> Jesse F. Hughes wrote: > Mathematics is about deductive consequence, and if the >> consequences of our axioms are counterintuitive or surprising, well, >> that's too bad for our intuitions (or perhaps we must in extreme cases >> reconsider our axioms). > > Why not trying the latter?
Because the great advances in mathematics post-Cantor show that the basic foundations are interesting? Good luck constructing a viable alternative. Once you have it (and you have the initial results showing that it is interesting, rich, useful, etc.), let me know.
> Were not distributions introduced like generalized density functions > as to cope with some problems of physics? This reminds me of > cosmetics rather than a fundamental recast. I tend to see Cantor's > paradise the eggshells of a mathematics of discrete numbers while > Hilbert was correct in that the infinitely large as well as the > infinitely small evade our comprehension of numbers. > > Let me suggest some practical corrections. > > You will certainly agree that any single finite value xn of a continuous > quantity x is per se completely irrelevant because it is surrounded by > infinitely many nearly identical neighbours.
Sorry for being dense (no pun intended), but I won't agree to that. I don't really know what it means.
> It simply has no comparable weight. Therefore it is not justified to > distinguish between x>xn and x>=xn.
Uh. Well. Maybe you should get them results *first* and then provide the motivational argument later. It doesn't seem to be going so well this way.
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