The inquiry naturally presents itself as to the real nature and limitations of geometrical intuition .... [I distinguish] between what I call the naive and the refined intuition. It is the latter that we find in Euclid: he carefully develops his system on the basis of well-formulated axioms, is fully conscious of the necessity of exact proofs, and so forth ....
The naive intuition, on the other hand, was especially active during the period of the genesis of differential and integral calculus. Thus Newton [did not ask] himself whether there might not be continuous functions having no derivative ....
At the present time we are living in a critical period similar to that of Euclid. It is my private conviction ... that Euclid's period must also have been preceded by a naive stage of development ....
In my opinion, the naive intuition is not exact, while the refined intuition is not properly intuition at all, but arises through the logical development from axioms considered as perfectly exact.
The first half of this statement [implies that] we do not picture in our mind an abstract mathematical point, but substitute something concrete for it. In imagining a line, we do not picture a length without breadth, but a strip of a certain width. [Abstractions] in this case are regarded as holding only approximately, or as far as may be necessary ....
I maintain that in ordinary life we actually operate with such inexact definitions. Thus we speak without hesitancy of the directions and curvature of a river or a road, although the line in this case certainly has considerable width ....
As regards the second half of my proposition, there are actually many cases where the conclusions derived by purely logical reasoning from exact definitions cannot be verified by intuition. To show this, I select examples from the theory of automorphic functions, because in more common geometrical illustrations our judgment is warped by the familiarity of the ideas ....
Let any number of non-intersecting circles 1, 2, 3, 4, ... , be given, and let every circle be reflected (i.e. transformed by inversion, or reciprocal radii vectors) upon every other circle; then repeat this operation again and again, ad infinitum.
The question is, what will be the configuration formed by the totality of all the circles, and in particular, what will be the position of the limiting points? There is no difficulty in answering these questions by purely logical reasoning, but the imagination seems to fail utterly when we try to form a mental image of the result ....
When the original points of contact happen to lie on a circle being excluded, it can be shown analytically that the continuous curve which is the locus of all the points of contact is not an analytical curve. It is easy enough to imagine a strip covering all these points, but when the width of the strip is reduced beyond a certain limit, we find undulations, and it seems impossible to clearly picture the final outcome.
Note that we have here an example of a curve with indeterminate derivatives arising out of purely geometrical considerations, while it might be supposed from the usual treatment of such curves that they can only be defined by artificial analytical series ....
Kopcke has [concluded] that our space intuition is exact as far as it goes, but so limited as to make it impossible for us to picture curves without tangents ....
Pasch believes - and this is the traditional view - that it is in the end possible to discard intuition entirely, basing all of science on axioms alone. This idea of building up science purely on the basis of axioms has since been carried still further by Peano, in his logical calculus .... I am of the [firm] opinion that, for the purposes of research it is always necessary to combine intuition with the axioms. I do not believe, for instance, that it would have been possible to derive the results discussed in my [previous] lectures, the splendid researches of Lie, the continuity of the shape of algebraic curves and surfaces, or the most general forms of triangles, without the constant use of geometrical intuition ....
What has been said above places geometry among the applied sciences. Let me make a few general remarks on these sciences. I should lay particular stress on the heuristic value of the applied sciences as an aid to discovering new truths in [pure] mathematics. Thus I have shown (in my little book on Riemann's theories) that the abelian integrals can best be understood and illustrated by considering electric currents on closed surfaces. In an analogous way, theorems concerning differential equations can be derived from the consideration of sound-vibrations, and so on ....
The ordinary mathematical treatment of any applied science substitutes exact axioms for the approximate results of experience, and deduces from these axioms the rigid mathematical conclusions. [But] it must not be forgotten that mathematical developments transcending the limit of exactness of the science are of no practical value.
It follows that a large portion of abstract mathematics remains without any practical application, the amount of mathematics that can be usefully employed in any science being in proportion to the degree of accuracy attained in that science ....
As examples of extensive mathematical theories that do not exist for applied science, consider the distinction between the commensurable and the incommensurable. It seems to me, therefore, that Kirchhoff makes a mistake when he says in his Spectral Analyse that absorption takes place only when there is an exact coincidence between the wave-lengths. I side with Stokes, who says that absorption takes place in the vicinity of such coincidences ....
All this raises the question of whether it would not be possible to create a, let us say, abridged system of mathematics adapted to the needs of the applied sciences, without passing through the whole realm of abstract mathematics .... [But no such] system ... is ... in existence, and we must for the present try to make the best of the material at hand.
What I have said here concerning the use of mathematics in the applied sciences [must] not be interpreted as in any way prejudicial to the cultivation of abstract mathematics as a pure science. Apart from the fact that pure mathematics cannot be supplanted by anything else as a means for developing the purely logical powers of the mind, there must be considered here as elsewhere the necessity of the presence of a few individuals in each country developed in a far higher degree than the rest. Even a slight raising of the general level can be accomplished only when some few minds have progressed far ahead of the average ....
Here a practical difficulty presents itself in the teaching, let us say, the elements of the calculus. The teacher is confronted with the problem of harmonizing two opposite and almost contradictory requirements. On the one hand, he has to consider the limited and as yet undeveloped intellectual grasp of his students and the fact that most of them study mathematics mainly with a view to the practical applications; on the other, his conscientiousness as a teacher and man of science would seem to compel him to detract in nowise from perfect mathematical rigor, and therefore to introduce from the beginning all the refinements and niceties of modern abstract mathematics. In recent years, university instruction, at least in Europe, has been tending more and more in the latter direction. [If a work like] Cours d'analyse of Camille Jordan is placed in the hands of a beginner a large part of the subject will remain unintelligible, and at a later stage, the student will not have gained the power of making use of the principles in the simple cases occurring in the applied sciences ....
It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction ....
I am led to these remarks by the consciousness of growing danger in Germany of a separation between abstract mathematical science and its scientific and technical applications. Such separation can only be deplored, for it would necessarily be followed by shallowness on the side of the applied sciences, and by isolation on the part of pure mathematics ....
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