Do breakthroughs in mathematics lead to breakthroughs in other scientific disciplines, vice versa, both, or is there no relationship?

Question

I asked this question in the Mathematics StackExchange, but I was told it might be better posted here in the Philosophy StackExchange.

I heard a professor say once that Einstein's mathematics led him to E=mc^2, and it was not observation of physical phenomena that inspired or led him to discover the equation. Yet, this abstract math accurately represents what others discovered with the atomic bomb.

I also know there are inventors who have observed certain phenomena in the natural world, and then "did the math" alongside experimentation to figure out how it worked and to mimic it in an invention.

Questions:

-Does abstract math always represent some physical reality, whether that physical reality has been discovered or will be discovered in the future?

-In human history, does math tend to lead to breakthroughs in other disciplines, vice versa, some of both, or is there no real relationship between a breakthroughs in math and breakthroughs in other disciplines?

Answer 1

Pure mathematics and physics are different disciplines aiming at different goals and using different methods of investigation. If one considers pure mathematics as a game which follows only the rules of logic, then mathematics is independent from physics. On the contrary, history has shown: Progress in physics needs the formalization of physical ideas by powerful mathematical concepts. And physics needs reasoning in the language of mathematics.

Towards your question 1: The theory of prime numbers, Riemann’s development of differential geometry, Grothendieck‘s investigation of scheme theory are examples of mathematical theories developed long before they had any application in physics.

Other examples like the invention of calculus by Newton and Leibniz show a close link between progress in physics and progress in mathematics. Here it was necessary to invent new mathematics to make progress in physics.

In the opposite direction: developments in String Theory during the last decades triggered a series of new mathematical developments.

Hence we have all kinds of interrelation between progress in mathematics and progress in physics. The theory of primes was long considered a prototype of pure mathematics without any applications. Today all algorithms in cryptography use sophisticated results from number theory. No one knows whether mathematics like scheme theory introduced by Grothendieck, will have applications in physics. The future will show.

Towards question 2: I do not see similar close relations between mathematics and disciplines different from physics. Statistics has many applications in social sciences. Nevertheless, I do not see any breakthrough in one or the other field due to these relations.

Aside: It seems an open question why mathematics applies to formalize our physical theories about the world we live in. This issue is the subject of Eugene Wigner's short article „The unreasonable effectiveness of mathematics in the natural sciences“. For the document see https://www.maths.ed.ac.uk/~v1ranick/papers/wigner.pdf

Answer 2

"Abstract mathematics" sometimes describes aspects of our physical world. In the cases where it does happen to do so, the implications of the math do indeed guide research in physics, where the connections between the math and the physics are most obvious. Here are some examples:

Galois invented the field of mathematics called group theory which was subsequently extended in very important ways by Abel- in particular, to include so-called nonabelian groups. Over a hundred years later, nonabelian group theory was discovered to be exactly the formalism needed to describe and unify the mathematical relationships between different families of subatomic particles in the highly-successful Standard Model of particle theory.

The mathematical concept of symmetry as it relates to group theory was used by Gell-Mann to predict the existence of new subatomic particles from first principles in a model called the eightfold way, which particles were then subsequently discovered in particle accelerator experiments- once the experimentalists knew where and how to look for them. This was the stuff of Nobel prizes.

Riemann invented the field of mathematics called non-euclidean geometry which posits the existence of curved spaces of arbitrary dimension. Many years later, when Einstein consulted his mathematician friend Grossman on how to mathematically describe gravity as a consequence of spatial curvature, Grossman informed him that exactly the right tool had already been worked out in detail by Riemann, and Einstein used the Riemannian formalism in his construction of general relativity.

Answer 3

I heard a professor say once that Einstein's mathematics led him to $E=mc^2$, and it was not observation of physical phenomena that inspired or led him to discover the equation. Yet, this abstract math accurately represents what others discovered with the atomic bomb.

You're teacher is kinda wrong, E=mc2, being an equation, obviously comes from math, but that's not what initiated Einstein's inquiry, but rather a free fall thought experiment and some inconsistencies in physics of the day

I also know there are inventors who have observed certain phenomena in the natural world, and then "did the math" alongside experimentation to figure out how it worked and to mimic it in an invention.

That's more realistic

-Does abstract math always represent some physical reality, whether that physical reality has been discovered or will be discovered in the future?

abstract math doesnt necessarily represent anything, that's why it's called abstract. However the founding rules of math & logic were made using our common sense and our common sense comes from our observation of physical reality, that's why math and reality tend to be compatible

-In human history, does math tend to lead to breakthroughs in other disciplines, vice versa, some of both, or is there no real relationship between a breakthroughs in math and breakthroughs in other disciplines?

