VOL.16 No. 65 OCTOBER 1966
pages 289 to 307[Some difficulty in reproducing logical symbols and lay-out in HTML]
[If too difficult to read, E-mail me, and I will take out unnecessary detail from the scanned text]
NOT `THEREFORE' BUT `BUT' *
Plato began it. After thinking about the nature of argument he concluded that the correct way of reasoning was the axiomatic way, and formulated the programme of axiomatization that Eudoxus and Euclid subsequently carried out. Since then the axiomatic method has been firmly established, not only as the method for mathematics, but as a paradigm to which all other disciplines should strive to be assimilated; and in this present century not only has axiomatization been carried through as completely as it can be, but the most determined efforts have been made to wish hypotheticodeductive schemata on to biology, economics, and even history.
The latter attempts are largely misconceived. More than that, even the nature of mathematics is being distorted and concealed by over-much emphasis on the axiomatic method. The axiomatic approach naturally generates a formalist philosophy of mathematics, and I join with many real mathematicians in finding Formalism ringing false as an account of what mathematics is and what mathematicians do. An exclusively axiomatic approach, as Plato himself foresaw, leads to the conclusion that one is manipulating entities which are in themselves essentially meaningless. For the axioms admit of alternative interpretations. They give only an implicit definition of the primitive terms they use. Thus we are led to conclude with the formalists that we are in mathematics merely manipulating
* I have to thank Mr. J. C. Manisty of Winchester College who made me understand infinitesimals; Dr. F.Waismann who set me thinking on the lines developed here; Dr. I. Lakatos, who anticipated me on many points and of whose ideas I have made great use here (see especially his ``Proofs and Refutations '', British Journal for the Philosophy of Science, May 1963) also to Dr. J.Ravetz, my onetime colleague, for many valuable discussions; and to my pupil, Mr. A. E. Dusoir, for asking the right questions.
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uninterpreted symbols according to such rules as we please to adopt, and
to exclaim, with Plato, ho_I gar arche men ho me oide, teleute de kai ta metaxu ex hou me oiden sumpeplektai, tis mechane ten toiauten homologian pote epistemen genesthai;1
The conclusion is repugnant for many reasons. The normal objections to Formalism, as a philosophy of mathematics, although not as a logical technique, need not be recounted here. The objection I want to make is that the axiomatic ideal makes mathematics out to be a monologue rather than a dialogue, and represents a two-personal interchange as a one-personal inference; it gives an extraordinarily mechanical view of the mathematician's mind, and represents the thinking of the mathematician as quite different from, and more solitary than, other forms of reasoning.
Most types of reasoning are unformalized and dialectical. There is no one fixed form of valid argument. Rather, there are arguments on the one side and arguments on the other, and the historian or the lawyer or the philosopher has to use his trained judgement to decide between conflicting judgements. The final say, such as there is, on what is or is not a good reason or a reasonable argument is reserved to the Reasonable Man, the sound and judicious expert in the field. And so our ways of arguing are more like conversations than proof-sequences. They proceed through claim and counter-claim, objection and rebuttal, each person stating his own side of the case, and trying to meet the arguments the other has adduced. The typical connective of argument is not `therefore' but `but'.2
Only in mathematics, expounded more geometrico, are we presented with a monolithic monologue, in which the reader is expected to play no part, and allowed to raise no queries. Only in mathematics does the ideology of the subject explicitly preclude an ultimate recourse to the consensus of informed opinion. In mathematics, according to the axiomatic account, all reasons must be statable and stated in advance, and must be one of a few specified sorts; and in this, perhaps the most difficult of all the disciplines, we find the paradoxical tenet held, that a proof, if really a proper proof, must be capable in principle of being checked and followed by the dimmest nitwit alive.
The pattern of the dialogue survives nonetheless in mathematical reasoning. In the earliest proofs the person addressed still retained the shadowy function of granting at the outset of the proof those data that the geometer required for his demonstration: but the programme of axiomatization was soon carried to the point where there were required only five postulates, which, it was half-hoped, could be had without having to be asked for. But although the interlocutor has since then been altogether banished, mathematics is in many of its features more intelligible if viewed as a conversation
2. See J. R. Lucas: "The Philosophy of the Reasonable Man", The Philosophical Quarterly, 1963, pp. 97-106.
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between friends trying to communicate mathematical insights to one another than if considered as a set of proof-sequences constructed by an isolated mind.
Hardy, not exactly an apostle for sloppy mathematics, once said:3 ``I have myself always thought of a mathematician as in the first instance an observer, a man who gazes at a distant range of mountains and notes down his observations. His object is simply to distinguish clearly and notify to others as many different peaks as he can. There are some peaks which he can distinguish easily, while others are less clear. He sees A sharply, while of B he can obtain only transitory glimpses. At last he makes out a ridge which leads from A, and following it to its end he discovers that it culminates in B. B is now fixed in his vision, and from this point he can proceed to further discoveries. In other cases perhaps he can distinguish a ridge which vanishes in the distance, and conjectures that it leads to a peak in the clouds or below the horizon. But when he sees a peak he believes that it is there simply because he sees it. If he wishes some one else to see it, he points to it, either directly or through the chain of summits which led him to recognize it himself. When his pupil also sees it, the research, the argument, the proof is finished.
The analogy is a rough one, but I am sure that it is not altogether misleading. If we were to push it to its extreme we should be led to a rather paradoxical conclusion; that there is, strictly, no such thing as mathematical proof; that we can, in the last analysis, do nothing but point; that proofs are what Littlewood and I call gas, rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils. This is plainly not the whole truth, but there is a good deal in it."
