A physicist, walking home at night, spots a mathematician colleague under a street lamp staring at the ground, "something wrong?" he asks; "I've dropped my keys" he replies, "whereabouts?" asks the physicist, keen to help. "Over there" says the mathematician pointing; "So why don't you look over there?" retorts the physicist, "the light is better here" says the mathematician.

Disclosure, I'm a mathematician.

> Hitchin agrees. “Mathematical research doesn’t operate in a vacuum,” he says. “You don’t sit down and invent a new theory for its own sake. You need to believe that there is something there to be investigated. New ideas have to condense around some notion of reality, or someone’s notion, maybe.”

This is kind of it I think. It's not just physics that drives interesting math, and it's not just recently that this relationship holds. Math is, IM humble O, the ultimate domain-specific language. It's a tool we use to model things, and then often it turns out that the model is interesting in its own right. Trying to model new things (ex. new concepts of reality) yields models that are interesting in new ways, or which recontextualize older models; and and so we need to reorganize, condense, generalize, etc; and so the field develops.

One of my physics lecturers at university made the offhand observation that the distinction between physics and mathematics is a twentieth-century idea: it wasn't made during the nineteenth century or before, and it seems to be disappearing in the twenty-first.

Try to make some innovative software product without talking to any user. You'll see why physics is good at crating new math.

Physics is also great for machine learning, though the approaches can be rather unintuitive. For example message passing and belief propagation in trees/graphs (Bayesian networks, Markov random fields etc.) for modeling latent variables are usually taught using the window/rainy weather marginal probability analogy and involves splitting out a bayesian/statistical equation into subcomponents via the marginalization chain rule. For physicists however, they tend to teach it using Ising models and magnetic spin, which is a totally different analogy.

A lot of the newer generative ML models are also using differential equations/Boltzmann distribution based approaches (state space models, "energy based" models) where the statistical formulations are cribbed wholesale from statistical physics/mechanics and then plugged into a neural network and autodiff system.

The best example is probably the Metropolis-Hastings algorithm which was invented by nuke people.

https://web.archive.org/web/20150603234436/http://flynnmicha...

As one of my Physics Professors once said, "Math is Physics without a purpose."

(I was once a reasonably successful Physicist, so I might be biased :D)

I'm not a physics or math whiz but isn't the relationship more of a virtuous cycle?

I think I read that the 20th century was a revolution because of the marriage between physics and math. Quarternions are key to relativity. Discrete math is littered all over quantum mechanics and the Standard Model. Like U(1) describes electromagnetism, SU(2) describes the weak force and SU(3) describes the strong nuclear force. In particular the mass of the 3 bosons that mediate the weak force is what led directly to the Higgs mechanism being theorized (and ultimately shown experimentally).

One of the great advances of the 20th century was that we (provably) found every finite group. And those groups keep showing up in physics.

The article mentions how string theory has led to new mathematics. This is really interesting. I'm skeptical of string theory just because there's no experimental evidence for "compact dimensions". It seems like a fudge. But interestingly there have been useful results in both physics and maths based on if string theory was correct.

Do we know if it’s better at creating new math than other fields? For example, computers sure created a lot of new math. Statistics was entirely driven by external pressure from medicine, social sciences, and business. Finance and economics created a lot of math around modeling and probability. And so on.

> Might there be certain laws of physics that are also “necessary” in the same way? In his paper, Molinini argues that the principle of conservation may be one such law. In physics, some properties of a system, such as energy or momentum, can’t change. A bicyclist freewheeling down a hill, for example, is converting her gravitational potential energy into movement energy, but the total amount of energy she and her bike have stays the same.

Arithmetic itself is a consequence of physical conservation: if you have a collection of four acorns, another collection of three acorns, then combine them without dropping an acorn, then you must have a collection of seven acorns. It is our deep physical understanding of space and causality which leads to simple arithmetic being intuitively true to most (if not all) vertebrates. (If the squirrel only got six acorns after combining then there must be a causal explanation for the quantitative discrepancy; another squirrel stole an acorn from the older stash, or maybe it fell in a hole.)

