Do Mathematician understand or like Physics?

[eljose]

Do Mathematician "understand" or like Physics?..

This is an strange point of view...when staying at university Physicist and Mathematician shared a common subject "Theoretical Mechanics" including Lagrangian and Hamiltonian Approach..it surprised me a bit that several math student had some problems understanding "mechanical" concepts (energy, and so on) What,s the opinion of a mathematician about physics?..why sometimes is difficult for a mathematician to understand a physical concept?..it,s very strange since they study more complicate concepts than we usually do such us "Topology" or "Differential Geommetry"...

 

[matt grime]

The main difficulity is sociological in some sense. You say that mathematicians don't understand simple physical concepts like energy, but can you define all these simple physical things properly? Mathematics is about definitions, not about what is intuitively obvious to you. I would say topology and diff. geom. are very simple concepts (with hard results) where as in physics it is not even clear what the definitions ought to be sometimes, e.g. do we want to study topological, conformal, quantum or some combination of the three field theories.

My rejoinder would be: why is it so hard for physicists (including youreself, Jose) to make themselves understood to mathematicians? We are simple folks, just give us the definitions and we'll work it out for ourselves. And, whilst we're on it, why is it so hard, if physics is so obvious, for physicists to prove anything is reasonably correct?

 

[fourier jr]

My rejoinder would be: why is it so hard for physicists (including youreself, Jose) to make themselves understood to mathematicians? We are simple folks, just give us the definitions and we'll work it out for ourselves. And, whilst we're on it, why is it so hard, if physics os os obvious, for physicists to prove anything is reasonably correct?

let's see if I'm the 1st to post it:

Physics is much too hard for physicists. -- David Hilbert

the biggest problem I had in physics was when dealing with 2 equations involving V (velocity i guess), it was THE SAME V... they weren't two different vs. so i didn't know i could solve for one & plug it into the other equation, etc. if i could start all over again i think i would do ok in physics, but i don't think i'd find it very interesting now because it's too concrete.

 

[mathwonk]

i have trouble understanding physics as matt suggested, because the terms are never explained clearly or precisely enough. i.e. the definitions are not given. or you might say because i don't have enough common sense.

I once tried to read some famous physicists' works like pauli, einstein, etc... they would pose various problems to "prove" and i never could prove them.. then when i read their answers they would always use some hypotheses which had not been given. pauli would say something like "well since space is homogeneous.." and I would say. hey why is space homogeneous, you did not assume that!


so physicists are not above making unwarranted assumoptions anytime they help the cause. now actually this is a good habit, and one mathematicians would well adopt, since this is also a good research strategy, i.e. keep making more hypotheses until you get something you can do.

but anyway, when i was a freshman i took physics 1 and there was a problem where a simple computation would give the result except that the computation made no sense, so i spent an hour or so making sense of it and explained what i meant by it, and the grader said "you are the first person in over 100 homeworks to make sense of this!" that was my one satisfying experience in physics.

so it was not long before i dropped out, as it would have taken too long for me to make sense of all the nonsense they were asking us to do, besides it was too hard to come up with all the appropriate unnamed assumptions.


so there. does that help? we are more literal than physicists.

 

[mathwonk]

remark, the brilliant differential geometer michael spivak just wrote a set of notes called elementary mechanics from a mathematicians viewpoint, trying to make sense of things like a pendulum, or a lever. he laughed when people assumed he meant he was puzzled by string theory or quantum field theory, he wanted to know what physicists meant by the most trivial phenomena, like 2 weights balancing each other on a seesaw.

 

[Kurdt]

Interesting that two people using basically the same toolkit can misubderstand each other so much. As a physicist I found it hard to understand a lot of pure mathematical forms because to me the courses were presented in an 'aimless' manner. By aimlessI mean that things were being presented such as identities etc without any link to anything in the real world. Of course later in physics subjects the same identity or whatever would be used to explain a physical system and it would all click then. Its a strange thing when I get 50% in mathematics modules and and 85's in physics modules when the only difference is that one is applied to situations in the real world.

I suppose its the point of view you're coming from though. There must be a mathematicians way of thinking and a physicists way of thinking. I'd love to learn how to do both!