This is no strange paradox being put forward. The Greek word for a proof, semeion, means just this, a pointer, or an indicator. One is pointing out a feature of the figure drawn in the sand to a friend to enable him to see for himself that what one says is true. In like spirit, Hardy and Littlewood are indicating to their colleagues certain truths about the natural numbers. They want to say enough to enable others to identify the mountain peak they have sighted, but not so much as to confuse those whose minds they are trying to guide. There is an assumption of friendliness rather than hostility basic to the communication they are attempting to achieve. It is not a case of having to compel an unwilling reader by coercive arguments to concede the truth of their conclusion: he is with them in wanting the conclusion to be true; it is only that he needs their help to be able to see it for himself. The object is to share an insight, not force a conclusion. They are trying to point out a pattern, indicate a Gestalt, literally in the case of geometry, metaphorically in the case of algebra, analysis or the theory of numbers. This surely is, conversely, our experience in trying to follow the proofs of others: we can go over them again and again, and in a
3.Mind, 1929, p.18. See also I, Lakatos, op. cit., pp. 125-6.
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sense see that each step follows from the last, but not really understand a proof at all, until suddenly it tumbles into shape and we have ``got it''. Different things trigger off the flash of understanding for different people; there is no one standard recipe for communicating the gist of a proof to any comer. The one standard procedure that there is, the proof-sequence, is the one that can be guaranteed as universally useless, for it alone ensures that no newcomer will be able to discern the wood for the trees.
The communication of mathematical truth occurs in a very different fashion from what the philosophers of mathematics would lead one to suppose. It is indeed far more a matter of persuading than convincing, and where too great a stress is laid on arguments having to be completely watertight, communication is hindered rather than helped. Verbal communications are valuable just because they are less inhibited than what appears in learned journals: one can use some metaphor about ``squeezing'' a function in between two limits, or "throwing away the tail'', which sounds all right when spoken, but would read awkwardly in print: I suspect that the reason why mathematicians are more attached to their blackboards than are the exponents of other disciplines, is just that blackboards are easy to rub out, and so allow a liberty of action which would be denied if they had to stand by everything they said.
Mathematical physicists are even more chary of mathematical finickiness than pure mathematicians. The aspiring physicist is often advised not to read mathematics as an undergraduate, because although he very much needs to be able to think in mathematical symbols, there is a danger of catching the infection of scholasticism from the purists' approach, and losing the far more valuable "physical intuition'' which enables him to pick out the really significant formulations, and neither to be deterred by the objections which a clever mathematician could obviously make to his argumentation nor to be sidetracked by elegant and amusing systems which lack real, physical, significance. The physicist is content to deal with non-watertight concepts: when he talks about a function he means a sensible function, and does not worry his head about the monstrosities that clever mathematicians think up as counter-examples to theorems propounded by their colleagues. Functions are continuous and differentiable, series and integrals converge within the relevant limits, singularities do not exist unless they are wanted, spaces are homogeneous and connected; and altogether it is a thoroughly sensible and reasonable world that the physicist deals with. He is not concerned to block up every loop-hole in advance: he says only enough for the purpose, the purpose of showing another physicist the line of argument. If some serious counter-example, not excluded by his original specification, turns up, he reserves the right to say then "I didn't mean that", and revise his original claim: but he does not attempt to screw up his argument in advance so that it shall be immune to every criticism. He is concerned to indicate the truth, not construct a proof-sequence.
Thus far Hardy is right, and if the casual reader of his and Littlewood's
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articles in the Proceedings of the London Mathematical Society in the years before 1929 finds it strained to describe those proofs as so much gas, nevertheless the main point is well taken that those proofs are rhetorical, in that their purpose is, properly, to carry along, and not to compel, the reader. Yet Hardy was nothing if not a rigorous thinker. Rigour in mathematics is as necessary as elsewhere, because mathematical judgements are not infallible, any more than any other judgements are. Although the first prerequisite for mathematical communication is a certain friendliness, at least in the sense that both parties should have in common a concern for mathematical truth, the second prerequisite, which must be satisfied if these communications are to be of any value, is that that friendliness should not be uncritical. One needs to be able to assure oneself that a mistake is not being made, and one seeks to communicate this assurance together with the original insight. Unfortunately, the fear of being wrong has in most academics become neurotic, and the presentation of discoveries has been turned inside out. All the effort goes in showing that, whatever has been said, at least it is not wrong, and no attention is paid at all to showing what is being said, or showing why it is worth saying. Instead of explaining first what is biting him and what his basic-idea is, and then meeting and answering the various objections which might be raised, the mathematical writer, in obedience to the ideal of axiomatization, tries to reverse the procedure. He first anticipates and answers all possible objections, and then forgets to explain why his claim was worth substantiating and what was interesting about it in the first place.
Nevertheless, although the orthodox presentation of proofs is too precipitate in considering objections, it is right that objections should be raised. For our intuitions are fallible. Most people have had the experience of thinking that they have proved a theorem, only to discover when they come to write it down that the " proof " was illusory. Occasionally we do not spot the invalidity of our "proof'', and it becomes the painful duty of a friend to undeceive us. It is only very, very seldom that a totally fallacious ``proof" manages to be accepted by the learned world for any length of time before the fallacy is pointed out. There is an enormous unanimity of informed opinion on the substantial validity of a mathematical proof, which hardly ever has to be corrected in the course of time.
