The Miracle of Applied Mathematics (2001) [pdf] | Hacker News Reader

The Miracle of Applied Mathematics (2001) [pdf]

colyvan.com

74 pointslainon15 comments

Mathematics wasn't unreasonable effective until the invention of the calculus. Indeed, mathematics-led physics was circling the drain with epicycles.

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.

Epicycles have decent predictive power as a description of planetary motion. They're just needlessly complicated.

Indeed it was always effective. Geometry was indispensable in architecture, mechanics, and geography; arithmetics, pretty much everywhere.

> decline of formalism as a credible philosophy of mathematics

I suppose someone who'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 only credible philosophy of mathematics. Hilbert's approach to axiomatic systems led to Gödel'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 "isn't it amazing? Don't you think it's unlikely?" which is nothing more than the fallacious "argument from incredulity."[4]

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's forth only a great risk.

My current understanding of the philosophy of science is that we have Popper'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 "Use mathematics to describe your models at your own risk! Management assumes no liability!"

[0]: https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...

[1]: https://plato.stanford.edu/entries/frege-hilbert/

[2]: https://plato.stanford.edu/entries/frege-theorem/

[3]: https://www.reddit.com/r/philosophy/comments/10iy83/why_did_...

[4]: https://en.wikipedia.org/wiki/Divine_fallacy

[5]: https://plato.stanford.edu/entries/popper/

Formalism is not a good philosophy of mathematics - because it is a rather extreme form of reductionism, and because as such it fails to address the issue of the "unreasonable effectiveness of mathematics."

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.

[1] https://plato.stanford.edu/entries/nominalism-mathematics/#M...

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.

Concerning "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?"

Abandoning the primary goal of "aesthetic considerations" would undoubtedly lead to such math becoming unusable, or even degenerating into non-math.

Therefore, I believe that in everybody's best interest math should stick to its own goal of exploring mere aesthetics.

You've missed a step here. No one thought to do math until people started to seriously worry about proportions, distance, quantity, time, etc of the natural world. Once you've set up those rules (and necessarily rigged them in favor of explaining the natural world) you can stay in the platonic realm, change the rules so they apparently stop reflecting reality, and rely on your sense of aesthetic, but then I think the onus is back on you to explain where this sense comes from.

is this why we have string theory but can't confine plasmas?

I consider applicability one of the primary aesthetic qualities in mathematics, but most mathematical applications are internal to mathematics or science, not about solving some highly specific practical problem like improving web search performance. Applications of complex analysis to proving the prime number theorem or to designing airfoils are in the same category for me aesthetically. They both enrich my understanding and appreciation for complex analysis. The applications might be more or less shallow. The applications of complex analysis to number theory go very deep. But potential theory is also a beautiful and theoretically rich subject that extends into higher dimensions, and the airfoil design example is striking because it relies on the non-rigidity of conformal mappings in 2D, which is the main feature of potential theory that fails to extend to higher dimensions. If you study any application, it will invariably improve your understanding of the original ideas.

> math is best done with total disregard for applicability

Except that has rarely been the case: mathematical theories are most often developed in the view of applications within mathematics itself.

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/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.

Quantum computers are not exactly what people imagine them to be. For one, they're not for all purposes but for niche uses by their very nature (e.g. that's not something we can invent our way out in the future).

And they depend on shitloads of regular algebra and arithmetic to create and program them.

In fact, Quantum Physics is a huge user of math.

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