Talk:The philosophy of mathematics

By Swami Sarvottamananda

‘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 afford to learn it,’ said the Mock Turtle 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 different 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 scribbled 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. There was a young girl on the street and he conceived the idea of asking 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 Daddy, Mummy thought you would forget!’ The world of mathematics is beautifully reflected in Alice’s Adventures in Wonderland—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 mathematician is typified by the absent-minded professor. In fact, the world of mathematics is an imaginary world, a creation of brilliant minds that live and thrive in it. Mathematics and the mathematicians who live in its abstract world, alike create a feeling of unworldliness in the common mind. Mathematics 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 philosophy of mathematics? And what specifically 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’. Thus, in essence, philosophy as a subject tries to supplement our knowledge by finding 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-shivam-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. Though many practising mathematicians do not think that philosophical issues are of particular relevance 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. This is attested to by the fact that both the ongoing scientific revolution and the concomitant phenomenal rise of technology borrow heavily from the progress in mathematics—a dependence that can be seen throughout the evolution of civilization by the discerning 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 find 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. Moreover, the foundations of mathematics are rock solid. Never has a mathematical position needed retraction. Even in physics, considered a glamorous field in present-day society due to its numerous applications, one finds 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 methodology. 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 respective methodologies more than the laws, theories, and hypotheses—that is, the content of physics or mathematics—that they discover or propound. That is one of the chief reasons why there is no crisis in scientific circles when one scientific theory fails and another takes its place.

Contrast this with the philosophies of old, particularly those which were not based on the firm foundation of logic. There 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 favorite 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 often seen in Indian philosophical dialectics—through intricate and abstruse arguments as well as ludicrously naïve squabbling—is not likely to surprise us. There will be much to gain if we in-corporate the logic of mathematics and the methodology of physics into our classical philosophies, and give up the esoteric dependence on classification, enumeration, categorization, and obfuscation. We need both the fine edifice of logic and the firm foundation of methodology, because most of the Indian darshanas are not mere speculative philosophies but are also empirical—they have many elements of philosophical realism. Of course, the contribution 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.

Defining the Term

In his Introduction to Mathematical Philosophy, Bertrand Russell takes the ‘philosophy of mathematics’ and ‘mathematical philosophy’ to mean one and the same thing. His argument is that formal philosophy is mathematics. And, because of the explanation given by him as well as similar arguments advanced by other influential people, traditionally, works on mathematical philosophy also deal with the philosophy of mathematics, and vice versa. But a more commonsensical differentiation between these terms may be made thus: Mathematical philosophy is essentially philosophy done mathematically, hence falling within the purview of math-magicians, whereas philosophy of mathematics deals with the philosophical issues in mathematics, 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 abstract 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. Philosophy of mathematics, hence, is truly the metaphysics of mathematics—meta-mathematics, the higher knowledge of mathematics. ‘Normal mathematics’, on the other hand, deals with the relatively mundane, 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 mathematics we are dealing with all that is abstruse and complicated. Nothing can be further from the truth. It is the simple facts and elementary theorems of mathematics that pose the greatest difficulty 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 examines and the classical philosophical domains to which they belong: • Are numbers real? (Ontology) • Are theorems true? (Rationalism) • Do mathematical theorems constitute knowledge? (Epistemology) • What makes mathematics correspond to experience? (Empiricism) • Is there any beauty in numbers, equations, or theorems? (Aesthetics) • Which mathematical results are astounding, elegant, or beautiful? (Aesthetics) • Is doing mathematics good or bad, right or wrong? (Ethics) • Can non-human beings do mathematics? (Philosophy of Mind) • Can machines do mathematics? (Artificial Intelligence)

It is customary to consider philosophical theories like mathematical realism, logical positivism, 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 endeavours to extract the form from a given piece of deductive argument and to verify the logic on the basis of the validity of form, rather than directly to interpret the content at every step. Thus, a favorite 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 empirical 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 possibility—that mathematical objects are definitely conceptual entities. But what does the word conceptual mean here? Conceptual, with respect to mathematical entities, means that they are hypothetical—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 example, 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 redefined 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. The lines of the Euclidean plane can be thought of as equatorial circles and points as poles on a spherical surface without any loss of understanding. Only the operations on lines and points will have to be redefined 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. There 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 external reality. In other words, our mathematical models 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. There can be different mathematical explanations for the same event and, conversely, there can also be different physical interpretations of the same model. This is illustrated in the example below where we try to model real-world addition. Let us define 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 from 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 different interpretations of the operators P and Q are not mutually compatible. Thus we see that the same model can be interpreted in three different ways.