Although the number of howlers to be admitted into the corpus of accepted mathematical truth is so small as to be negligible---as Hardy says, "The history of mathematics shows conclusively that mathematicians do not evacuate permanently ground which they have conquered once. There have been many temporary retirements and shortenings of the line, but never a general retreat on a broad front''4---the number of minor errors is legion. Often theorems and definitions, although substantially correct, have turned out to need some refinement and adjustment. It is common for a theorem to be stated and " proved ", and for it then to be discovered that
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the proof is "not quite" valid----a certain unrecognized assumption was being made which usually but not always holds good; or the theorem is true only under slightly more stringent conditions than those stated----and the theorem then has to be more carefully phrased so that it is true. Little by little we load into the theorem more and more provisos----"provided that the function is continuous", "provided it is everywhere differentiable'', ``provided that none of the integrals diverge within the given interval'', `` excluding the degenerate cases where the conic becomes two coincident or two parallel straight lines"----until we have blocked up every loop-hole there is. The historical development of the theorem has been very much in the pattern of a dialogue: first the initial insight and statement, then an objection, followed by a modification of the original claim in order to meet the objection; then another objection and another modification, and so on. Fourier first had the insight that any periodic function could be expressed as a sum of cosine and sine curves whose periods were all sub-multiples of that of the original function: but his "proof" made many assumptions which he did not justify, and it is easy to pick holes in his "proof" and his "theorem": a modern treatment is very much more careful; there is a cluster of theorems about various classes of functions which can be expanded under various types of condition.5 The modern treatment is much more exact: but what it has gained in rigour, it has lost in clarity; or rather, one needs first to get the original idea, and then expose it to all sorts of criticism, and screw it up in the light of the points made. And it is for this second, and secondary, process that the axiomatic method comes into its own. It has little creative value, but unrivalled critical power, because by breaking down the proof into a sequence of very simple steps, and by subsuming every move under some rule of inference, and listing every premiss required, it can reveal what assumptions are being made, and can hope to spot illegitimate inferences or unfounded assumptions. Hilbert's axiomatization of Euclidean geometry revealed, what nobody had hitherto realized, that many more assumptions were being made than those listed by Euclid; in particular, the assumption that between any two points on a line there always was a third: and Hilbert's critique of Euclid has enlarged our understanding of geometry. Nevertheless, it is only a critique. Criticism refines, but, with few exceptions, neither creates nor destroys. One could still have, as one did have before Hilbert, Euclidean geometry incompletely axiomatized, whereas one could not have had Hilbert's axiomatization unless there had already been Euclidean geometry for it to be an axiomatization of. The order of genius among geometers is Pythagoras, Eudoxus, Euclid, Hilbert: the order of understanding, for most people, likewise follows the historical order, and not that of an axiomatic development-one needs first to understand the key theorems of Euelidean geometry before one sees the point of some of the fiddling lemmas that lead up to them. I believe both these inversions of the axiomatic order to be epistemologically significant.
5. I owe this example to Dr. I. Lakatos.
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Not only theorems but concepts are refined by the course of mathematical criticism. Real numbers, complex numbers, infinitesimals and infinite series have all had to be more and more carefully defined to meet the objections of acute critics. The history of analysis in particular has been the search to construct definitions which will catch our vague intuitive notions such as `slope', `continuous', `function', and express them in precise and workable form. In each case there is first the intuitive notion, and then various attempts to define it in simple terms, resulting in closer and closer approximations, and ultimately in an adequate definition. Pythagoras had the idea that /2 was a number, but he, or a pupil of his, saw that it could not be defined as a ratio of two integers or rational numbers. Aristotle, although feeling queasy, was prepared to deal with real numbers,6 and Eudoxus gave axioms for their use, whereafter they were used for two thousand years, and it was not until Dedekind and Cantor that satisfactory definitions were devised. Complex numbers, involving \/-1, were similarly used first, and made respectable only much later. The best example of all is afforded by the notion of an infinitesimal. Bishop Berkeley was quite right in his criticisms.7 The infinitesimals of Newton and Leibniz were self-contradictory and confused. But if Berkeley was right, so also were Newton and Leibniz. The criticisms of the critics were well founded and just, but the insights of genius were valid and valuable all the same; and in the end, Weierstrass was able to construct a definition which both expressed the notion that Newton and Leibniz were groping after, and was invulnerable to attacks by the critics.
The notion of an infinitesimal illustrates my thesis in another, deeper way. Not only is it a prime example of the dialectical rather than axiomatic development of a mathematical concept, but in its final analysis it reveals how even in mathematical arguments the form of a dialogue is sometimes maintained. Weierstrass' account of an infinitesimal, like all epsilon proofs in analysis, can easiest be understood as a dialogue or game between two persons, where the one invites the other to say how near to the limiting value he wants to get, and then guarantees to find a value of the independent variable, which will do the trick. Thus when I say that the slope of the curve y=x2at the point (1, 1) is 2, 1 mean that I challenge you to lay down your limits of tolerance; and however finely you draw them, I shall find a value of x + dx, such that the slope of the chord between the corresponding point on the curve (x + dx, y + dy) and the point (1, 1) will differ from 2 by an amount that lies within your limits of tolerance. If you demand accuracy to within .001, I shall offer dx = .004, which will give a point on the curve (1.0004, 1.00080016), whose chord from (1, 1) will have a slope .00080016 divided by .0004 = 2.0004, which is within the limits you laid down. This is what I mean when I say ``dy/dx = 2". This apparently monologous statement can be best explicated in terms of a dialogue in which I
6. Nicomachean Ethics, V, 3-5
7. Principles of Human Knowledge, §§ 122-132(Everyman ed., pp. 177-82).
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can guarantee always to have the last word. The concept of an infinitesimal is, so to speak, the projection of a two-personal interchange onto a onepersonal discourse, and it may be that much of the difficulty there has been in elucidating this concept has been due to the unconscious assumption by mathematicians that all mathematical discourse must be one-personal, with the consequent loss of structure imposed on what are naturally two-personal concepts.
