Are proofs essential in understanding mathematics in physics?

[Schrodinger's Dog]

I've noticed a lot of mathematics on this forum revovles around proofs, but have not really come across any in depth proofs as such and so am unfamilliar with much of the topics discussed here, and in fact some I can't really follow.

I got to thinking though at which point does it become required to see the proof of something in physics in general, for example I can quite easilly understand how a differential works from looking at say the proof for x^2, which is simple to follow when broken down. But I'm not entirely sure I need to prove much of the mathematics I'm learning and it's fine once you know the underlying principles and simple proofs to just assume it works in all cases.

What I'm really asking is how important are proofs of maths in physics, and at what point does it become important to really know how to prove say why binomial distibution works. Or is it only really necessary at the highest levels to have a deep understanding and at lower levels it's more important to be able to fluently use the mathematical constructs. The reason I ask is I find myself asking why it is necessary to prove for example that $$nx^{n-1}$$ works in all cases when simply applying it and knowing how generally differentials work is enough?

Is their a deal of proofs in the learning of physics at say degree level or is this approached more in mathematics?

 

[arildno]

Why is it so horrible to actually understand what you are doing?
Why this fear of logic?

 

[Schrodinger's Dog]

Why is it so horrible to actually understand what you are doing?
Why this fear of logic?

You misunderstand I am only asking whether it's more important to concentrate on the method early on, and when it becomes more important to look at proofs, which frankly with the level I'm at I only need a cursory understanding of. For example I need to know why X^2 = 2x and how differentials work but I do not need to rigorously prove that all the tabled differentials work, it may well be better to get to grips with the meat of calculus before really getting deep in, if in fact I ever need to go that far.

 

[arildno]

Well, I'd say that to understand the general proofs of calculus are more important than being able to make up proofs for specific function choices.

For example, if you know how to prove for general functions f and g why the chain rule holds,
then that is the important thing.

In general, I'd say that from the viewpoint of maths, to understand existence proofs are more important than finding an actual value that works (unless you're in the business of doing tight estimates, of course.)

 

[ssd]

In my opinion, when you are looking into some problem with solution unknown, say a reserch problem, you have to keep proofs of proven things at back of your mind, and keep their results floating at the forefront of the mind. Ultimately an unknown chemistry of thinking process will lead you to solution. It is a fact that alternative cycles of heavy thinking and recess works the best.
On the contrary of what you posed, I think the process of rethinking on mere trifles gives a clear vision of the subject.

 

[Schrodinger's Dog]

Well, I'd say that to understand the general proofs of calculus are more important than being able to make up proofs for specific function choices.

For example, if you know how to prove for general functions f and g why the chain rule holds,
then that is the important thing.

In general, I'd say that from the viewpoint of maths, to understand existence proofs are more important than finding an actual value that works (unless you're in the business of doing tight estimates, of course.)

Yes I looked at why the chain theory works in Hottenany's thread about the basics of differentiation(kudos for that btw, I'm not ebntirely sure I followed it all, but it was very well written none the less)

Actually there is a good reason for this question, I'm between courses at the moment and so that I don't get out of practice I've been busy swatting and refining what I learn't in the last year, when I came across a lot of proofs while looking through the maths forum, I began to worry that maybe I was focusing too much on just plugging in numbers and following the rules of algebra and calculus and not enough on worrying about why generally they work, then when I though about it I decided I am not that well practiced and perhaps I need to focus on refining my skills before I get into the why's and wherefores beyond why generally y=mx+c type proofs of differentials, or the area under the curve proofs of integrals which I already know the basic concepts of anyway.

After having a conversation with someone he said he never really got into the proofs behind a lot of the maths when he studied physics back in the fifties, he was more interested in understanding and application than why it worked in general, and it was the mathematicians who were more rigorous about the foundations of maths? So I thought Ok, I do think it's important to get invovled with why something works, but I see what your saying let's not go overboard and lose focus on what we're looking into, so theirs a fine line between getting to carried away with why something works and not analysing how it may be applied more practically.

