Nobody complains about physicists' math?

[jostpuur]

One might think that you can find anything on the internet, but I haven't found any site where somebody would be complaining about physicists' way of using mathematics. I wonder why. Wouldn't physicists math be an easy thing to make fun of?

 

[Mororvia]

If it describes reality, why would it be made fun of?

 

[zoobyshoe]

Wouldn't physicists math be an easy thing to make fun of?

http://www.aip.org/history/einstein/images/ae17.jpg

"What do you mean, funny? Let me understand this cause, I don't know maybe it's me, I'm a little ****ed up maybe, but I'm funny how? I mean, funny like I'm a clown, I amuse you? I make you laugh... I'm here to ****in' amuse you? What do you mean funny, funny how? How am I funny?"

 

[Jimmy Snyder]

You mean like 'Quantum physics is classical physics in the limit as zero approaches h?'

 

[mgb_phys]

In physics 0/0 or infinity/infinity cancels - it's obvious!

 

[jostpuur]

This (or things like this) happened often in physics lectures. A physicists wants to use the derivative rule of composite functions

<br /> D_u f(x_1(u), x_2(u)) = (\partial_1 f) x&#039;_1(u) + (\partial_2 f) x&#039;_2(u)<br />

but of course he wouldn't use already proven simple rigorous theorem. Instead he does this with with "differentials". First the lecturer assumes that this is clear

<br /> d f = \frac{\partial f}{\partial x_1} dx_1 + \frac{\partial f}{\partial x_2} dx_2<br />

and then divides by du, and gets

<br /> \frac{df}{du} = \frac{\partial f}{\partial x_1}\frac{dx_1}{du} + \frac{\partial f}{\partial x_2} \frac{dx_2}{du}<br />

which is the desired result.

Q: What precisely is the dx and du?

A: Well they are some kind of infinitesimal quantities, but there's no need to be rigorous here, because this is physics.

Q: Why was the result derived like that?

A: Well mathematicians would have probably done this more complicatedly with epsilons and deltas, but a simpler proof such as this is sufficient for us, because this is physics.

 

[Gilligan08]

http://www.aip.org/history/einstein/images/ae17.jpg

"What do you mean, funny? Let me understand this cause, I don't know maybe it's me, I'm a little ****ed up maybe, but I'm funny how? I mean, funny like I'm a clown, I amuse you? I make you laugh... I'm here to ****in' amuse you? What do you mean funny, funny how? How am I funny?"

Goodfellas? What movie is that from?

 

[jostpuur]

The conclusion

<br /> f(x)=1+Ax+O(x^2)\quad\implies\quad f(x)=e^{Ax}<br />

is among the most unbelievable ones.

For example when you solve a quadratic equation, you can write the expression

<br /> x^2 + Ax<br />

in a form

<br /> (x+A/2)^2 - A^2/4.<br />

Here you write the old expression in a different new form. Similarly, once a particle physicist is given a function

<br /> f(x)=1 + Ax+O(x^2),<br />

he can write it in a form

<br /> f(x)=e^{Ax}.<br />

 

[ZapperZ]

This (or things like this) happened often in physics lectures. A physicists wants to use the derivative rule of composite functions

<br /> D_u f(x_1(u), x_2(u)) = (\partial_1 f) x&#039;_1(u) + (\partial_2 f) x&#039;_2(u)<br />

but of course he wouldn't use already proven simple rigorous theorem. Instead he does this with with "differentials". First the lecturer assumes that this is clear

<br /> d f = \frac{\partial f}{\partial x_1} dx_1 + \frac{\partial f}{\partial x_2} dx_2<br />

and then divides by du, and gets

<br /> \frac{df}{du} = \frac{\partial f}{\partial x_1}\frac{dx_1}{du} + \frac{\partial f}{\partial x_2} \frac{dx_2}{du}<br />

which is the desired result.

Q: What precisely is the dx and du?

A: Well they are some kind of infinitesimal quantities, but there's no need to be rigorous here, because this is physics.

Q: Why was the result derived like that?

A: Well mathematicians would have probably done this more complicatedly with epsilons and deltas, but a simpler proof such as this is sufficient for us, because this is physics.

1. Does this violate any part of mathematics, within the confines of what it is being used for?

2. Does this give a consistently correct description of the system it is describing?

Zz.