The controversies of the Finitists8 and Intuitionists are likewise illuminated, if we consider mathematics as a dialogue rather than a monologue. We then can elucidate the fine-structure of the concepts of `All' and `Some', and corresponding differences in our notions of infinity and existence. In a monologue we cannot distinguish between `All' and `Any' and `Every', whereas in a dialogue we can. Modal concepts, of possibility and necessity, have long posed difficulties for monologously-inclined logicians, because there is no obvious way of distinguishing what one does do from what one could or must do. But a dialogue provides the setting for a challenge, which constitutes a natural and powerful tool for determining what one can what one cannot, and what one cannot but, do.
We have first propositions which can be established very simply-by proofs that are almost like monologues. For example, the analytic statement `All red things are coloured' rests upon the meanings of the words `red' and `coloured'. The word `all' plays very little part: we have no strong notion of infinity or totality of all red things: we merely offer our listener a rule, which he can apply as and when he likes, that if anything is red, it is coloured, and he does not have to think about the things at all, but only about the meaning of the words `red' and `coloured'.9
There is nothing mathematical, either, about syllogistic arguments. If I conclude that All Greeks are mortal from the premisses that All men are mortal, and that All Greeks are men, I am not doing mathematics, nor do I have to be able to count all Greeks or all men.
Nor, thirdly, is there anything very mathematical in arguments by Complete Enumeration. If an undergraduate reports that all the girls at a party were Girton girls, he need not have counted them, or engaged in any mathematical argument. All that is necessary is that he should have known or discovered that certain ladies were members of Girton College, and that there were no other ladies there.
The two arguments which are used by mathematicians and which do raise problems are the argument by Generalization and the argument by
8. I use the word `Finitist' in its natural, strict sense, not in any of the senses used by Hilbert or elucidated by Kreisel. No human mathematicians, so far as I know, are Finitists, though some philosophers think they should be. Heyting has suggested to me (in private conversation) that the best example of a Finitist in my sense is a computer.
9. It should be noted that we do not need any principle of individuation here to be able to use the word `all'. I can know that All red things are coloured without knowing what constitutes one red thing (as opposed to two or more red things), or being able to count red things and say how many there are. Set Theory has suffered from assuming that to use the word `All' is necessarily also to have a principle of individuation,
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Mathematical Induction. I shall (going a little beyond what ordinary usage warrants) call these the argument from Any to All and the argument from Every to All.
The argument from Any to All is characteristic of early Geometry. The Geometer considers a particular figure, but a typical one, which is therefore representative of all similar ones. Aristotle called such an argument ekthesis.10 Locke11 and Berkeley12 had difficulty in giving a coherent account of it in terms of visual images. Modern quantification theory represents it by the rule of generalization:
Experienced logicians find no difficulty with the rule of generalisation: but beginners, especially beginners in Algebraic Geometry, often find it confusing, because they are unclear what terms are being regarded as variables at different stages of the argument. The same term seems to oscillate between being a constant and being a variable, and this engenders an air of equivocation which renders the whole argument suspect. The issues can be clarified if we regard the whole argument as a dialogue and consider that part of the inference-patterns of All given by the rule13
where t is an individual constant or an individual variable (and, of course, no free occurrence of x in F(x) is in a well-formed part of F(x) of the form (t)G(t) ). This is the rule which legitimates the inference
All men are mortal
This man is mortal
or (where it is known that Socrates is a man)
All men are mortal
Socrates is mortal
All men are mortal
Man is mortal
or, more colloquially, but less formally,
All men are mortal
A man must die
The argument from Any to All can be put into dialectical form as a challenge, a challenge to produce a counter-example. I maintain that All men are mortal. You deny it. I challenge you to produce a counter-example. Clearly, if you could produce a counter-example, a man who was not a mortal, a c such that ~F (e), then you would have proved that Not all
10. See, e.g., Prior Analytics 1:41:4-5, 49b33ff, or 1:8:3, 30a5ff. But the interpretation is disputed; see W. and M. Kneale. The Development of Logic, p. 77.
11. Essay concerning Human Understanding, Bk. IV, c.VII, Section 9.
12. Principles of Human Knowledge, Introduction, §§ XII-XVI (Everyman ed., pp. 100-105).
13. In many systems of Symbolic Logic, this is a derived rule, derived from Modus Ponens and the axiom |- (x)F(x)=)F(t).
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men are mortal; we must have, at least as a derived rule of inference, the rule
~(x) F (x)
(Modus tollendo tollens).
So I challenge you to produce a counter-example. You name any man you like----in Latin, quemvis or quemlibet----and I will show that even in your most favourable case, the man selected is nonetheless mortal. I invite you to make any move you like, and claim that I have an "end-game" which, whatever move you make, will lead to my conclusion.
I do not establish my case merely by challenging you, and your happening not to be able to produce a counter-example. I have to show that a counterexample is inherently impossible; that if you were to try to take up my challenge, you would be bound to fail. "Suppose", I have to say to a reluctant but unconvinced challenger, "you tried to produce a counterexample, since it would be of such-and-such general type, it could not be a counter-example after all". I have to show, that is, that I have got an endgame which will work for any case we may take as an example. But this, by hypothesis, I do have. What I have now shown, in addition, is that possessing an argument which applies to Any instance, I am entitled to draw a conclusion about All.
