REFLECTIONS ON PHILOSOPHY The Philosophy of Mathematics Swami Sarvottamananda (Continued from the previous issue) The Mathematical Method understood the nature of math- ments which are neither true nor false. For example, ematical concepts, we now need to brief- interrogatory and exclamatory statements are nei- Hly examine the mathematical method. ther true nor false. Also there is this classic example What is the method by which we arrive at the truth of a paradoxical self-referential statement, which is or falsity of mathematical statements? neither true nor false: In a mathematical system, we have axioms, P: e statement P is false which are facts taken to be obviously true (‘If a We have referred to the term theorem. Now is the is less than b, then a is not equal to b’ is one such time to de ne it. What is a theorem? A theorem is axiom—the axiom of linear order), and some non- nothing but a proposition for which there is a for- facts (which we shall call non-axioms) by the help mal proof. What then is meant by proof? A proof of which we prove or disprove theorems. Proving is simply a sequence of deductive steps governed by theorems means deriving them from known axi- well-de ned logical rules that follow from a set of oms. If we are able to deduce a theorem starting axioms. An axiom, of course, is a proposition that is from these basic axioms, then we say that the theo- given to be unconditionally true. e following de- rem is true. duction illustrates the rule of specialization, which However, proving the falsity of a theorem is dif- is one of the many rules of logic: ferent. If we are able to derive a non-axiom from a All men are mortal. proposition, then that proposition is false—a non- Socrates is a man. theorem. So, here we go the other way round—we erefore, Socrates is mortal start from the theorem itself, not from non-axioms. us, a mathematical system is a set of axioms Hence proving the truth and falsity of theorems are and non-axioms with prede ned rules of deduc- not mirror processes. tion, which are also referred to as rules of inference. e underlying assumption of this method is e rules of deduction or rules of inference are noth- that we cannot derive a non-fact from facts. Such ing but rules that add, remove, modify, and substi- a system is called consistent. If a system is inconsist- tute operators and symbols. ent, it is ‘trivially complete’; that is, every statement, Let us try an exercise to understand how the true or false, is derivable in an inconsistent system. rules of inference work. Suppose we have been giv- An inconsistent system, therefore, is of little prac- en the following rules of addition, removal, and tical use. substitution of symbols and (the other symbol remains there as in the starting axiom). e start- Propositional Logic ing axiom is , and x and y are variables: In the study of mathematical method we also need (i) x → x (Derive from ) to study propositional logic. Propositions play an (ii) x → xx ( Derive important part in mathematical proofs. What is a from ) proposition? A proposition is a statement which is (iii) xy → xy (Derive from ) either true or false. Note that there are certain state- (iv) xy → xy (Derive from ) PB October 2007 e Philosophy of Mathematics 45 Now try constructing the theorem starting participate in the exciting activity of mathematical only with the axiom using the above rules of discovery. inference. Is it possible to derive ? e crux of the matter discussed above is that Important Branches of Mathematics in mathematics, as well as in logic, the operators, Among the important branches of mathematics, the constants, and the functions can all be viewed, number theory, set theory, geometry, and logic are as in this example, as symbols which are added, re- historically very old. e oldest civilizations—the moved, and substituted by prede ned rules of in- Indian, Greek, Chinese, Egyptian, and Babylo- ference, without ascribing any interpretation to nian—had all developed these branches, in one them. Gödel exploited this fact beautifully in prov- form or other, for general use. is is substanti- ing his famous theorem on incompleteness. ated by the fact that without a fair understanding of geometry the remarkable architectural and civil- Is Mathematics a Uniquely Human Activity? engineering feats for which these civilisations are Since doing mathematics involves intricate reason- famous would not have been possible. Even such ing and abstract thinking, it is oen thought to be elementary constructions as a rectangular wall or a a very creative process requiring a lot of intuition. eld, or the more intricate hemispherical dome, re- Kant was of the opinion that since mathematics re- quire at least a rudimentary knowledge of geomet- quires human intuition it cannot possibly be done rical constructions. Incidentally, ancient Greeks by non-humans. But several later philosophers have gave much importance to geometry, whereas In- shown that it really does not require any human in- dians gave up geometry for abstract mathematics tuition to understand a mathematical proof. Find- during the Buddhist period. ing a proof for an open research problem, though, As far as number and set theories are concerned, might be an altogether di erent matter—comput- no one really knows when humans developed these. ers have failed till date to automatically generate Numbers surely came with the need for counting. proofs for