I always like comparing this triad of the mathematical model, the objective world, and our interpretation to that of śabda, artha, and jñāna—word, object, and meaning. The mathematical model of the world is equivalent to śabda, the world to artha, and the interpretation to jñāna. This 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 first notice is that mathematical 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 definition 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. They 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 different value? Would the physical world appear different if the speed of light were different? When we say that ‘The 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 objections, another universe where the speed of light is different, 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 physical 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 imbued with realism. Therefore, in its traditional five-step syllogism (pañcāvayava anumāna), it is mandatory to cite a real-life example (drstānta) while drawing an inference from given premises. This step is much like deducing a specific instance from a general principle. And because of this thoroughly realistic 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 entity 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. Though Nyaya too, as a system of formal logic, has its own hypothetical concepts, its grounding in the real world restricts its conceptual flexibility. Hence, Nyaya as a logical system is able to deduce only a subset of the truths which mathematical logic is able to derive.

Having understood the nature of mathematical concepts, we now need to briefly examine the mathematical method. What is the method by which we arrive at the truth or falsity of mathematical statements?

In a mathematical system, we have axioms, which are facts taken to be obviously true (‘If a is less than b, then a is not equal to b’ is one such axiom—the axiom of linear order), and some nonfacts (which we shall call non-axioms) by the help of which we prove or disprove theorems. Proving theorems means deriving them from known axioms. If we are able to deduce a theorem starting from these basic axioms, then we say that the theorem is true.

However, proving the falsity of a theorem is different. If we are able to derive a non-axiom from a proposition, then that proposition is false—a nontheorem. So, here we go the other way round—we start from the theorem itself, not from non-axioms. Hence proving the truth and falsity of theorems are not mirror processes.

The underlying assumption of this method is that we cannot derive a non-fact from facts. Such a system is called consistent. If a system is inconsistent, it is ‘trivially complete’; that is, every statement, true or false, is derivable in an inconsistent system. An inconsistent system, therefore, is of little practical use.

Propositional Logic

In the study of mathematical method we also need to study propositional logic. Propositions play an important part in mathematical proofs. What is a proposition? A proposition is a statement which is either true or false. Note that there are certain statements which are neither true nor false. For example, interrogatory and exclamatory statements are neither true nor false. Also there is this classic example of a paradoxical self-referential statement, which is neither true nor false:

P: The statement P is false

We have referred to the term theorem. Now is the time to define it. What is a theorem? A theorem is nothing but a proposition for which there is a formal proof. What then is meant by proof? A proof is simply a sequence of deductive steps governed by well-defined logical rules that follow from a set of axioms. An axiom, of course, is a proposition that is given to be unconditionally true. The following deduction illustrates the rule of specialization, which is one of the many rules of logic:

All men are mortal. Socrates is a man. Therefore, Socrates is mortal

Thus, a mathematical system is a set of axioms and non-axioms with predefined rules of deduction, which are also referred to as rules of inference. The rules of deduction or rules of inference are nothing but rules that add, remove, modify, and substitute operators and symbols.

Let us try an exercise to understand how the rules of inference work. Suppose we have been given the following rules of addition, removal, and substitution of symbols I and U (the other symbol M remains there as in the starting axiom). The starting axiom is MI, and x and y are variables:

(i) xI → xIU (Derive MUUIIIU from MUUIII)

(ii) Mx → Mxx ( Derive MUUIIIUUIII from MUUIII)

(iii) xIIIy → xUy (Derive MUUU from MUUIII)

(iv) xUUy → xy (Derive MIII from MU

Now try constructing the theorem MU starting only with the axiom MI using the above rules of inference. Is it possible to derive MU?

The crux of the matter discussed above is that in mathematics, as well as in logic, the operators, the constants, and the functions can all be viewed, as in this example, as symbols which are added, removed, and substituted by predefined rules of inference, without ascribing any interpretation to them. Gödel exploited this fact beautifully in proving his famous theorem on incompleteness.

Is Mathematics a Uniquely Human Activity?