 

[jim mcnamara]

I think somebody's cage got rattled, hmm, Zz?

 

[ZapperZ]

Someone is in a cage?

Zz.

 

[Integral]

AFAIK nearly all mathematicians laugh at the way Physicists do math. Of course, not nearly as hard as they laugh at the way engineers do math.
But this is so unsurprising that it is not discussed outside of Math dept coffee rooms.

 

[Mk]

AFAIK nearly all mathematicians laugh at the way Physicists do math.

Hey, physicists are the ones that always have to draw pictures before they can understand a problem

 

[zoobyshoe]

Goodfellas? What movie is that from?

Yeah, That's Newton doing the Joe Pesci line from "Goodfellas".

 

[robphy]

AFAIK nearly all mathematicians laugh at the way Physicists do math.

Maybe that's why some of us are physicists.
Somehow... maybe with luck or maybe with physical intuition... in spite of the physicist's sloppy and impatient mathematics, the physicist often gets the answer correct. (Certainly there are cases when the mathematics is just plain wrong and leads to nonsense... for [almost] everybody.)

I have encountered [overly-]mathematical colleagues who are too wrapped up in the math that they miss the physics under discussion.

I certainly appreciate the mathematical physicist who can use enough mathematics to show the physics [and its abstract structure]... but who is able [when pressed] to deliver the mathematical details.

 

[FredGarvin]

AFAIK nearly all mathematicians laugh at the way Physicists do math. Of course, not nearly as hard as they laugh at the way engineers do math.
But this is so unsurprising that it is not discussed outside of Math dept coffee rooms.

Yeah. And we laugh at mathematicians at the way they try to talk to girls.

 

[Chi Meson]

Yeah, That's Newton doing the Joe Pesci line from "Goodfellas".

Wow, Newton's pretty good. Has he been in any movies himself? Didn't he host SNL way back?

 

[zoobyshoe]

Wow, Newton's pretty good. Has he been in any movies himself? Didn't he host SNL way back?

He used to work the Catskills every summer, and had an off-Broadway one man show one season that Johnathan Demme was interested in translating to film, but the deal fell apart in negotiation due to Newton's temper.

He does a great Brando, too:

http://www.corrosion-doctors.org/Biographies/images/Newton.jpg

"Bonasera. Bonasera. What have I ever done to make you treat me so disrespectfully? If you had come to me in friendship then this scum that ruined your daughter would be suffering this very day. And if by chance a man like yourself should makes enemies then they would become my enemies. And then they would fear you."

 

[G01]

In physics 0/0 or infinity/infinity cancels - it's obvious!

I think I actually canceled infinity over infinity once on a optics test, and got full credit!

 

[NeoDevin]

to find a flux per unit area through an infinite plane we can do

\frac{1}{A}\int_S\Phi da

Where S is the entire infinite plane, and A is it's area.

 

[Pythagorean]

so is there a mathematical proof for why 0! = 1?

 

[animalcroc]

One might think that you can find anything on the internet, but I haven't found any site where somebody would be complaining about physicists' way of using mathematics. I wonder why. Wouldn't physicists math be an easy thing to make fun of?

I think that's way overrated.
"rigor" in math came after the giants who developed modern math in the first place, and they're "giants".

 

[Dragonfall]

Of course, not nearly as hard as they laugh at the way engineers do math.

I don't laugh at the way they do math. I'm terrified at it. Think about it the next time you're crossing a bridge or ride an elevator.

 

[Thrice]

Ah physics, wherein every function equals the first term of its taylor series.

 

[ice109]

so is there a mathematical proof for why 0! = 1?

http://www.adonald.btinternet.co.uk/Factor/Zero.html

Ah physics, wherein every function equals the first term of its taylor series.

most annoying thing ever

 

[Hurkyl]

This (or things like this) happened often in physics lectures. A physicists wants to use the derivative rule of composite functions

<br /> D_u f(x_1(u), x_2(u)) = (\partial_1 f) x&#039;_1(u) + (\partial_2 f) x&#039;_2(u)<br />

but of course he wouldn't use already proven simple rigorous theorem. Instead he does this with with "differentials". First the lecturer assumes that this is clear

<br /> d f = \frac{\partial f}{\partial x_1} dx_1 + \frac{\partial f}{\partial x_2} dx_2<br />

and then divides by du, and gets

<br /> \frac{df}{du} = \frac{\partial f}{\partial x_1}\frac{dx_1}{du} + \frac{\partial f}{\partial x_2} \frac{dx_2}{du}<br />

which is the desired result.

