Thread: Pure Mathematics or Theoretical Physics

Hi,

I've started my Calculus sequence and I intend to gain at leasts a masters in some branch of mathematics.

I've been trying to decide on which area of study I want to pursue and I believe I have it narrowed down to pure mathematics or physics, with an emphasis on theory. I think mathematics is amazing. I imagine the universe as a story written in a book. The language that the book is written in is math, so if I want to understand the story, I have to learn the language. I feel a lot like Werner Heisenberg when replying to Arnold Sommerfeld on the importance of applied physics and 'trivial tasks': "Even so, I am much more interested in the underlying philosophical ideas than in the rest."

Anyway, concerning the study of pure mathematics and physics, as well as the typical type of work to go into, I was wondering if anyone here would mind offering their opinion or personal experience.

Thank you! Take care.
/alan

1. Physics means labs.
2. Math, short of a PhD, needs, for job purposes, something else to apply to. Computers, physics, economics, teaching, etc. Physics not so much - with a masters in physics, you can work as a lab tech.
3. With math, you're learning sophisticated trivialities. It's all just a game: some human makes up some definitions and rules, and then mathematicans go and find the necessary consequences. Physics is about REALITY. Physics isn't a matter of logical necessity, but an exploration and discovery of the contigencies of our actual reality. Since logic is non-ampliative, math ultimately gives you no *actually* new knowledge. The only true epistemic meat to be had comes from science. So in that sense, math is a game, physics is serious.
4. Physics requires a fair amount of judgement of relevance ("eh - that's so small an effect we can ignore it to this scale"), doesn't deal in numbers so much as measurments within some error bars. Physics gives you leeway to be crude around the edges and imprecise within reason. In math, a 20,000 line proof, with only a single tiny error on line 14,872, becomes worthless.
5. Physicists butcher math. If you like watching people write things like "dx = 2 cm", then physics is a winner.

2. Math, short of a PhD, needs, for job purposes, something else to apply to. Computers, physics, economics, teaching, etc. Physics not so much - with a masters in physics, you can work as a lab tech.
3. With math, you're learning sophisticated trivialities. It's all just a game: some human makes up some definitions and rules, and then mathematicans go and find the necessary consequences. Physics is about REALITY. Physics isn't a matter of logical necessity, but an exploration and discovery of the contigencies of our actual reality. Since logic is non-ampliative, math ultimately gives you no *actually* new knowledge. The only true epistemic meat to be had comes from science. So in that sense, math is a game, physics is serious.

I can argue for the opposite. When you learn physics, you might as well be learning history or politics: in a different world, the laws of physics could conceivably be far different. You're just learning something about this particular physical reality.... on the other hand, logic is universal, we can't imagine a world with different mathematics.

well i see it like this: the universe is a very complicated place. we have two things we can do "to learn more about it":

1. we can organize our ideas about it, using abstraction. this is very powerful. of course, some one has to explore the consequences of any given abstract principle, to make sure "it makes sense".

2. we can just essentially catalog raw data, saving it for further use. the more detail, the better.

people who "apply mathematics" tend to do more of 2, than 1, using abstract principles as "seasoning" on a diet of "concrete facts". no one knows why this works so well, but the advent of technology is solid proof that it indeed produces results.

people who do "pure mathematics" tend do to more of 1 than of 2. the examples "illustrate" general principles, which are often held to be "universally true". this is a much more philosophical approach.

at one extreme, you have "lab monkeys", who do little more than collect data, using math to tabulate it, perhaps. they are focused on something in particular, some niche of exploration in minute. and the at the other end, you have people exploring what may be either "universal truth" or, at the very least, "the internal logical structure of our minds" (we cannot say for sure the way we THINK about things is ACCURATE. we'll never be "non-human", and will always in some sense be at the mercy of the way our brains process thought and perception).

is math "just a game"? it IS possible. but given the implications for science and technology (including not only physics, but chemistry, biology, economics, computer science, etc.) it's a pretty serious game, played for high stakes (the very engine of a good part of our culture).

why do mathematicians prove things? it's because there are true statements that aren't self-evident. apparently, we don't even "know what we know": some things we can be led to SEE are true, but are not "apparently true" from the outset. in other words: there are "short-cuts to understanding".

here is an example: suppose you are told to add all the numbers from 1 to n (where n might be 1,000 for example).

well, you could take out your scratch-paper (or calculator) and start adding:

1+2 = 3
3+3 = 6
6+4 = 10
10+5 = 15

....and so on, until you got the answer.


or you could observe, we have the following pairs (supposing for simplicitly that n is even):

1+n = n+1
2+(n-1) = n+1
....
(n/2) + (n - ((n/2)-1)) = n/2 + 2n/2 - n/2 + 2/2 = (2n+2)/2 = n+1

and deduce that since we have n/2 pairs that each add up to n+1, the sum is (n/2)(n+1). now what just happened there? we found "a short-cut to truth". when you SEE it like that, you realize you could have done that from the START, instead of just blindly adding all the numbers, but it's not obvious until you see it done. in some sense "we already knew it was true", we just "never thought of it that way".

looking at problems in a new light is important: often, solution strategies that were there the whole time occur to us, that wouldn't have before. does this really count as "new knowledge"? who cares? it certainly means more EFFECTIVE knowledge.

************

to the original poster: the choice you are contemplating will most likely affect most of your adult life (where you work, where you live, which people become your friends, and so on). do not feel you have to rush it. try to take at least two mechanics courses and an E-M field course in physics, and at least up to linear algebra (and/or abstract algebra, if you can afford to wait that long) before you decide. after calculus, math starts taking on a different "flavor" (or rather, many different kinds of flavor, since it starts to "branch off") and not of all these may be to your liking. also, physics will require a lot of lab work, and "non-theoretical stuff" that may seem to be like drudgery (if not, if you love it, that's great).

in other words, do what you love to do. you need some more experience in each to be sure. the first 2 years of college are designed to give you that experience.

I was in your situation once, facing the same dilema. What solved it was the realization that there are miriads of variations of theories in physics, and, in the best case scenario, only one of them is true (talk about playing games...); while in mathematics, all you discover in a consistent axiom system is eternal. It will always be true.

Thank you very much!

I appreciate all of your comments.


Take care.
/alan