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On the Nature of Mathematical Reasoning
I The very possibility of mathematical science seems an insoluble contradiction. If this science is only deductive in appearance, from whence is derived that perfect rigour which is challenged by none? If, on the contrary, all the propositions which it enunciates may be derived in order by the rules of formal logic, how is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new, and if everything must spring from the principle of identity, then everything should be capable of being reduced to that principle. Are we then to admit that the enunciations of all the theorems with which so many volumes are filled are only indirect ways of saying that A is A?
No doubt we may refer back to axioms which are at the source of all these reasonings. If it is felt that they cannot be reduced to the principle of contradiction, if we decline to see in them any more than experimental facts which have no part or lot in mathematical necessity, there is still one resource left to us: we may class them among à priori synthetic views. But this is no solution of the difficulty—it is merely giving it a name; and even if the nature of the synthetic views had no longer for us any mystery, the contradiction would not have disappeared; it would have only been shirked. Syllogistic reasoning remains incapable of adding anything to the data that are given it; the data are reduced to axioms, and that is all we should find in the conclusions.
No theorem can be new unless a new axiom intervenes in its demonstration; reasoning can only give us immediately evident truths borrowed from direct intuition; it would only be an intermediary parasite. Should we not therefore have reason for asking if the syllogistic apparatus serves only to disguise what we have borrowed?
The contradiction will strike us the more if we open any book on mathematics; on every page the author announces his intention of generalising some proposition already known. Does the mathematical method proceed from the particular to the general, and, if so, how can it be called deductive?
Finally, if the science of number were merely analytical, or could be analytically derived from a few synthetic intuitions, it seems that a sufficiently powerful mind could with a single glance perceive all its truths; nay, one might even hope that some day a language would be invented simple enough for these truths to be made evident to any person of ordinary intelligence.
Even if these consequences are challenged, it must be granted that mathematical reasoning has of itself a kind of creative virtue, and is therefore to be distinguished from the syllogism. The difference must be profound. We shall not, for instance, find the key to the mystery in the frequent use of the rule by which the same uniform operation applied to two equal numbers will give identical results. All these modes of reasoning, whether or not reducible to the syllogism, properly so called, retain the analytical character, and ipso facto, lose their power.
II The argument is an old one. Let us see how Leibnitz tried to show that two and two make four. I assume the number one to be defined, and also the operation x11—i.e., the adding of unity to a given number x. These definitions, whatever they may be, do not enter into the subsequent reasoning. I next define the numbers 2, 3, 4 by the equalities:
(1) 11152; (2) 21153; (3) 31154, and in the same way I define the operation x12 by the relation; (4) x125(x11)11.
Given this, we have:
2125(211)11; (def. 4). (211)115311(def. 2). 31154(def. 3). whence 21254 Q.E.D.
It cannot be denied that this reasoning is purely analytical. But if we ask a mathematician, he will reply: “This is not a demonstration properly so called; it is a verification.” We have confined ourselves to bringing together one or other of two purely conventional definitions, and we have verified their identity; nothing new has been learned. Verification differs from proof precisely because it is analytical, and because it leads to nothing. It leads to nothing because the conclusion is nothing but the premisses translated into another language. A real proof, on the other hand, is fruitful, because the conclusion is in a sense more general than the premisses. The equality 21254 can be verified because it is particular. Each individual enunciation in mathematics may be always verified in the same way. But if mathematics could be reduced to a series of such verifications it would not be a science. A chess-player, for instance, does not create a science by winning a piece. There is no science but the science of the general. It may even be said that the object of the exact sciences is to dispense with these direct verifications.
III Let us now see the geometer at work, and try to surprise some of his methods. The task is not without difficulty; it is not enough to open a book at random and to analyse any proof we may come across. First of all, geometry must be excluded, or the question becomes complicated by difficult problems relating to the role of the postulates, the nature and the origin of the idea of space. For analogous reasons we cannot avail ourselves of the infinitesimal calculus. We must seek mathematical thought where it has remained pure—i.e., in Arithmetic. But we still have to choose; in the higher parts of the theory of numbers the primitive mathematical ideas have already undergone so profound an elaboration that it becomes difficult to analyse them.