So I came up with the notion, perhaps erroneously that those who study physics might often be more interested in application than pure maths. Where as the mathematicians may love rattling through problems and finding rules. The question is which is it more important to do for a physics student or are they more or less of equal value? Is my friend right that physics is less geared towards rigorous pure proofs than perhaps mathematics? He told me a lot of students at university were told to concentrate more on applications than into spending too much time on proofs, has this changed and if so why might that be?

Sorry I'm waffling a bit but I hope you get what I mean? Where's the fine line?

 

[arildno]

Please note that hardly any physicist that has made significant contributions to their fields has been "suckers" at the math.

Their "lack" of rigour is far more akin to the lack of rigour in mathematicians who has a great "intuition", than with their inability and lack of appreciation of the importance of pure maths.

Sure you can become a competent computer who is good at applications but not proofs, but you won't ever become an inventive and great physicist by this method .

 

[Schrodinger's Dog]

Please note that hardly any physicist that has made significant contributions to their fields has been "suckers" at the math.

Their "lack" of rigour is far more akin to the lack of rigour in mathematicians who has a great "intuition", than with their inability and lack of appreciation of the importance of pure maths.

Sure you can become a competent computer who is good at applications but not proofs, but you won't ever become an inventive and great physicist by this method .

I see, agreed that giants were also very well versed in mathematics, are you a mathemetician or a phsysisist though? And if your a physisist at what point did you start to really rigorously go into the pure maths of pure maths?

I have left it too late to do anything groundbreaking in physics, to be honest I'll be more than happy with a degree, if I was twenty one I might consider studying it further but frankly I haven't got all the time in the world(anyway that's not the issue)

What I'm trying to find in my mangled way is what is a good balance between proof and application for a student? I'm sure focusing almost entirely on mathematical rigour would be a waste of time. As would just learning how to be a machine as you put it. Perhaps this should be moved to the academic and career guidance sections? I was kind of interested in finding out how physics is taught at degree level now though as well, and whether there is still less of an onus on mathematical proofs or whether it had slid the other way? I'm sure you can see why this is important, when I do start the degree I want to know what I should be paying particular attention to and what I shouldn't. What sort of importance general rules have in the broader scope and what sort of things I can leave aside to advance my physics knowledge about the concepts and the maths rather than concentrating too much on why $$\pi = 3.14$$ if you see what I mean.

 

[Werg22]

I think it depends on the use of the mathematics you are learning. If all you need is the application, then proofs are not of paramount importance. However, if you're into more theoretical disciplines, proofs are as important as the subject itself. Proofs allow you to view the concepts from angles in a way that is essential for thorough understanding. If I had to nail it, I’d say proofs strengthen your bases.

 

[robert Ihnot]

Werg22: I’d say proofs strengthen your bases.

I think Werg22 has got a very good point. Clearly people like Hawking and Einstein had to know a lot of math. However, for some purposes that is not the case. Mathematicians like Euler and DeMovier were more interested in results than any kind of rigor, which if it really was needed was an after thought. Remember Newton told Halley that he had found the path of a comet, which surprised Halley, who told him to publish--even at Halley’s expense. Thus was born Principia. On the other hand, the inventor of the Faucault pendulum wasn’t much at school work, but he was a very good experimenter. Similarly with the gyroscope, I don't think any math had been worked out on that, but Faucault was able to construct one. Then there was the eccentric Cavantage, who feared the sight of a
woman, who was terribly rich and died richer, while spending all his time alone on scientific experiments, the principal one being "The weighing of the Earth."

So the idea is probably to learn what you can, find out what you do best, and take it from there.

 

[matt grime]

Don't confuse physics or maths as was practised by Newton and Euler, with that of today's scientists. No mathematician is going to be allowed to get by without proper proof, and no physicist would be allowed to shove wooden spikes in his eyes to find out about the spectrum of light. And the way modern funding works, you ain't going to get very far if you can't justify why the research you want to perform is worth doing, and there might be some maths involved in that (verifying it being the experiment).

