Some Coloured Remarks on the Foundations of Mathematics in the 20th Century

What is Neologicism?

Bulletin of Symbolic Logic ◽

10.2178/bsl/1140640944 ◽

2006 ◽

Vol 12 (1) ◽

pp. 60-99 ◽

Cited By ~ 11

Author(s):

Bernard Linsky ◽

Edward N. Zalta

Keyword(s):

20Th Century ◽

Set Theory ◽

The Other ◽

Early 20Th Century ◽

Foundations Of Mathematics ◽

The One

Logicism is a thesis about the foundations of mathematics, roughly, that mathematics is derivable from logic alone. It is now widely accepted that the thesis is false and that the logicist program of the early 20th century was unsuccessful. Frege's [1893/1903] system was inconsistent and the Whitehead and Russell [1910–1913] system was not thought to be logic, given its axioms of infinity, reducibility, and choice. Moreover, both forms of logicism are in some sense non-starters, since each asserts the existence of objects (courses of values, propositional functions, etc.), something which many philosophers think logic is not supposed to do. Indeed, the tension in the idea underlying logicism, that the axioms and theorems of mathematics can be derived as theorems of logic, is obvious: on the one hand, there are numerous existence claims among the theorems of mathematics, while on the other, it is thought to be impossible to prove the existence of anything from logic alone. According to one well-received view, logicism was replaced by a very different account of the foundations of mathematics, in which mathematics was seen as the study of axioms and their consequences in models consisting of the sets described by Zermelo-Fraenkel set theory (ZF). Mathematics, on this view, is just applied set theory.

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WEYL REEXAMINED: “DAS KONTINUUM” 100 YEARS LATER

Bulletin of Symbolic Logic ◽

10.1017/bsl.2020.23 ◽

2020 ◽

Vol 26 (1) ◽

pp. 26-79

Author(s):

ARNON AVRON

Keyword(s):

20Th Century ◽

Mathematical Analysis ◽

Foundations Of Mathematics ◽

Mathematical Ideas ◽

Modern System ◽

Famous Book ◽

Hermann Weyl ◽

Mathematics And Physics

AbstractHermann Weyl was one of the greatest mathematicians of the 20th century, with contributions to many branches of mathematics and physics. In 1918, he wrote a famous book, “Das Kontinuum”, on the foundations of mathematics. In that book, he described mathematical analysis as a ‘house built on sand’, and tried to ‘replace this shifting foundation with pillars of enduring strength’. In this paper, we reexamine and explain the philosophical and mathematical ideas that underly Weyl’s system in “Das Kontinuum”, and show that they are still useful and relevant. We propose a precise formalization of that system, which is the first to be completely faithful to what is written in the book. Finally, we suggest that a certain set-theoretical modern system reflects better Weyl’s ideas than previous attempts (most notably by Feferman) of achieving this goal.

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