naturally, in the course of the day, he had an insight into a problem that he had been pondering over. He reached into his pocket, found a piece of paper on which he furiously scrib-bled some notes, thought the matter over, decided there was a fallacy in his idea, and threw the piece of paper away. At the end of the day, he went home (to the old Cambridge address, of course). When he got there he realized that they had moved, that he had no idea where they had moved to, and that the piece of paper with the address was long gone. Fortunately inspiration struck. Tere was a young girl on the street and he conceived the idea of ask-ing her where he had moved to, saying, ‘Excuse me, perhaps you know me. I’m Norbert Weiner and we’ve just moved. Would you know where we’ve moved to?’ To this the young girl replied, ‘Yes Dad-dy, Mummy thought you would forget!’ The world of mathematics is beautifully reflect-ed in Alice’s Adventures in onderland—a world of ideas, where absurdity is a natural occurrence. Mathematics takes us to a world of ideas away from the ordinary, so much so that the archetypal mathe-matician is typifed by the absent-minded professor. In fact, the world of mathematics is an imaginary world, a creation of brilliant minds who live and thrive in it. Mathematics, and the mathematicians who live in its abstract world, alike create a feel-ing of unworldliness in the common mind. Math-ematics is itself abstract; more so is the philosophy of mathematics—the subject of the present article. Why Study the Philosophy of Mathematics? Before we enter the subject, we must answer some questions: What is the utility of studying the phi-losophy of mathematics? And what specifcally is the utility in the context of a journal dedicated to Vedanta? The word philosophy is derived from the Greek philo-sophia, ‘love of wisdom’. Tus, in essence, phi-losophy as a subject tries to supplement our knowl- edge by fnding out what is knowable and what is not; just as, in essence, logic as a subject deals with what is provable and what is not, ethics with what is right and what is wrong, aesthetics with what is beautiful and what is ugly, and religion with what is good and what is evil. Vedanta deals with what is real and what is unreal, and asserts satyam-shiv-am-sundaram as a triune entity—that which is real is also good and beautiful. So if we view Vedanta from this angle, then it is religion, ethics, aesthetics, and philosophy—all rolled into one. The philosophy of mathematics deals with meta-physical questions related to mathematics. It discusses the fundamental assumptions of mathematics, enquires about the nature of mathematical entities and structures, and studies the philosophical implications of these assumptions and structures. Tough many practising mathematicians do not think that philosophical issues are of particular rel-evance to their activities, yet the fact remains that these issues, like any other issue in life, do play an important role in shaping our understandings of reality as also in shaping the world of ideas. Tis is attested to by the fact that both the ongoing scien-tifc revolution and the oncomitant phenomenal rise of technology borrow heavily from the progress in mathematics—a dependence that can be seen throughout the evolution of civilization by the dis-cerning mind. The importance of mathematics can be judged by the fact that it is used in every walk of life—and this is no overstatement. It is invariably present wherever we fnd the touch of rational thought. It is the ubiquitous guide that shapes and reshapes our thoughts and helps us in understanding ideas and entities, both abstract and concrete. Moreo-ver, the foundations of mathematics are rock solid. Never has a mathematical position needed retrac-tion. Even in physics, considered a glamorous feld in present-day society due to its numerous applications, one fnds scientists backing out from positions they held some years earlier. But it is not so in mathematics. Once a mathematical truth is discovered, it seems to remain a truth for eternity. Why is this so? Contrary to common belief, the real importance of mathematics does not rest in the fantastic theorems discovered; it is in the way mathematics is done—the mathematical process or methodol- ogy. It is this that is the matter of our careful scrutiny. Physics has its own methodology too, which is of equal importance. Tough it may not appear obvious, both streams stress equally their respec- tive methodologies more than the laws, theories, and hypotheses—that is, the content of physics or mathematics—that they discover or propound. Tat is one of the chief reasons why there is no crisis in scientifc circles when one scientifc theory fails and another takes its place. Contrast this with the philosophies of old, particularly those which were not based on the frm foundation of logic. Tere the methodologies, the facts and theories, the lives and teachings of the proponents, and, to a lesser extent, the mythologies and cosmologies, were so intermingled, with no clear cut demarcations between