I shall first give a classification of mathematics into three distinct levels according to the degree to which the notion of 'infinite' is used in the various branches of mathematics. The first and lowest level is represented by elementary number theory, i.e. by the theory of numbers that does not make use of techniques from analysis. The infinite occurs here in its simplest form. An infinite sequence of objects, in this case the natural numbers, is involved. Several other branches of mathematics are logically equivalent to elementary number theory, viz., all those theories whose objects can be put into one-to-one correspondence with the natural numbers and which are therefore 'denumerable'. Almost the whole of algebra belongs here - the rational numbers, the algebraic numbers, also polynomials, can after all be proved to be denumerable - so does combinatorial topology, for example, i.e., that part of topology which deals only with objects whose properties are describable by finitely many data. The well-known four-colour problem belongs here. All these theories are, logically speaking, entirely equivalent. It is therefore sufficient to deal with elementary number theory only; the theorems and proofs in the remaining theories can be reinterpreted as number-theoretical theorems and proofs by a correlation of their objects with the natural numbers. To the four-colour problem, for example, there corresponds indeed an equivalent number-theoretical problem, although our special interest in it derives of course solely from its intuitive topological formulation.
The second level of mathematics is represented by analysis. As far as the application of the concept of infinity is concerned, the essentially new feature here is the fact that now even individual objects of the theory may themselves be infinite sets. The real numbers, i.e., the objects of analysis, are after all defined as infinite sets, as a rule as infinite sequences of rational numbers. In this connection it makes no difference whether the particular definition chosen is that by nested intervals, or Dedekind cuts, or by some other means. The whole theory of complex functions also belongs at this level; nothing essentially new is here added. The third level of application of the concept of infinity, finally, is encountered in general set theory. Admitted as objects here are not only the natural numbers and other finitely describable quantities, as the first level, as well as infinite sets of these, as at the second level, but, in addition, infinite sets of infinite sets and again sets of such sets, etc., in the utmost conceivable generality.
The given classification subsumes every branch of mathematics. As far as geometry is concerned, for example, it no longer presents any special problems today in connection with the concept of infinity. What might appear to be such problems either belong to physics, or occur in an equivalent form in analysis; the difterent geometries can after all always be interpreted in terms of logically equivalent models from analysis.
There are essentially two fundamentally different interpretations of the nature of infinity in mathematics, and I shall now go on to describe them. I shall call them the 'actualist' ((an sich)) interpretation and the 'constructivist' ((konstruktiv)) interpretation of infinity. The former is the interpretation of classical mathematics as we have all learned it at university. Several mathematicians have adopted the constructivist view - although not always to the same extent - among them Kronecker, Poincaré Brouwer and Weyl. These names alone indicate that we are dealing with a direction of opinion that must indeed be taken seriously. I shall try to bring out the essence of the constructivist view vis-à-vis the actualist interpretation; in the short time available this can be done only imperfectly, especially since it must be kept in mind that by its very familiarity, the actualist interpretation has become second nature to us and that it is not easy to adopt, for once, a quite different way of thinking.
I shall begin with the antinomies of set theory. Here we have a situation in which actualist considerations have led to an absurdity which could not have resulted from the constructivist interpretation of the matter. For on the basis of the quite general concept of a set indicated earlier it is also possible to form, for example, the concept of the 'set of all sets'; this is a correctly defined set. Yet contradictions quite understandably follow from it: The set of all sets must after all contain itself as an element and in a certain sense - easily made precise - it must therefore be larger than itself, and this can obviously not be so. Upon closer examination it becomes easily apparent how the absurdity comes about : Strictly speaking, the 'set of all sets' must not itself be considered as belonging to the sets; it is a subsequent formation, as it were, which produces an entirely new collection from a given totality of sets. This is in fact the constructivist view of the situation: New sets may, as a matter of principle, be formed only constructively one by one, on the basis of alreadv constructed sets. According to the actualist view, on the other hand, all sets are defined in advance by the abstract concept of a set and are therefore already available 'as such' ((an sich)), quite independently of how individual sets may be selected from them by means of special constructions. This view had led to the antinomy.
If we were to try to express the essence of the constructivist view in as general a principle as possible, we would formulate it about as follows: 'Something infinite must never be regarded as completed, but only as something becoming, which can be built up constructively further and further.' I recall Gauss's well-known dictum that 'the use of an infinite quantity as something completed is never permissible in mathematics'.
