People have asked me, "How can you, a nominalist, do work in set theory and logic, which are theories about things you do not believe in?" ... I believe that there is value even in fairy tales and the study of fairy tales. [A Tarski quoted in A. Burdman-Feferman, S. Feferman: "Alfred Tarski – life and logic", Cambridge Univ. Press (2004) p. 52]

On the most fundamental level, the foundations of mathematics are much shakier than the mathematics that we do. Most mathematicians adhere to foundational principles that are known to be polite fictions. For example, it is a theorem that there does not exist any way to ever actually construct or even define a well-ordering of the real numbers. There is considerable evidence (but no proof) that we can get away with these polite fictions without being caught out, but that doesn't make them right. Set theorists construct many alternate and mutually contradictory "mathematical universes" such that if one is consistent, the others are too. This leaves very little confidence that one or the other is the right choice or the natural choice. [W.P. Thurston: "On proof and progress in mathematics", arXiv (1994) p. 10]

Most mathematicians however accept the message as the final say that modern mathematics consists of idle chatter. And it is miraculous to see how they feel happy in this remaining ruin. [R. Taschner: "Das Unendliche ist nur ein Wort – oder irren die Mathematiker womöglich?", Die Presse, Wien (18 Jan 2012)]

We need to question infinity. I'm betting that we also need to let go of it. [M. Tegmark "What scientific idea is ready for retirement? – Infinity", Edge.org (2014)]

A constructivist version of a mathematical theory is adequate for all the applications to be made of the theory within natural science: [N. Tennant: "Logic, mathematics, and the natural sciences", ResearchGate (2007)]

Bertrand Thébault Con’s about transfinite numbers: After a century of "existence": no useful scientific applications. [B. Thébault: Was Leopold Kronecker right about transfinity?, BoldRIFT (2020)]

Calculus does not need any concept of infinity in order to provide limits, derivatives, integrals, and differentials. [J. Trek: "Finitists", sci.math (5 Feb 1999)]

T. Tsujishita: "Alternative mathematics without actual infinity", arXiv (2012)

Thats quite a blow that mathematics deals with different structures!

LoL

WM schrieb am Donnerstag, 23. Dezember 2021 um 11:05:18 UTC+1:
> People have asked me, "How can you, a nominalist, do work in set theory and logic, which are theories about things you do not believe in?" ... I believe that there is value even in fairy tales and the study of fairy tales. [A Tarski quoted in A. Burdman-Feferman, S. Feferman: "Alfred Tarski – life and logic", Cambridge Univ. Press (2004) p. 52]
>
> On the most fundamental level, the foundations of mathematics are much shakier than the mathematics that we do. Most mathematicians adhere to foundational principles that are known to be polite fictions. For example, it is a theorem that there does not exist any way to ever actually construct or even define a well-ordering of the real numbers. There is considerable evidence (but no proof) that we can get away with these polite fictions without being caught out, but that doesn't make them right. Set theorists construct many alternate and mutually contradictory "mathematical universes" such that if one is consistent, the others are too. This leaves very little confidence that one or the other is the right choice or the natural choice. [W.P. Thurston: "On proof and progress in mathematics", arXiv (1994) p. 10]
>
> Most mathematicians however accept the message as the final say that modern mathematics consists of idle chatter. And it is miraculous to see how they feel happy in this remaining ruin. [R. Taschner: "Das Unendliche ist nur ein Wort – oder irren die Mathematiker womöglich?", Die Presse, Wien (18 Jan 2012)]
>
> We need to question infinity. I'm betting that we also need to let go of it. [M. Tegmark "What scientific idea is ready for retirement? – Infinity", Edge.org (2014)]
>
> A constructivist version of a mathematical theory is adequate for all the applications to be made of the theory within natural science: [N. Tennant: "Logic, mathematics, and the natural sciences", ResearchGate (2007)]
>
> Bertrand Thébault Con’s about transfinite numbers: After a century of "existence": no useful scientific applications. [B. Thébault: Was Leopold Kronecker right about transfinity?, BoldRIFT (2020)]
>
> Calculus does not need any concept of infinity in order to provide limits, derivatives, integrals, and differentials. [J. Trek: "Finitists", sci.math (5 Feb 1999)]
>
> T. Tsujishita: "Alternative mathematics without actual infinity", arXiv (2012)

On Thursday, December 23, 2021 at 5:05:18 AM UTC-5, WM wrote:

> We need to question infinity. I'm betting that we also need to let go of it.

Which reminds me... Did you ever come up a definition of a finite set in your goofy little system, Mucke? I can see why you might hesitate since any workable definition would lead immediately to the definition of sets that are NOT finite, e.g. the set of natural numbers.

Dan

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