Since doing mathematics involves intricate reasoning and abstract thinking, it is often thought to be a very creative process requiring a lot of intuition. Kant was of the opinion that since mathematics requires human intuition it cannot possibly be done by non-humans. But several later philosophers have shown that it really does not require any human intuition to understand a mathematical proof. Finding a proof for an open research problem, though, might be an altogether different matter—computers have failed till date to automatically generate proofs for even very simple non-trivial mathematical problems. This is not to suggest that proving mathematical theorems is a uniquely human activity incapable of computer simulation—it is simply a matter of selective processing power. Computers cannot distinguish between boring mathematical truths and interesting mathematical results and keep happily churning out one mathematically uninteresting result after another, ad infinitum.

Mathematical thinking, in fact, is apparently not unique to humans. Rudimentary mathematical understanding is also seen in other animal species. And, of course, computers are ‘doing’ mathematics all the time. If one is to argue that finding and discovering mathematical truths rather than understanding proofs constitutes the test of mathematical intelligence—and computers fail this test—then it may be pointed out that this will also place the majority of humans at par with machine intelligence, because the vast majority of humans do not participate in the exciting activity of mathematical discovery.

Important Branches of Mathematics

Among the important branches of mathematics, number theory, set theory, geometry, and logic are historically very old. The oldest civilizations—the Indian, Greek, Chinese, Egyptian, and Babylonian—had all developed these branches, in one form or other, for general use. This is substantiated by the fact that without a fair understanding of geometry the remarkable architectural and civil-engineering feats for which these civilisations are famous would not have been possible. Even such elementary constructions as a rectangular wall or a field, or the more intricate hemispherical dome, require at least a rudimentary knowledge of geometrical constructions. Incidentally, ancient Greeks gave much importance to geometry, whereas Indians gave up geometry for abstract mathematics during the Buddhist period.

As far as number and set theories are concerned, no one really knows when humans developed these. Numbers surely came with the need for counting. Most civilizations seem to have been formally using numbers right from their inception. It was needed for commerce, and in earlier tribal societies to quantify one’s possessions.

Set theory is more fundamental than number theory, for it deals with classification rather than counting. Formal logic was a later development. But its rudiments were probably coeval with the development of language—with the need to coherently and intelligently communicate one’s opinions, arguments, and deductions to others. In fact, logic and language are so interlinked that many consider logic to be merely a linguistic construct. Historically, both Nyaya and Aristotelian philosophy had formalized logic for their respective civilizations, the Indian and the Greek.

Number Theory

Let us begin with numbers. We may ask: What is the nature of numbers? Are numbers real? In the Nyaya and Vaisheshika philosophies, for instance, numbers are real entities, belonging to one of the seven categories of real entities. However, there are conceptual difficulties if we grant numbers an objective reality. Consider the following: We have two books. So, we have books and we have also the number two. Let us add another pair of books to our collection. Does it destroy the number two and create the number four? Or does the number two transform into the number four? Suppose we add two notebooks, to distinguish them from the original pair of books. Then we have got a pair of twos as well as a four. None of the original numbers is destroyed or transformed and yet a new number is created. The ancient Buddhists were therefore not wrong in pointing out that numbers are in fact mental concepts. They do not have any existence outside the mental world.

Furthermore, mathematicians say that numbers can also be thought of as properties of sets, being their sizes (though the Buddhists would not feel comfortable with this either). Numbers as properties of sets were called cardinal numbers by George Cantor in contrast to ordinal numbers which represented positions in a series (first, second, and so on). Again, these are not to be taken as real properties, for there is an equally long-standing debate on substances and their properties. Essentially, therefore, numbers are abstract properties of equally abstract sets. Or, with greater ingenuity, the abstract concept of set itself can be thought of as representing numbers—not just the properties of sets but the sets themselves. Thus, we may have: { } = 0 {ϕ} = 1 {ϕ, {ϕ}} = 2 {ϕ, {ϕ}, {ϕ,{ϕ}}} = 3, and so forth.

Does anyone find this remarkable example illuminating or fascinating! All the same, this is what we meant by our statement that mathematical entities are not real but are merely conceptual entities.

Historically, the notion of numbers was formalized in the following succession. The notion of natural numbers (1, 2, 3, …) was developed first ‘God created the natural numbers; everything else is man’s handiwork’, the German mathematician Leopold Kronecker had famously observed. The incorporation of zero as a number was the great contribution of the Indian subcontinent. The natural numbers are complete as far as the operations of addition and multiplication are concerned—if we add or multiply two natural numbers we get another natural number. However, the class of natural numbers is not complete with respect to subtraction (you don’t get a natural number if you subtract 3 from 2). So if the result is to belong to the set of numbers, we need to extend the list of natural numbers to include negative numbers. The result is the set of integers.