The argument from Any to All, although common in mathematics, is not peculiar to it. The argument from Every to All----the argument by Mathematical Induction----is regarded as the mathematical argument par excellence.14 Suppose we have a predicate of natural numbers, P(x), and we have proved
(i) |- P(0)
(ii) |- P(x) =) P(x + 1)
then we claim that P (x) holds for every value of x among the natural numbers. Without having recourse to any new principle, we can tell our listener that if he will choose any number he likes, say n, we will prove P (n), using only methods he has already accepted. Thus if he chooses x = 3, and demands that we prove P(3), we do so as follows
|- P(0) (i) already obtained above.
|- P(0) =) P(1) substituting 0 for x in (ii) above.
|- P(1) Modus Ponens
|- P(1) =)P(2) substituting 1 for x in (ii) above.
|- P(2) Modus Ponens
|- P(2) =) P(3) substituting 2 for x in (ii) above.
|- P(3) Modus Ponens
And so similarly for any other value of x he may pick on. It is quite clear that we can prove P(x) for any n whatsoever; but if we are actually to
14. See H.Poincaré, Science and Hypothesis (Dover, New York), pp. 1-19.
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give the proof in the above manner, we must know for which value we are being asked to prove P(x). For although it will be the same sort of proof for P(4), P(27), P(257), etc. as for P(3), it will not be exactly the same proof. It will be a chain proof just the same, but it will have more links in it. This contrasts with the proof of, for example,
|- P(x) v P(x). =) P(x) in ordinary quantification theory.
This holds for any and every value of x, and moreover it can be proved for each particular value of x by an exactly similar proof. We can prove it for x = 3 thus:
|- pv p =) p Axiom
|- P(3) v P(3). =) P(3) Rule of Substitution P(3) for p
and for x = 4, thus:
|- p v p =) p Axiom
|- P(4) v P(4). =) P(4) Rule of Substitution P(4) for p
and so for x = 27, x = 257, etc. in exactly the same way with exactly the same number of steps, viz. two.
This intuitive difference is reflected in the formal fact that whereas in the latter case we can, without more ado, simply assert
|- P(x) v P(x). =) P(x)
in the former case all we can say is that
for every natural number n, |- P(n),
In the one case we are stating a theorem, and have a formal proof of it in the other we are stating a metatheorem, that we can prove as a theorem any one formula you care to choose of a certain type.
It needs only very limited intelligence on the part of the listener to see, after a few examples, that if once a mathematician has been able to prove
(ii) P(x) =) P(x + 1)
then he will be able to prove P(x) for any value of x, and therefore that P(x), and, if we allow bound variables as well, (x)P(x) will be true. A listener who was unable to see this, or a formal system which did not allow this to be proved, would be somewhat lacking. Logicians call this inadequacy omega-incompleteness. We can forestall omega-incompleteness by having as an additional rule of inference
P(x) =) P(x + 1)
(x). P(x) - P(x + 1)
or, what comes to the same thing, an additional axiom
|- P(0) =) .(y)(P(y) =) P(y + 1)) =) (x)P(x).
This is the Rule or Principle, or Axiom or Postulate, of Mathematical Induction. It has the effect of enabling a dialogue argument with an intelligent
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listener to be converted into a monologue argument in which nothing is required of the listener except to follow the steps and recognize the applicability of a rule of inference at each step. The intelligent listener not only finds, as does the unintelligent one, that each particular case that he queries is thereupon proved by the mathematician, but goes on to see that this always will be so. He therefore ceases to query and concedes. And this readiness to concede is what is formulated as the Rule of Mathematical Induction.
Although it needs only very limited intelligence to "see", even without a formally stated Rule or Axiom of Mathematical Induction, that the Principle of Mathematical Induction holds good, it is difficult to give the rationale of the argument. One "sees" that one can go on "ad infinitum", so as to cover every case in turn, thus covering all, but what is this "seeing"? It is seeing that it would be hopeless to try to meet my challenge by naming a number as a counter-example to my claim that All numbers possess the property in question. The argument from Every to All is thus far exactly like the argument from Any to All. Only, whereas in the case of Any it did not matter which instance you chose----the argument would be just the same----in the case of Every it does matter. With Any it is the same "end-game" whatever move you make: with Every the actual details of the "end-game" do depend-slightly-on your move. And therefore with Any I can show you the argument beforehand----what would be the argument, if you were to attempt to take up my challenge and produce a counter-example----because it is entirely general and does not depend on any particular features of the case. You only have to recognize its generality, and that it does apply to any case. Whereas, with the argument by Mathematical Induction from Every to All, the proof does differ (in length at least) according to the number named: and therefore, if you refuse to take up my challenge at all, I cannot get going and give you a general argument antecedently to any particular case, which will apply to whatever case you might choose, irrespective of which case you actually select for consideration. I am therefore one degree more dependent on your being willing to take up my challenge, at least hypothetically, if I am to convince you that the argument from Every to All is valid. For what I have to justify my claim is not a straight argument applicable to any case, but a schema of argument. And this is one level more abstract than ordinary Generalisation, one degree more difficult to communicate.