If this principle of interpreting the infinite constructively is accepted, then differences vis-à-vis the actualist interpretation of classical mathematics manifest themselves not only in the theory of sets, but already in the realm of elementary number theory. I shall now discuss these differences in greater detail. ln elementary number theory, we encounter the infinite only in its simplest form, viz., in the form of the infinite sequence of the natural numbers. According to the actualist interpretation, we may regard this sequence as a completed infinite totality, whereas the constructivist interpretation allows us to say only this: We can progress further and further in the number sequence and always construct new numbers, but we must not speak of a completed totality. A proposition such as 'all natural numbers have the property B', for example, has in each case a somewhat different sense. According to the actualist interpretation, it says: The property B holds for any number that may somehow be singled out from the complete totality of numbers. According to the constructivist interpretation we may say only this: Regardless of how far we progress in the formation of new numbers, the property B continues to hold for these new numbers.
In practice, this difference in interpretation is here, however, immaterial. A proposition about all natural numbers is normally proved by complete induction, and this inference certainly appears to be in harmony also with the constructivist interpretation; particularly since complete induction is after all based on the idea of our progressing in the number sequence. The situation is different in the case of exsistential propositions. The proposition 'there exists a natural number with the property B' says, according to the actualist interpretation: 'Somewhere in the completed totality of the natural numbers there occurs such a number.' According to the constructivist interpretation such an assertion is of course without sense. But this does not mean that under this interpretation existential propositions must be rejected outright. If a definite number N, for which the property B holds, can actually be specified, then even under this interpretation can we speak of the existence of such a number; in reality, the existential proposition now no longer refers to the infinite totality of numbers; it would after all suffice to speak only of the numbers from 1 to N. The existence proofs that occur in practice are indeed mostly such that an example can actually be given. However, proofs are also possible where this is not the case, viz., indirect existence proofs: It is assumed that there is no number for which the property B holds. If this assumption leads to a contradiction, it is inferred that the number for which the property B holds exists after all. It may then happen that an effective procedure for actually producing such a number is altogether unobtainable. From the constructivist point of view, such a proof must consequently be rejected. Another technique of proof which likewise becomes unacceptable from this point of view and which is usually quoted in this connection, is the application of the 'law of the excluded middle' to propositions about infinitely many objects. According to the constructivist interpretation, for example, we cannot even say: 'A property B holds for all natural numbers or it does not hold for all natural numbers.' The rejection of the law of the excluded middle seems particularly paradoxial, at first, but it is only a necessary consequence of the principle of interpretating the infinite potentially. After all, this law is based on the idea of the completed number sequence. This must not be interpreted to mean that the constructivists regard this law as altogether false; from their point of view it is more correct to regard it as being withhout sense. It thus makes no sense whatever to even speak of the totality of numbers as something completed, precisely because 'In reality', the number sequence is never completed, all that is given is an indefinitely extendable process of progression.
In practice these forms of inference, which are nonadmissible according to the constructivist interpretation, hardly ever occur in elementary number theory. The situation is different in analysis and set theory. Here the differences between the two interpretations are essentially the same as those described for the natural numbers; I shall therefore not discuss them further. In the case of analysis and set theory, however, the significance of the difference is considerably greater with the result that from the constructivist point of view extensive parts of analysis and almost all of set theory cannot be accepted.
In this connection, attention should be drawn to the fact that the delimitation between what is constructively permissble and what is not cannot be defined unequivocably in certain borderline cases and that the opinions of the different mathematicians representing this point of view are not identical. Yet these differences are not important enough to the picture as a whole to warrant a more detailed discussion. Words like 'intuitionist' (Brouwer) and 'finitist' (Hilbert) denote such somewhat different constructivist points of view.
Now the cardinal question becomes this: Which of the two interpretations is actually correct? Both are defended. On the one hand, we have the intuitionists under the leadership of Brouwer with the totally radical thesis that all of mathematics which is incompatible with the constructivist point of view must be discarded. On the other hand, the majority of mathematicians are understandably reluctant to make such a sacrifice. The antinomies, so they say, are indeed founded on inadmissible formation of concepts, but such concepts can be avoided by a proper delimitation; the whole of analysis and a fortiori number theory, so they claim, is entirely unobjectionable. Unfortunately, the delimitation of the inadmissible inferences can be carried out in basically different ways without their necessarily leading to a definite common point, and I must say that to me the clearest and most consequential delimitation seems to be that given by the principle of interpreting the infinite constructively.
We should nevertheless be reluctant to discard the extensive nonconstructive part of analysis which has, among other things, certainly stood the test in a variety of applications in physics.