
The Miracle of Applied Mathematics (2001) [pdf] - lainon
http://www.colyvan.com/papers/miracle.pdf
======
thanatropism
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.

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

------
olooney
> 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...](https://en.wikipedia.org/wiki/Formalism_\(philosophy_of_mathematics\)#Hilbert's_formalism)

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

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

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

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

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

~~~
Koshkin
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."

------
threatofrain
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?

------
hackinthebochs
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...](https://plato.stanford.edu/entries/nominalism-
mathematics/#ModStr)

------
lincpa
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.

------
neokantian
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.

~~~
s_m_t
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.

------
breatheoften
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.

~~~
coldtea
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.

