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Axioms | |
 | Catalog / Culture / Science / Mathematical Sciences / Foundations and Topology / Axioms |
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Mathematical Problems by David Hilbert
Philosophy of problems, relationship between mathematics and science, role of proofs, axioms and formalism. [eng] |
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54: General topology
Topology is the study of sets on which one has a notion of "closeness" -- enough to decide which functions defined on it are continuous. Thus it is a kind of generalized geometry (we are still interested in spheres and cubes, for example, but we might consider them to be "the same", yet distinct from a bicycle tire, which has a "hole") or a kind of generalized analysis (we might think of the functions f(x)=x^2 and f(x)=|x| as being "the same", and yet distinct from f(x)=signum(x)=x/|x|, which has a discontinuity). [eng] |
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The Mathematical Axioms of Quasi-Bayesian Theory
The Mathematical Axioms of Quasi-Bayesian Theory. [eng] |
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Kant and Mathematical Knowledge
In the Preface to the Second Edition of the Critique of Pure Reason, Kant proposes to save metaphysics from its random groping by instigating a revolution analogous to those which set mathematics and physics upon the secure path of science. The true method, he claims, rests in the realization that reason can only have certain knowledge of that which is necessarily presupposed a priori by reason itself. [eng] |
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