If once you will make a move and name a number, I have an end-game, a finite end-game, which will lead to mate. The end-game is finite, because although there are an infinite number of numbers, each number is itself only finite. Although we have an infinite progression of numbers, for every individual number our argument can be put in the form of a regress which is necessarily only finite (I can prove my claim for n, provided I can prove it for n--1; and I can prove it for n--1 provided I can prove it for n--2; and so on, until I come down to 0 (or 1), where I have already established
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it). The dialectical challenge reverses the direction of the burden of proof. The burden of proof is still on my shoulders, but instead of my taking on the impossible task of scaling an infinite ascent and proving my claim for all of infinitely many cases, I get you to propose to me the manageable task of descending from any given stage and showing how well-grounded my argument is there. And this I always can do, in a finite number of steps. For although the natural numbers go on without ever coming to an end, they do have a definite beginning. And the argument by Mathematical Induction secures its credit by trading on this fact.
The Finitists are sceptical about some universal A (and E) propositions, All individuals possess such-and-such a property (or No numbers do): the Intuitionists are sceptical about some existential O) (and I proposition, Some individual possesses such-and-such a property (or Some individual does not). Brouwer's Fixed Point Theorem shows that in any closed surface----such as the surface of the tea in a tea-cup----which has been "stirred"----every point has undergone a continuous transformation----not all the points will turn out to have moved; there will be one point which will be unchanged by the transformation. But Brouwer could not say which this "fixed point" was. His proof showed only that the assumption that all the points move would lead to a contradiction. He had proved that Not All the points move, but he came to question the inference from this to the, more obviously existential, proposition that `Some point(s) do(es) not move'. For Brouwer and his followers we are justified in asserting `Some As are B' or `Some As are not B' only if we can (or at least in principle could) produce an example of an A which was B (or, for the O proposition, was not B). The only inferences they regard as validly leading to propositions expressed by the use of the existential quantifier are
F(c) and ~F(c)
(Vx)F(x) (Vx)~F (x)
[V here represents the existential quantifier]
where c is an individual constant. They do not regard as valid the two inferences admitted by classical logicians
~[(x)~F(x)l and ~[(x)F(x)l
We can read the right-hand inference of the latter pair as the inferences from Not All to Some Not. There is no natural reading of the left-hand inference in English, but we can draw useful distinctions in Latin, between quidam, nonnullus, and nescioquis.
Quidam in Latin means `Someone, I know who, but it is not relevant to my present discourse'. "A certain man went down from Jerusalem to Jericho''. ``Someone, who had bought one of them, said the gearbox gave a great deal of trouble." If I use the word quidam I invite the question `Who ?', and give you to understand that if you were to ask it, I could answer it. Quidam, the Intuitionists say, is what Some should mean. Nonnullus, means, as it says, that it is not the case that no one is or does
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something. It is the perfect verbal representation of the classical definition of the existential quantifier
(Vx) F(x) = df. ~(x)~F(x).
Nonnullus neither invites the question `Who?' nor wards it off. Nescioquis does ward it off. It makes a minimal claim. With it I assert that there is someone who is or does something, but I cannot say who it is. My grounds for making the assertion must be of an indirect kind. I apologize for not being able to give fuller details.
The Intuitionists are clearly right in drawing attention to the differences of meaning between quidam, nonnullus and nescioquis, and in preventing us from slurring these differences over, and sliding too easily from one term to another. Their claim, however, that the word `some' and the existential quantifier should be reserved for what is expressed in Latin by the word quidam, and that we can never argue, in those cases where we cannot say quidam, from nonnullus to nescioquis, is counter-intuitive.
Let us again consider a dialogue, only, as we are dealing with the I and 0 corners of the Square of Opposition instead of the A and E corners, a more complicated dialogue. Suppose I say that Some point (in a closed surface undergoing a continuous transformation) is fixed (the I form), or Some point is not moved (the O form); there are then two dialogues which may ensue. There is first the quidam dialogue, in which you challenge me to say which point is fixed, or which point does not move. In many cases, though not this one, I can meet the challenge and cite a suitable example. In such cases I have established my claim, and more. I can say Some because I can say quidam. In this case, however, I cannot cite an example, and cannot say quidam. Rather than concede, I start a second dialogue by counter-challenging you. Do you maintain, I ask, that it is not the case that some point is fixed ? Do you deny that some point does not move ? If you do, I shall show that you are wrong; if you do not, because you cannot, you should no longer question my original assertion.
The argument with the Intuitionists, like the argument with the Finitists, is an argument by challenge; only whereas I challenge the Finitist to produce counter-examples, which he finds he cannot do, I challenge the Intuitionists to maintain a counter-thesis. But the only way of saying that I cannot say quidam is to say nullus, and if you say this, I can show you that you are wrong.
In order to maintain that it is not the case that some point is fixed, you have to maintain that I could not ever produce a point, any point, which was fixed. You must deny that there is any point that does not move. Since my original assertion could, by the first dialogue, be established by examples, your counter-thesis, which I am challenging you to assert, is vulnerable to counter-examples. And therefore to maintain your counterthesis, you have got to maintain that there are no counter-examples. You therefore must say that there is no fixed point; you must maintain that every point moves, But if once you do this, then, in virtue of the proof
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already discovered,15 you are checkmate. So you cannot take up my counterchallenge after all. So your own challenge to my original assertion fails, and my original assertion holds the field.
How does the argument I use against the Intuitionist differ from the argument which could be used to defend an omega-inconsistent system from the use of the argument by Mathematical Induction? In an omega-inconsistent system it is claimed
although for every n, |- ~F(n).
That is, some number nonnullus, is alleged not to possess the property F, although we can never say that some particular number quidam, does not possess the property. It might seem that the second dialogue which shifts the onus of disproof on to the Intuitionist could be used with equal effect to prevent the Classical Mathematician eliminating an omega-inconsistency.
