Talk:Arithmetization of analysis
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problematic article
editThis 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)
WikiProject class rating
editThis article was automatically assessed because at least one WikiProject had rated the article as stub, and the rating on other projects was brought up to Stub class. BetacommandBot 03:47, 10 November 2007 (UTC)
What to do?
editYes, this is an important concept in the history of math & phil of math -- lots of work has been done. However, the text of this article (at this date) appears to have been lifted from the following web location:
... or is it the other way round, viz., that this is the source for that article? HH. In any case, it would be possible to deal with the problems noted by others above by incrementally building up the history of the arithmetization process/program/project, were there a need for it. But is there? Whayes43 (talk) 20:31, 26 May 2014 (UTC)