The average teacher or for that matter researcher of mathematics rarely bothers herself with the philosophy or epistemology of mathematics. Those who love mathematics are attracted to it perhaps for the following reason: it is a subject that seems driven mainly by logic and reason, not subject to opinion or interpretation. The research mathematician is also motivated by considerations such as the beauty and elegance of Mathematics, or the relevance and importance of her work. A good student of mathematics at school enjoys the subject in part because he does not need to resort to memorization and can derive everything from first principles. Of course there is always the thrill of solving a problem and knowing that one can actually prove that it is the correct solution. The very same students, however, when presented with examples of so called ‘logical paradoxes’ (one of which I will present later in this article), seem disconcerted! They experience the same discomfort when they encounter, say, the concept of different infinities. It seems to shake their sense of certainty in Mathematics.
It may be worthwhile as mathematics teachers to explore and understand something of the nature of Mathematics as a body of knowledge. It would also be fruitful to examine the issues and limitations that lie in this area. The purpose of this article is to show that while Mathematics is intimately linked with logic and reason, on the one hand this link gives it tremendous power, but on the other hand it also leads to serious and fascinating philosophical uncertainties. Along the way I will briefly introduce two unique features in the epistemology of mathematics –impossibility and undecidability. In the epilogue, I will explore what motivates Mathematicians in spite of such philosophical uncertainties in the very foundations of Mathematics.
I must confess right at the start that I am not an expert in the areas of philosophy, epistemology or the foundations of mathematics. My approach is akin to an amateur tourist in the vast landscape of Mathematics, coming upon some rather puzzling features and wanting to share these with a captive audience! Mathematics teachers might like to traverse some of these landscapes on their own and share their discoveries with their students. At the end of the article, youwill find a numbered list of readings referred to in the text.
Logic, the handmaiden of mathematics
Our story begins with the Greeks, for whom mathematics was very closely linked with philosophy and logic. For Plato, philosophy was the love of knowledge, and he wanted to discover the truth behind the fog of perception and opinion. Mathematics (more specifically geometry) was the best place to discover truths free of sensory experience and shifting perceptions. This was largely due to Euclid, who in his famous work the Elements, had had the genius to axiomatise geometry. Starting with axioms, or what he called self-evident truths, Euclid was able to prove several amazing theorems using only these axioms and the rules of logic. As long as we are agreed upon what is ‘logically correct’ anyone can understand Mathematics. Therefore the proof due to the Greeks that v2 is irrational, has stood the test of time for 2000 years and is unlikely to be disputed as long as we humans agree on the laws of logic. This is a crucial point: while a mathematical fact may be discovered by some deep intuition, it has to be established by cold logic and reason, what is known as ‘proof’ in mathematics. In fact proofs are so important to mathematicians, that Bertrand Russell narrates a story in which G. H. Hardy (the famous British Mathematician) once told him that “.... or If he (Hardy) could find a proof that I (Russell) was going to die in five minutes he would of course be sorry to lose me, but this sorrow would be quite outweighed by the pleasure in the proof! I entirely sympathised with him andwas not at all offended.”
With logic as its handmaiden and abstraction at its core, Mathematics has been extremely powerful as a system and a tool of science. A good illustration of this power comes from the idea of impossibility. In no other discipline am I aware of such a notion. The most popular examples come again from Greek attempts to solve three famous problems from antiquity. Using only a straight edge and compass, one is asked to trisect any angle, construct a circle with the area of a given square, and construct a cube with a volume twice that of a given cube. After nearly 2200 years of attempting these three problems, the work of two mathematicians in the nineteenth century proved conclusively that these are all impossible to do! Here I must stress that one can prove, using logic, that it is not a matter of how hard one works or how clever one is, but given the constraints that one must use only straight edge and compass it is impossible to perform the above three constructions. Surely this is unique to mathematics (see Suggested Reading 9 for more details).
An interesting question is – why do we humans share this sense of what is logically correct? It seems to transcend cultural and religious conditioning, and almost seems hardwired into our brains. The first person to attempt an explanation for this phenomenon was Immanuel Kant, who spoke of two components to our perception of nature – the a priori (already present before any experience) and the a posteriori (after experience). Could the ability of humans to have a consensus on logical validity be one of the a priori categories of thought– something an individual already possesses at birth? Today, using evolutionary theory, we might restate Kant’s position in the following way. Over a period of millennia, a number of faculties useful for survival in our environment (including the ability for logical thought) have evolved in the human brain. Neurobiologists believe that, our predecessor, Homo erectus to some extent used the cerebral faculty of logical reasoning in carving their stone tools. Thus although a posteriori for the human species, logical thinking is essentially a priori for the individual human being (read 4 for more on this interesting topic). And as far as the link with Mathematics in concerned, the great mathematician David Hilbert has said, “Kant’s a priori theory contains anthropomorphic dross from which it must befreed. After we remove that, only that a priori remains which is also the foundationof pure mathematical knowledge.”
