Is There A Crisis in Mathematics?
In 1924 Rolin Wavre published: Y a-t-il une crise des mathématiques? A propos de la notion d'existence et d'une application suspecte du principe du tiers exclu, Rev. de métaph. et de mor. 31 (1924), 435-470. A summarised translation into English of this article by Alice Ambrose was published as: Rolin Wavre, Is There A Crisis in Mathematics?, Amer. Math. Monthly 41 (8) (1934), 488-499. We give here a version.
Is There A Crisis in Mathematics? With reference to the notion of existence and a doubtful application of the law of the excluded middle
By Rolin Wavre, University of Geneva.
"Men do not understand each other because they do not speak the same language and because there are languages which are not learned."
In saying that, Poincaré sought to describe the irreconcilable nature of that clash of temperament manifested in the very heart of mathematics at the appearance of Cantorian ideas and logistic. The clash has taken form under various captions. Du Bois-Reymond called the divergent tendencies empiricist and idealist; Poincare, pragmatist and realist; Brouwer, intuitionist and formalist. We shall here use intuitionist for the one, but according to circumstances, idealist, formalist, or even realist for the other.
The intuitionist is at present identified by his extreme caution. Being anxious to attain the greatest intelligibility and clarity, he challenges such propositions as Zermelo's axiom of selection or such reasoning as does not seem rigorous. The idealist claims to depart in nothing from the rigor which the intuitionist rightly demands, and this without evidencing the least distrust with regard to those modes of reasoning - even on the contrary conceding their perfect legitimacy and applying them literally and without restriction. Russell, Hilbert, Zermelo, and Hadamard are idealists; Lebesgue, Borel, and Baire are intuitionists.
The clash is very much more evident now than in the past, as a result of the several publications, on the one hand of Brouwer and Weyl, two new and undeniably revolutionary intuitionists, and on the other of Hilbert. We should like to summarize here the essence of these publications and show how clear-cut the clash becomes in connection with the notion of existence and a doubtful application of the law of excluded middle. We shall see in particular why, and in what system of definition, the intuitionist denies us the right to say:
Two mathematical points are either coincident or distinct.
Two functions defined for the same values of the variable coincide for all those values or else there exists one for which they do not coincide.
That the intuitionist formulates such paradoxes in the name of truth seems curious. But the paradox is bound up with the words and not with the ideas.
Weyl, however, believes he has discerned at the basis of the theory of real numbers a vicious circle which endangers its value, and that he should therefore declare that mathematics is undergoing a crisis. Brouwer and Weyl have attempted to reconstruct the theory of sets and that of functions on new foundations, carefully avoiding the suspected law and vicious circle. In this they have carried distrust as to certain modes of reasoning, whose legitimacy has created no doubt for three centuries, further perhaps than the French school of the theory of sets.
Hilbert is not of the opinion that it is the turn of mathematics, after physics, to go through a revolution. Observing that the consequences of the idealistic attitude would never lead to the slightest contradiction, however far they be pushed, he considers that the principles suspected by the intuitionists, although not having perfect evidence, are nevertheless legitimate. Unlike Brouwer and Weyl, Hilbert does not believe in the necessity of reconstructing the foundations of mathematics, for he prescribes a radical remedy in order to prevent the crisis and to legitimate a frankly idealistic attitude. This means is the axiomatic method. In substance he does this: to the axioms accepted by the intuitionists, he adjoins a new proposition called the axiom of the transfinite which contains in itself all the doubtful principles united. And from this would be deduced the doubtful applications of the law of excluded middle. But to justify such a procedure, it must be demonstrated that this total system of axioms does not imply contradiction. By this artificial union of its idealistic foundations present- day mathematics would not forfeit the renown of being the discipline whose truth is above suspicion.