Again, we see that the class of integers is not complete with respect to division. So the set of numbers is further extended to include ratios—rational numbers. The word rational here is derived from ‘ratio’ and not ‘reason’. Next we get surds or irrational numbers, when we extend the set of numbers to include limits, sums of series, square roots, trigonometric functions, logarithms, exponential functions, and so on. This gives us real numbers. Actually, this gives us only a subset of real numbers because these constitute only what are called computable numbers (which can be computed to any desired degree of precision by a finite, terminating algorithm). Not all real numbers can be so constructed. To be mathematically precise, we need to see each real number as a partition which divides the set of numbers into two groups A and B. If the partition is such that there is a largest element of A or a smallest element of B then the (partitioning) number is rational. But if there is neither a largest number in A nor a smallest number in B then the divisive number is irrational. This is the concept of ‘cuts’ developed by the celebrated mathematician Richard Dedekind.

The other day I was arguing with a friend that every real number can be seen as a decimal expansion which can be computed one digit after another using a suitable algorithm. I was, however, wrong. Alan Turing has proved this mind-boggling truth that not all real numbers are computable.

People found out very quickly that negative numbers could not have real square roots. In order to make the set of numbers complete even with the operation of determining square roots, the domain of real numbers was again extended to that of complex numbers, which are nothing but the sum of a real number and an imaginary number (i.e. a number expressed as a multiple of √−1). The historical choice of the names imaginary and complex was, however, unfortunate. For this makes one think that complex numbers are not numbers at all. One could on the other hand look at complex numbers as a dyad such that the subset of this dyad with the second term as zero is actually the set of real numbers. Moreover, all algebraic operations that can be carried out using real numbers can also be applied to the complex number dyads when these are suitably redefined. This interpretation is much more appropriate than the one commonly taught in schools. It is also worth noting that the class of complex numbers is ‘complete’ in the sense that if we apply any normal operator or any common function to complex numbers we always get a complex number.

With the introduction of complex numbers, one would think that the number system was at peace. But that was not to be, for serious trouble was brewing with the inclusion of the concept of infinity. There is a common misconception that there is one and only one mathematical infinity. And the people who seem to be more prone to this misconception are people from a Vedantic background! I wish to point out that here we are not merely thinking of +∞ and −∞, or even ‘radial infinites’ in the complex plane. It was George Cantor who proved that there are numerous infinities in relation to numbers. As a matter of fact, while the set of integers and of rational numbers are countably infinite, the set of real numbers is uncountably infinite. (Countability or denumerability refers to being able to be counted by one-to-one correspondence with the infinite set of all positive integers.) More remarkably, Cantor was able to prove that even uncountably infinite sets have different cardinalities: that if N₀ is an infinite set then there exists a set (N₁) which can be proved to be larger than this set, and this process can be extended to obtain infinites with still greater cardinality. Cantor’s treatment of infinities, however, was abstract rather than constructive. And this cost him an appointment at Berlin University—though his work was mathematically sound—as Kronecker, a firm believer in constructions, opposed him. Mathematicians, after all, are also human!

Zeno’s Paradox

Besides the problem of infinity, mathematicians working with numbers had also to tackle the problems of limits and series. To appreciate the problem with series, we consider one of Zeno’s paradoxes—a set of problems devised by Zeno of Elea to support Parmenides’s doctrine that ‘all is one’. This doctrine asserts that, contrary to the evidence of our senses, the belief in plurality and change is mistaken, and, in particular, motion is nothing but an illusion. This is much like the Buddhist doctrine of ksanikavāda.

‘Achilles and the Tortoise’ is the most famous of these paradoxes. Fleet-footed Achilles, of Battle-of-Troy fame (in Homer’s Iliad), and a tortoise are participating in a race. Achilles is reputed to be the fastest runner on earth; and the tortoise is one of the slowest of living beings. However, according to Zeno, Achilles can never win the race if the tortoise is given but a little head start. This is how it happens: Suppose the tortoise is, say, ten feet ahead of Achilles. In an instant Achilles covers the distance of ten feet. But during that instant the tortoise has already advanced a short distance. Again in another bound Achilles covers that small distance, but to his dismay, during that time the tortoise has advanced still more, and so on. Thus, Achilles can never possibly catch up with the tortoise.

But this clearly is nonsense. In reality, things never happen like that. This is actually a graphic description of the problem of the sum of an infinite series of decreasing terms which yields a finite value. Of course, not every such series will yield a finite value. The harmonic series (1 + ½ + ⅓ + ¼ + …) is one such.

