The Philosophy of Mathematics

1. The Philosophy of Mathematics • The philosophy of mathematics is the branch of philosophy whose task is to reflecton, and account for the nature of mathematics. This is a special case of the task ofepistemology which is to account for human knowledge in general. • The philosophyof mathematics addresses such questions as: • What is the basis for mathematicalknowledge? • What is the nature of mathematical truth? • What characterises the truthsof mathematics? • What is the justification for their assertion? • Why are the truths ofmathematics necessary truths?

2. The Nature of Mathematical Knowledge Traditionally, mathematics has been viewed as the paradigm of certain knowledge. Euclid erected a magnificent logical structure nearly 2,500 years ago in his Elements,which until the end of the nineteenth century was taken as the paradigm forestablishing truth and certainty. Newton used the form of the Elements in hisPrincipia, and Spinoza in his Ethics, to strengthen their claims to systematically expound truth. Thus mathematics has long been taken as the source of the mostcertain knowledge known to humankind.

3. Knowledge is classified on the basis of the grounds for its assertion. A prioriknowledge consists of propositions which are asserted on the basis of reason alone,without recourse to observations of the world. Here reason consists of the use ofdeductive logic and the meanings of terms,typically to be found in definitions. Incontrast, empirical or a posteriori knowledge consists of propositions asserted on thebasis of experience, that is, based on the observations of the world (Woozley, 1949).

4. Mathematical knowledge is classified as a priori knowledge, since it consists ofpropositions asserted on the basis of reason alone. Reason includes deductive logicand definitions which are used, in conjunction with an assumed set of mathematicalaxioms or postulates, as a basis from which to infer mathematical knowledge. Thusthe foundation of mathematical knowledge, that is the grounds for asserting the truthof mathematical propositions, consists of deductive proof.

5. The Absolutist View of Mathematical Knowledge The absolutist view of mathematical knowledge is that it consists of certain andunchallengeable truths. According to this view, mathematical knowledge is made upof absolute truths, and represents the unique realm of certain knowledge, apart from logic and statements true by virtue of the meanings of terms. Many philosophers, both modern and traditional, hold absolutist views of mathematical knowledge

6. The absolutist view of mathematical knowledge encountered problems at thebeginning of the twentieth century when a number of antinomies and contradictions were derived in mathematics (Kline, 1980; Kneebone, 1963; Wilder,1965). Russell produced his well-known paradox by defining the property of ‘not being an element of itself. Frege’s law allows theextension of this property to be regarded as a set. But then this set is an element ofitself if, and only if, it is not; a contradiction. Frege’s Law could not be dropped without seriously weakening his system, and yet it could not be retained.

7. The outcome of this crisis was thedevelopment of a number of schools in the philosophy of mathematics whose aimswere to account for the nature of mathematical knowledge and to re-establish its certainty. The three major schools are known as logicism, formalism andconstructivism (incorporating intuitionism). The tenets of these schools of thoughtwere not fully developed until the twentieth century, but Korner (1960) shows thattheir philosophical roots can be traced back at least as far as Leibniz and Kant.

9. Logicism is the school of thought that regards pure mathematics as a part of logic. Themajor proponents of this view are G.Leibniz, G.Frege (1893), B.Russell (1919), A.N.Whitehead and R.Carnap (1931). At the hands of Bertrand Russell the claims oflogicism received the clearest and most explicit formulation. There are two claims: 1. All the concepts of mathematics can ultimately be reduced to logicalconcepts, provided that these are taken to include the concepts of set theoryor some system of similar power, such as Russell’s Theory of Types. 2. All mathematical truths can be proved from the axioms and rules of inferenceof logic alone.

10. Whitehead and Russell (1910–13) were able to establish the first of the two claimsby means of chains of definitions. However logicism foundered on the second claim. Mathematics requires non-logical axioms such as the Axiom of Infinity (the set of allnatural numbers is infinite) and the Axiom of Choice (the Cartesian product of afamily of non-empty sets is itself non-empty).

11. Thus not all mathematical theorems and hence not all the truths of mathematics can bederived from the axioms of logic alone. This means that the axioms of mathematics arenot eliminable in favour of those of logic. Mathematical theorems depend on anirreducible set of mathematical assumptions. Indeed, a number ofimportant mathematicalaxioms are independent, and either they or their negation can be adopted, without inconsistency (Cohen, 1966). Thus the second claim of logicism is refuted

12. Formalism In popular terms, formalism is the view thatmathematics is a meaningless formalgame played with marks on paper, following rules. Traces of a formalist philosophy ofmathematics can be found in the writings of Bishop Berkeley, but the majorproponents of formalism are David Hilbert (1925), early J.vonNeumann (1931) andH.Curry (1951).