THE INTUITIONIST POINT OF XIEWI do not conceal the difficulty encountered in trying to express the exact meaning of the relation Brouwer establishes between mathematics and logic. However, I shall try to set off the essentials in the thought of the great Dutch mathematician. In 1907 Brouwer made bold to reverse the roles which Russell and Couturat would have had mathematics and logistic play. Instead of its being the second which accounts for the first, it is on the contrary the part of mathematics to comprise logistic and even traditional logic. The Aristotelian logic, born of natural classification, would be adequate to the theory of finite collections and would not go beyond it; it would be concerned exclusively with the relations of whole and part. Its complete self-evidence is the cause of the a priori character conferred on it; but, taken in by this a priori character, we would have called traditional logic to a function it is incapable of exercising.
Mathematics and Logic
Mathematics and Logic
Traditional logic can at most claim to conform discourse to rules, but language itself becomes more and more inadequate to the genuine understanding of the facts of present-day mathematics; thus the word "all," for example, despite the subtle distinctions of Russell, has an imprecise meaning as soon as one is concerned with "all" the objects of an infinite collection. The mathematical intuition is on the contrary, guarantee of the autonomy of his science and is not bound to respect always and everywhere in its translation into discourse the rules of syntax which traditional logic prescribes. Logistics, on the other hand, would be only the algebra of the language by which the reasoning is translated. But a logical construction of mathematics independent of the mathematical intuition is impossible, according to Brouwer, because one would obtain thereby only a verbal construction irrevocably divorced from the science. Further, this would be a vicious circle, for logic itself requires the fundamental intuition of mathematics.
According to Brouwer the field of application of traditional logic would be limited to finite collections. The crux of the issue then is the right to reason on finite and infinite sets in the same manner. For Brouwer does not admit the law of excluded middle, expressed in abstracto and universally in the form: "A thing either is or is not" or "A proposition is either true or false."
The Question of LogicThe laws of contradiction and of excluded middle express two fundamental relations between an attribute A and its contradictory, non-A, both well defined relatively to the same object, such as even and odd for an integer. The law of contradiction forbids attribution of A and non-A at the same time to the same object. The law of excluded middle compels assertion of one or the other. They can be regarded as defining the words "well defined attribute" or 'attribute non-A" or simply negation; bound up as they are with the definitions, there can be no question of doubting them. The sole question at issue is to know when they are applicable and to what attributes. In more precise terms, it is a matter of recognizing whether two distinct attributes A and B are such that one is entitled to make one the logical equivalent of the negation of the other. It is a matter of knowing, for example, whether for an integer the attribute of being factorable is well defined and whether the attribute of being prime is by definition equivalent to non-factorable. If it is by definition, these laws would apply with full right; if not, a special examination of the two attributes becomes necessary, and it is only in virtue of an indirect evidence that the laws do apply. In the numerous examples which mathematics furnishes of well defined attributes, or which one considers as such, only an intuitive, extra-logical evidence permits us to consider two attributes A and B as the negation one of the other.
The following example is suggestive and of great importance in the present issue: Let us call a fundamental aggregate a sequence of integers: 1, 3, 4, 5, 7, for example. Are the following propositions, which state an attribute of the sequence, related as A and non-A?
a. All the numbers of the sequence are odd.No one doubts that they are so related, but that is in virtue of indirect evidence. I run through the sequence; this done, at some determinate position I either have or have not encountered an even number, and I cannot at the same time have found and not found one. I then have the right, from the point of view of their logical function, to assimilate a to non-b and b to non-a. The logistician, the formalist, makes "there exists" equivalent by definition to "not all." That is his right. But in doing so, he introduces a new axiom or a new definition.
b. There exists in the sequence an even number.
b. There exists in the sequence an even number.
To the question: Does there exist an even number in the sequence?, whether I refer to the intuitive sense of the word "exist" or to its logistic sense, I cannot refuse to answer with yes or with no; yes, if I affirm b, no, if I affirm a. Generalised, this becomes:
1. To the question: Does there exist in a fundamental aggregate a number having a well defined attribute A?, I can only reply with yes or no. This is the application of the law of excluded middle which we had in mind.Likewise the following undoubted proposition, "The numbers of the sequence 1, 3, 4, 5, 7 which are odd form a new set 1, 3, 5, 7," becomes when generalized:
2. In a fundamental aggregate, a well defined attribute A suffices to characterize a sub-class of the elements which possess it.With Brouwer, we call this principle 2 Zermelo's axiom of inclusion. It states that in a sequence of integers a definition by intension is equivalent to a definition by extension. The formalists believe that they have the right to apply principles 1 and 2 without any restriction to the case where the fundamental aggregate is composed of an infinity of elements, such as the natural series of positive integers. The intuitionists refuse this right.
