Do physics books butcher the math?

  • #1
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Thread was split of from this thread: https://www.physicsforums.com/showthread.php?t=761954

Only read them if you get annoyed at how the physics books butcher the math and if you want to see how the math should actually be done.
Oops, I think you meant to say "Only read them if you are disatisfied with how physicists elegantly reduce the mathematics to be as simple as it needs to be rather than the grotesque overcomplications invented by mathematicians on account of the fact that their subject epistemically misunderstands what mathematics actually is."
 
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  • #2
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Oops, I think you meant to say "Only read them if you are disatisfied with how physicists elegantly reduce the mathematics to be as simple as it needs to be rather than the grotesque overcomplications invented by mathematicians on account of the fact that their subject epistemically misunderstands what mathematics actually is."

No, I think butchering is the right word :tongue: But hey, mathematicians and physicists are different. I totally accept that physicists don't use math very rigorously and that many steps are shady and sometimes criminal. What matters is that they get accurate predictions for experiments. Mathematicians however must rely on logic and they have to make sure that every step is justified and well defined. So both mathematicians and physicists use math, but have different goals. I just prefer the mathematicians perspective on things.
 
  • #3
atyy
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But what about Euler summing all the positive integers to -1/12 ?
 
  • #4
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But what about Euler summing all the positive integers to -1/12 ?

Yes, amazing result. But totally wrong and nonrigorous from modern standards. It can be made rigorous however.

This shows the genius of Euler actually. He did a lot of shady things with series which are always a bit wrong from modern standards. But he had an amazing intuition and always did get the right result. Any less good mathematician who would attempt the same thing would get total nonsense.
 
  • #5
dextercioby
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I also call it butchering. The most successful theory of physics (quantum electrodynamics) is sadly not mathematically sound.
 
  • #6
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I also call it butchering. The most successful theory of physics (quantum electrodynamics) is sadly not mathematically sound.

The fact that it is so extraordinarily successful speaks volumes about how meaningless mathematical soundness actually is.
 
  • #7
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The fact that it is so extraordinarily successful speaks volumes about how meaningless mathematical soundness actually is.

Mathematical soundness is indeed not necessary for physical theories to work and to be accurate. Physicists can do their job perfectly without being rigorous (and they might even do it better).

However, I think mathematical soundness is a philosophical property of a theory that is desirable. If a theory is mathematically rigorous, then it means we understand it completely and we have reduced it to pure logic. On the other hand, QED is not mathematically rigorous, which means to me that we understand how to do the calculations, but not really why the calculations work

A stupid example, but it is of course well known that we can calculate areas by inversing differentiation. It works fine and gives us all the results. But a theory that stated this principle without any justification, would be incomplete for me. The fundamental theorem of calculus shows us exactly why differentiation and areas are linked. So while the theory would work perfectly well without somebody ever proving the theorem, it would not be philosophically satisfactory.

I know you don't think much of mathematicians and mathematical theory. You are satisfied with knowing you can predict everything. However, you cannot deny that making a theory mathematically rigorous is something that humans should attempt to do. It is in our nature to understand the theory as well as we can, and a nonrigorous theory would not be as well understood as a rigorous one. The rigorization of a theory might not yield any applications, but I think it is wrong to do science only with the applications in mind. One should do it to try and understand nature better.
 
  • #8
atyy
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I also call it butchering. The most successful theory of physics (quantum electrodynamics) is sadly not mathematically sound.

Why not just put a high energy cut off, and put the theory in finite volume?
 
  • #9
atyy
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Yes, amazing result. But totally wrong and nonrigorous from modern standards. It can be made rigorous however.

This shows the genius of Euler actually. He did a lot of shady things with series which are always a bit wrong from modern standards. But he had an amazing intuition and always did get the right result. Any less good mathematician who would attempt the same thing would get total nonsense.

So physicists are like Euler, except that where he had exceptional intuition to guide him, physicists have experiments - ie. if it predicts experimental results, then the theory is a candidate for being made rigourous. I think doing it the other way round is much harder - if we did not know that QCD was experiemtally successful, it wouldn't make any sense to set rigourous Yang Mills as a Clay problem. So from this point of view, couldn't it be argued that physics is not so different from mathematics, but part of it?
 