We need "beermaking is unreasonably good at creating new stats"

"Mathematics is the part of physics where experiments are cheap." (V.I. Arnol'd)

Could this be a case of physics being more "tangible", thus leading to more obvious paths? Like, if you only study pure maths and you stay in your field, can you really point to a concrete direction for where to look for new stuff? In physics, your job is literally to study how the universe already behaves, so you have a frame of reference to take inspiration from and the efforts are a bit more concentrated. In fact, since models don't describe reality perfectly, you can always observe where it fails to know in which direction to attack. In maths, on the other hand, everything that is proven is correct forever. The model is reality. So it seems to be more difficult to find criticalities to look for. It's more of a "I wonder if this property does or doesn't hold" ordeal, which seems much more vague. Just a question.

Discoveries are made and measurements are taken with the tools available.

The measurements, theories, and currently understood or applicable math may not match up with observations.

People ponder and discover, then attempt to explain the observations and measurements with a new theory. If the theory pans out, a deeper explanation of that theory is necessary and that's where the new math's at.

It's not that physics is good at creating math. Physics is good at describing our observations /with/ math. That's kind of its whole job.

Next time you look at raindrops in a puddle, try to imagine how you would describe those movements scientifically. One needs math for that.

Sometimes the available tools and math are sufficient for a thorough explanation, and sometimes one needs to invent a universe of math to describe a tiny fluctuation.

I’m not sure how it could be otherwise. On some level mathematics is a description of reality that we can use to compute things in reality.

For example, pi is the ratio of a circle’s circumference to its diameter. It’s just what a circle is in two dimensions. The value of pi isn’t any more mysterious or connected to physics than the existence of this thing called a circle. If you have some other Euclidean shapes you’ll have other ratios and values that have other relationships to other things in physical reality.

And if reality was different, hence the physical laws were different then the math would be different.. and the beings in that world might wonder why their math and physics were so interconnected.

I think it’s the other way: math is unreasonable good at describing physics!

Imagine a universe where the laws are best described in iambic hexameters under the condition that the last letters of the stanzas form specific words.

The ancients held some believes like that: kabala, astrology and the like. How wonderfully absurd it must have felt to them that the answer was something even more removed from reality.

> “Bad” physics can sometimes lead to good math.

cf. string theory

The subject of study of physics is "physical quantity" which is defined as a number with a unit. Physical quantity doesn't have to be a "physical" quantity. So physics does not study exclusively physical objects. I think this is how mathematics and physics are related, mathematics does not deal with units (except unity).

I understand that one huge reason for Ed Witten's optimism about strong theory is this very fact. That, in his terms, the process of building out string theory has led to the uncovering of so much "buried treasure" in the form of novel developments of maths.

Of course it's not anything like a proof but something that bolsters an intuition.

See also perhaps the article "The Unreasonable Effectiveness of Mathematics in the Natural Sciences":

* https://web.archive.org/web/20210212111540/http://www.dartmo...

* https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness...

Physics research gets funded because of applications, existing or promising, for curiosity, fundamental science, the economy, medicine, and national security, etc. Since math can help physics research, that research is funded and motivated to make applications of math, old or new. Math alone is less involved with applications.

If you want to bring physics and mathematics closer together, then string theory is a rather bad example to follow.

> the mathematics of known physics is just a tiny fraction of all the mathematics out there

But the opposite is also true: the physical reality that has been explained by mathematical thinking is just a tiny fraction of all the reality out there.

I've long thought physics is a subsection of maths and reality is a mathematical object that exists the same way that prime numbers or the Mandelbrot set do. Hence the unreasonable effectiveness of one in the other.

Feynman on mathematicians vs physicist

https://youtu.be/obCjODeoLVw?si=2akBzyo-fC2j90OH

Entertaining viewpoint

https://archive.ph/yUiis

We have lived long enough to see Physics become Modern Math and create no new provable Physics

Obviously, physics never creates any math but discovers it, there are no new things under the sun.

This is the well known litany from string theorists to try and justify the inordinate amount of money threw at them to get back nothing of physical value: no falsifiable prediction.

Instead of reasoning on the worth of the effort spent in this direction to investigate nature (a very tangible companion) they try to steer the discourse toward this nonsense. We spent >50 years listening to these tales and the time has long passed since we are required to stop playing with these smoke and mirrors.

String theory seems more like a branch o mathematics than anything else.

“Physicists are much less concerned than mathematicians about rigorous proofs. ... That allows physicists to explore mathematical terrain more quickly than mathematicians.”

The end.

There's no magic here.