even very simple non-trivial mathemati- Most civilizations seem to have been formally using cal problems. is is not to suggest that proving numbers right from their inception. It was need- mathematical theorems is a uniquely human activ- ed for commerce, and in earlier tribal societies to ity incapable of computer simulation—it is simply quantify one’s possessions. a matter of selective processing power. Comput- Set theory is more fundamental than number ers cannot distinguish between boring mathemati- theory, for it deals with classi cation rather than cal truths and interesting mathematical results and counting. Formal logic was a later development. keep happily churning out one mathematically un- But its rudiments were probably coeval with the interesting result aer another, ad innitum. development of language—with the need to coher- Mathematical thinking, in fact, is apparently ently and intelligently communicate one’s opinions, not unique to humans. Rudimentary mathemati- arguments, and deductions to others. In fact, logic cal understanding is also seen in other animal spe- and language are so interlinked that many consider cies. And, of course, computers are ‘doing’ math- logic to be merely a linguistic construct. Histori- ematics all the time. If one is to argue that nding cally, both Nyaya and Aristotelian philosophy had and discovering mathematical truths rather than formalized logic for their respective civilizations, understanding proofs constitutes the test of mathe- the Indian and the Greek. matical intelligence—and computers fail this test— then it may be pointed out that this will also place Number Theory the majority of humans at par with machine intel- Let us begin with numbers. We may ask: What is ligence, because the vast majority of humans do not the nature of numbers? Are numbers real? In the PB October 2007 46 Prabuddha Bharata Nyaya and Vaisheshika philosophies, for instance, ‘God created the natural numbers; everything else numbers are real entities, belonging to one of the is man’s handiwork’, the German mathematician seven categories of real entities. However, there Leopold Kronecker had famously observed. e are conceptual di culties if we grant numbers an incorporation of zero as a number was the great objective reality. Consider the following: We have contribution of the Indian subcontinent. e natu- two books. So, we have books and we have also ral numbers are complete as far as the operations of the number two. Let us add another pair of books addition and multiplication are concerned—if we to our collection. Does it destroy the number two add or multiply two natural numbers we get an- and create the number four? Or does the number other natural number. However, the class of natural two transform into the number four? Suppose we numbers is not complete with respect to subtrac- add two notebooks, to distinguish them from the tion (you don’t get a natural number if you subtract original pair of books. en we have got a pair of from ). So if the result is to belong to the set of twos as well as a four. None of the original numbers numbers, we need to extend the list of natural num- is destroyed or transformed and yet a new number bers to include negative numbers. e result is the is created. e ancient Buddhists were therefore set of integers. not wrong in pointing out that numbers are in fact Again, we see that the class of integers is not mental concepts. ey do not have any existence complete with respect to division. So the set of outside the mental world. numbers is further extended to include ratios—ra- Furthermore, mathematicians say that numbers tional numbers. e word rational here is derived can also be thought of as properties of sets, being from ‘ratio’ and not ‘reason’. Next we get surds or their sizes (though the Buddhists would not feel irrational numbers, when we extend the set of num- comfortable with this either). Numbers as proper- bers to include limits, sums of series, square roots, ties of sets were called cardinal numbers by George trigonometric functions, logarithms, exponential Cantor in contrast to ordinal numbers which repre- functions, and so on. is gives us real numbers. sented positions in a series ( rst, second, and so on). Actually, this gives us only a subset of real numbers Again, these are not to be taken as real properties, because these constitute only what are called com- for there is an equally long-standing debate on sub- putable numbers (which can be computed to any stances and their properties. Essentially, therefore, desired degree of precision by a nite, terminating numbers are abstract properties of equally abstract algorithm). Not all real numbers can be so con- sets. Or, with greater ingenuity, the abstract con- structed. To be mathematically precise, we need to cept of set itself can be thought of as representing see each real number as a partition which divides numbers—not just the properties of sets but the the set of numbers into two groups A and B. If the sets themselves. us, we may have: partition is such that there is a largest element of A { } = or a smallest element of B then the (partitioning) {ϕ} = number is rational. But if there is neither a largest {ϕ, {ϕ}} = number in A nor a smallest number in B then the {ϕ, {ϕ}, {ϕ,{ϕ}}} = , and so forth. divisive number is irrational. is is the concept of Does anyone nd this remarkable example illumi- ‘cuts’ developed by the celebrated