Set Theory

Now that we are on paradoxes, let us start our discussion of set theory with Russell’s paradox. In set theory, we have finite sets as well as infinite sets. For infinite sets it is possible that a set contains itself. {ϕ,{ϕ}, {ϕ,{ϕ}}, {ϕ,{ϕ},{ϕ,{ϕ}}}, …} is one such set. Keen observers would have noted that this is the number ‘infinity’ in the ‘illuminating’ example of a previous section. Now call a set abnormal if it contains itself. Define a set R of all ‘normal’ sets: ‘the set of all sets that do not contain themselves as members’. Now ask the question: Is R normal or abnormal? We see that this question cannot be answered in either the affirmative or the negative.

The ‘axiomatic set theory’ was developed to address such paradoxes by incorporating an ‘axiom of choice’ within the theory. But this goes beyond the scope of our discussion, although it may be mentioned in passing that a surprising corollary to this theory is the fact that a universal set—the hypothetical set containing all possible elements—does not exist.

In practice, sets are normally related to groups and collections of objects in the external world. Here too, a similar question, as with numbers, arises: are sets real? In Indian philosophical thought too, the same question appears repeatedly. The Buddhists, for instance, argue that the axe which is a combination of the handle and the blade does not exist ‘in itself ’. It is absurd, they say, to call an axe a family heirloom of great value if its blade is changed just five times and its handle just fourteen times.

This question of absurdity, however, does not arise in mathematics because sets as well as their constituent members are all hypothetical entities—conceptual objects which are granted no intrinsic reality.

Geometry

In contrast to sets and numbers, it is easy for us to see that geometrical objects are conceptual. But it was not so for the Greeks—they took their geometry seriously exactly for the opposite reason: they thought geometry was real.

Take, for instance, the case of a point and a line in a plane. What is a point? A point, as every schoolchild knows, is a geometrical object that does not have any length or breadth (all its dimensions are zero). And what is a line? A line is a geometrical object that has only length but no breadth. These very definitions make it obvious that true points and lines cannot exist in the real world distinct from our mental constructions.

Credit goes to Euclid for formalizing the field of geometry into a body of axioms and theorems. Though his treatment of the subject was fully conceptual, it took a really long time—two thousand years—for people to see that these concepts do not quite match the real world. All this time everyone had been mistakenly assuming that the world is Euclidean. Geometrical results seemed to fit our experiential world so very nicely that people failed to see that they could be unreal. Nevertheless, with the advent of Einstein’s theories of relativity—both special and general—the realization dawned that the world is in fact non-Euclidean; it is more accurately described in terms of several Riemannian (or elliptic) geometries.

Another point to note is that, in formalizing geometry, we try to arrive at proofs which do not appeal to our intuition or visual sense but are logically correct. For though original mathematical insights are often derived through intuition, these ‘insights’ also run the risk of being proved wrong. Even the great Euclid—though he was well aware of this and therefore tried very hard to avoid intuitive judgments—himself committed a few mistakes in his proofs, because these proofs relied on the way he drew the illustrations. All the same, this does not take away any of the credit due to him in recognizing what is correct mathematical procedure. And certainly the momentous task of formalizing the great body of geometry already known at his time was not an easy task by any standard.

Logic

Mathematical logic is the final edifice of mathematics. And every logical system has to deal with the question of completeness and consistency. Completeness means that every true statement must be verifiable, must have a proof. Consistency is slightly different: it means that we should not be able to ‘prove’ false statements as true, that is, false statements must not have valid proofs in the theory in question.

At the beginning of the twentieth century, David Hilbert posed the ultimate problem of logic to mathematicians—to prove the consistency of mathematics as a system. This challenge came to be fondly called the Hilbert programme. Hilbert observed:

When we are engaged in investigating the foundations of a science, we must set up a system of axioms which contains an exact and complete description of the relations subsisting between the elementary ideas of that science. The axioms so set up are at the same time the definitions of those elementary ideas; and no statement within the realm of the science whose foundation we are testing is held to be correct unless it can be derived from those axioms by means of a finite number of logical steps. Upon closer consideration the question arises: Whether, in any way, certain statements of single axioms depend upon one another, and whether the axioms may not therefore contain certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that shall be altogether independent of one another.