  • #10
Fredrik
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Oops, I think you meant to say "Only read them if you are disatisfied with how physicists elegantly reduce the mathematics to be as simple as it needs to be rather than the grotesque overcomplications invented by mathematicians on account of the fact that their subject epistemically misunderstands what mathematics actually is."
I couldn't disagree more with this view. Physics books are filled with math that's been "simplified" to the point where it's impossible to understand because the definitions are painfully inadequate. It's been almost 20 years now, but the way physicists explained tensors to me still makes me angry every time I think about it.
 
  • #11
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So physicists are like Euler, except that where he had exceptional intuition to guide him, physicists have experiments - ie. if it predicts experimental results, then the theory is a candidate for being made rigourous. I think doing it the other way round is much harder - if we did not know that QCD was experiemtally successful, it wouldn't make any sense to set rigourous Yang Mills as a Clay problem. So from this point of view, couldn't it be argued that physics is not so different from mathematics, but part of it?

Yes, I agree. There are very intimate links between physics and mathematics. It can even be claimed that one is part of the other. If you haven't read this, you should: http://pauli.uni-muenster.de/~munsteg/arnold.html It's definitely controversial, but he has a point.

But yes. Any discovery in mathematics is usually done in a very informal manner first. It is only later that it is rigorized and that the relevant axioms are invented. So in that sense, mathematics is an experiment science, since we always look at concrete examples first and see what happens there. The way they teach mathematics in high school or university (give a sequence of axioms and definitions and then derive lemmas and useful theorems) is totally inverse to how it was discovered. The theorems are usually discovered first, then the lemmas, then the definitions and axioms are made.

Also, the following quote is nice:

Jean Bourgain, in response to the question, "Have you ever proved a theorem that you did not know was true until you made a computation?" Answer: "No, but nevertheless it is important to do the computation because sometimes you find out that more is there than you realized."
 
  • #12
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Mathematical soundness is indeed not necessary for physical theories to work and to be accurate. Physicists can do their job perfectly without being rigorous (and they might even do it better).

However, I think mathematical soundness is a philosophical property of a theory that is desirable. If a theory is mathematically rigorous, then it means we understand it completely and we have reduced it to pure logic. On the other hand, QED is not mathematically rigorous, which means to me that we understand how to do the calculations, but not really why the calculations work

A stupid example, but it is of course well known that we can calculate areas by inversing differentiation. It works fine and gives us all the results. But a theory that stated this principle without any justification, would be incomplete for me. The fundamental theorem of calculus shows us exactly why differentiation and areas are linked. So while the theory would work perfectly well without somebody ever proving the theorem, it would not be philosophically satisfactory.

I know you don't think much of mathematicians and mathematical theory. You are satisfied with knowing you can predict everything. However, you cannot deny that making a theory mathematically rigorous is something that humans should attempt to do. It is in our nature to understand the theory as well as we can, and a nonrigorous theory would not be as well understood as a rigorous one. The rigorization of a theory might not yield any applications, but I think it is wrong to do science only with the applications in mind. One should do it to try and understand nature better.
This implies that it might be possible to make QFT mathematically rigorous.

I am suspicious that this is possible. To you (I think? I do not want to caricature you mistakenly) y' = a*y is an object which has an existence independent of its applications. Because it is an independent object, it makes sense to climb the ladder of abstraction and think about such objects generally.

To me however it is meaningless until one begins to talk about an application. The equation is in the image of something found in reality. Fascinatingly this particular equation applies to backterial growth rates as well as it applies to nuclear decay, which are very different things, but this does not imply to me that this symbolic representation of real phenomena should suddenly spring to life and have a meaningful independent existence.

This explains why it is extremely challenging, as an example, to derive meaningful abstract relations between non-linear ODE's. They are symbolic representations of real world phenomena, and once the restriction of linearity is removed they have significantly greater freedom, as a symbolic language, to describe real phenomena, and as real phenomena from astrophysics to economics are varied and lack sweeping general principles so too do nonlinear ODE's. It also explains why it is easy to pen down an equation that describes nothing at all, and by describing nothing at all, is meaningless.

That is why I think it is plausible that no rigorous formulation may ever be composed for QFT, and also why rigorous formulations of physical theories typically add little to the progress of physics. Since mathematics is made in the image of real world phenomena and extremely high energy scales are so absurd to our intuition, that mathematics, which is the product of an object (the human brain) which evolved for and exists in a low energy environment, could be invented which is sensible to us seems unlikely. If not impossible, it is at least unnecessary. Again, in my
universe mathematics is only ever a reflection of something in reality; if that reality is fundamentally incomprehensible to us, the mathematics probably will be as well.