In exposition of the arguments for the respective positions, consider the following imaginary dialogue:
The idealist: Either there exists a factorable number in the sequence mn = 22n+38 + 1 (n = 1, 2, 3, ... ) or else one such does not exist.
The Intuitionist: I can only say one exists by exhibiting such a number, say m1000, which is factorable; and I can only deny it by deducing from the definition of the numbers mn that they are all prime. But I do not see that the rejection of one of the parts of the alternative compels me to affirm the other; as I cannot exclude a priori every tertium, I refuse to be reduced to your alternative.
The Idealist: Suppose I take successively the numbers 1, 2, 3, ...; then either I shall or shall not come across a number n giving rise to a factorable number mn. My encounter either will or will not occur.
The Intuitionist: To deny that all the numbers m are prime does not imply that there exists a determinate one, e.g., m263, which is factorable. And in order to be certain of not having met one, you must have exhausted the series of integers, and that you will never do. You will take several steps in the series, and can perhaps say one exists, but you will never take enough to make a denial. I should not deny that the number does or does not exist if the existence of an object in an infinite set were a well defined attribute of the set; but existence is not a well defined attribute. There are perhaps several modes of existing.
Let us introduce the two following propositions which state two attributes of the fundamental aggregate:
a. All the numbers of the aggregate possess the attribute A.As soon as an infinity of objects is concerned, the word "all" is suspect to the intuitionist. The proposition a can only have the precise meaning which its demonstration confers on it. This meaning will vary according to the demonstration. The most complete meaning would be such that it would follow just from the definition of the numbers of the set that they all have the attribute A. The "there exists" can only have meaning if the object said to exist is actually found. "There exists a number having such an attribute" signifies for the intuitionist: "Here is a number possessing this attribute." "All" from the formalist and logistic point of view means: and this, and this, ... and this. Likewise, "there exists means: or this, or this, ... or this. They are logical product and sum. But if the fundamental aggregate is infinite, the handling of such product and sum requires precautions as to convergence analogous to those with the infinite products and series of analysis. Intuitionists (such as Weyl) seem to confer meaning only on the implication b implies non-a, which suffices to establish that propositions a and b cannot both be affirmed. But as the inferences non-b implies a and non-a implies b, maintained by formalists, are doubtful or no longer meaningful, they see no need of their being related by the law of excluded middle. Consider the intuitionist meaning of general and existential demonstrations and the fact that the expressions "not all" and "there does not exist" have a doubtful meaning, and one will perhaps no longer be surprised at the paradoxes of the most extreme intuitionists. Even Hilbert recognizes that these phrases are devoid of a clear and immediate meaning. The intuitionists refuse to make an alternative of the affirmation of a universal affirmative or of a singular negative, when an infinite class is in question; of saying, for example, either all the integers of a class are factorable or else there exists a determinate one which is prime.
b. There exists a number of the aggregate possessing the attribute non-A.
b. There exists a number of the aggregate possessing the attribute non-A.
Here are two examples illustrative of Brouwer's attitude:
- Let m = φ(n) and m' = φ'(n) be two laws of correspondence between a positive integer n and two integers m and m'. Let us say that the two functions φ and φ' are identical if the numbers m and m' are equal for all values of n, and that the two functions are different if there exists a value of n giving two unequal numbers m and m'. To demonstrate the identity of the two functions, we should have to be able to reduce the one to the other algebraically or analytically; to demonstrate their difference we should have to discover a number n giving rise to two distinct numbers m and m'. Considering what such discovery entails, it perhaps will not seem surprising that the functions are not a priori identical or different.