I will not mention "other disciplines" as it's too vague. But merely talk about science. And I will define science as the scientific method. The scientific method requires comparing an hypothesis or a theory (often expressed with math), with reality through an experiment. Both can progress independently from each other:

-math breakthroughs come from math not anything else (because you can only solve a math problem with math)

-physics breakthroughs do not come from math breakthroughs (but rather finding a proper model for a given phenomenon), but math tools are often developped in response to a need for modeling physics

Answer 4

Avoiding the discussion Was mathematics invented or discovered?, this is an approach considering a Kantian perspective: it is not only rational, and not only empirical, it is both: a) math is discovered, it indirectly depends on empirical knowledge, and at the same time, b) math is invented, it is a priori knowledge.

In such case:

Does abstract math always represent some physical reality, whether that physical reality has been discovered or will be discovered in the future?

In physics, it is accepted that g=9.8m/s² and not that g=10m/s²: physics is a description of physical reality, it does not represent physical reality "always", only when discoveries allow it (for example, we've found that g=9.8m/s², probably after lot of trial and error, using not only other values like g=10m/s², but also other approaches, not based on acceleration). Same happens with mathematics and metaphysical reality. It is possible that before r²=x²+y², Pythagoras also considered r=x+y.

In human history, does math tend to lead to breakthroughs in other disciplines, vice versa, some of both, or is there no real relationship between a breakthroughs in math and breakthroughs in other disciplines?

Both. First case: Albert Einstein's math (which he mistrusted) lead to discover that the space is expanding, so, math allowed in this case to developments in other disciplines of knowledge. Second case: there was no mathematics to describe quantum facts, so Paul Dirac started by inventing a notation, which allows a whole formalism; in this second case, empirical discoveries lead to math developments.

Answer 5

I heard a professor once say that Einstein's mathematics led him to E=mc^2.

Actually, forty five years before Einstein, a British mathematician and physicist, William Clifford, after reading Riemanns lectures on the new concept of manifold and curvature declared that all forces will be shown to be an aspect of curvature. His insight is broadly true, not only for Einstein's gravity but for all the other forces in the standard model.

So this was a case in which mathematics led to a breakthrough in physics. But the story doesn't end there. Riemann's novel notions of geometry began with Gauss who was his supervisor. And Gauss discovered non-Euclidean geometry after reading Kants Critique of Pure Reason fives times. What did Kant write that inspired Gauss to discover this? Well, he wrote that it was not a priori obvious that the angles in a triangle should add upto 180 degrees. That they do so is, in Kantian terms, because it is a synthetic a priori truth. For a mathematician of Gauss's calibre, this clue was enough to set Gauss on his journey towards non-Euclidean geometry.

In fact, Einstein also name-checked Hume as an inspiration for his development of special relativity which reconceptualised our physical understanding of the relationship of space and time because his natural philosophy threw a critical eye on the nature of causality. This critique was also important for Kant too. However, this critique did not begin with Hume but with al-Ghazali.

So there you have two instances of probing philosophical questions in natural philosophy bringing about breakthroughs in mathematics and physics and also from mathematics to physics.

It can happen the other way around: that is breakthroughs in the sciences can lead to revolution in the arts. It's well known that Einsteins special relativity was the inspiration for Braque and Picasso's discovery of cubism and it was also the inspiration of Dali's famous painting, The Persistence of Memory. And the ideology of progress encapsulated in Baconian science led to Italian futurism and it's offshoots like Vorticism and so on.

Does abstract maths always represent some physical reality?

Well, it's unlikely that the large cardinal axioms from set theory are going to figure directly in any physical theory soon. I would also say that prime numbers, which are a large part of number theory, are also not going to turn up in physical theory in a natural way. They haven't done so, so far; and attempts to use them have usually an air of artificiality and ade really attempts to use physical tools to answer mathematical questions.

But more deeply speaking, as Aristotle already pointed out, the reason why mathematics does so well in describing physical laws is that both are aspects of neccessity - as well as logic. This answers Wigners question about the "unreasonable effectiveness of mathematics in physics". It's not "unreasonable" at all, and a small amount of reflection, or an acquaintance of the appropriate philosophical literature will show why. But then again, asking contemporary mathematicians or physicists to be acquainted with anything outside their narrow specialisms seems to be a bit of a big ask these days - especially in philosophy - and which is one reason why good questions like Wigner's are dogmatically resurrected from time to time with no appreciation that they have already been answered. And in this case, two and a half millenia ago.