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So, I have been reading up on the works by Kenneth Wilson, mainly his 3 statements that he concluded to be true about our universe. His first: 'There exists a hierarchy to our universe'. From this i understand that he means to convey that the different disciplines are built upon the former. I.e, chemistry upon physics and physics upon maths. If i have understood this right then Ken is conveying that, fundamentally, physics is built upon some fundamental mathematics and that's why everything is the way that it is. I.e, the reason there are a certain number of fundamental forces can be boiled down to some pure mathematics that is just 'true' (Like 2 + 2 = 4, albeit more complicated). Is there a branch of physics that deals specifically with trying to find how these fundamental physical laws link to pure mathematical concepts? I have been struggling to find textbooks on the topic, providing my thought process is correct.

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  • $\begingroup$ Do you have some context for that quote? Given Wilson's landmark work with the renormalization group, in the absence of more information I would interpret "there exists a hierarchy to our universe" to be a reference to physical length/energy scales. $\endgroup$
    – J. Murray
    Jun 22 at 3:00
  • $\begingroup$ I was watching a colloquia given by David Tong on the work of Wilson. It was an admittedly brief segment. I think that this is definitely the main context behind this quote, however I think to avoid confusion in the audience he simplified the idea. $\endgroup$
    – Aidan Daly
    Jun 23 at 10:44
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    $\begingroup$ I am even more sure now that Tong was using an analogy, and was actually talking about physical scales (which is the central idea of Wilson's work). See e.g. the section "Nature is Organized by Scale" which begins on the bottom of page 3 of these notes, in which he does the same thing. $\endgroup$
    – J. Murray
    Jun 23 at 13:09
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If I recall correctly Feynman's The Character of Physical Law (1965) covered this pretty well. The format is accessible to laypeople.

To answer your implied question about the nature of the relationship between math and physics:

No mathematical statement describes the world. Math is the process of starting with a set of arbitrary axioms and determining other statements that must be true given those axioms, and there is neither a need to test those statements against observation nor the possibility of ever doing so. Put another way, math is the study of the nature of internally consistent systems.

The universe is, as far as anybody can tell, internally consistent. Therefore, if we can make up some arbitrary axioms that seem to match all of our observations, we can apply what we know about the nature of internally consistent things (math) and determine other things that would be true if our axioms were right. By refining our axioms, we can start making really good predictions. By making really good observations, we can start guessing really good axioms. That process is physics.

So, physics (and the other sciences) are not built on a foundation of mathematics, but a foundation of observations. However, all studies of internally consistent things are inherently mathematical because mathematics is the study of internal consistency itself.

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