Talk:Arithmetization of analysis
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This article apparently treats an important idea in the history of math-foundational thought; see for example Simpson, who asserts:
- Another important development was the "arithmetization of analysis" (Weierstrass, Dedekind). Thus it was no longer necessary to regard real numbers and continuous functions as basic, unanalyzed concepts; instead they could be reduced to the natural numbers. This made possible the axiomatization of analysis in terms of second order arithmetic (carried out systematically by Hilbert and Bernays).
From a modern perspective, though, the treatment is kind of misleading, because the real numbers are not in fact reduced to natural numbers, but rather to sets of natural numbers, a fundamentally richer and more complicated notion. A related problem is the discussion of
- the more extreme philosophical position that all of mathematics should be derivable from logic and set theory, ultimately leading to Hilbert's program, Gödel's theorems and non-standard analysis.
where "logic" and "set theory" seem to be more or less conflated, suggesting, say, that the reals should be a purely logical, analytic notion, because sets are supposed to be. But in fact sets have, if anything, less claim to analyticity than do the reals. This, of course, is clear only in retrospect and there's no warrant to hold Weierstrass or Dedekind accountable for it, so from a historical perspective the treatment is probably correct. That doesn't save it from being potentially confusing to a contemporary reader. --Trovatore 18:11, 1 April 2006 (UTC)
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