The difference lies in the first dialogue. Although I was not in fact able to produce the counter-example demanded, I was not debarred in principle from so doing. If I had produced an instance, no disaster would have befallen me. I should not have landed myself in an inconsistency. The defender of an omega-inconsistent system is debarred from producing instances. He cannot for the life of him name a number that does not possess the property F. I only happened to be unable to say which the fixed point was. I always may later discover a way of finding out where it is in every case----I already can in some cases. Therefore although we both are unable to say quidam, and both have to say nescioquis, his nescio is essential, mine only an apologetic concession.
The logic of the situation can be seen more clearly if we allow three rounds of play. I make an existential claim----there is a fixed point. You challenge me, and we have the quidam dialogue. If I win this round, I have won the game. In this case, I lose the first round. The second round then gives you a chance of winning the game, by showing that if I had tried to take up your challenge, I should have been bound to fail; that is, you attempt to show that any example I care to choose (from your point of view, quivis: from mine "quivolo") will prove fatal to my original contention. If you win this round, you win the game. I can no longer claim nescioquis, if quivolo has been refuted. If, however, you do not win the second round, the game continues to a third round, with nescioquis still in the field. In this third round I force you to take the initiative again, this time, since examples have not got either of us anywhere, in terms of a universal proposition as counter-thesis. omega-inconsistency is eliminated in round two. Intuitionism, more sophisticated, survives to the third.
Normal treatments of Intuitionism concentrate on the special rules in Intuitionist logic for negation and disjunction. Intuitionist logic does not
15. The proof of the Fixed-Point Theorem is like that of the Fundamental Theorem of Algebra, viz. that every polynomial has a (possibly complex) root. In each case we show that there must be some singularity somewhere in the plane, since otherwise a proof would go through whose conclusion is clearly false.
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have the Law of Double Negation
|- ~~p =) p
or the Law of the Excluded Middle
|- p v ~p
nor will it allow us to assert any disjunction
|- p v q
unless we are in a position to assert one or other of the alternatives by itself; unless, that is, we have either
Intuitionists thus will not allow the classical arguments by Reductio ad Absurdum or by Dilemma. What is at stake in these (interconnected) refusals becomes much clearer if we view argument as a two-personal interchange than if we represent it as a one-personal inference. The context of challenge and counter-challenge provides the natural setting for the Law of Double Negation, and the argument by Dilemma reflects the fact that sometimes one party to an argument may be forced to take the initiative and declare his hand.
Our Concept of Negation has two roots. It stems partly from that of consistency, partly from that of contradiction. Every symbolic system must eschew some sorts of inconsistency, else all things are permissible, and therefore nothing carries any weight, and the symbols cease to be symbols and become mere marks or noises. What distinguishes marks and noises which are symbols from those that are not, is that with the former there are some rules for their use, allowing them to be used in certain contexts, and laying down that they shall not be used in other contexts. In particular, in logistic systems and most formalizations of arithmetic we need to distinguish between theorems and non-theorems; for if we do not have this distinction, anything may be asserted, and there is no point in asserting one propositional form rather than any other. A logistic system must be "absolutely consistent"16 if it is not to be utterly pointless and trivial. Absolute consistency is, as it were, a topological concept. It requires that not every well-formed formula be a theorem, that is that there is a boundary between the class of theorems and a class of well-formed formulae which are not theorems. An inconsistent system fails because it fails to distinguish between theorems, which may be asserted, and nontheorems, which may not. In an inconsistent system "everything is permitted", so that there is no point in asserting any thing in particular, just because there is nothing special about being asserted which anything else might lack. Chalepa ta kala:17 it is only in a system in which not everything is equally easy that there is any point in asserting anything, or any value or achievement in having proved it. Therefore if a system is to avoid point-
I6. See Alonzo Church: Introduction to Mathematical Logic, Vol. 1, Ch. 1, Section 17, p. I 08. "A logistic system is absolutely consistent if not all its sentences and propositional forms are theorems"
17. "Worthwhile achievements are difficult to accomplish"
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lessness, it must be absolutely consistent. There must be some distinction among the well-formed formulae, the meaningful sentences so to speak, separating them at least into two classes, one more assertible than the other. Therefore at the least we need consistency.
If we have consistency, do we have negation? In a sense, obviously Yes. Consistency is but one facet, the chief facet, of the requirement that we keep to the rules, that we cannot just say anything we Eke. Some things are forbidden, seine combinations are verboten, and in saying this we are already giving a sense to the word `Not'. In any developed system,18 we can go further, and pick out one (not necessarily primitive) logical constant, with the properties required for Church's first, intuitively most perspicuous, definition of consistency.19
A second approach to negation lies through contradiction. Contradiction is a prerequisite not of Formal Logic, which is essentially a monologue, but of ordinary language, and any dialogue that is a communication between two persons, not the rule-governed calculus of only one. If we have any sort of conversation between two people, it is essential that either should be able to say "No" to the other, and contradict what he is saying. Else there is no point in talking, If there is no possibility of disagreement, if whenever I say anything, you have to agree with it, then there is no need for speech or conversation. For either there is no reciprocity----what I say goes, but not what you say-----and the dialogue collapses into a monologue. with me the only one to be able to speak the truth; or all parties to a conversation are equally infallible, and nobody is ever in a position to gainsay what another says because nothing anybody says can ever be wrong. But then, in this case too, there is no need for communication, because everybody knows all there is to know already. Communication is only necessary between different, finite, fallible centres of knowledge and ratiocination, where each may know some things and have thought out some things right, but each may be ignorant of some things, and is liable to have got some things wrong. Under such conditions communication makes sense. But under such conditions, it is always possible, and will sometimes be the case, that there is a difference of opinion, and one communicator will wish to dissent from what another communicator has said. And this requires something with the force of the word `No' by means of which the one can contradict the other.
