Is There a Fundamental Difference Between Proof-Based Math and Applied Physics?

[gibberingmouther]

i guess a lot of physics could be considered applied math, though the more you know about how nature works, including how more abstract subjects in math work, probably the better will be your overall understanding.

so there are problems of the type you have in a physics textbook - these are usually word problems, or problems to show you know how to perform the calculations required in physics subjects like engineering. often the textbook will show how to derive the formulas used, or as i do, you could look on the internet.

an abstract book in, say, calculus (or analysis? don't know much about that) will have strict logical definitions, like the confusing definition for what a limit is, and there may be lots of proofs. the book may challenge the reader to do some of his own proofs.

i seem to be able to understand my college level physics textbook, as far as the end of the 8th chapter. but i was curious, i took calculus a couple years ago, and today i decided to review some of the theorems. I've done proofs before in a class called discrete math, and i seemed to be okay at doing them, but they were almost formulaic and I'm not good at doing proofs in other contexts. i can be effusive when it comes to writing about some subjects, but math is not really one of them. it's like the text might say "prove this" and my mind would not really have much to say on the matter unless it's something that I've seen done before in a similar context.

Andrew Wiles wrote a 129 page proof for Fermat's last theorem (according to the Wikipedia). i mean that boggles my mind. physics isn't easy i guess but is there anything in physics that is comparable to that, in terms of difficulty? math is all about logic, while it seems like physics is more concerned with just using the mathematical models that can be built with the toolset provided by mathematicians. do i have this right? I'm not sure what my question is, i just want to know more about how deep you can delve in these subjects, and how they relate to each other.

 

[fresh_42]

i'm not sure what my question is,..

Honestly? Me neither.

... i just want to know more about how deep you can delve in these subjects,..

Which? Math or physics?

... and how they relate to each other.

If you take a closer look on quantum field theory, you will discover a lot of pretty pure math. Good theoretical physicists are usually also good mathematicians. But they use math on a different level than mathematicians do. E.g. coordinates are all over the place in physics and the average mathematician hates them. Physicist use math as a doctor uses a drug. It's not necessary to understand how they function on a molecular level, that's what pharmacologists do. It is similar here. A sequence $(a_n)_{n\in \mathbb{N}}$ converges to a limit $L$ if for every $\varepsilon > 0$ there is a number $N_\varepsilon$ such that $|a_n-L| < \varepsilon$ for all $n > N_\varepsilon$. That's the mathematical point of view. In physics on the other hand, it often (not always) is sufficient to know: The bigger the index $n$ gets, the closer the sequence $a_n$ gets to the limit $L$. It is basically the same thing, and if it comes down to a rigorous paper, the mathematical definition will be needed. Until then, the description is as good. Apart from that, the daily scientific business is not what Andrew Wiles did, but instead a steady communication and dialogues, even across the disciplines, especially between mathematicians and physicists. And although Wiles solved Fermat's last theorem mainly on his own, he, too, had colleagues he talked to - and with whom he repaired his first incomplete version. It is therefore difficult to draw lines where in real life there aren't any.

 

[gibberingmouther]

thank you fresh_42, that was some of the additional information i was looking for. i figured it would work something like that, but it's hard to get insight into how academia/industry works when you're just a student.

 

[jamalkoiyess]

Until now all the math that I have seen being used in physics is just taking a rigorous mathematical idea and applying it loosely to the physical scene to describe it. This is what calculus is in mechanics, rates of change happen to explain well the mechanics of nature. However, I have not yet seen (myself in MY education) any actual rigorous mathematical proof using the rigor I saw in my Analysis and Algebra courses. It was disappointing as I was expecting those in physics for some reason. But it seems as if it isn't that beneficial to what I have seen. No one in physics cares if the function of the position of a particle is continuous pointwise or uniformly continuous. They only care that it is continuous enough to make sense and allow predictions.
I think that this is a glaring difference between math and physics. Most people think that physics is just math with some meaning in it. But it is not that. It is better described as a science where math has been proven to be stupidly efficient and helpful, but it does not care about math. Physics exited without math and it still could, but it would not be as powerful. And that tells you why proofs such as those are not existent in physics. Most of the time the proof that a physicist needs for some mathematical issue was already done and if it wasn't and he made it, it would be as if a butcher is sharpening his knife, making it more useful and powerful, but that does not mean that his trade is based on sharpening knives.

Disclaimer: This is sometimes offensive to some mathematicians, and it makes them feel as if their math is just a tool. It is not! its way more, but explaining that will take another essay