現代数学の系譜11 ガロア理論を読む32 [無断転載禁止]©2ch.net

[過去ﾛｸﾞ] 現代数学の系譜11 ガロア理論を読む32 [無断転載禁止]©2ch.net (700ﾚｽ)
上下前次1-新
通常表示 512ﾊﾞｲﾄ分割ﾚｽ栞
抽出解除ﾚｽ栞

このｽﾚｯﾄﾞは過去ﾛｸﾞ倉庫に格納されています｡
次ｽﾚ検索歴削→次ｽﾚ栞削→次ｽﾚ過去ﾛｸﾞﾒﾆｭｰ

103: 現代数学の系譜11 ガロア理論を読む [sage] 2017/05/22(月) 14:05:54.68 ID:0yI+BCI1

>>101 補足

おれの読んだところでは
１．Georg Cantor, starting in 1874 now called naive set theory
２．Russell's paradox, Berry's paradox and the Burali-Forti paradox were discovered in Cantor's naive set theory, the issue became a heated topic among mathematicians.
３．One position was the intuitionistic mathematics that was advocated by L. E. J. Brouwer, which rejected the existence of infinite objects until they are constructed.
　（Brouwerの直観主義と言われるが、構成主義（rejected the existence of infinite objects until they are constructed）とも解される）
４．Another position was endorsed by David Hilbert: finite mathematical objects are concrete objects, infinite mathematical objects are ideal objects
　　More formally, Hilbert believed that it is possible to show that any theorem about finite mathematical objects that can be obtained using ideal infinite objects can be also obtained without them.
　　Hilbert's views are also associated with formalist philosophy of mathematics.
　（今日の日本では、ヒルベルトの形式主義と教えられるが、ヒルベルトは”finite mathematical objects”で”ideal infinite objects”と思っていた・・）
５．In the years following Godel's theorems, as it became clear that there is no hope of proving consistency of mathematics,
　（Godelの理論で、それは出来ないことが証明された）

ということかいな（＾＾；

http://rio2016.5ch.net/test/read.cgi/math/1495369406/103

108: 現代数学の系譜11 ガロア理論を読む [sage] 2017/05/22(月) 15:18:33.36 ID:0yI+BCI1

>>103 追加引用

https://en.wikipedia.org/wiki/Finitism
Finitism
(抜粋)

History

Today most classical mathematicians are considered Platonist and readily use infinite mathematical objects and a set-theoretical universe.[citation needed]

Views regarding infinite mathematical objects

Aristotle especially promoted the potential infinity as a middle option between strict finitism and actual infinity (the latter being an actualization of something never-ending in nature, in contrast with the Cantorist actual infinity consisting of the transfinite cardinal and ordinal numbers, which have nothing to do with the things in nature):

But on the other hand to suppose that the infinite does not exist in any way leads obviously to many impossible consequences: there will be a beginning and end of time, a magnitude will not be divisible into magnitudes, number will not be infinite. If, then, in view of the above considerations, neither alternative seems possible, an arbiter must be called in.

??Aristotle, Physics, Book 3, Chapter 6

http://rio2016.5ch.net/test/read.cgi/math/1495369406/108

上下前次1-新書関写板覧索設栞歴

ｽﾚ情報赤ﾚｽ抽出画像ﾚｽ抽出歴の未読ｽﾚ AAｻﾑﾈｲﾙ

ぬこの手ぬこTOP 0.022s