
Immanuel Kant, in his Critique of Pure Reason, [has asserted that] ... we conduct ourselves passively when we receive impressions through intuition and actively when we deal with them in our thought. Furthermore, according to Kant, we must distinguish between two ingredients of intuition. One ... arises from experience ... such as colours, sounds, smells, hardness, softness, roughness, etc. The other is a pure a priori part independent of all experience ..: [Kant believed that] geometry, as it has been taught since ancient times, deals with the properties of the space that is fully and exactly presented to us by pure intuition ....
However plausible these ideas may at first seem, and however well they corresponded to the state of science in Kant's day, their foundations have been shaken by the course that science has taken since then ....
[These quotes] narrow the subject to geometry and intuition, and attempt to show how it came about that, even in the branch of mathematics which would seem to be its original domain, intuition gradually fell into disrepute and at last was completely banished ....
One of the outstanding events in this development was the discovery [by Weierstrass of] curves that possess no tangent at any point. [That is,] ... it is possible to imagine a point moving in such a manner that at no instant does it have a definite velocity. [This] directly affects the foundations of differential calculus as developed by Newton (who started with the concept of velocity) and Leibniz (who started the socalled tangent problem) .... The standard curves that have been studied since early times: circles, ellipses, hyperbolas, parabolas, cycloids, etc. [have tangents everywhere. However,] the graph of the function t cos(1/t) demonstrates that a curve does not have to have a tangent at every point. It used to be thought that intuition forced us to acknowledge that such a deficiency could occur only at isolated and exceptional points of a curve [and] that a curve must possess an exact slope, or tangent, at an overwhelming majority of points [The Weierstrass function goes beyond that. By replacing lines with sawtooth curves, one obtains a simplified variant, the Takagi function ...] Its character entirely eludes intuition: indeed, after a few repetitions of the segmenting process, the evolving figure has grown so intricate that intuition can scarcely follow, and it forsakes us completely as regards the curve that is approached as a limit. The fact is that only logical analysis can pursue this strange object to its final form. Thus, had we relied on intuition in this instance, we would have remained in error, for intuition seems to force the conclusion that there cannot be curves lacking a tangent at any point .... [To avoid such advanced] branches of mathematics, I propose to examine an occurrence of failure of intuition at the very threshold of geometry. Everyone believes that ... curves are geometric figures generated by the motion of a point. But ... Peano ... proved that the geometric figures that can be generated by a moving point also include entire plane surfaces. For instance, it is possible to imagine a point moving in such a way that in a finite time it will pass through all the points of a square and yet no one would consider the entire area of a square as simply a curve .... This motion cannot possibly be grasped by intuition; it can only be understood by logical analysis.
[For] a second example of the undependability of intuition even as regards very elementary geometrical questions, think of a map showing three countries.
Intuition seems to indicate that corners at which all three countries come together ... can occur only at isolated points, and that at the great majority of boundary points on the map only two countries will be in contact. Yet Brouwer showed how a map can be divided into three countries in such a way that at every boundary point all three countries will touch one another ....
Intuition cannot comprehend this pattern, although logical analysis requires us to accept it. Once more intuition has led us astray. Intuition seems to indicate that it is impossible for a curve to be made up of nothing but end points or of branch points. This intuitive conviction as regards branch points was refuted [when] Sierpinski proved that there are curves all of whose points are branch points ....
Because intuition turned out to be deceptive in so many instances, and because propositions that had been accounted true by intuition were repeatedly proved false by logic, mathematicians became more and more skeptical of the validity of intuition. They learned that it is unsafe to accept any mathematical proposition, much less to base any mathematical discipline on intuitive convictions. Thus, a demand arose for the expulsion of intuition from mathematical reasoning, and for the complete formalization of mathematics. That is to say, every new mathematical concept was to be introduced through a purely logical definition; every mathematical proof was to be carried through strictly by logical means. The task of completely formalizing mathematics, of reducing it entirely to logic, was arduous and difficult; it meant nothing less than a reform in root and branch ....
Let us now summarize. Again and again we have found that, even in simple and elementary geometric questions, intuition is a wholly unreliable guide. It is impossible to permit so unreliable an aid to serve as the starting point or basis of a mathematical discipline ...
But what are we to say to the often heard objection that only conventional geometry is usable, for it is the only one that satisfies intuition? My first comment on this score ... is that every geometry ... is a logical construct. Traditional physics is responsible for the fact that until recently the logical construction of threedimensional Euclidean, Archimedean space has been used exclusively for the ordering of our experience. For several centuries, almost up to the present day, it served this purpose admirably; thus we grew used to operating with it. This habituation to the use of ordinary geometry for the ordering of our experience explains why we regard this geometry as intuitive, and every departure from it unintuitive, contrary to intuition, and intuitively impossible. But as we have seen, such intuitional impossibilities, also occur in ordinary geometry. They appear as soon as we no longer restrict ourselves to the geometrical entities with which we have long been familiar, but instead reflect upon objects that we had not thought about before ....
The theory that the earth is a sphere was also once an affront to intuition. However, we have got used to the idea, and today it no longer occurs to anyone to pronounce it impossible because it conflicts with intuition.
If the use of [new] geometries for the ordering of our experience continues to prove itself so that we become more and more accustomed to dealing with these logical constructs; if they penetrate into the curriculum of the schools, if we, so to speak, learn them at our mother's knee, as we now learn threedimensional Euclidean geometry, then nobody will think of saying that these geometries are contrary to intuition. They will be considered as deserving of intuitive status as threedimensional Euclidean geometry is today. For it is not true, as Kant urged, that intuition is a pure a priori means of knowledge, but rather that it is force of habit rooted in psychological inertia.
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