Writing about the historical relationship between math and physics

Question

I'm not sure if this is best suited for Writing SE, or perhaps one of the science SE's. In any case, feel free to move this if fitting.

I've been tasked with writing a 20-30 page paper on the history of physics from a mathematical perspective. That is, to discuss the way math has influenced physics, and physics has influenced mathematics, throughout history and up to the present day. It must cover the 'technical' aspects as well, discussing (where possible) the actual ideas in a way that can be understood by a general population.

There are no other requirements.

My issue is, I'm not quite sure where to start or how to structure this work. Obviously, I cannot provide a comprehensive history in a mere 20-30 pages, so maybe I should cover 3-5 'big ideas'? Also, since no strict timeline was given (except that it should reach up to the modern day), I'm not sure if I should go all the way back to the time of Newton (or earlier), or stay more modern (say, going back to Einstein).

Since it's such a long paper, I'd like to decide on a structure now instead of jumping in (as I usually would) and finding it doesn't work well halfway through.

Answer 1

From a theoretical physics perspective alone you should separate it into:

1. Classical Physics
2. Relativity
3. Quantum

This structure also alludes to a chronological/historical order since the last 2 are more recent than the first.

Then you will have to concentrate only on the mathematical principles (mathematical physics):

1. Calculus:

basic - kinematics and more for math in classical physics see this

tensor - everything especially quantum an relativity see other math in quantum physics and string theory as well as math in relativity

2. Differential equations see applied mathematics

3. Statistics and probability - statistical mechanics.

4. Algebra - used in most equations - this is relatively minor

5. Possibly combinatorics? Basically anything else important

You will need to know the history of the areas of physics that use math and the history of that math, but once you have explained the history of calculus and differential equations, most topics in physics can be relate to it. Warning: if you write about the discovery of calculus then it won't be necessary to bring it up every time calculus is used.

Answer 2

I would suggest that a somewhat technical but highly relevant example would be the way in which the development of the concept of Uniform convergance was driven by the development in physics of Fourier series and Fourier analysis. This is a case where advances in physics required new math, drove the development of a new mathematical concept, and that concept was then used to help describe and further advance the physics.

An earlier case might be how the development of conic sections permitted the formulation of Newtonian physics.

An interesting later one might be how cryptography (particularly public-key cryptography)drove advances in number theory and factoring, previously thought to be purely abstract adn of no practical value.

A classic but again rather technical example would be how tensors were essential to Eisenstein relativity calculations. A later relates one would be how manifold matrices were essential to developer forms of string theory. See The Elegant Universe for a very clear adn not too technical account.

It might also be worth reviewing Feynman's The Character od Physical Law.

Honestly, this topic could be better treated in 300 pages than 30, so it will probably be essential to pick a few good clear examples and limit discussion to those.