Another interesting question – is there only one kind of logic, i.e., the traditional two-valued logic we’ve been speaking of here? In a two-valued logical system, we assume the ‘principle of the excluded middle’, wherein a statement and its negation cannot both be true at the same time. Another way of putting this is that a statement can possess only one of two possible truth statuses–true of false. The statement “You are reading this article” is either true or false. Interestingly however, starting with the Jain philosophers in India years ago, people have come up with other logical systems which are not two-valued, where statements can be indeterminate and uncertain. In these systems, it ispossible for a statement to take on one of three, four, or an infinite number of possible truth-values! The amazing thing, however, is that given a set of axioms and any consistent logical system (not only the two-valued system Euclid used), mathematical theorems and proofs can be generated. It is even conceivable that a science using mathematics based on multi-valued logic would yield a workable picture of the world. However, as far as I am aware, most Mathematics, and the science based upon it, does use two-valued logic only. In fact, giving up the principle of the excluded middle means giving up many rich results in Mathematics and hence also the concepts in science that depend on these mathematical results. In any case, when we refer to logic in this article, we shallassume that it is two-valued.
So far we have been discussing the intimate tie-in between Mathematics and logic, and the power this must bestow on Mathematics. Once a mathematical result is logically proven, it remains true for eternity and is unaffected by changing paradigms. Here now is the next part of the story, which illustrates the primacyof mathematical logic over what we believe reality is!
Enter, Non-Euclidean Geometry
We now move to mathematicians’ discoveries in the 1800s of geometries other than the standard Euclidean geometry that had held centre-stage since the Greeks. These discoveries led to many serious questions about the nature of mathematical reality; yet they reaffirmed the power of the axiomatic systemand logic.
You will recall that Euclid had created axioms or self-evident truths on which his geometry was based. Specifically, he used five axioms or postulates, all of which were inspired by what he saw in space around him. The first axiom for instance states: “A straight line may be drawn between any two points”. Most mathematicians were also convinced that the first four of his axioms were indeed self-evident truths. However the fifth postulate was not so straightforward. As stated by Euclid himself this axiom is rather tedious, but other mathematically equivalent versions have been used, this one in particular: “Given a line and a point not on the line, only one line can be drawn through the point parallel tothe given line.”
Mathematicians from the time of Euclid himself were convinced that only the first four axioms were essential, and that the fifth could be derived from the other four, making it a theorem instead of an axiom. In fact Euclid too did not use this fifth axiom to derive the first 28 results in the Elements. But like many problems in mathematics that seem innocuous and easy to state at the start, this one turned out to be a tough nut to crack. All attempts to establish the fifth postulate as a theorem failed. Then, a novel approach was tried. Mathematicians like Gauss, Riemann, Bolyai and Lobachevsky tried the following: suppose one assumes that the negation of the fifth axiom is true and one proceeds with generating results, one should soon come upon a contradiction which is the result of that wrong assumption. This would hence establish the fifth axiom as true! (By the way, this method of proof is called reductio ad absurdum and it isvitally dependent on two-valued logic).
Now there are two possible negations for the fifth axiom –one is: “Given a line and a point not on it, no lines can be drawn through the point parallel to the given line. “ The other negation is: “Given a line and a point not on it, infinitely many lines can be drawn through the point parallel to the given line.” To the amazement of the mathematicians, both these assumptions led to no contradictions! In fact, each assumed negation led to a self-consistent, non- Euclidean geometry in its own right. The first is called Riemannian geometry and the second is called Lobachevskian geometry. But unlike Euclidean geometry these two geometries, while logically sound, had no known basis in realityaround us.
The discovery of non-Euclidean geometries had both a shattering effect (Gauss hesitated to publish these results because he felt that the world was not ready for them) and a liberating effect on Mathematics. On the one hand, Mathematics or at least geometry no longer necessarily reflected reality. But on the other hand, mathematicians were now free to create Mathematics without bothering about its relevance and application to the real world. A new and very important question was raised after the discovery of these geometries. If Euclidean geometry no longer represented the exemplar ofcertainty in human knowledge, then what did?