Q: What precisely is the dx and du?

A: Well they are some kind of infinitesimal quantities, but there's no need to be rigorous here, because this is physics.

Q: Why was the result derived like that?

A: Well mathematicians would have probably done this more complicatedly with epsilons and deltas, but a simpler proof such as this is sufficient for us, because this is physics.

If one of my mathematical colleagues wrote that derivation, I would have considered it perfectly rigorous -- it's an elementary manipulation of differential forms, coupled with a typical but harmless¹ ambiguity in notation. Alas, it's a seems that the scientific community has a strong disdain for mathematican sophistication, and you're stuck with rubbish explanations like this.


1: At least, it's harmless if you know what's going on...

 

[Hurkyl]

Similarly, once a particle physicist is given a function

<br /> f(x)=1 + Ax+O(x^2),<br />

he can write it in a form

<br /> f(x)=e^{Ax}.<br />

If f(x)=1 + Ax+O(x^2), then it is exactly right that f(x)=e^{Ax}+O(x^2).

 

[jostpuur]

If one of my mathematical colleagues wrote that derivation, I would have considered it perfectly rigorous -- it's an elementary manipulation of differential forms, coupled with a typical but harmless¹ ambiguity in notation. Alas, it's a seems that the scientific community has a strong disdain for mathematican sophistication, and you're stuck with rubbish explanations like this.1: At least, it's harmless if you know what's going on...

This trickery is irrational, because the equation

<br /> df = \frac{\partial f}{\partial x_1} dx_1 + \frac{\partial f}{\partial x_2} dx_2<br />

comes out from nowhere. If you want to use mathematics as a tool, why not just take the chain rule as it is, and then use it? The physicists could also merely write

<br /> \frac{df}{du} = \frac{\partial f}{\partial x_1} \frac{dx_1}{du} + \frac{\partial f}{\partial x_2} \frac{dx_2}{du}<br />

and say "this is a known result, and we can use it".

Why start with something else, and then do some kind of pseudo proof for the chain rule? And why insist, that this pseudo proof was the easier way? That's basically deriving the derivative out of Taylor series.

 

[jostpuur]

If f(x)=1 + Ax+O(x^2), then it is exactly right that f(x)=e^{Ax}+O(x^2).

The physicists can also get the precise

<br /> f(x)=e^{Ax}<br />

from this. The thing is, that if some expression involves exponential function, there's no way preventing physicist somehow messing with it.

Suppose a physicist wants to postulate, that some spinor transforms, in rotations, according to action given by matrices

<br /> e^{-i\theta\cdot\sigma/2}<br />

The physicist will not postulate this directly (because then everything would still be rigor). Instead he postulates that the transformation matrices are

<br /> 1 - \frac{i}{2}\theta\cdot\sigma + O(|\theta|^2)<br />

and then derives the full transformation, and does the derivation somehow wrong, but gets the desired result.

I don't understand what is achieved by this.

 

[Jimmy Snyder]

so is there a mathematical proof for why 0! = 1?

It's defined that way for convenience, not as the result of any proof. Surely a mathematician would be justified in laughing at someone who tried to prove definitions.

 

[ZapperZ]

This trickery is irrational, because the equation

<br /> df = \frac{\partial f}{\partial x_1} dx_1 + \frac{\partial f}{\partial x_2} dx_2<br />

comes out from nowhere. If you want to use mathematics as a tool, why not just take the chain rule as it is, and then use it? The physicists could also merely write

<br /> \frac{df}{du} = \frac{\partial f}{\partial x_1} \frac{dx_1}{du} + \frac{\partial f}{\partial x_2} \frac{dx_2}{du}<br />

and say "this is a known result, and we can use it".

Why start with something else, and then do some kind of pseudo proof for the chain rule? And why insist, that this pseudo proof was the easier way?

But you still haven't told me what is WRONG with it. All you have done is argue it based on a matter of TASTES.