them, that systems stood or fell as a whole. It was a favourite technique of opposing schools of thought to point out a single fallacy or discrepancy somewhere in a gigantic work: that was enough to invalidate the whole philosophy. Seen in this light, the strange method of proving the supremacy of one’s philosophy that is ofen seen in Indian philosophical dialectics—through intricate and abstruse arguments as well as ludicrously naïve squabbling—is not likely to surprise us. Tere will be much to gain if we in-corporate the logic of mathematics and the meth-odology of physics into our classical philosophies, and give up the esoteric dependence on classifca-tion, enumeration, categorization, and obfuscation. We need both the fne edifce of logic and the frm foundation of methodology, because most of the Indian darshanas are not mere speculative philoso-phies but are also empirical—they have many ele- ments of philosophical realism. Of course, the con-tribution of the Indian philosophies in the realm of mind and abstract thought is enormous. Equally important are the bold proclamations of the rishis about consciousness and transcendental realities, which are beyond criticism. Defning the Term In his Introduction to Mathematical Philosophy, Ber-trand Russell takes the ‘philosophy f mathematics’ and ‘mathemati-cal philosophy’ to mean one and the same thing. His argument is that formal philosophy is mathe-matics. And, because of the expla-nation given by him as well as similar arguments advanced by other influential peo-ple, traditionally, works on mathematical philosophy also deal with the philosophy of mathematics, and vice versa. But a more commonsensical diferentiation between these terms may be made thus: Mathematical phi- losophy is essentially philosophy done mathemati-cally, hence falling within the purview of math- ematicians, whereas philosophy of mathematics deals with the philosophical issues in mathemat- ics, something that is to be done by philosophers. Philosophy of mathematics, as we treat the subject in this article, is indeed philosophy taking a look at mathematics, and therefore is not the same as mathematical philosophy. Thus, we shall only try to look at answers to ab-stract questions related to mathematics—the form, language, and content of mathematics; the nature of mathematical concepts; and the truth and reality of mathematical discoveries and inventions. Philos-ophy of mathematics, hence, is truly the metaphys-ics of mathematics—meta-mathematics, the higher knowledge of mathematics. ‘Normal mathematics’, on the other hand, deals with the relatively mun- dane, the concrete, the useful, and the visible. The Subject Matter Let me clarify a misconception. We are apt to think that when we talk about the philosophy of math-ematics we are dealing with all that is abstruse and complicated. Nothing can be further from the truth. It is the simple facts and elementary theo-rems of mathematics that pose the greatest dif-culty to philosophical understanding, by virtue of their fundamental nature, a nature with essential properties which we unknowingly take for grant-ed. To illustrate the point, we list here some of the questions that philosophy of mathematics exam-ines and the classical philosophical domains to which they belong: • Are numbers real? (Ontology) • Are theorems true? (Rationalism) • Do mathematical theorems constitute knowl- edge? (Epistemology) • What makes mathematics correspond to experi- ence? (Empiricism) • Is there any beauty in numbers, equations, or the- orems? (Aesthetics) • Which mathematical results are astounding, el- egant, or beautiful? (Aesthetics) • Is doing mathematics good or bad, right or wrong? (Ethics) • Can non-human beings do mathematics? (Phi- losophy of Mind) • Can machines do mathematics? (Artificial Intelligence) It is customary to consider philosophical the-ories like mathematical realism, logical positiv- ism, empiricism, intuitionism, and constructivism when studying the philosophy of mathematics. But we shall try to steer clear of these murky depths here. Nature of Mathematics Mathematics is a formal and not empirical science. What is a formal science? A formal science endeav-ours to extract the form from a given piece of de-ductive argument and to verify the logic on the basis of the validity of form, rather than directly to interpret the content at every step. Tus, a favour-ite technique to prove the fallacy of an argument is to substitute hypothetical axioms in its form so that it leads to an obvious absurdity—reductio ad absurdum. Another important distinguishing feature of a formal science such as mathematics is the use of the deductive method in its arguments, unlike empiri-cal sciences such as physics which use the inductive method to arrive at generalizations. Nature of Mathematical Entities Are mathematical entities real? If they are not real, then whatever name we choose to call them by—abstract or conceptual—the fact remains that they exist only in our mind, a figment of our imagination—not unlike our feelings, though possibly a bit different.