mathematician nating or fascinating! All the same, this is what we Richard Dedekind. meant by our statement that mathematical entities e other day I was arguing with a friend that are not real but are merely conceptual entities. every real number can be seen as a decimal expan- Historically, the notion of numbers was for- sion which can be computed one digit aer another malized in the following succession. e notion using a suitable algorithm. I was, however, wrong. of natural numbers (, , , …) was developed rst. Alan Turing has proved this mind-boggling truth PB October 2007 e Philosophy of Mathematics 47 that not all real numbers are computable. merability refers to being able to be counted by People found out very quickly that negative one-to-one correspondence with the in nite set of numbers could not have real square roots. In order all positive integers.) More remarkably, Cantor was to make the set of numbers complete even with able to prove that even uncountably in nite sets the operation of determining square roots, the do- have di erent cardinalities: that if E is an in nite main of real numbers was again extended to that of set then there exists a set (E) which can be proved complex numbers, which are nothing but the sum to be larger than this set, and this process can be of a real number and an imaginary number (i.e. a extended to obtain in nites with still greater car- number expressed as a multiple of √−). e histor- dinality. Cantor’s treatment of in nities, however, ical choice of the names imaginary and complex was, was abstract rather than constructive. And this cost however, unfortunate. For this makes one think him an appointment at Berlin University—though that complex numbers are not numbers at all. One his work was mathematically sound—as Kroneck- could on the other hand look at complex numbers er, a rm believer in constructions, opposed him. as a dyad such that the subset of this dyad with Mathematicians, aer all, are also human! the second term as zero is actually the set of real numbers. Moreover, all algebraic operations that Zeno’s Paradox can be carried out using real numbers can also be Besides the problem of in nity, mathematicians applied to the complex number dyads when these working with numbers had also to tackle the prob- are suitably rede ned. is interpretation is much lems of limits and series. To appreciate the prob- more appropriate than the one commonly taught lem with series, we consider one of Zeno’s para- in schools. It is also worth noting that the class of doxes—a set of problems devised by Zeno of Elea complex numbers is ‘complete’ in the sense that to support Parmenides’s doctrine that ‘all is one’. if we apply any normal operator or any common is doctrine asserts that, contrary to the evidence function to complex numbers we always get a com- of our senses, the belief in plurality and change is plex number. mistaken, and, in particular, motion is nothing but With the introduction of complex numbers, an illusion. is is much like the Buddhist doctrine one would think that the number system was at of kṣaṇikavāda. peace. But that was not to be, for serious trouble ‘Achilles and the Tortoise’ is the most famous was brewing with the inclusion of the concept of of these paradoxes. Fleet-footed Achilles, of Bat- in nity. ere is a common misconception that tle-of-Troy fame (in Homer’s Iliad), and a tortoise there is one and only one mathematical in nity. are participating in a race. Achilles is reputed to be And the people who seem to be more prone to this the fastest runner on earth; and the tortoise is one misconception are people from a Vedantic back- of the slowest of living beings. However, according ground! I wish to point out that here we to Zeno, Achilles can never win the race if the tor- are not merely thinking of +∞ and −∞, toise is given but a little head start. is is how or even ‘radial in nites’ in the complex it happens: Suppose the tortoise is, say, ten plane. It was George Cantor who proved feet ahead of Achilles. In an instant Achil- that there are numerous in nities in les covers the distance of ten feet. But dur- relation to numbers. As a mat- ter of fact, while the set of integers and of rational numbers are countably innite, the set of real numbers is uncount- Zeno’s Paradox: Given a head start, ably innite. (Countability or denu- CHANDRA the tortoise is always the winner PB October 2007 48 Prabuddha Bharata ing that instant the tortoise has already advanced family heirloom of great value if its blade is changed a short distance. Again in another bound Achilles just ve times and its handle just fourteen times. covers that small distance, but to his dismay, dur- is question of absurdity, however, does not ing that time the tortoise has advanced still more, arise in mathematics because sets as well as their and so on. us, Achilles can never possibly catch constituent members are all hypothetical entities— up with the tortoise. conceptual objects which are granted no intrinsic But this clearly is nonsense. In reality, things reality. never happen like that. is is actually a graphic description of the problem of the sum of an in nite Geometry series of decreasing terms which yields a nite value. In contrast to sets and numbers, it is easy for us to Of course, not every such series will yield a nite see that geometrical objects are conceptual. But it value. e harmonic series ( + ½ + ⅓ + ¼ + …) is