It is therefore at the beginning of Arithmetic that we must expect to find the explanation we seek; but it happens that it is precisely in the proofs of the most elementary theorems that the authors of classic treatises have displayed the least precision and rigour. We may not impute this to them as a crime; they have obeyed a necessity. Beginners are not prepared for real mathematical rigour; they would see in it nothing but empty, tedious subtleties. It would be a waste of time to try to make them more exacting; they have to pass rapidly and without stopping over the road which was trodden slowly by the founders of the science.
Why is so long a preparation necessary to habituate oneself to this perfect rigour, which it would seem should naturally be imposed on all minds? This is a logical and psychological problem which is well worthy of study. But we shall not dwell on it; it is foreign to our subject. All I wish to insist on is that we shall fail in our purpose unless we reconstruct the proofs of the elementary theorems, and give them, not the rough form in which they are left so as not to weary the beginner, but the form which will satisfy the skilled geometer.
Definition of Addition I assume that the operation x11 has been defined; it consists in adding the number 1 to a given number x. Whatever may be said of this definition, it does not enter into the subsequent reasoning.
We now have to define the operation x1a, which consists in adding the number a to any given number x. Suppose that we have defined the operation x1(a21); the operation x1a will be defined by the equality: (1) x1a5[x1(a21)]11. We shall know what x1a is when we know what x1(a-1) is, and as I have assumed that to start with we know what x11 is, we can define successively and “by recurrence” the operations x12, x13, etc. This definition deserves a moment’s attention; it is of a particular nature which distinguishes it even at this stage from the purely logical definition; the equality (1), in fact, contains an infinite number of distinct definitions, each having only one meaning when we know the meaning of its predecessor.
Properties of Addition Associative. I say that a1(b1c)5(a1b)1c; in fact, the theorem is true for c51. It may then be written a1(b11)5(a1b)11; which, remembering the difference of notation, is nothing but the equality (1) by which I have just defined addition. Assume the theorem true for c5g, I say that it will be true for c5g11. Let (a1b)1g5 a1(b1g), it follows that [(a1b)1g]115[a1(b1g)]11; or by def. (1)—(a1b)1(g11)5a1(b1g11)5a1[b1(g11)], which shows by a series of purely analytical deductions that the theorem is true for g11. Being true for c51, we see that it is successively true for c52, c53, etc.
Commutative. (1) I say that a11511a. The theorem is evidently true for a51; we can verify by purely analytical reasoning that if it is true for a5g it will be true for a5g11.* Now, it is true for a51, and therefore is true for a52, a53, and so on. This is what is meant by saying that the proof is demonstrated “by recurrence.”
(2) I say that a1b5b1a. The theorem has just been shown to hold good for b51, and it may be verified analytically that if it is true for b5b, it will be true for b5b11. The proposition is thus established by recurrence.
* For (g11)115(11g)11511(g11).—[Tr.] Definition of Multiplication We shall define multiplication by the equalities: (1) a315a. (2) a3b5[a3(b-1)]1a. Both of these include an infinite number of definitions; having defined a31, it enables us to define in succession a32, a33, and so on. Properties of Multiplication Distributive. I say that (a1b)3c5(a3c)1(b3c). We can verify analytically that the theorem is true for c51; then if it is true for c5g, it will be true for c5g11. The proposition is then proved by recurrence.
Commutative. (1) I say that a31513a. The theorem is obvious for a51. We can verify analytically that if it is true for a5a, it will be true for a5a11.
(2) I say that a3b5b3a. The theorem has just been proved for b51. We can verify analytically that if it be true for b5b it will be true for b5b11. IV This monotonous series of reasonings may now be laid aside; but their very monotony brings vividly to light the process, which is uniform, and is met again at every step. The process is proof by recurrence. We first show that a theorem is true for n51; we then show that if it is true for n21 it is true for n, and we conclude that it is true for all integers. We have now seen how it may be used for the proof of the rules of addition and multiplication—that is to say, for the rules of the algebraic calculus. This calculus is an instrument of transformation which lends itself to many more different combinations than the simple syllogism; but it is still a purely analytical instrument, and is incapable of teaching us anything new. If mathematics had no other instrument, it would immediately be arrested in its development; but it has recourse anew to the same process—i.e., to reasoning by recurrence, and it can continue its forward march. Then if we look carefully, we find this mode of reasoning at every step, either under the simple form which we have just given to it, or under a more or less modified form. It is therefore mathematical reasoning par excellence, and we must examine it closer.