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[¶5] Mathematics and physics are the two theoretical cognitions of reason that are supposed to determine their objects a priori, the former entirely purely, the latter at least in part purely but also following the standards of sources of cognition other than reason.
[¶6] Mathematics has, from the earliest times to which the history of human reason reaches, in that admirable people the Greeks, traveled the secure path of a science. Yet it must not be thought that it was as easy for it as for logic - in which reason has to do only with itself - to find that royal path, or rather itself to open it up; rather, I believe that mathematics was left groping about for a long time (chiefly among the Egyptians), and that its transformation is to be ascribed to a revolution, brought about by the happy inspiration of a single man in an attempt from which the road to be taken onward could no longer be missed, and the secure course of a science was entered on and prescribed for all time and to an infinite extent. The history of this revolution in the way of thinking - which was far more important than the discovery of the way around the famous Cape - and of the lucky one who brought it about, has not been preserved for us. But the legend handed down to us by Diogenes Laertius - who names the reputed inventor of the smallest elements of geometrical demonstrations, even of those that, according to common judgment, stand in no need of proof - proves that the memory of the alteration wrought by the discovery of this new path in its earliest footsteps must have seemed exceedingly important to mathematicians, and was thereby rendered unforgettable. A new light broke upon the first person who demonstrated the isosceles triangle (whether he was called "Thales" or had some other name). For he found that what he had to do was not to trace what he saw in this figure, or even trace its mere concept, and read off, as it were, from the properties of the figure; but rather that he had to produce the latter from what he himself thought into the object and presented (through construction) according to a priori concepts, and that in order to know something securely a priori he had to ascribe to the thing nothing except what followed necessarily from what he himself had put into it in accordance with its concept.
Summary
Mathematics and physics are also examples of disciplines attained security as sciences. Mathematics attained security by attending to the properties that we introduce, a priori, into our own judgments and from which the a priori necessity of the mathematical cognitions extends.
Commentary
After lengthy praise of the significance of the revolution in mathematics, Kant discusses a crucial aspect of math's success: the necessity of our judgments in mathematics rest on what we have contributed to the construction of their concepts.
Kant recognized that advances in sciences followed the recognition that we must guide our study rather than depend on experience to lead us on its own. When mathematics became a secure science it made use of general solutions to problems a proiri and left behind imprecise measures such as using precalculated charts to approximate answers. Perhaps most memorable are the advances of ancient Greek geometry which must be limited in dealing with shapes so far as we experience them, but can judge universally and with necessity with concepts of figures such as circles and triangles constructed by us in advance.
As an example we can consider the first proposition in Euclid's Elements: to construct an equilateral triangle upon a given line. The proof follows by constructing more figures whose properties we understand in advance, namely circles. Two circles are constructed both with their center upon different ends of the line and with radii equal to the length of the entire given line. Because we know that the radii of the circles are the same as each other, and the same as the line they are constructed upon, we can find the place these circles intersect and connect each end of the line to this intersection point. The purpose of all this is to point out how the geometer does not sit idle, but actively contributes not only constructions to the proof, but the concepts that allow for the necessity resulting judgment. (An example such as this can also be used to point out how the judgments in mathematics are synthetic.)
Questions
Does Kant take mathematics to have been invented?
Concerning the debate of whether mathematics is invented or discovered, Kant appears to be on the side of invention; Kant even calls all knowledge from the construction of concepts mathematical knowledge. A brief argument for this can be drawn from how mathematics could attain a secure status. Mathematics judges with necessity and so these judgments can never be a posterioiri, but always a priori. The concepts of mathematics may have been inspired by reflection on experience in many cases, but many of these could not have been drawn from experience. The pure concepts of mathematics also require no exhibition in experience for their validity, for example, I do not need to discover a perfect circle or equilateral triangle for my concept of a circle to find acceptance - even in the application to calculations in experience. These concepts would then either be innate or constructed. If they are innate we would not learn mathematics as much as become conscious of it in ourselves, but this is not so, therefore these concepts are constructed by us.
To the argument that because everyone agrees on mathematics it must be discovered rather than invented, Kant's response would seem to be this: an alternative for the consistency in the judgments of mathematicians is also through the commonality of the form of intuition through which the concepts are constructed. The agreement between the mathematicians doesn't imply some inherent a priori insight into the order of the universe, but ultimately a form of social agreement around having a common sense.
Terminology
theoretical cognitions of Reason (theoretische Erkentnisse der Vernunft), pure (rein)
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