When one says definitions are not given I suppose its usually that the definition is something a physicist would work with commonly and knows anyway without reference to a text or anything else, or its something that can be easily derived by considering the physical model that's being decribed but i can see why that would frustrate you guys. Apologies on behalf of physicists everywhere :).

 

[DeadWolfe]

I should point out that there certainly are some mathemeticians I know who, despite being highly comptetent in pure mathematics, nonetheless enjoy physics.

 

[alias25]

I get so frustrated with physics sometimes but I love Physics! and I can do maths which I enjoy lots, sometimes more than physics (I think I am more of an astronomer than physicist, I don't know, I am confused.)
Physics and Maths rock!
ok I am a complete geek i think everyone got that picture..hmm.

 

[mathwonk]

re: post #6: i agree, this is the distinction between being sure a statement is correct, and understanding what it means. as mathematicians we do not care if the statement is totally abstruse as long as it is consistent with the hypotheses, no matter where they came from.

i.e. in math we try to get things technically correct, and in physics they try to get a meaningful answer that explains the phenomena, even if the argument is logically fuzzy and technically wrong. I.e. a physicist believes that if the conclusion agrees with the real world data, then the reasoning must be right in some cosmic sense. and eventually the mathematicians will figure out why.

actually history teaches that the physicists, being guided by the real world example, are as likely or more so to come to the correct conclusion than the mathematicians who rely on logic.

 

[Kurdt]

Yeah I loved my quantum mechanics course where dirac just invented his delta function and pure mathematicians had a heart attack wondering what on Earth it was.

 

[mathwonk]

you wil notice that many of my posts here are physicist like, in that i try to explain what the mathematics means viscerally rather than what the symbols mean, or why the computations are correct.e.g. when someone asked elsewhere about gradients versus tangent vectors, i tried to explain the temperature distribution in the vicinity of a radiant source as viewable either as flow lines of heat radiating from the source (gradient), or as spherical regions of constant temperature surrounding the source (tangent planes to spheres). but the poster wasn't having any and ignored me.

 

[mathwonk]

recall that lefschetz, one of the great mathematicians, was originally an engineer, until he burned his hands off in an accident. then he became a wonderful topologist. mostly famous for his insightful ideas and techniques, since many of his actual arguments were wrong. but his theorems tended to be right, and to shed huge light on geometric problems.

 

[mathwonk]

for example i am posting some comments on cayley hamilton theorem in the linear algebra forum, concerning a proof that is too good to be true. i.e. it is trivial, but hard to make sense of. but it is so simple, a physicist should realize that it must be correct from some point of view, and it is up to us to find the right point of view.

mathematicians on the other hand are so uptight about precision, that we go to extreme lengths to give an undeniably correct proof, even if it is tedious, lengthy, and unnatural.

i.e. cayley hamilton says, if chA(X) is the characteristic polynomial of A, then chA(A) = 0. how could anything be simpler? it must be true for a simple reason.

but just try to find a proof of cayley hamilton in a math book that both explains why it is true, and is simple. but apparently the goal of finding such a proof is not important to most math textbook writers.


maybe the easiest proof is in artin and essentially bourbaki, i.e. the theorem is true for diagonalizable matrices, and those are dense in all matrices, so it is true for all matrices. but the density statement requires a little work.

the thing that bothers me is that most books give the cramers rule formula that immediately implies cayley hamilton, but do not explain this fact.

 

[3trQN]

i.e. cayley hamilton says, if chA(X) is the characteristic polynomial of A, then chA(A) = 0. how could anything be simpler? it must be true for a simple reason.

but just try to find a proof of cayley hamilton in a math book that both explains why it is true, and is simple. but apparently the goal of finding such a proof is not important to most math textbook writers.

Would the simplest proof come from calculus?

 

[matt grime]

Almost certainly not, not even with the widest interpretation of the terms. I doubt any proof will be supplied purely by considering analytic things.

 

[Dragonfall]

Mathematicians may be uptight about proofs, but physicists are even more so about lab reports.

 

[mathwonk]

the simplest proof i know is this: if f(X) is the characteristic polynomial of the matrix A, then f(X).I = (X.I-A).g(X) where g is the classical adjoint of (X.I-A). (cramers rule)

then the non commutative root factor theorem says that since since (XI.-A) is a left factor of f(X), then A is a left root of f. I.e. f(A) = 0.

but how many of us have seen the non commutative root factor theorem?