It is common to acknowledge only the second posibility—that mathematical objects are def-nitely conceptual entities. But what does the word conceptual mean here? Conceptual, with respect to mathematical entities, means that they are hypo-thetical—they may or may not have any correlation with reality. In that case, these entities could be represented and interpreted in any number of ways. This fact has surprising consequences. For exam-ple, if numbers are represented by some well-structured sets—as we shall do in the section on number theory—and the operations addition, subtraction, multiplication, and division are redefned for these sets, then the sets themselves may be thought of as numbers without any loss of generality. Yet another
example is that of spherical geometry. Te lines of the Euclidean plane can be thought of as equato- rial circles and points as poles on a spherical surface without any loss of understanding. Only the opera-tions on lines and points will have to be redefned so that Euclidean axioms still hold true. But what are the consequences of mathematical entities being conceptual? On Concepts being Hypothetical mathematical universe consists of conceptual objects alone. Tere is no direct relation between mathematical entities and the phenomenal objects of the empirical world. And these mathematical objects are only indirectly correlated to existent objects and interpreted as such by the human mind. For a given system of mathematical truths, we try to interpret factual truths of the external world in such a way that they ft the mathematical model we have developed. And it may not be possible to match every mathematical model with some exter-nal reality. In other words, our mathematical mod-els and external objective reality are connected only by our interpretation of the model. Nevertheless, it is worth noting that there is no one-to-one relation-ship between these two domains. Tere can be dif-ferent mathematical explanations for the same event and, conversely, there can also be diferent physical interpretations of the same model. Tis is illustrated in the example below where we try to model real-world addition. Let us defne two operators P and Q, such that we have the following relations: • 1 P 1 Q 2 • 1 P 2 Q 3 • 2 P 3 Q 5, and so on. Given the above axioms, the operators P and Q could be interpreted as plus and equal to respectively; thus 1 plus 1 equals 2. Are other interpretations of P and Q possible? Yes. P and Q may also be interpreted as equal to and subtracted fom respectively so that 2 P 3 Q 5 could be read as 2 equals 3 subtracted from 5. Again, Q could also be interpreted as greater than or equal to instead of equal to, in which case the above statement would read 2 plus 3 is greater than or equal to 5. In each of these cases we have a reason-able interpretation of the axioms, though the dif-ferent interpretations of the operators P and Q are not mutually compatible. Tus we see that the same model can be interpreted in three diferent ways. I always like comparing this triad of the math-ematical model, the objective world, and our inter- pretation to that of śabda, artha, and jñāna—word, object, and meaning. Te mathematical model of the world is equivalent to śabda, the world to artha, and the interpretation to jñāna. Tis is the way in which mathematical concepts relate to the objects of experience through an interpretation of events that is entirely a product of our thinking. Mathematics and Physics Let us now compare the theories of mathematics and physics. What we frst notice is that mathemat-ical truths are necessary truths, that is to say, truths deducible from axioms, and true in each and every alternative system (or universe) where the axioms hold. In other words, mathematical truths are true by defnition and not incidentally. Immanuel Kant, the celebrated German philosopher, called them a priori truths. Empirical truths, on the other hand, are a posteriori truths, only incidentally true. All physical facts are, surprisingly, only incidentally true. Tey may not be true in an alternative world or in an alternative physical system. For example, take the speed of light. Physicists tell us that the speed of light is a constant, nearly 300,000 km/sec. Now why should the speed of light be this value? Can it not be a diferent value? Would the physical world appear diferent if the speed of light were diferent? When we say that ‘Te speed of light is nearly 300,000 km/sec’ is not a necessarily true statement, then we mean that we can postulate, without fear of any technical objec-tions, another universe where the speed of light is diferent, say, 310,000 km/sec. Of course, that world would be unlike ours and is not known to exist, but this line of thinking gives us a hint that there is no a priori reason for physical constants to have the immutable values that characterize them—however real they may be for us. In fact, Vedanta boldly proclaimed a long time ago that the physi-cal universe does not have any a priori reason for its existence, and Buddhist thought has also followed this great tradition. Here it may be of interest to draw a comparison with Nyaya, the traditional Indian system of logic. Nyaya is an empirical philosophy and is fully im-bued with realism. Terefore, in its traditional fve-step syllogism (pañcāvayava anumāna), it is man-datory to cite a real-life example (dṛṣṭānta) while drawing an inference from given premises. Tis step is much like deducing a specifc instance from a gen-eral principle. And because of this thoroughly real-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. Tough Nyaya too, as a system of formal logic, has its own hypothetical concepts, its grounding in the real world restricts its conceptual fexibility. Hence, Nyaya as a logical system is able to deduce only a subset of the truths which math-ematical logic is able to derive. (To be concluded) Abstract thinking: Is doing mathematics an exclusively human trait?