That's like arguing that one shouldn't use a screw driver to open the lid of a can of paint, because the screw driver was designed to be used in a certain way. You seem to forget the important point here : it works!

That phrase "it works" has always been severely undermined. Yet, it is THE most powerful argument there is. As long as the usage of the "tool" does not break any "laws" (I didn't use the screw driver to murder someone to get to that paint can to paint my house), then the claim that it works validates its usage. That is why I asked you what is mathematically wrong with it. I'm not talking about "canceling" 0/0, which would be breaking mathematical "laws", and in fact, results in something that doesn't work (look at the Fraunhoffer diffraction pattern). I'm talking about shortcut in notations that you are highlighting here.

Most physicists use mathematics as a tool, whether you like it or not. We need to know what the tools are, and how to use it correctly within its limits of validity. Once we know that, HOW we use it really shouldn't be the sore point of mathematicians. Physicists really do not have the patience nor the inclinations to focus on the "tools". If we do, what's the use of mathematicians?

In the experimental facility that I work at, we have this stainless steel plate that's mounted on a low ceiling by 4 bolts. The ends of the bolts stick down from the plate. While it is high enough for most of us not to hit it, someone around 6' tall or taller could hit his/her head on it. The safety regulation requires us to do something about it, and this could include shaving the bolt to a shorter length, putting in a cover over the whole contraption, etc... But we came out with something easier. Puncture 4 tennis balls, and stick the protruding ends of the bolt into the tennis balls. The bright, fluorescent color provides the advance warning to anyone approaching that area, and even if someone hits his/her head on it, it would not hurt. It was a quick, easy, and CHEAP solution to a problem. Yet, we had used something for what it wasn't meant to be used. Even our safety inspector was impressed. Why? Because IT WORKS!

The ability of physicists to adapt the mathematics to suit their needs, without really violating any mathematical laws, shows their creativity and imagination to solve the problem at hand. I'm sure this is done in many other fields as well, especially in engineering. I really don't understand why this would be a subject of ridicule or to be laughed at. In fact, I would think that the ability to take something and use it in a different manner while still maintaining its validity, is something that should be admired.

Zz.

 

[Jimmy Snyder]

Ah physics, wherein every function equals the first term of its taylor series.

The sum of the first and second terms no?

 

[ZapperZ]

It's defined that way for convenience, not as the result of any proof. Surely a mathematician would be justified in laughing at someone who tried to prove definitions.

Are you sure that this is not derived via the Gamma function?

Zz.

 

[Jimmy Snyder]

My favorite story about such laughings comes from Feynman (who else?). I'm mangling his story, but here goes:

A small group of math majors was looking at a french curve and pondering its shape. A french curve is a piece of plastic (or other material) with an edge that looks somewhat spirally. It's used by draftsmen in drawing curved lines. Because of the spirally edge, it approximates arcs of any radius in a range. The mathematicians knew this, but pondered why it had the particular shape that it had. Was it optimal in some way. Feynman told them that this particular shape had the following property. In no matter what orientation you hold it, the tangent at the bottom was parallel to the ground. The math majors satisfied themselves that this was true and thanked him for his help.

What he told them would be true of any curve. That's the fundamental theorem of calculus, that the slope of a curve at an extremum is zero.

 

[Jimmy Snyder]

Are you sure that this is not derived via the Gamma function?

Zz.

Yes. 0! is defined to be 1. The Gamma function is a late comer.

 

[Hurkyl]

The factorial function is what it is. What is "definition", "derivation", "calculation", "demonstration", "theorem" or whatever is merely an artifact of the way its being presented.

Axiomizations for theories are like spanning sets for vector spaces -- they are useful for presentation and calculation, but otherwise irrelevant. (In fact, both are examples of the more general notion of a "set of generators")

(The above is for a developed theory. Research is a different story)

 

[yasiru89]

Do physicists often get their answers correct? Funny I'd expect a remotely tangible and unified theory of reality if that were the case.
There is little unification in physics which on the other hand is a strong point of mathematics.