There are all manner of physicists, and mathematicians. I don't know how many of either would say that mathematical proof is important in physics. But there is a difference between proof and understanding, though the former helps the latter. It is sufficient to know 'what something is getting at' without proving it properly.

As an example, if f(x) is a fuunction, then all you ought to care about its derivative is that it is a function f', that satisfies

f(x+e)=f(x)+ef'(x) + terms quadratic and higher in e.

It is thus a linearization about x. It is then straight forward to see what the derivative of x^n is and why - it is just the binomial expansion of (x+e)^n.

It is also easy to justify the product rule, chain rule, and quotient rule with this without getting too bogged down with showing things precisely converge to zero at the required rate.

Please note these are derivatives, not differentials. Differentials are entirely different.

 

[HallsofIvy]

In fact, don't confuse physics with mathematics!

" What I'm really asking is how important are proofs of maths in physics."

The answer- not very. If you understand the proof of a mathematical statement that may help you think about how it applies to "non-standard" situations. In terms of learning physics, that is given in the physics books. Some physics students may not like being told "this is how that mathematics is applied" without being told precisely why and so turn to mathematics texts (more power to them!) but that is not necessary. A research physicist might wish to know whether or not some abstuse field of mathematics can be applied to a specific area of physics (like differential geometry to relativity or group theory to quantum theory) but he is more likely to walk down the hall to the mathematics department and ask for advise than to try to learn the mathematics himself. The mathematicians need to know the proofs, the physicists need to know the results.

 

[MathematicalPhysicist]

The mathematicians need to know the proofs, the physicists need to know the results.

that's an ugly picture you're drawing here, but it's true for the most part of students, but i cannot say the same for those who have earned a degree in mathematics and physics.
and because i also learn physics i see this in a course i have in physics which is called introductory mathematics for physicists1, i still haven't learned calculus 3, which is concerned with triple integrals, double integrals, green, stokes theorem etc, and i already need to calculate tripled integrals, i can easily confirm your assertion, it's hard to calculate those without a rigourous treatment of the subject.
the problem is you also learning mechanics which for them you need to use those tools.
pitty, pitty )-:

 

[matt grime]

ti still haven't learned calculus 3, which is concerned with triple integrals, double integrals, green, stokes theorem etc, and i already need to calculate tripled integrals, i can easily confirm your assertion, it's hard to calculate those without a rigourous treatment of the subject.

I doubt anything could be further from the truth. Doing an integral is purely mechanical, and needs no rigorous understanding of analysis. (Unless one wants to use complex integrals and residue theorems, parhaps).

 

[MathematicalPhysicist]

i have to reject your assertion, perhaps the techniques of integration is mechanical, but for example you need to calculate the volume inscribed between two bodies, it's hard to find the limits of the integral you want to find, and in class from no where the lecturer starts taliking about taking the minimum out of the two functions of the bodies, without discussing rigourously why.

 

[matt grime]

i have to reject your assertion, perhaps the techniques of integration is mechanical,

absolutely - everyone is entitled to their opinion

but for example you need to calculate the volume inscribed between two bodies, it's hard to find the limits of the integral you want to find, and in class from no where the lecturer starts taliking about taking the minimum out of the two functions of the bodies, without discussing rigourously why.

And here we see why it is 'opinion'. You consider that to be some kind of rigorous treatment. I don't. Rigorous would be some proofs like every monotone function is integrable, indeed, what a Riemann integral is. What you have written is not something I would even consider as needs to be proved.

 

[HallsofIvy]

This may be a difference in interpretation of the word "rigorous". Certainly, I was not saying that prospective physicists should not learn calculus in detail. What I was saying was that physicists do not need to learn the details of Lesbesque measurable functions, for example, or other intricacies of mathematical analysis. They have enough to do, for God's sake, learning physics! Which, as I said before, will include learning some linear algebra, differential geometry, and group theory. Not just memorize formulas: learn the basice concepts but not necessarily learn how to prove all of their properties.