In the preceding paragraphs, I have mentioned several facts which lend some support to the latter vlew. On the other hand, the tremendous wealth of successful applications of classical analysis in physics, to mention only one aspect of greatest significance, weighs heavily against this conclusion. In the following, I shall try to make clear how someone, even upon acknowledging the fundamental thesis of constructivism, can still reach the conclusion that actualist analysis should be retained and continued.
Hilbert himself has here shown the way: viz., by the method of 'ideal elements'.
I.e.: propositions which talk about the infinite in the sense of the actualist interpretation are regarded as 'ideal propositions', as propositions which do not really mean at all what the words in them purport to mean, but which can be of greatest value in rounding off a theory, in facilitating its proofs, and in making the formulation of its results more straightforward. In projective geometry, for example, ideal points are introduced for the same reason, with the advantage that many theorems are simplified which would otherwise be plagued by exceptions. In the bargain, we must of course accept the fact that in some cases the sense of a theorem is now no longer the usual one. The following is asserted, for example: 'Two straight lines always have a point in common'. If the straight lines happen to be parallel, however, then they plainly have no point in common in reality. The procedure is harmless, because it has been precisely specified what, in such exceptional cases, is to be understood by the notion of a 'point', which has now a wider sense.
We might consider still another example which, in its relation to physics, seems to provide even more striking analogies to the relationship between constructivist mathematics and actualist mathematics:
I am thinking of the occasional attempt to construct a 'natural geometry', i.e., a geometry which is better suited to physical experience than the usual (Euclidean) geometry, for example. In this natural geometry, the theorem 'precisely one straight line passes through two distinct points' holds only if the points are not lying too close together. For if they are lying very close together, then several adjacent straight lines can obviously be drawn through the two points. The draftsman must take these considerations into account; in pure geometry, however, this is not done because here the points are idealized. The extended points of experience are replaced by the ideal, unextended, 'points' of theoretical mathematics which, in reality, have no existence. That this procedure is benefical is borne out by its success: It results in a mathematical theory which is of a much simpler and considerably smoother form than that of natural geometry, which is continually concerned with unpleasant exceptions.
The relationship between actualist mathematics and constructivist mathematics is quite analogous: Actualist mathematics idealizes, for example, the notion of 'existence' by saying: A number exists if its existence can be proved by means of a proof in which the logical deductions are applied to completed infinite totalities in the same form in which they are valid for finite totalities; entirely as if these infinite totalities were actually present quantities. In this way the concept of existence therefore inherits the advantages and the disadvantages of an ideal element: The advantage is, above all, that a considerable simplification and elegance of the theory is achieved - since intuitionist existence proofs are, as mentioned, mostly very complicated and plagued by unpleasant exceptions -, the disadvantage, however, is that this ideal concept of existence is no longer applicable to the same degree to physical reality as, for example, the constructive concept of existence.
As an example, let us consider the equation a . x = b over the real numbers. According to the actualist interpretation, the following may simply be asserted: The equation has a root, as long as a is not equal to 0. The intuitionist, however, says: The equation has a root if it has been determined that a is different from 0. It may happen, however, that from the way a is given it cannot be anticipated whether a is equal to 0 nor whether a is different from 0.
In this case the question of the existence of a root remains open. It must certainly be admitted that this interpretation corresponds more to the position of the physicist who may have to determine the coefficient a from an experiment which is not precise enough to establish with certainty a difference between a and 0.
The question now arises: What use are elegant bodies of knowledge and particularly simple theorems if they are not applicable to physical reality in their literal sense? Would it not be preferable in that case to adopt a procedure which is more laborious and which yields more complicated results, but which has the advantace of making these results immediately meaningful in reality ? The answer lies in the success of the former procedure: Again consider the example of geometry. The great achievements of mathematics in the advancement of physical knowledge stem precisely from this method of idealizing what is physically given and thereby simplifying its investigation. In any application of the general results to reality, their special status due to this idealization must, of course, be kept in mind and a corresponding reinterpretation must be carried out. This is where applied mathematics has its realm of activity.
For the sake of comparison, I quote from Heyting and Weyl:
Heyting, the intuitionist, says in one place:
'From the formalist standpoint the aim of physics can be characterized
as the mastering of nature. If this aim can be achieved by formal methods' -
i.e., by actualist mathematics - 'then no argument is tenable against them'.
In the dispute between Brouwer and Hilbert, Weyl formulates
his position as follows:
'In studying mathematics for its own sake one should follow Brouwer and
confine oneself to discernible truths into which the infinite enters only as an
open field of possibilities; there can be no motive for exceeding these bounds.
lit the natural sciences, however, a sphere is touched upon which is no longer
penetrable by an appeal to visible self-evidence, in any case; here cognition
necessarily assumes a symbolic form. For this reason it is no longer necessary,
as mathematics is drawn into the process of a theoretical reconstruction
of the world by physics, to be able to isolate mathematics into a realm of
the intuitively certain: On this higher plane, from which the whole of science
appears as a unit, I agree with Hilbert.
I am under the impression that certain fundamental intuitionist concepts, e.g., the concept of existence or that of a real number, are strictly speaking already 'idea] elements'. Yet this may remain debateable; it's difficult to discuss and not that important. In any case, it would not mean that the application of such concepts also requires a consistency proof, they are after all applied only in such a way that their precise constructive sense always remains apparent [ ... ]. The same is true of the 'ideal points' in projective geometry; the situation is different in the case of the ideal concepts of actualist mathematics which - looked at from the constructivist point of view - do not involve any inherent 'sense' at all, but which, in spite of this, are used as if, by their very wording, they were endowed with such a sense.
While the constructivists, on the one hand, are thus conceding a purpose to actualist mathematics, it seems reasonable that the constructivist point of view, should, on the other hand, be given a greater role in mathematics than it has at present. In foundational research it is already customary to carry the proofs out along constructivist lines wherever possible, not only because of their greater indisputability - this is not always the aim of a proof - but also because of the greater tangible content of the result. For it is clear that a constructivist existence proof means more than an indirect actualist proof. Particularly in elementary number theory and, generally, in all theories dealing only with finitely, describable objects, it is natural to take as a basis the constructivist point )f view". fn the past this was done quite automatically, in any case; the genuinely naive reasoning in which no special attention is paid at all to the methods of proof, is by nature chiefly constructive, i.e., it shuns the 'infinite'. In these areas, moreover, the application of transfinite actualist forms of inference serves hardly any practical purpose. Not so in the realm of the continuum, in analysis and geometry: Here the actualist approach celebrates its triumphs; here the constructivist approach is inferior in practise.
In conclusion, it can therefore be said: The constructivist ('intuitionist', 'finitist') mathematics constitutes an important realm within the whole of mathematics because of its great self-evidence and the particular significance of its results. Yet no compelling reasons exist why all parts of analysis that are based on the actualist interpretation should be radically rejected; on the contrary, they are afforded a great significance in their own right, above all in view of their physical applications.