# Unreasonable effectiveness of mathematics (Wigner)

**URL:** <https://boards.straightdope.com/t/unreasonable-effectiveness-of-mathematics-wigner/972403>\
**Category:** In My Humble Opinion\
**Created:** [September 28, 2022, 5:39pm UTC](https://boards.straightdope.com/t/unreasonable-effectiveness-of-mathematics-wigner/972403 "2022-09-28T17:39:44Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![xtenkfarpl](https://avatars.discourse-cdn.com/v4/letter/x/c2a13f/32.png) [@xtenkfarpl](https://boards.straightdope.com/u/xtenkfarpl)\
**Post date:** [September 28, 2022, 5:39pm UTC](https://boards.straightdope.com/t/unreasonable-effectiveness-of-mathematics-wigner/972403/1 "2022-09-28T17:39:44Z")

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Creating a new topic here because I don’t want to muddy the waters in a former technical discussion about infinitesimals.

The problem is the infinities that arise in quantum theory (and to a lesser extent in general relativity).

There are tricks (renormalisation) that sort-of work for quantum electrodynamics, but nobody has found a good solution for the strong force.

Paul Dirac objected to renormalisation, saying something like “In mathematics we ignore quantities which are sufficienly small. But we don’t ignore infinite quantities just because we don’t like them”.

So the question: is real number math (aka analysis) really the right mathematical framework for physics? Or do we need new math, as well as new physics to get to a Theory of Everything (if such a thing exists).

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**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [September 28, 2022, 6:34pm UTC](https://boards.straightdope.com/t/unreasonable-effectiveness-of-mathematics-wigner/972403/2 "2022-09-28T18:34:46Z")

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Just like working with infinitesimals, you can work with renormalization at a “physics” level of rigour and calculate actual numbers, but just like with the other subject, that does not mean there is no mathematical understanding of what is going on. For instance the following text has a lot of precise (but technically formulated) results

> **[Renormalization-QFT.pdf](https://www-fourier.ujf-grenoble.fr/~eherscov/Articles/Renormalization-QFT.pdf)**
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> 953.30 KB

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [October 1, 2022, 7:58pm UTC](https://boards.straightdope.com/t/unreasonable-effectiveness-of-mathematics-wigner/972403/3 "2022-10-01T19:58:40Z")

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Physicists don’t actually work with real numbers. I don’t know if mathematicians have a name for the things we do work with, and most physicists never bother worrying about the distinction, but what we always actually work with is distributions of numbers, of widths that depend on the precision of our measurements.

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**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [October 1, 2022, 8:17pm UTC](https://boards.straightdope.com/t/unreasonable-effectiveness-of-mathematics-wigner/972403/4 "2022-10-01T20:17:36Z")

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> [@Chronos](#):
>
> Physicists don’t actually work with real numbers

I think some do, though. IIRC I took a class once where space-time was some sort of manifold. Even quantum mechanics had Hilbert spaces and von Neumann algebras and stuff. And

> [@Chronos](#):
>
> what we always actually work with is distributions of numbers

Are these distributions real-valued? Complex-valued? If not, how are real (or complex) numbers avoided?

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**Author:** ![Chronos](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/chronos/32/134_2.png) [@Chronos](https://boards.straightdope.com/u/Chronos)\
**Post date:** [October 2, 2022, 1:48am UTC](https://boards.straightdope.com/t/unreasonable-effectiveness-of-mathematics-wigner/972403/5 "2022-10-02T01:48:54Z")

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Usually not complex-valued (and even when they are, a complex number is just a pair of real numbers, anyway). But it’s impossible to distinguish between, say, real-valued and rational-valued distributions.