Logic Turns the Tables
The next episode of our story shows how the basis of Mathematics in logic ran into serious difficulties due to the work of Gödel in the 1930s. Finding that geometry no longer guaranteed certainty, Mathematicians turned to number and arithmetic as the foundation for mathematics. In this attempt, they came upon the notion of a set, and in particular infinite sets. A set is an arbitrary collection of objects, and an infinite set is one which contains an infinite number of these objects. Frege was a logician of those times who showed that ‘set theory’ was in fact equivalent to logic. For example, the statement in set theory that “A is a subset of B” is equivalent to the logical statement “If A then B”. The attempt of logicians, who included people like Bertrand Russell, was to show that Mathematics was nothing but a branch of logic. This would secure the foundations of mathematics in a way the Greeks would have surely approved of! However, by a strange coincidence, it was Russell himself who came upon the first stumbling block. He discovered the now famous Russell’s paradox, which questioned the very notion of a set. Given that a set is a very simple notion, a collection of arbitrary elements, it should be possible to state whether a given element belongs in a given set or not. Russell’s paradox showed that this was not so simple after all. He created an infinite set in such a way that it was impossible to state whether a particular element belonged to the set or not. This obviously created problems for Russell and company! A layman’s version of this is the Barber’s Paradox. It poses the following question – suppose there is a barber in an army camp who follows strict orders to “shave all those people who do not shave themselves”. Then does he shave himself, or not? That is, does he belong in the set of all those people who do not shave themselves, or not? I would urge the reader to try answering this question, and you will see the problemfor yourself. It can neither be answered yea nor nay!
When Frege learned of Russell’s paradox just as he was about to publish his second book on the logical foundations of arithmetic, this is what he had to say in an appendix to the book: “Arithmetic totters, a scientist can hardly encounter anything more undesirable than to have the foundation give way just as the work is finished. In this position I was put by a letter from Mr. Bertrand Russell as the work was nearly through the press.” Russell himself was not too happy with the situation. In order to avoid these paradoxes, he and Whitehead had to create a new system of logic, detailed in a book called Principia Mathematica, in which it takes 362 pages to show that 1+1=2. Russell’s commentary on this is very telling:
“I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in Mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in Mathematics, it would be in a new field of Mathematics, with more solid foundations than those that had hitherto been thought secure. But as the work proceeded, I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was no more secure than the elephant, and after some 20 years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making Mathematical knowledge indubitable”.
There were a couple of other attacks on this problem of making the foundations of Mathematics secure. We will focus on the attempt of the great mathematician Hilbert who was mentioned earlier. In his words, “The goal of my theory is to establish once and for all the certitude of Mathematical methods. The present state of affairs where we run up against paradoxes is intolerable. Just think, the definitions and deductive methods which everyone learns, teaches and uses in mathematics, the paragon of truth and certitude, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?”
Hilbert’s attempt was to show that any axiomatic system (i.e. a system where a proof of a theorem can be represented by a formal derivation and can be mechanically verified if one started from some axioms of the system) can be shown to be both complete and consistent. The Principia Mathematica of Russell and Whitehead would be one such system. In the context of the Principia, completeness would mean that every true statement in number theory could be proved within the framework of the Principia. And consistency would mean that a statement and its negation could not both be true at the same time.
Unfortunately for Hilbert and for mathematics (!), in 1931 a 25 year old Austrian mathematician, Kurt Gödel, published a paper entitled ‘On Formally Undecidable Propositions of Principia Mathematica and Related Systems’. Gödel showed that all attempts to prove that arithmetic is free from contradiction are doomed to failure. His theorem says, in effect, that in any formal system such as the Principia, there will always be true statements that cannot be proved using only the axioms listed in the system. Thus the system will be incomplete – unless of course you allow it to be inconsistent! In other words, it is impossible to demand both consistency and completeness at the same time in an axiomatic system such as the Principia.
Gödel’s work basically put an end to attempts to make mathematical foundations firm. To this day we have to live with this uneasy feeling! Moreover Gödel had now introduced a new concept in Mathematics – the notion of undecidability. Certain questions in Mathematics have now become undecidable. When faced with questions which have a ‘yes or no’ answer, the answer, it turns out, depends on what system of set theoretic axioms you choose. For one system the answer may be yes, and for another the answer no! The most famous example is the ‘continuum hypothesis’ (a bit too technical to go into in this article).
The question we now face is – why do mathematicians continue doing Mathematics, in spite of a certain failure to secure the very foundations of Mathematics? In fact, since Gödel’s result, the world has seen some glorious Mathematics including the solution of the famous Fermat’s last theorem. In attempting to answer this question, I shall touch upon three strands–the nature of Mathematics, its applicability and the beauty of Mathematics.
