The Miracle of Applied Mathematics (2001) [pdf]

Mathematics wasn&#x27;t unreasonable effective until the invention of the calculus. Indeed, mathematics-led physics was circling the drain with epicycles.<p>As it turns out, the calculus has a good correspondence with the idea of dynamical systems. No surprise that it was invented together with modern physics and to serve modern physics.

&gt; decline of formalism as a credible philosophy of mathematics<p>I suppose someone who&#x27;s writes a paper about mathematics anticipating physics would not be a formalist, but to say formalism is not credible is a bridge too far. As far as I know, Hilbert formalism[0] is the <i>only</i> credible philosophy of mathematics. Hilbert&#x27;s approach to axiomatic systems led to Gödel&#x27;s work and ZFC; on the contrary, rigorous attempts by non-formalist to assign meaning to logic and mathematics have largely collapsed after (for example) being unable to define numbers in terms of logic[2] (Frege) or to be consistent with its own principles[3] (Russel). And non-rigorous attempts, such as those described in this paper, have never then gone further than handwaving and saying &quot;isn&#x27;t it amazing? Don&#x27;t you think it&#x27;s unlikely?&quot; which is nothing more than the fallacious &quot;argument from incredulity.&quot;[4]<p>In a sense formalism is the Cartesian skepticism of the philosophy of mathematics: a secure position to which you can always return if pressed, and from which one sally&#x27;s forth only a great risk.<p>My current understanding of the philosophy of science is that we have Popper&#x27;s Falsificationism[5] on the one hand, mathematical formalism on the other, and a gap in the middle with a warning label stuck on it that says &quot;Use mathematics to describe your models at your own risk! Management assumes no liability!&quot;<p>[0]: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Formalism_(philosophy_of_mathematics)#Hilbert&#x27;s_formalism" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Formalism_(philosophy_of_mathe...</a><p>[1]: <a href="https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;frege-hilbert&#x2F;" rel="nofollow">https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;frege-hilbert&#x2F;</a><p>[2]: <a href="https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;frege-theorem&#x2F;" rel="nofollow">https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;frege-theorem&#x2F;</a><p>[3]: <a href="https:&#x2F;&#x2F;www.reddit.com&#x2F;r&#x2F;philosophy&#x2F;comments&#x2F;10iy83&#x2F;why_did_logical_positivism_fail_what_is_the&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.reddit.com&#x2F;r&#x2F;philosophy&#x2F;comments&#x2F;10iy83&#x2F;why_did_...</a><p>[4]: <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Divine_fallacy" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Divine_fallacy</a><p>[5]: <a href="https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;popper&#x2F;" rel="nofollow">https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;popper&#x2F;</a>

I think math is applicable for the same reason that scientific theories are applicable; both are stark abstractions compared to the amount of parameters at play, so why they able to model the phenomena?

Modal structuralism[1] is a tragically underappreciated interpretation of mathematics that leaves nothing mysterious about the correspondence of math and nature.<p>[1] <a href="https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;nominalism-mathematics&#x2F;#ModStr" rel="nofollow">https:&#x2F;&#x2F;plato.stanford.edu&#x2F;entries&#x2F;nominalism-mathematics&#x2F;#M...</a>

From an AI point of view, All subjects are mathematically dressed in camouflage. For a person with good mathematical thinking and multiple interests, you can learn a lot of subjects in a very short period of time and integrate them.

In my own impression, math is best done with total disregard for applicability, which is to remain entirely accidental.<p>Concerning &quot;The problem is epistemic: why is mathematics, which is developed primarily with aesthetic considerations in mind, so crucial in both the discovery and the statement of our best physical theories?&quot;<p>Abandoning the primary goal of &quot;aesthetic considerations&quot; would undoubtedly lead to such math becoming unusable, or even degenerating into non-math.<p>Therefore, I believe that in everybody&#x27;s best interest math should stick to its own goal of exploring mere aesthetics.

Arithmetic has been important for the progress of science to date — but I wonder if it will remain so in the future. It seems to me like should quantum computers&#x2F;programmable quantum simulations ever arrive — arithmetic will disappear from theory as constituent operations of nature (which include in their very observability, limits on knowledge and boundaries on errors) become the language in which theory is written and from which predictions are made.