Indeed physicists and engineers can be amusing in their use of mathematics. For the trivialties if nothing else. However around the 1900s any pure mathematician would have been appalled by the manner physicists used mathematics just as rich new ideas were being elaborated upon. Hilbert sought to change this and it is through his part as much as any other(Poincare, von Neumann, etc.) that we have such stable(if not all that absolutely descriptive) theories of quantum phenomena, etc.
In physics, particularly elementary physics rigour is given second place to something they call intuitive obviousness. This leads to indecent representation of approximation(where such simple things as the constant use of the equality sign in the place of approximation make me want to throw up) and lax(to the point of being incorrect) definitions.
The importance of mathematics in the proverbial 'equation' is when we finally understand a significant part of a certain theory we can have a rigorous mathematical representation of it.
Mathematics is a judge of our understanding of something regardless of what an intuitive grasp may mean

 

[NoTime]

I don't laugh at the way they do math. I'm terrified at it. Think about it the next time you're crossing a bridge or ride an elevator.

Things don't fall down all that often
Most of all that is improper maintenance.

Aren't you glad that physicists have nothing to do with the operation of the universe

 

[ZapperZ]

Do physicists often get their answers correct? Funny I'd expect a remotely tangible and unified theory of reality if that were the case.
There is little unification in physics which on the other hand is a strong point of mathematics.

Indeed physicists and engineers can be amusing in their use of mathematics. For the trivialties if nothing else. However around the 1900s any pure mathematician would have been appalled by the manner physicists used mathematics just as rich new ideas were being elaborated upon. Hilbert sought to change this and it is through his part as much as any other(Poincare, von Neumann, etc.) that we have such stable(if not all that absolutely descriptive) theories of quantum phenomena, etc.
In physics, particularly elementary physics rigour is given second place to something they call intuitive obviousness. This leads to indecent representation of approximation(where such simple things as the constant use of the equality sign in the place of approximation make me want to throw up) and lax(to the point of being incorrect) definitions.
The importance of mathematics in the proverbial 'equation' is when we finally understand a significant part of a certain theory we can have a rigorous mathematical representation of it.
Mathematics is a judge of our understanding of something regardless of what an intuitive grasp may mean

Then maybe you'd like to tackle the mathematics of many-body physics without using any kind of intuition whatsoever and no amount of approximation. When you can do that and derive something similar to the Fermi Liquid theory, or superconductivity, then maybe what you said here have some validity.

Zz.

 

[Hurkyl]

Then maybe you'd like to tackle the mathematics of many-body physics without using any kind of intuition whatsoever and no amount of approximation. When you can do that and derive something similar to the Fermi Liquid theory, or superconductivity, then maybe what you said here have some validity.

Zz.

Maybe I'm just tired, but this doesn't sound like it relates at all to what yasiru89 said. He didn't say that one shouldn't approximate -- he's complaining about the practice of not acknowledging that an approximation was used. Nor did he say that one shouldn't use one's intuition.

 

[ZapperZ]

Maybe I'm just tired, but this doesn't sound like it relates at all to what yasiru89 said. He didn't say that one shouldn't approximate -- he's complaining about the practice of not acknowledging that an approximation was used. Nor did he say that one shouldn't use one's intuition.

Er.. but that doesn't make any sense either. Where exactly are such approximation not mentioned? Just because some sloppiness in substituting an approximation into an equality? That's it?

I think we need to give these physicist A BIT of credit in terms of intelligence to know when such approximations are being made. When I read about the "mean field theorem", I know perfectly well what kind of approximation is being made, even when the potential is written as an equality.

If this is what is making someone cringe to the point of throwing up, I think the problem here is elsewhere and not with the physics/physicists.

Zz.

 

[yasiru89]

It seems necessary to spell out 'definition' to most in this forum- it is key to basis and hence understanding.
Zz, Hurkyl- what jostpuur is trying to say is: 'did the physicists pull the infinitesimals out
of their a&$#?'
A combinatorial proof of 0! = 1 is elementary and even obvious. Therein it justifies itself as more than a convenience. Think how many ways you can pick 0. Time to hit the books again. The fact that this coincides perfectly with the generalised form- the Gamma function merely establishes a connexion that allows Gamma to encompass the factorial.
When Hurkyl says 'the factorial is what it is' what the hell does he mean? Or is that unimportant?
When validities are overstretched your 'tools' need attention and lack of it is by no means 'creative', it is a blunder of extreme proportions.
Mathematicians really have tried, by rectifying certain results by reducing them to the rigorous theory of limits(most of those that failed have been rejected) and bringing a representation theory basis using Regularisations to the renormalisations, but while these may be exceptional cases(the uses in number theory and groups) most others aren't really our problems.