But above all I wish to designate the following as the most important among the numerous questions which can be asked with regard to axioms: To prove that they are not contradictory, that is, a definite number of logical steps based on them can never lead to contradictory results.

The questions of consistency and completeness are important because if mathematics as a system were both complete and consistent, then it could well yield an easy path to new discoveries by way of a method to automatically discover mathematical theorems, what with superfast computers with supermemory and super processing power as tools.

Kurt Gödel, however, proved that mathematics is in fact incomplete. He further showed that the consistency of mathematics cannot be proven from within the field of mathematics itself, or to be precise, from within Peano’s axiomatization of the number theory. So with this dual stroke he delivered a terrible blow to the human quest for ‘knowing everything’.

In brief, Gödel’s theorems have the following twin consequences: First, there exist true statements which do not have any proof, and second, even if we have a proof for such a statement, we do not also know (by means of a valid proof) that its converse is not true. The wording and formulation of the second part is important as it makes a distinction between the truth of a statement and having a proof thereof.

A question may naturally arise at this juncture: Is Gödel’s incompleteness theorem applicable to every logical system? Turing is credited with extending the results of Gödel’s theorem to the field of computation. He has shown the non-existence of several kinds of computational procedures that could have helped us circumvent the implications of Gödel’s theorem, enabling us to find the truth and falsity of statements in a circuitous way. Thus, he was able to draw our attention to the far-reaching consequences of Gödel’s incompleteness theorem. In short, this theorem brings under its purview every kind of logical system—ancient or modern or postmodern—that is powerful enough to deduce facts. It only leaves out trivial theories like those based on first-order predicate calculus (logic).

So it would not be correct to say that Gödel’s incompleteness theorem applies only to formal logic or axiomatic mathematics, and not to the Nyaya or Buddhist logical systems, because these systems also involve predicates and possess deductive (anumāna) power.

Mathematics, Mind, and Maya

Let me conclude with some personal reflections:

First, mathematics has to constantly fight off utilitarians who accuse it of a lack of concern with reality—at least pure mathematics does not concern itself with applications. In fact, many pure mathematicians think that applied mathematics—being more interested in the results than in the process—is a degradation, and hence no mathematics at all.

It is a mundane fact that less-advanced disciplines further their cause with the assistance of more advanced ones. The latter, however, can keep advancing only by keeping intact their pristine purity. Thus even though others may use mathematics, mathematics stands to lose if it starts catering to the demands of other disciplines: the only way for mathematics to advance is by concentrating on its lofty aims. Thus it should be left to other disciplines to find the applications for and uses of mathematics, so that pure mathematics remains pure.

Second, Gödel was able to prove that there exist true theorems for which there is no proof. Some take this as proof of the superiority of the human intellect—after all, we know indirectly about the truth of these theorems even though they cannot be proved. This is not correct. Gödel only showed that both the theorems and their converse have no proof, and so if a system is consistent, one of them is bound to be true. Thus we have, by inference, a true theorem which does not have a proof. But we do not know specifically which of the two (the theorem or its converse) is true. A further corollary to his theorem is that only inconsistent systems are trivially complete. And our hopes of omniscience are further dampened when we remember that the consistency of a system is impossible to prove from within the system itself.

Third, Vedanta as a system of philosophy is an empirical system. However, the only empirical facts that it sticks to with heart and soul are the reality of Brahman, the unreality of samsara, and the oneness of Atman, the individual soul, and Brahman, the supreme Reality. These are empirical truths according to Vedanta because Vedanta firmly holds that Atman, Brahman, and maya are mere statements of facts—a posteriori truths, truths that need to be experienced or realized. However, as the world is granted only a conceptual reality—as a construct of the cosmic mind (hiranyagarbha)—Vedanta remains within the purview of empirical sciences only very loosely. Strictly speaking, then, Vedanta as a system with a single composite empirical fact—brahma satyam jaganmithyā jiva brahmaiva nāparah; Brahman is real, the world unreal, and the individual soul is no different from Brahman— which is not provable by sensory perceptions, becomes a system independent of physics and mathematics alike. Nevertheless, care should be taken, when we talk (as Vedantists) either about the world that is a product of maya or when we use a deductive process to infer the unity of existence and the unreality of the world, for then there is noescape from the sciences, both empirical and formal— physics and mathematics. Within the realm of maya, Vedanta cannot go against the findings of physics and mathematics.

• Originally published as "The Philosophy of Mathematics" by Prabhuddha Bharata September 2007 . October 2007 editions. Reprinted with permission.