was not so for the Greeks—they took their geom- one such. etry seriously exactly for the opposite reason: they thought geometry was real. Set Theory Take, for instance, the case of a point and a Now that we are on paradoxes, let us start our dis- line in a plane. What is a point? A point, as every cussion of set theory with Russell’s paradox. In set schoolchild knows, is a geometrical object that theory, we have nite sets as well as in nite sets. For does not have any length or breadth (all its dimen- in nite sets it is possible that a set contains itself. {ϕ, sions are zero). And what is a line? A line is a geo- {ϕ}, {ϕ,{ϕ}}, {ϕ,{ϕ},{ϕ,{ϕ}}}, …} is one such set. Keen metrical object that has only length but no breadth. observers would have noted that this is the number ese very de nitions make it obvious that true ‘in nity’ in the ‘illuminating’ example of a previous points and lines cannot exist in the real world dis- section. Now call a set abnormal if it contains itself. tinct from our mental constructions. De ne a set R of all ‘normal’ sets: ‘the set of all sets Credit goes to Euclid for formalizing the eld that do not contain themselves as members’. Now of geometry into a body of axioms and theorems. ask the question: Is R normal or abnormal? We see ough his treatment of the subject was fully con- that this question cannot be answered in either the ceptual, it took a really long time—two thousand a rmative or the negative. years—for people to see that these concepts do not e ‘axiomatic set theory’ was developed to ad- quite match the real world. All this time everyone dress such paradoxes by incorporating an ‘axiom of had been mistakenly assuming that the world is choice’ within the theory. But this goes beyond the Euclidean. Geometrical results seemed to t our scope of our discussion, although it may be men- experiential world so very nicely that people failed tioned in passing that a surprising corollary to this to see that they could be unreal. Nevertheless, with theory is the fact that a universal set—the hypo- the advent of Einstein’s theories of relativity—both thetical set containing all possible elements—does special and general—the realization dawned that not exist. the world is in fact non-Euclidean; it is more ac- In practice, sets are normally related to groups curately described in terms of several Riemannian and collections of objects in the external world. (or elliptic) geometries. Here too, a similar question, as with numbers, aris- Another point to note is that, in formalizing ge- es: are sets real? In Indian philosophical thought ometry, we try to arrive at proofs which do not ap- too, the same question appears repeatedly. e peal to our intuition or visual sense but are logically Buddhists, for instance, argue that the axe which is correct. For though original mathematical insights a combination of the handle and the blade does not are oen derived through intuition, these ‘insights’ exist ‘in itself ’. It is absurd, they say, to call an axe a also run the risk of being proved wrong. Even the PB October 2007 e Philosophy of Mathematics 49 great Euclid—though he was well aware of this and tions which can be asked with regard to axioms: therefore tried very hard to avoid intuitive judge- To prove that they are not contradictory, that is, a ments—himself committed a few mistakes in his de nite number of logical steps based on them can proofs, because these proofs relied on the way he never lead to contradictory results. drew the illustrations. All the same, this does not e questions of consistency and completeness take away any of the credit due to him in recogniz- are important because if mathematics as a system ing what is correct mathematical procedure. And were both complete and consistent, then it could certainly the momentous task of formalizing the well yield an easy path to new discoveries by way of great body of geometry already known at his time a method to automatically discover mathematical was not an easy task by any standard. theorems, what with superfast computers with su- per-memory and super processing power as tools. Logic Kurt Gödel, however, proved that mathemat- Mathematical logic is the nal edi ce of mathe- ics is in fact incomplete. He further showed that matics. And every logical system has to deal with the consistency of mathematics cannot be proven the question of completeness and consistency. Com- from within the eld of mathematics itself, or to be pleteness means that every true statement must be precise, from within Peano’s axiomatization of the veri able, must have a proof. Consistency is slightly number theory. So with this dual stroke he deliv- di erent: it means that we should not be able to ered a terrible blow to the human quest for ‘know- ‘prove’ false statements as true, that is, false state- ing everything’. ments must not have valid proofs in the theory in In brief, Gödel’s theorems have the following question. twin consequences: First, there exist true state- At the beginning of the twentieth century, Dav- ments which do not have any proof, and second, id Hilbert posed the ultimate problem of logic even if we have a proof for such a statement, we to mathematicians—to prove the consistency of do not also know (by means of a valid proof ) that mathematics as a system. is challenge came to its converse is not true. e wording and formula- be fondly called the Hilbert programme. Hilbert tion of