 

[yasiru89]

I find finesse and subtlety lost. If this is the manner in which physicists treat things aren't ALL your efforts wasted if the universe(or multiverse) were a subtle creature?

 

[ZapperZ]

It seems necessary to spell out 'definition' to most in this forum- it is key to basis and hence understanding.
Zz, Hurkyl- what jostpuur is trying to say is: 'did the physicists pull the infinitesimals out
of their a&$#?'
A combinatorial proof of 0! = 1 is elementary and even obvious. Therein it justifies itself as more than a convenience. Think how many ways you can pick 0. Time to hit the books again. The fact that this coincides perfectly with the generalised form- the Gamma function merely establishes a connexion that allows Gamma to encompass the factorial.
When Hurkyl says 'the factorial is what it is' what the hell does he mean? Or is that unimportant?
When validities are overstretched your 'tools' need attention and lack of it is by no means 'creative', it is a blunder of extreme proportions.
Mathematicians really have tried, by rectifying certain results by reducing them to the rigorous theory of limits(most of those that failed have been rejected) and bringing a representation theory basis using Regularisations to the renormalisations, but while these may be exceptional cases(the uses in number theory and groups) most others aren't really our problems.

If it is a blunder of extreme proportions, and it is so common that it is making you sick, then you should, by now, have a ton of papers and rebuttals to all those physics papers that actually are based on such mathematical errors. Am I correct?

Or are these, as I've said before, not really mathematical errors, but rather simply a matter of TASTES? Again, all I've seen in this thread is not really actually mathematical errors, which as physicists, even we don't want to do, but rather sloppiness in usage or notations. This is what is making you want to throw up? Really?!

If these are mathematical errors, and I'm a mathematician, I would have not waited a second to write a rebuttal to such things. Look at the Laughlin wavefunction in his Nobel Prize winning PRL paper on the fractional quantum hall effect and I can point out to you several "approximations" that he made based nothing more than intuitions. And yes, he wrote those as "equality", thank you. I'd like to see you write a rebuttal arguing to the "extreme blunder" in that paper.

Again, as I've suspected, my earlier argument that IT WORKS, is completely being ignored, as if that means absolutely nothing.

Zz.

 

[Hurkyl]

The "blunder of extreme proportions" is the general disdain for any sort of mathematical sophistication. Frankly, I'm dumbfounded that you would actually praise someone who gave a lecture like the one jostpuur described. This isn't "adapting mathematics to suit their needs" -- this is "those weirdos down the hall have known all about this stuff for a long time, but by golly I'm going to make you guess at what's going on..." and quite probably "...because I don't know what's going on".

Again, as I've suspected, my earlier argument that IT WORKS, is completely being ignored, as if that means absolutely nothing.

It doesn't mean as much as you seem to think it means. Just because something works doesn't mean it's a good way of doing things. Eschewing any sort of mathematical sophistication impairs students because they have to learn many ideas through osmosis, impairs experts because they can't adequately convey their intuition to others or to themselves, and impairs the entire subject because it deters experts in other fields from pursuing an interest.

 

[yasiru89]

Sloppiness in notation and more importantly definition simply cannot be overlooked. I'm not telling you that you should stop approximating, you should for most realistic purposes- that's why Poincare bothered to introduce a rigorous theory of asymptotics and why the big O notation is widely used. It is not at fault, however while,

f(x) = a + bx + O(x^2)
is true,
f(x) = a + bx , is simply incorrect
f(x) ~ a + bx , is more proper. If say a+bx = g(x)
then obviously,
f(x)~ a+bx = g(x) and one may continue working on things from there.

Sloppiness is indeed a serious thing and the simple observation that 'it works' does not endear these methods any basis.

We underestimate the importance of language and notation. While a notion is the most important in itself and should not find notation a hurdle (as per Gauss), if communication is our precept(or post) then it is of utmost importance.

And I am very sorry, while I trust the paper was most interesting and probably not exactly 'faulty' in computations, it just isn't my problem. I am simply saying as some others are that when we have something we need to say where it came from with 'definition' and lack of that coupled with sloppy use of notation can really mess things up - for you not me.