- Fermat's theorem that the sum of two nth powers of two positive integers is never equal to the nth power of another integer as soon as n is greater than 2 has no known demonstration. One is tempted to say: if the proposition is false, I can assure myself of it by a finite number of trials on the integers. For then, there exist three integers and a power larger than 2 giving rise to the equality, and I can order my experiences in one series in such a way as to be certain of finding the numbers in question within a finite range in the series. Brouwer absolutely refuses to argue this. For him, even the demonstration that Fermat's theorem led to a contradiction would not imply the existence of four numbers invalidating it.
As Lebesgue has already said, we can only prove the existence of a mathematical entity by constructing it. The "there exists" would be only a check, without value in itself, so long as we cannot find the bank. And perhaps we shall never find it to convert its nominal value into its effective one. The "there exists" of the idealist is only an incitation to formulate a true judgment; ideal existence is worthless so long as it is not converted into actual existence. It is also to be noted that general judgments have meaning only through the fact that they imply an indefinite number of singular judgments; they are only true because they are constantly verifiable.
The Mathematical Continuum Conceived as a BecomingAccording to the classical conception, the mathematical continuum is only the aggregate of real numbers, all given as its ultimate elements. With Brouwer appears a new conception. Without, we hope, distorting his meaning, we shall here follow a path a little different from his in the exposition of the intuitionist view. We shall identify the real number and the decimal expansion. The irrational number and its expansion are infinitely more common than the rational number. The irrational number has no repetend, and we shall never conceive it except approximately inasmuch as we cannot think simultaneously of an infinity of numbers (the numbers of the decimal expansion). We shall never have anything but a finite number of figures of the expansion before us. To define a number with precision, this expansion must be given with as many figures as we wish.
We have now two distinct questions to consider:
1. By what process can a real number be defined?
2. What idea can be formed of the continuum?
2. What idea can be formed of the continuum?
- Since an infinity of numbers cannot be given one by one, we must have recourse to a law of generation of the decimal expansion, m = f(n), which prescribes that the nth decimal be equal to m (0 ≤ m ≤ 9). Such a law of generation will be given, for example, by the arithmetic process of extracting the square root. Only such a law can define an irrational number, and in a general way, we can say that a real number is a law f of generation of a decimal expansion.
- One would be tempted to reply to the second question as to the first, giving oneself a law g of generation, no longer of the decimal expansion, but of the real numbers themselves, of the laws f. Proceeding thus, one would construct the continuum. It would be denumerable and considerably mutilated. But this will not do. The laws f must remain entirely general, so that they shatter the frame in which a law g which presided over their genesis would confine them. Nothing ought to restrict the freedom of choice of the laws f. Hence if we wish to preserve for the continuum the richness of which it allows, we must avail ourselves of this freedom. We are here led to the following definition: The mathematical continuum is the decimal expansion in the freedom of its genesis. The idea of the continuum then comes to that of a series of arbitrary choices, or better, to that of a free sequence, in the sense that nothing prejudges, after n choices, what will be the next.
Strictly speaking, the real number is in the continuum in this sense, that being free to choose the decimals as I please I can choose them conformably to a law f. But it would be absolutely meaningless to make of the continuum a simple aggregate of real numbers and to see in it a relation analogous to that of a whole and its parts. One cannot rise from the number to the continuum as from the elements to the class, and see nothing in the class other than the aggregate of elements. And then the numbers are not all given; one must be satisfied with a representation of particular numbers by a law f and not imagine a given anterior to this representation and containing them all.
Should we wish to parcel out the continuum at any cost, the ultimate elements with which we should end would not be points, but intervals; we should not pass beyond these, and the interval is a new continuum. From the intuitionist point of view, the stopping place must be the interval, for we can have before us only a finite number of figures of the decimal expansion. The real number is sequence of intervals (i, n) each enclosing the next, their length approaching zero as one progresses in the sequence. It is in a way a passage to the limit of a series of intervals, an approximation whose error we are sure of making as small as we wish. But we shall never make this passage to the limit; we shall never do away with this error. The interval still remains infinitely divisible. Weyl thinks this conception can claim to do justice in a lasting way to the notion of becoming.