Thus, for there to be any possibility of communication, language must, like a calculus, be consistent, and therefore contain the equivalent of the word `Not': for there to be any need of communication, there must be a possibility of disagreement, and language therefore must enable us to contra-
18. This excludes, for example, Hilbert's Positive Propositional Calculus, which has no primitive logical constants from which we can develop any concept of negation.
19. Loc. cit. A logistic system is consistent with respect to a given transformation by which each sentence or propositional form A is transformed into a sentence or propositional form A', if there is no sentence or propositional form A such that |-A and |-A'. " Typically we have a logical constant, Negation, ` ~ ', which provides the requisite transformation; and the corresponding rule that we never have |- A and |- ~A.
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dict one another, and therefore have the equivalent of the word `No'.
`No', though not necessarily `Not', must yield the rule of double negation. For if a person contradicts me, and I am not persuaded to change my mind, I shall want to contradict him and re-affirm my original assertion. The Law of Double Negation is a "dialectical" necessity between two parties, symmetrically placed and enjoying parity of epistemological esteem.
The Intuitionist's rejection of the Law of Double Negation thus reveals him as occupying a special epistemological position of disparity, and as a non-participant in the normal activity of argument, a non-player in the mathematics game. The Intuitionist is a Sceptic, always demanding proofs, never offering any grounds for his doubts, never altering any alternative counter-assertions in place of those he wants to put in question. Sometimes, of course, it is reasonable only to query, and not to deny categorically. But in most disciplines, if I presume to doubt, I must back up my unbelief with some justification for my incredulity; and I am always liable to be asked, " Well, then; what is your account of the matter?". And if I always only query, and never affirm or deny, and if I always put the burden of proof on you, and never am prepared to shoulder it myself, then I am destroying parity between yourself and me, and am degrading our conversation from being a dialogue into being a mere monologue. This is the sin of the Intuitionist. He is a Promethean Sceptic, bringing a Classical God down to earth. Not for him the apologetic confession of ignorance of the non-omniscient human mathematician, "There is a fixed point, although I do not know which one it is". If God knows that there is a fixed point, He knows also which point it is: therefore let Him tell us which it is. And it is not for us to consider alternatives, or think what else could be the case if the original assertion is not accepted. We, as listeners, should take an entirely passive role. We may question, but should not propose. We should not consider hypotheses, but merely wait to be convinced.
Games, although competitive, are also co-operative, enterprises. They cannot continue if one of the players refuses to play. The Finitist and the Intuitionist refuse to play fairly with the mathematician. They are prepared to challenge the mathematician, but not to take up the challenges he offers them. By going completely inert and dumb, each can cripple the mathematician's argument. If either will accept challenges, or make objections or air doubts, then the argument will be a dialogue, and can continue. But if the Intuitionist, or the Finitist, always refuses to say anything when his turn for speaking comes along, then by converting mathematics into a monologue, he will prevent its ever having purchase enough to achieve a conclusion. So long as he keeps his mouth shut, I can get no further forward. It takes two to make an argument. The inarticulate sense, felt by most mathematicians, that Intuitionists are not playing the game, is exactly right. Reciprocity is being denied, in that they will issue, but will not accept, challenges. In normal mathematics, however, as in other forms of argument, the structure of the dialogue survives. If I am to contest your
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claim I must have some substance in my opposition. Idle questioning carries little weight. I am, of course, entitled to ask you for your reasons for believing what you affirm, but by the same token, you are entitled to ask me my reasons for doubting your claim: and if I maintain a set policy of refusing either to accept any reasons for believing or to give any for not believing, then I am not engaged in a serious attempt to discover where the truth lies, but am merely wasting your time. I am practising eristike rather than dialektike. By refusing to accept challenges I can be sure of never being proved wrong: but it is only by sticking out my neck that I can hope to view new truth.
Plato knew this, but he failed to follow the consequences through. It was his theory of argument that went wrong. He was responding to the pressure of the Sophists. It was not enough to probe with Socrates, or lead Glaucon to an apprehension of the truth: he must refute Callicles, beard Thrasymachus, compel the assent of Dion even in his unwilling moods. Argument therefore must be armour-plated, proof against every sceptical query, every sophistical doubt. His deep understanding of mathematics in the Meno, the Phaedo and the Republic, as the search for, and study of, underlying patterns----Gestalten, eide----is distorted to accommodate an increasingly stringent and less generous notion of rationality. The dialogues cease to be actual arguments with equals and become first seminars with yes-ful youths, and later lectures in all but name; and mathematics ceases to be the co-operative endeavour of friends, and becomes first the communication of insights from master to pupil, and finally the self-sufficient exercise of lonely thinkers, proof against every objection the most sophisticated sophist might raise, just as, in the face of the eristike of others, dialektike ceases to be conversation and becomes the most cogent and coercive form of deductive logic. We can understand how Plato came to be more concerned with criticism than creativity; and we sympathize with his developing his idea of reasoning in such a way as to stress the universality of reason as against all comers; but we can only regret the gradual, and often reluctant, change from the Plato who talked with Socrates to the Plato who laid down the Laws.
J. R. LUCAS
Merton College, Oxford.