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 dene 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-dened 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 predened 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 predened 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 dierent 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 classication 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.