13. The formalist thesis comprises two claims. 1. Pure mathematics can be expressed as uninterpreted formal systems, in whichthe truths of mathematics are represented by formal theorems. 2 .The safety of these formal systems can be demonstrated in terms of theirfreedom from inconsistency, by means of meta-mathematics. Kurt Godel’s Incompleteness Theorems (Godel, 1931) showed that the programmecould not be fulfilled.

14. Constructivism The constructivist strand in the philosophy of mathematics can be traced back atleast as far as Kant and Kronecker (Korner, 1960). The constructivist programme isone of reconstructing mathematical knowledge (and reforming mathematicalpractice) in order to safeguard it from loss of meaning, and from contradiction. Tothis end, constructivists reject non-constructive arguments such as Cantor’s proofthat the Real numbers are uncountable, and the logical Law of the Excluded Middle.

15. The best known constructivists are the intuitionists L.E.J.Brouwer (1913) and A.Heyting (1931, 1956). More recently the mathematician E.Bishop (1967) has carriedthe constructivist programme a long way, by reconstructing a substantial portion of Analysis, by constructive means. Various forms of constructivism still flourish today, such as in the work of the philosophical intuitionist M.Dummett (1973, 1977).

16. Constructivists claim that both mathematical truths and the existence ofmathematical objects must be established by constructive methods. This means thatmathematical constructions are needed to establish truth or existence, as opposed tomethods relying on proof by contradiction. For constructivists knowledge must be established through constructive proofs, based on restricted constructivist logic, andthe meaning of mathematical terms/objects consists of the formal procedures bywhich they are constructed.

17. Another problem for the constructivist view, is that some of its results areinconsistent with classical mathematics. Thus, for example, the real numbercontinuum, as defined by the intuitionists, is countable. This contradicts the classicalresult not because there is an inherent contradiction, but because the definition ofreal numbers is different. Constructivist notions often have a different meaning from the corresponding classical notions.

19. The Fallacy of Absolutism We have seen that a number of absolutist philosophies of mathematics have failed to establish the logical necessity of mathematical knowledge. Each of the three schoolsof thought logicism, formalism and intuitionism (the most clearly enunciated form ofconstructivism) attempts to provide a firm foundation for mathematical truth, by deriving it by mathematical proof from a restricted but secure realm of truth.

20. The fact that three schools of thought in the philosophy of mathematics havefailed to establish the certainty of mathematical knowledge does not settle thegeneral issue. It is still possible for other grounds to be found for asserting thecertainty of mathematical truth. Absolute truth in mathematics still remains amathematics also have a powerful impact on the way mathematics is taught (Davis, 1967; Cooney, 1988; Ernest, 1988b, 1989c).

21. Criteria for an Adequate Philosophy ofMathematics a proposed philosophy ofmathematics should account for: (i) Mathematical knowledge: its nature, justification and genesis, (ii) The objects of mathematics: their nature and origins, (iii) The applications of mathematics: its effectiveness in science, technology andother realms. (iv) Mathematical practice: the activities ofmathematicians, both in the presentand the past.

22. A Further Examination of Philosophical Schools A. The Absolutist Schools B. Progressive Absolutism C. Platonism D. Conventionalism E. Empiricism

23. the logicist, formalist and intuitionist schools areabsolutist. We have given an account of the failure of the programmes of these schools, and indeed refuted in general the possibility of absolutism in the philosophy of mathematics.

24. In contrast, progressive absolutist philosophies: 1. accommodate the creation and change of axiomatic theories; 2. acknowledge that more than purely formal mathematics exists, formathematical intuition is needed as the basis for theory creation; and hence 3. acknowledge human activity and its outcomes, in the creation of newknowledge and theories.

25. C. Platonism Platonism is the view that the objects of mathematics have a real, objective existence in some ideal realm. It originates with Plato, and can be discerned in the writings ofthe logicists Frege and Russell, and includes Cantor, Bernays (1934), Hardy (1967)and Godel (1964). Platonists maintain that theobjects and structures of mathematics have a real existence independent of humanity,and that doing mathematics is the process of discover ing their pre-existingrelationships. According to platonism mathematical knowledge consists ofdescriptions of these objects and the relationships and structures connecting them.

26. The conventionalist view of mathematics holds that mathematical knowledge andtruth are based on linguistic conventions. In particular, that the truths of logic andmathematics are analytic, true by virtue of the meanings of the terms involved. Amoderate form of conventionalism, such as that of Quine (1936) or Hempel (1945),uses linguistic convention as the source of basic mathematical truth on which theedifice of mathematics is constructed. According to this view linguistic conventionsprovide the basic, certain truths of mathematics and logic, and deductive logic(proofs) transmits this truth to the remainder of the body of mathematical knowledge, thus establishing its certainty.