Another objection to the atomistic conception which sees in the continuum only an aggregate of points appears in the following considerations: Whether the elements be points or intervals, can one say of a given couple whether they are identical or different? Now this one can say of intervals, but not of points, whereby we are brought back to the question of logic. Two points are given by two laws m = f(n), and m' = f'(n), that is, by two functions of integers, and we cannot affirm them to be either identical or different. The main point of the matter is in the rejection of the application of the law of excluded middle, which brings us to this: it is false to say, "either a given point is interior to an infinity of intervals or it is exterior to one of them." The realist-atomist says: Two points are identical or distinct. With that, he places himself in the presumed given, the aggregate of points, and not in the act generating such points, which is alone intelligible. He is heedless of the fact that a point is only defined by an indefinite sequence of integers, by a process implying an infinity of operations. If this series of operations could be thought in its entirety and not in its law of generation, the decision would perhaps not fail to be forthcoming. But this exhaustion is inconceivable.
The Intuitionist ReconstructionIf, as Leibniz thought, systems are false by what they deny and true by what they affirm, I cannot escape the duty of sketching here the most important points of the intuitionist reconstruction.
The intuitionist, who rejects Zermelo's axiom of inclusion, cannot define the set by means of an attribute characteristic of all its elements; it must be constructed. To construct, in the narrow sense of the word, would consist in exhibiting the elements one by one; but here again one would find oneself within too restricted limits. One would risk confinement in the domain of the law g which I spoke just now. A construction of sets is possible, starting from the free sequence. Now the free sequence or sequence in becoming consists in choosing arbitrarily a first number, then a second arbitrarily, and so on as long as desired. Let us call such a sequence s. The fundamental mathematical operation which is substituted for the notion of set and of function is a law of construction c(s, k) making a determinate number correspond to each sequence s and to each order number k in the sequence. This well defined number could indeed depend very well not only on the kth number of the sequence, but on the k first ones. The expression mk = nl + n2 + ... + nk furnishes an example of such a law. One can make k vary in such a series as well as change the series. This law of correspondence c(s, k) Brouwer calls a set, Weyl a function. A very special case of the function would be the correspondence between two integers of the series S, that is, a law Nk = c(S, k). That would be a functio discreta, to use Weyl's phrase. The law N = 2k is an example; it defines the set of even numbers. In the general case the law c makes a new series correspond to each given infinite series. Weyl calls this dependence a functio continua. But it is not equivalent to the continuous function of analysis. That, from the intuitionist point of view, is as Weyl shows, a functio discreta.
THE IDEALIST POINT OF VIEW OF HILBERTOne willingly supposes that reasoning in which one starts from such evidence as certain mathematical axioms and continues according to the rules of the strictest logic could not lead to a contradiction. To start from evidence, to reason logically, is that a guarantee of never being inconsistent? As contradiction is the worst thing which a mathematician can encounter, it is important to demonstrate that however far one pushes mathematico-logical deductions, a contradiction will never arise. And this task is all the more urgent since the axioms are not all equally evident, especially those which the intuitionists reject.
The Absence of Contradiction among the Axioms of Arithmetic
The Absence of Contradiction among the Axioms of Arithmetic
Hilbert was fully aware of this, and this task has been thrust upon him all the more forcibly since the time when he claimed to reinstate the inferences doubtful to the intuitionists, by means of a new axiom containing all the doctrine of Zermelo and Cantor, plus four axiom-definitions of the notions all, not all, there exists, there does not exist - propositions whose evidence is not at all immediate.
The problem of non-contradiction among the axioms is stated as follows: Given a system of axioms, to show that from this two propositions reducible to the form A and non-A will never be deduced. In his axiomatic of geometry, Hilbert has fully succeeded in solving this problem so far as the axioms of geometry are concerned, but it was through postulating that the axioms of arithmetic implied no contradiction. At the international congress of mathematics in 1904 he tried to establish that the latter in turn were exempt from contradiction. Poincaré made essentially two objections to this attempt of Hilbert:
- Your sole means of showing that you will never meet a contradiction would be to establish that, if at the nth deductive operation no contradiction appears, it will not appear in the following deduction either; and then to reason by complete induction so as to be sure of never meeting one. It must at least be postulated that the axiom of induction taken in itself does not lead to contradiction.