 

[robphy]

Maybe the best way to think about this discussion is that
the physicist is trying to tell a story about the real world [as best as he or she can] with the aid of a mathematical language.

It may be the case that physicist's story isn't told as precisely as it could be... but it impatiently gets the key points and the main conclusions [arguably, "the good stuff"] across. (The lack of imprecision also provides opportunity for others so inclined to fill in the gaps.) On those [rare?] occasions when the key points and the conclusions are led astray by sloppy misapplication of mathematics, certainly someone will come along and correct it... leading to a revised story with a cautionary tale.

 

[Gokul43201]

The physicist does not have the desire or the patience to rigorously establish that the math used is correct. If the math turns out wrong, an experiment will eventually come along to demonstrate this. And if it turns out to be correct, a mathematician will eventually come along to demostrate this. [1]

Much more often than not, math that has been used based on an intuition for its soundsness has eventually been shown to be sound. I think this way of dealing with the math has been helpful, to keep the theory progressing at the pace that it has. If the physicists stopped before differentiating every step function, things would move along a lot more slowly.

[1] recalling the words of my many-body theory prof

 

[ZapperZ]

Sloppiness in notation and more importantly definition simply cannot be overlooked. I'm not telling you that you should stop approximating, you should for most realistic purposes- that's why Poincare bothered to introduce a rigorous theory of asymptotics and why the big O notation is widely used. It is not at fault, however while,

f(x) = a + bx + O(x^2)
is true,
f(x) = a + bx , is simply incorrect
f(x) ~ a + bx , is more proper. If say a+bx = g(x)
then obviously,
f(x)~ a+bx = g(x) and one may continue working on things from there.

Sloppiness is indeed a serious thing and the simple observation that 'it works' does not endear these methods any basis.

We underestimate the importance of language and notation. While a notion is the most important in itself and should not find notation a hurdle (as per Gauss), if communication is our precept(or post) then it is of utmost importance.

And I am very sorry, while I trust the paper was most interesting and probably not exactly 'faulty' in computations, it just isn't my problem. I am simply saying as some others are that when we have something we need to say where it came from with 'definition' and lack of that coupled with sloppy use of notation can really mess things up - for you not me.

That is why I asked you to look at many-body physics and derive the Fermi Liquid Theory or superconductivity. How exactly does one deal with a gazillion particle, all making such interactions?

When we write the potential for such many body problem, while we write an equality, no one in their right mind would ever, EVER, consider that this is exact. Maybe mathematicians who have no clue on what "many-body physics" is may get all knotted up when we write such a thing, but no physicists would be fooled into thinking that anything in there is "exact". That's like asking someone to keep all the significant figures that one gets out of a calculator!

When one deals with QFT and all those Feynman diagrams, one HAS to decide what LEVEL of interactions one has to include. Thus, the self-energy term, for example, has to be truncated using some criteria. Often, this criteria is based on what degree of accuracy that we can measure. It is useless to include all the high order interactions when there's no way those can be measured, or when such interactions produce no significant effect on anything. Thus, we can easily be justified to simply write the self-energy as an equality without even having to acknowledge that we ignore the higher-order terms. Simply by defining how one truncates the interaction is more than sufficient. That's why we show feynman diagrams in the first place! This is not sloppiness. This is physics!

Again, if this is such a major problem with the mathematics, write a rebuttal. All I have seen so far are nothing more than a matter of tastes. If these are errors in mathematics, then the results that are obtained should not be correct because they have made a serious logical error. I haven't seen such a thing being brought up in journal rebuttals. Either mathematicians who have problems with these are not speaking up, or they simply like to make fun of physicists but could not put their money where their mouths are. For something that is purportedly to be very prevalent, there seems to be nothing in the journals that points to such glaring errors. Why is that?

Zz.

 

[Pythagorean]

Math and physics shouldn't really be compared like this.

They're two different things. To physics, math is a tool, not a holy grail. We use mathematical language because we can express things to each other that way. We have other tools though, like experiment and qualitative analysis.

Satyendra Nath Bose only stumbled upon Bose-Einstein statistics because of a mathematical mistake he made during a lecture that ended up matching experimental results.