the second part is important as it makes a observed: distinction between the truth of a statement and When we are engaged in investigating the foun- having a proof thereof. dations of a science, we must set up a system of A question may naturally arise at this juncture: axioms which contains an exact and complete de- Is Gödel’s incompleteness theorem applicable to scription of the relations subsisting between the every logical system? Turing is credited with ex- elementary ideas of that science. e axioms so set tending the results of Gödel’s theorem to the eld up are at the same time the de nitions of those ele- mentary ideas; and no statement within the realm of computation. He has shown the non-existence of the science whose foundation we are testing is of several kinds of computational procedures that held to be correct unless it can be derived from could have helped us circumvent the implications those axioms by means of a nite number of logi- of Gödel’s theorem, enabling us to nd the truth cal steps. Upon closer consideration the question and falsity of statements in a circuitous way. us, arises: Whether, in any way, certain statements he was able to draw our attention to the far-reach- of single axioms depend upon one another, and ing consequences of Gödel’s incompleteness theo- whether the axioms may not therefore contain rem. In short, this theorem brings under its pur- certain parts in common, which must be isolated if one wishes to arrive at a system of axioms that view every kind of logical system—ancient or mod- shall be altogether independent of one another. ern or postmodern—that is powerful enough to But above all I wish to designate the following deduce facts. It only leaves out trivial theories like as the most important among the numerous ques- those based on rst-order predicate calculus (logic). PB October 2007 50 Prabuddha Bharata So it would not be correct to say that Gödel’s in- a true theorem which does not have a proof. But completeness theorem applies only to formal logic we do not know speci cally which of the two (the or axiomatic mathematics, and not to the Nyaya theorem or its converse) is true. A further corollary or Buddhist logical systems, because these sys- to his theorem is that only inconsistent systems are tems also involve predicates and possess deductive trivially complete. And our hopes of omniscience (anumāna) power. are further dampened when we remember that the consistency of a system is impossible to prove from Mathematics, Mind, and Maya within the system itself. Let me conclude with some personal reections: ird, Vedanta as a system of philosophy is an First, mathematics has to constantly ght o empirical system. However, the only empirical facts utilitarians who accuse it of a lack of concern with that it sticks to with heart and soul are the reality reality—at least pure mathematics does not con- of Brahman, the unreality of samsara, and the one- cern itself with applications. In fact, many pure ness of Atman, the individual soul, and Brahman, mathematicians think that applied mathematics— the supreme Reality. ese are empirical truths ac- being more interested in the results than in the cording to Vedanta because Vedanta rmly holds process—is a degradation, and hence no mathemat- that Atman, Brahman, and maya are mere state- ics at all. ments of facts—a posteriori truths, truths that It is a mundane fact that less-advanced disci- need to be experienced or realized. However, as plines further their cause with the assistance of the world is granted only a conceptual reality—as more advanced ones. e latter, however, can keep a construct of the cosmic mind (hiranyagarbha)— advancing only by keeping intact their pristine pu- Vedanta remains within the purview of empirical rity. us even though others may use mathematics, sciences only very loosely. Strictly speaking, then, mathematics stands to lose if it starts catering to Vedanta as a system with a single composite em- the demands of other disciplines: the only way for pirical fact—brahma satyaṁ jaganmithyā jiva brah- mathematics to advance is by concentrating on its maiva nāparaḥ; Brahman is real, the world unreal, loy aims. us it should be le to other disciplines and the individual soul is no di erent from Brah- to nd the applications for and uses of mathematics, man—which is not provable by sensory percep- so that pure mathematics remains pure. tions, becomes a system independent of physics Second, Gödel was able to prove that there ex- and mathematics alike. Nevertheless, care should ist true theorems for which there is no proof. Some be taken, when we talk (as Vedantists) either about take this as proof of the superiority of the human the world that is a product of maya or when we use intellect—aer all, we know indirectly about the a deductive process to infer the unity of existence truth of these theorems even though they cannot and the unreality of the world, for then there is no be proved. is is not correct. Gödel only showed escape from the sciences, both empirical and for- that both the theorems and their mal—physics and mathematics. converse have no proof, and so Within the realm of maya, if a system is consistent, one Vedanta cannot go against of them is bound to be true. the findings of physics us we have, by inference, and mathematics. P
‘Vedanta cannot go against ‘Only in the realm of maya!’ the ndings of physics and mathematics!’
PB October 2007