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timeletsI've added numbering and spacing to emphasize Kant's key ideas wrt philosophy and mathematics:
Logic, as a preorder, is useful because it deals consistently with structures that have a beginning (see Kant's problem #1P). Once we find or assume a system that has a beginning, we can apply a logic to it and be guided along the chain of implications (see Kant's #1M). etc.
If we can't find a beginning, we can fall back to the question of existence and start our logic from there (see #2P), running into opposite directions.
“The questions:
1P) whether the world has a beginning and a limit to its extension in space;
2) whether there exists anywhere, or perhaps, in my own thinking Self, an indivisible and indestructible unity — or whether nothing but what is divisible and transitory exists;
3) whether I am a free agent, or, like other beings, am bound in the chains of nature and fate;
4) whether, finally, there is a supreme cause of the world, or all our thought and speculation must end with nature and the order of external things
— are questions for the solution of which the mathematician would willingly exchange his whole science; for in it there is no satisfaction for the highest aspirations and most ardent desires of humanity. ”
Nay, it may even be said that the true value of mathematics — that pride of human reason — consists in this:
1M) that she guides reason to the knowledge of nature — in her greater as well as in her less manifestations — in her beautiful order and regularity —
2) guides her, moreover, to an insight into the wonderful unity of the moving forces in the operations of nature, far beyond the expectations of a philosophy building only on experience;
3) and that she thus encourages philosophy to extend the province of reason beyond all experience,
4) and at the same time provides it with the most excellent materials for supporting its investigations, in so far as their nature admits, by adequate and accordant intuitions."
Immanuel Kant, “The Critique of Pure Reason.”
Logic, as a preorder, is useful because it deals consistently with structures that have a beginning (see Kant's problem #1P). Once we find or assume a system that has a beginning, we can apply a logic to it and be guided along the chain of implications (see Kant's #1M). etc.
If we can't find a beginning, we can fall back to the question of existence and start our logic from there (see #2P), running into opposite directions.