There are two dominant schools of thought about the nature of Mathematics: one is the Platonist or Realist (deriving from Plato) and the other is the Formalist. The Platonists believe that mathematical objects exist independent of us and inhabit a world of their own. They are not invented by us but rather discovered. A mathematician is seen as an explorer in this world of mathematics observing and discovering mathematical truths. Moreover, mathematicians possess a special ability to access this world. Surely if you possess such a worldview, and are endowed with this ability to contact ‘mathematical reality’ it would provide a very strong motivation to do mathematics! A large majority of mathematicians subscribe to this view (a survey claims it is as high as 65%) and seem to ignore some of the problems that arise out of such a view. For example, the logical paradoxes that we encountered earlier must somehow inhabit this platonic reality, and moreover one must almost have a mystical belief in the existence of such a reality and one’s ability to contact it.
Formalists on the other hand believe that there are no such things as mathematical objects. Mathematics consists of axioms; definitions and theorems invented by mathematicians and have no meaning in themselves except that which we ascribe to them. In other words mathematics is like a game of chess. Hardy, himself a staunch Platonist, describes it very well: “The axioms correspond to the given positions of the pieces; the process of proof to the rules for moving them; and the demonstrable formulae to all possible positions which can occur in the game”. It is hard to believe that any working mathematician would spend an entire life manipulating meaningless symbols! Perhaps a more realistic situation is described by a well-known mathematician, J.A. Dieudonné, of the famous Bourbaki school. “We believe in the reality of mathematics, but of...” course when philosophers attack us with their paradoxes we rush to hide behind formalism and say, “Mathematics is just a combination of meaningless symbols, ” and then we bring out Chapters 1 and 2 on set theory. Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. This sensation is probably an illusion, but is very convenient.”
Interestingly both the Platonists and the Formalists fail to explain what the famous physicist Eugene Wigner called in a 1959 lecture, “The unreasonable effectiveness of mathematics in the natural sciences”. Philosophers, mathematicians and scientists have been grappling with the question–why does mathematics have this uncanny ability to describe the natural world? What is spooky is that very often, when a physicist is looking for the right language to describe some aspect of nature he has discovered, he will find that the appropriate mathematics already exists. When Einstein came up with his revolutionary ideas he found that it was actually Riemannian geometry that best described space! What a strange turn of events. Wigner himself ends his lecture by saying, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of Physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning.” I believe it is this ‘unreasonable effectiveness’ that motivates many mathematicians. Some scientists believe (see 7) that due to evolutionary processes our brains are only able to make models of the world of ‘middle dimensions’, and to deal with the very large and very small we need mathematics as a sixth sense as it were.
To conclude this article, I would like to suggest that perhaps the strongest motivation of all in pursuing mathematics is experienced at an emotional level. All mathematicians, no matter what their philosophical inclination, will agree that mathematics is beautiful and that in the process of creating mathematics they experience a sense of ‘illumination spreading throughout the brain’. Alain Connes, a Fields medallist (the highest honour one can receive in mathematics), describes this sensation as follows: “But the moment illumination occurs, it engages the emotion in such a way that it’s impossible to remain passive or indifferent. On those rare occasions when I’ve actually experienced it, I couldn’t keep tears from coming to my eyes.”
Is it that the beauty and uncanny applicability of Mathematics bewitches us to such an extent that we fail to see some of the serious limitations of reason and logic?
- John D. Barrow, Pi in the Sky- Counting, Thinking, and Being, Clarendon Press, Oxford, 1992.
- Jean-Pierre Changeux and Alain Connes, Conversations on Mind, Matter, and Mathematics, Princeton University Press, 1995.
- Philip J. Davis and Reuben Hersh, The Mathematical Experience, Penguin Books, 1980.
- Max Delbruck, Mind from Matter? An Essay on Evolutionary Epistemology, Blackwell Scientific Publications, 1986.
- G.H. Hardy, A Mathematician’s Apology, Canto Books, Cambridge University Press, 1992.
- Douglas R Hofstadter, Gödel, Escher and Bach: An Eternal Golden Braid, Vintage Books edition, 1980.
- N. Mukunda, Existence and reality in mathematics and natural science, Current Science, Vol. 73, No 3, 10 August 1997, pp 236 -241.
- S. Sarukkai et al, Special section on mathematics with the theme ‘the unreasonable effectiveness of mathematics’, Current Science, Vol. 88, No 3, 10 February 2005, pp 360 - 423.
- S.Jagadeeshan, Whoever said nothing is impossible? Three problems from antiquity, Resonance (Journal of science education), Vol. 4, March 1999, pp. 25- 38.
- 10. E.P. Wigner, The unreasonable effectiveness of mathematics in the natural sciences, Commun. Pure Appl. Math., 1960, 13.