You may point out that non-relativistic quantum mechanics is completely rigorous or can be formulated in such a manner (to my knowledge anyway, i've never investigated it). But much of it is done in non-rigorous ways; think of the horrible delta function in everything from projection operators to the physicist's functional derivative. Even more troubling, the mathematics makes no intuitive sense, even in cases as basic as how physicists combine probabilites with the squared norm.
 
  • #13
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This implies that it might be possible to make QFT mathematically rigorous.

I am suspicious that this is possible. To you (I think? I do not want to caricature you mistakenly) y' = a*y is an object which has an existence independent of its applications. Because it is an independent object, it makes sense to climb the ladder of abstraction and think about such objects generally.

No, I don't think that at all. A mathematical object should not be divorced from its applications. The applications add a lot to the mathematics and give us ways to deal with it.

But every mathematics there is comes from some kind of application (even if the link is not very clear anymore). If there is no link with known applications or other known mathematics, then only a very few people will bother to study it. I personally am not interested in studying something that has no interesting applications.

You may point out that non-relativistic quantum mechanics is completely rigorous or can be formulated in such a manner (to my knowledge anyway, i've never investigated it). But much of it is done in non-rigorous ways; think of the horrible delta function in everything from projection operators to the physicist's functional derivative. Even more troubling, the mathematics makes no intuitive sense, even in cases as basic as how physicists combine probabilites with the squared norm.

Sure, much of quantum mechanics is done in very nonrigorous ways. From a mathematical point of view, most QM books are horrible. Quantum Mechanics has even been discovered in a very nonrigorous way. Still, we were able to completely rigorize it. And I think that rigorization is very interesting and reveals quite a lot. Indeed, if one does not know the rigorous version of quantum mechanics, then one will easily fall into various traps: http://arxiv.org/abs/quant-ph/9907069

So I am almost certain that the same thing will happen to QED.

Also, the mathematics makes perfect intuitive sense to me. I don't virtually nothing of physics (and certainly not of quantum mechanics). But it has happened often that my knowledge of math alone was sufficient to talk about quantum mechanics to physicists and even help them with issues they had. So the mathematical intuition I created while studying the math is not so far off from the physical intuition of the physicists. Not that I say that every physicist should start learning pure math, but I thought that was rather remarkable.
 
  • #14
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"Also, the mathematics makes perfect intuitive sense to me."

Fascinating. I'd love an intuitive explanation of quantum mechanics. Can we start with the Schroedinger equation? Why is that the equation describing the time evolution of a wave function?

EDIT: You've said the math makes sense. Well, sure, once you start with the fact that it is a PDE and you know what a PDE is everybody who understands PDE's "understands" the Schroedinger equation. But I would venture to guess that a true understanding of the mathematics would coincide with an understanding of why the mathematical superstructure was built the way it is.
 
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  • #15
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"Also, the mathematics makes perfect intuitive sense to me."

Fascinating. I'd love an intuitive explanation of quantum mechanics. Can we start with the Schroedinger equation? Why is that the equation describing the time evolution of a wave function?

EDIT: You've said the math makes sense. Well, sure, once you start with the fact that it is a PDE and you know what a PDE is everybody who understands PDE's "understands" the Schroedinger equation. But I would venture to guess that a true understanding of the mathematics would coincide with an understanding of why the mathematical superstructure was built the way it is.

Sure. And the Schrodinger equation can be very neatly understood by using mathematics. For example, C*-algebras and operator theory gives me a very intuitive explanation of quantum mechanics and the Schrodinger equation. It shows that it's just classical mechanics which is made noncommutative.
 
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  • #16
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Why would you want to make classical mechanics commutative (I'm also baffled that it would be commutative; momentum and position operators commute in classical mechanics but not in quantum mechanics, but I know nothing of C* algebras)?
 
  • #17
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Why would you want to make classical mechanics commutative (I'm also baffled that it would be commutative; momentum and position operators commute in classical mechanics but not in quantum mechanics, but I know nothing of C* algebras)?

Sorry, it should have been noncommutative. Classical mechanics is just the commutative version of QM in the C*-algebra formalism.
 