27. The Problem of Existence in Mathematics Among the theorems of classical mathematics, one finds countless existence assertions. For example: ‘There exists areal number which is such that its cube minus seven is zero’ and ‘There exists a set whose members are just those setsthat have as members only finite cardinal numbers’. Now what is the significance of asserting that some mathematicalentity (or some sort of mathematical entity) exists? And what does it mean? How, in short, are we to understandexistence assertions in mathematics? Philosophers might reasonably ignore such questions did they not also believe that mathematics is true, i.e. that theassertions of mathematics are for the most part true. Of course, not all people believe that mathematics is true. AParisian mathematician once asserted, during a lecture I was giving, that in France children are not taught that theassertions of mathematics are true—instead, they are taught that these assertions are good!

28. The assertion ‘It was discovered that there really are black swans’ naturally gives rise to the question ‘Where are these black swans to be found?’, but the assertion ‘It wasdiscovered that there are continuous functions that are nowhere differentiable’ gives rise to no analogous question. Itseems to make no sense, in the case of mathematics, to ask ‘Where are these functions?’ Evidently, mathematicalentities do not exist anywhere in physical space. Thus, the classical mathematician is regarded as asserting the existenceof “abstract” entities not to be found in the physical world. The mathematician who believes the theorems of classical mathematics to be true, according to this position, ought to believe that there really are such entities as naturalnumbers, functions, sets, ordered pairs, and the like. The view that emerges then is that of the mathematicianinvestigating a realm of entities that cannot be seen, felt, heard, smelled, or tasted, even with the most sophisticatedinstruments.

29. But if this is so, how can the mathematician know that such things exist? We seem to be committingourselves to an impossible situation in which a person has knowledge of the properties of some objects even thoughthis person is completely cut off from any sort of causal interaction with these objects. And how does the mathematician discover the various properties and relationships of these entities that the theorems seem to describe?By what powers does the mathematician arrive at mathematical knowledge? In short, how is mathematical knowledgepossible? As we shall soon see, those who espouse the kind of view under consideration attempt to provide plausible answers to these queries.

30. Matematika???

31. these are a few ofthe ingredients in the mathematical image: (1) Mathematical results are certain (2) Mathematics is objective (3) Proofs are essential (4) Diagrams are psychologically useful, but prove nothing (5) Diagrams can even be misleading (6) Mathematics is wedded to classical logic (7) Mathematics is independent of sense experience (8) The history of mathematics is cumulative (9) Computer proofs are merely long and complicated regular proofs (10)Some mathematical problems are unsolvable in principle

32. Certainty The theorem proving the infinitude of primes seems established beyond a doubt. The naturalsciences can’t give us anything like this. In spite of its wonderful accomplishments, Newtonian physics hasbeen overturned in favour of quantum mechanics and relativity. And no one today would bet too heavily onthe longevity of current theories. Mathematics, by contrast, seems the one and only place where we humanscan be absolutely sure we got it right.

33. Let’s begin with a nice example, the proof that there are infinitely many prime numbers. If asked for atypical bit of real mathematics, your friendly neighbourhood mathematician is as likely to give this example as any. First, we need to know that some numbers, called ‘composite’, can be divided without remainder orbroken into factors (e.g. 6=2×3, 561=3×11 ×17), while other numbers, called ‘prime’, cannot (e.g. 2, 3, 5, 7,11, 13, 17, …). Now we can ask: How many primes are there? The answer is at least as old as Euclid and iscontained in the following. Theorem: There are infinitely many prime numbers. Proof: Suppose, contrary to the theorem, that there is only a finite number of primes. Thus, therewill be a largest which we can call p. Now define a number n as 1 plus the product of all the primes: n=(2x3x5x7x...xp)+1

34. Is n itself prime or composite? If it is prime then our original supposition is false, since n is larger thanthe supposed largest prime p. So now let’s consider it composite. This means that it must be divisible(without remainder) by prime numbers. However, none of the primes up to p will divide n (since wewould always have remainder 1), so any number which does divide n must be greater than p. Thismeans that there is a prime number greater than p after all. Thus, whether n is prime or composite, oursupposition that there is a largest prime number is false. Therefore, the set of prime numbers isinfinite.