- You must at least admit that the series of integers does not imply contradiction. After these criticisms of Poincaré, Hilbert's attempt might seem vain. Hilbert avoids the first objection provided one understands clearly what he wishes to establish. He does not seek to demonstrate that should deductions be infinite in number they will all be exempt from contradiction, but simply that one will not actually meet such a contradiction. Man is capable of only a finite number of actual deductions; so if he meets a contradiction, it is after a finite number of operations. And if one succeeds in establishing that this circumstance cannot occur, the problem will be humanly solved.
- Each word must be rid of the dangerous richness of ordinary language - of what any notion, as that of number or of implication, for example, conjures up before imagination. To this end it is best to reduce each notion to an easily recognizable sign.
- Mathematical axioms and logical principles must be represented by complexes of signs.
- A symbolic representation must be given of the elementary deduction itself. The latter is the hypothetical syllogism of the propositional calculus: if A (is true) and if A implies B, then B (is true). All mathematical demonstrations would be reduced to this unique procedure repeated a convenient number of times. The premises A and A implies B would spring directly from axioms, or would be propositions previously established.
- A criterion of contradiction must be given, also representable by a sign complex, for example, a ≠ a.
The Axiom of the TransfiniteIn his memoir of 1922 Hilbert shows how, by his axiomatic method, he is in a position to reinstate the principles doubtful in the eyes of Brouwer and Weyl.
Let E be a set assumed to be well defined, e any element of the set and A any property well defined relatively to each element of the set. It is postulated that there exists an element t of the set such that if it has the property A, that implies that all the elements likewise possess it. For example: If E is the set of men and A the property of being corruptible, the object t would be a man of such inviolable incorruptibility that if he is corruptible; all men are. Hilbert's idea is this: it will not always be possible to discover the element whose existence is postulated, but one can without risk of error act as if this discovery had already taken place. In other words, ideal existence would have the same logical value as actual existence.
ConclusionIn his axiom of the transfinite set-up to reinstate suspected principles, Hilbert tries to give logical value to ideal existence, which the intuitionist refuses to recognize. The divergences of viewpoint which were manifest in connection with the axiom of selection cannot fail to spring up again as soon as the admission of that axiom is in question.
Ideal existence is for the intuitionist only a false window for the logical symmetry of propositions, bearing on the finite set on the one hand and on the in- finite set on the other, a fiction of the logicians imagined not to save logic-it is not in danger-but to extend its domain. Now it is only an intuitive analysis of each particular case which determines whether one happens to be under its jurisdiction. The intuitionist proceeds from the representation to the object represented, while the idealist does not hesitate to deal with a system of things, such as the set of real numbers, of which he is incapable of giving a representation. Intuitionistic existence is just that representation, while idealistic existence seems in the final count to vanish in the idea of the non-contradictory. The intuitionist is more prudent; he seems to enjoin us not to assert existence of an indefinable object, or even of a non-constructible one, just as the physicists of the relativistic school enjoin us not to invent hypotheses which do not correspond to a physical set-up which is at least imaginable if not practically realisable. Mathematics ought perhaps to give up the classic conception of the continuum. But the free sequence and the operations on it are a new canton open to it.
The intuitionists through their great concern for rigour and evidence occupy the stronger position and run the least risk. But so long as they have not put their finger on a sort of tertiumn datur, they cannot convince the idealists of the necessity of slackening their pace. And it indeed seems improbable that idealists will arrive at any contradiction. It seems to me that we shall not give up the language and even the reasoning of the idealist, but shall require intuitionist verifications. Often in the past, intuitionists (notably Lebesgue) have succeeded in replacing idealist demonstrations, in which the axiom of selection was invoked, by others where one dispensed with it. It will perhaps be the same with reference to the doubtful application of the law of excluded middle. The divergence of temperament, as Brunschvicg has so well shown, has its roots deep in history. It has not constituted, properly speaking, a great danger. And even today, when the intuitionist is more uncompromising, the word "crisis" in the foundations of mathematics is inappropriate.