Wikipedia:Reference desk/Archives/Mathematics/2007 July 2

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July 2

Russell's Paradox

In the article "Reductionism" under the heading "Set-Theoretic Reductionism" it states the following: "[Gottlob Frege] then proposed his own form of reductionism, logicism, which in turn was famously disproven by Russell's Paradox." According to wikipedia "Logicism is one of the schools of thought in the philosophy of mathematics, putting forth the theory that mathematics is an extension of logic and therefore some or all mathematics is reducible to logic." Nowhere in these 2 articles is there any proof to how logicism is disproven by Russell's paradox. So I ask, how is logicism, as stricly defined by the above excerpt, false?67.70.129.135 03:49, 2 July 2007 (UTC)

I wouldn't say that logicism was, strictly speaking, disproved by Russell's paradox. But it was definitely dealt a severe blow. Frege's logicist project for set theory attempted to identify sets with purely logical entities (extensions of definable properties). That didn't work, or at the very least, Frege's version of it didn't work.

There was still a logicist school after Russell; it was still possible to hope that the natural numbers, at least, might be reduced to logic, even if sets could not be. Many mathematical logicians now feel that this hope was rendered untenable by Gödel's incompleteness theorems. But there's no unanimous agreement on that, either -- Torkel Franzen didn't think so, and even today there are logicians who call themselves "neo-logicist".

These are not the sorts of questions that are really susceptible to final proof or disproof in the ordinary mathematical sense. However, mathematical discoveries do shed light on how difficult these metamathematical positions are to hold, how much sense they make, and how much they accomplish. --Trovatore 05:41, 2 July 2007 (UTC)

Thanks for pointing out this problem in the reductionism article. I took a look at it and tried to figure out how to fix the "set-theoretic reductionism" section, but decided it was so badly flawed that all I could do was delete it. Could be an interesting addition if someone could source it and explain it clearly and accurately, but I saw nothing worth saving in the existing text and didn't have a source that would allow me to rewrite it. My specific criticisms are on talk:reductionism#"set-theoretic reductionism". --Trovatore 08:32, 4 July 2007 (UTC)