35. Objectivity Whoever first thought of this theorem and its proof made a great discovery.

36. Proof is essential With a proof, the result is certain; without it, belief should be suspended. That might beputting it a bit too strongly. Sometimes mathematicians believe a mathematical proposition even thoughthey lack a proof. Perhaps we should say that without a proof a mathematical proposition is not justified and should not be used to derive other mathematical propositions. Goldbach’s conjecture is an example. It saysthat every even number is the sum of two primes. And there is lots of evidence for it, e.g. 4=2+2, 6=3+3, 8=3+5, 10=5+5, 12=7+5, and so on. It’s been checked into the billions without a counter-example. Biologistsdon’t hesitate to conclude that all ravens are black based on this sort of evidence; but mathematicians (while they might believe that Goldbach’s conjecture is true) won’t call it a theorem and won’t use it to establishother theorems—not without a proof.

38. Diagrams There are no illustrations or pictures in the proofs of most theorems. In some there are, butthese are merely a psychological aide. The diagram helps us to understand the theorem and to follow theproof—nothing more. The proof of the Pythagorean theorem or any other is the verbal/symbolic argument. Pictures can never play the role of a real proof.

39. Misleading diagrams Pictures, at best, are mere psychological aids; at worst they mislead us—often badly. Consider the infinite series which we can illustrate with a picture The sum of this series is 2π/6=1.6449… In the picture, the sum is equal to the shaded area. Let’s suppose we paint the area and that this takes one can of paint.

41. Classical logic Notice the structure of the proof of the irrationality of . We made a supposition. Wederived a contradiction from this, showing the supposition is false. Then we concluded that the negation ofthe supposition is true. The logical principles behind this are: first, no proposition is both true and false (noncontradiction)and second, if a proposition is false, then its negation is true (excluded middle). Classicallogic, it seems, is an essential tool of mathematics. Without this tool, much of traditional mathematicswould crumble.

43. Sense experience All measurement in the physical world works perfectly well with rational numbers. Letting the standard metre stick be our unit, we can measure any length with whatever desired accuracy our technical abilities will allow; but the accuracy will always be to some rational number (some fraction of a metre). In other words, we could not discover irrational numbers or incommensurable segments (i.e. Lengthswhich are not ratios of integers) by physical measurement. It is sometimes said that we learn 2+2=4 bycounting apples and the like. Perhaps experience plays a role in grasping the elements of the naturalnumbers. But the discovery of the irrationality of was an intellectual achievement, not at all connected to sense experience.

44. Cumulative history The natural sciences have revolutions. But amathematical result, once proven, lasts forever. There are mathematical revolutions in the sense of spectacular results which yield new methods to work with and which focus attention in a new field—but no theorem is ever overturned. The mathematical examples I have so far discussed all predate Ptolemaicastronomy, Newtonian mechanics, Christianity and capitalism; and no doubt they will outlive them all.

45. Computer proofs Computers have recently played a dramatic role in mathematics. One of the mostcelebrated results has to do with map colouring. How many colours are needed to insure that no adjacentcountries are the same colour? Theorem: Every map is four-colourable. See: five_most_famous_problem

46. Some mathematical problems are unsolvable in principle Solving problems There are lots of things we might ask, but have little chance of answering: ‘Does Godexist?’ ‘Who makes the best pizza?’ These seem perfectly meaningful questions, but the chances of findinganswers seems hopeless. By contrast, it seems that every mathematical question can be answered and everyproblem solved. Is every even number (greater than 2) equal to the sum of two primes? We don’t know—now, but that’s because we’ve been too stupid— so far. Yet we are not condemned to ignorance aboutthe way we are about the home of the best pizza. It’s Goldbach’s conjecture the sort of question that we should be able to answer, and in the long run we will.

47. Having said this, a major qualification is in order. In fact, we may have to withdraw the claim. So far, inlisting the elements of the mathematical image we’ve made no distinction among mathematicians, philosophers and the general public. But at this point we may need to distinguish. Recent results such asGödel’s incompleteness theorem, the independence of the continuum hypothesis and others have led manymathematicians and philosophers of mathematics to believe that there are problems which are unsolvable inprinciple. The pessimistic principle would seem to be part of the mathematical image.

48. What’s the greatest discovery in the history of thought? Of course, it’s a silly question—but it won’t stop me from suggesting an answer. It’s Plato’s discovery of abstract objects. Most scientists, and indeed mostphilosophers, would scoff at this. Philosophers admire Plato as one of the greats, but think of his doctrine of the heavenly forms as belonging in a museum.

50. The Original Platonist We notice a similarity among various apples and casually say, There is something they have in common.’ But what could this something they have in common be? Should we even take such a question literally? Plato did and said the common thing is the form of an apple. The form is a perfect apple, or perhaps a kindof blueprint. The actual apples we encounter are copies of the form; some are better copies than others. Adog is a dog in so far as it ‘participates’ in the form of a dog, and an action is morally just in so far as it participates in the form of justice.