  • #18
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Sorry, it should have been noncommutative. Classical mechanics is just the commutative version of QM in the C*-algebra formalism.

Sure, but what about nature at the quantum scale necessitates non-communativity of the position/momentum operators?

One possible axiom is the uncertainty principle, but this just raises a deeper question: why is there some fundamental limit on the uncertainty of these operators? Why propose it in the first place?
 
  • #19
Fredrik
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Fascinating. I'd love an intuitive explanation of quantum mechanics. Can we start with the Schroedinger equation? Why is that the equation describing the time evolution of a wave function?
The problem is that these things are only intuitive to people who understand the mathematics really well. I'm only half-way there myself, but I understand enough to say that a person who understands topology, measure theory, integration theory, Hilbert spaces and operator algebras well enough to understand exactly in what sense QM is a generalization of probability theory, has a far better understanding of QM than a typical physicist.
 
  • #20
Fredrik
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Sure, but what about nature at the quantum scale necessitates non-communativity of the position/momentum operators?
To ask this is to ask why the universe is such that quantum theories makes more accurate predictions than classical theories. The only thing that can answer that is a better theory. (Better than all quantum theories). If we had such a theory, you'd probably be asking the same question about that theory instead.
 
  • #21
atyy
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Regarding QFT: why not make it rigourous just by putting it on a lattice in finite volume?

According to the Copenhagen interpretation, in QM we always divide the world into classical and quantum parts. Since the quantum part is only a subset of the universe, finite volume should be ok. Then by putting it on a lattice, we ensure that there are no ultraviolet divergences, since everything is now QM. So in this way, couldn't one make QFT rigourous?

I do understand that in 2+1 D one can make QFT rigourous without putting it on a lattice, and it is intellectually interesting to ask whether one can do the same for any 3+1 D QFTs, such as Yang Mills, which non-rigourous asymptotic freedom calculations suggest should be possible. But given that our experiments can't tell whether there is a fine lattice or not, why not just use the lattice and stop saying that QED is not rigourous?
 
  • #22
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Sure, but what about nature at the quantum scale necessitates non-communativity of the position/momentum operators?

One possible axiom is the uncertainty principle, but this just raises a deeper question: why is there some fundamental limit on the uncertainty of these operators? Why propose it in the first place?

It can be seen as an experimental fact, can't it?
I'm not claiming you can use math to invent new kinds of physics without ever needing experiments. I'm just saying that the mathematics of QM is quite intuitive to me and makes many parts of the physics of QM also intuitive to me. I do not claim to be an expert in QM or that I could invent it.
 
  • #23
Fredrik
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Regarding QFT: why not make it rigourous just by putting it on a lattice in finite volume?
...
But given that our experiments can't tell whether there is a fine lattice or not, why not just use the lattice and stop saying that QED is not rigourous?
This type of approach seems reasonable to me too. I don't know why exactly it's so important to find a rigorous version of QED on ##\mathbb R^4##. I guess one reason is that if we e.g. use a box of finite volume with periodic boundary conditions to remove the infinities, we'd have one version of QED for each box size. This would be pretty ugly, but I think each of those theories would be fully rigorous, and experimentally indistinguishable from each other, if we only consider large enough box sizes.

On the other hand, I think a rigorous QED on ##\mathbb R^4## might turn out to be pretty awesome. It could e.g. improve our understanding of QFTs in a way that's similar to how non-commutative probability theory has improved our understanding of QM.
 
  • #24
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This thread is getting extremely off topic and I think it is very unfair to the OP.

Not everyone does physics for reproducible experimental results. A good number of people do it to work with, discover, and appreciate mathematical beauty and elegance in nature. As much as it might sting to admit it, pure math in the hands of mathematicians is absolutely indispensable for this if only to put a physical theory in a more coherent, compactified, and elegant theoretical framework. This is probably why GR is orders of magnitude more elegant than QFT even though QED is infinitely superior in experimental accuracy and reproducible/testable results. This does matter to some people and you can even find articles on it. As far as learning physics goes it is probably comletely irrelevant. Micromass' statements in post #2 are sound in that regard and there is quite literally nothing more to say than what he already said.
 
  • #25
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This thread is getting extremely off topic and I think it is very unfair to the OP.

Yes, I'll split the thread in two parts.
 

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