Mathematicians' contributions to physics

[Demystifier]

But their theory seem interesting, and being myself a mathematician, I think this is only good for physics (most physicists seem to be unaware of the real problems in physics. They just choose to ignore them and consider that everything is "clear" for them).

Are you suggesting that mathematicians are sometimes better than physicists in spotting the real problems in physics? (Just asking.)

 

[malawi_glenn]

A spin-off discussion in this thread https://www.physicsforums.com/threa...ic-interactions-by-babin-and-figotin.1050807/
A quote from member @coquelicot (now deleted) was suggesting that most physicsts are unaware of the "real" problems with physics and that they [the physicsts] choose to ignore them.

Apart from the logical problem that you can not choose to ignore something that you are unaware of, I thought that here we could discuss if/how mathematicians are needed in physics. For intance list examples of where mathematicians have entered the field of physics and made significant contributions. Also examples where such endevaours have not been fruitful. Also examples in which the original idea was nonsense physically, but was still fruitful - like Weyl which was mentioned in a post by vanhees (see below).
Also we could discuss areas in physics which (still) lacks mathematical rigour.

This is, of course, nonsense. There are brillant books on theoretical physics written by mathematicians. Historical examples are Weyl's, Raum, Zeit, Materie and von Neumann's book on the mathematical foundations of quantum mechanics.

What these example also demonstrate is that you must be careful when it comes to the physics part. Weyl had the superficially brillant idea to gauge the scale invariance of the free gravitational field in GR to describe the electromagnetic field as the corresponding gauge field. The only disadvantage is that it's physically impossible, because it contradicts the simple fact that the spatial and temporal scales of charged matter doesn't depend on its electromagnetic history, as both Einstein and Pauli immediately pointed out to Weyl. Nevertheless the idea is indeed brillant, because the principle of making global symmetries local lead to a tremendous success in model building in connection with relativistic quantum field theory and the understanding of the fundamental interactions in terms of the Standard Model of elementary particle physics, which heavily builds on this idea of "gauge invariance".

As the thread starter, one should always contribute something yourself. I will do so with this quote by Hilbert
Physics is becoming too difficult for physicists.

What are your thoughts on this?

 

[coquelicot]

A spin-off discussion in this thread https://www.physicsforums.com/threa...ic-interactions-by-babin-and-figotin.1050807/
A quote from member @coquelicot (now deleted) was suggesting that most physicsts are unaware of the "real" problems with physics and that they [the physicsts] choose to ignore them.

Actually, I've edited the question and replaced "most physicists" by "many physicists", which is closer to my own thoughts.

 

[malawi_glenn]

Actually, I've edited the question and replaced "most physicists" by "many physicists", which is closer to my own thoughts.

What are the REAL problems in physics?

 

[Swamp Thing]

Emmy Noether ?

 

[Haborix]

Aside from the usual problem with discussions like these, i.e., that the physics being discussed is really a very narrow subset of physics, it seems the problem is more a lack of new physical insight combined with a lag in experimental data. Mathematicians have often come in later and made the physics richer and more coherent. That's a long-winded way of saying I'm skeptical that mathematics is the primary problem for progress in fundamental physics.

 

[vanhees71]

I think, it's just a matter of specialization. A mathematician has different interests and thus aims at answering different questions than physicists.

An example is relativistic quantum field theory. The physicists have a framework, which they find pretty "successful", because it describes all (known) elementary particles and their interactions in terms of the Standard Model with the exception of the gravitational interaction. The latter is not (yet?) implemented in this "physicists' QFT framework" in a satisfactory way.

Now nobody can deny that there are serious mathematical problems with how physicists treat relativistic quantum field theories by just introducing operator-valued distributions and mutliply them without asking, which sense this makes, because it's an ill-defined operation. The result is that almost from the beginning one gets divergent quantities. E.g., already the total energy of non-interacting fields in the vacuum state is divergent, but this is obviously due to the indefiniteness of the ordering of the field operators when writing down the formal expression for the energy density, and this divergence can be subtracted and one gets a well-defined finite energy for all physical states of the field, and usually one makes the energy eigenvalue of the vacuum state 0.

Now the physicists go on and look at interacting field theories. Not being able to solve the corresponding non-linear field-operator equations exactly they do "perturbation theory", introducing the "interaction picture". This ends in a desaster, because going beyond tree level in the corresponding Feynman diagrams, i.e., to higher-order loop corrections, which are the most interesting calculations needed to get the high-precision results in, e.g., QED (anomalous magnetic moment of electrons or muons, the lamb shift of hydrogen spectral lines,...). Also there it turned out that with the right prescription of how to get rid of the divergences, lumping them into unobservable "bare field-normalization factors, masses, and coupling constants) and working with the corresponding finite physical (renormalized) quantities, leads to a consistent scheme, which is empirically highly successful.

On this "perturbative level" of definition of relativistic QFTs one can make things quite rigorous by being more careful when mutliplying field operators by using "smeared field operators" that can be multiplied in a meaningful way. That's the Epstein-Glaser approach, and well explained in the book by Scharf, Finite Quantum Electrodynamics.

On the other hand for mathematicians all this is, of course, very unsatisfactory, because it's not clear, whether quantum field theories can be formulated in a proper clean way without all the tricks and beyond perturbation theory. This lead to the development of what's called "axiomatic quantum field theory" with various branches. The problem with this is that it seems that not much has been achieved, which goes beyond the free fields and toy models in lower space-time dimensions. There are, however, interesting results like the fact that the physicists's beloved interaction picture does not exist from this rigorous mathematical point of view, and indeed the physicists intuitively know that and first define things in a finite-volume box and at the very end take this volume to infinity in a specific way for the quantities (like S-matrix elements), where this procedure leads to plausible results.

In some sense there exist now two camps in the QFT community: the pragmatic physicists, being satisfied (maybe with a bad conciousness by some) with the "robust mathematics" leading to empirically successful descriptions of observable quantities like scattering cross sections, particle lifetimes, etc. and the "axiomatic-QFT mathematicians", which are not at all satisfied with this.

In any case, in my opinion, it's worth while to have in mind both points of view, because on the one hand, one wants to understand the phenomenology from a theoretical-physicists'-point-of view using mathematically somewhat "mediocre" techniques of calculation, but on the other hand, thinking deeper about the mathematical problems with this naive approach and maybe finding some remedy of all the problems, may lead to better theories, which one day may be also applicable to "realistic" interacting-field theories in 4 spacetime dimensions and maybe finally also lead to a satisfactory description also of gravitation (general relativity) within a unified quantum (field?) theoretical framework.

 

[coquelicot]

To begin with, I think that:
Many physicists are also excellent mathematicians, and many mathematicians are quite good in physics.

As a mathematician, I could witness that when I was learning at the university, there were many physicists in the class learning with us topology, group theory etc.

These two disciplines always gone together. At the beginning, there were no mathematicians and no physicists, they were most often both of them. Archimedes, Fermat, Descartes, Pascal, Robertval, Newton, Lagrange, Legendre, Gauss, and many, many other. In a letter of Abel, I could read once: "I'm the only one in Europe that deals only with mathematics".

Admittedly, by our days, you have to specialize in a sub-sub-sub discipline of these sciences.
Nevertheless, the links are sufficiently strong to allow mathematicians to contribute to physics, especially fundamental and original ideas (that's the domain were we are really good after all).

This being said, in addition to the reply of Vanhees71, I would like to give an example at a very, very elementary level, where most physicists are unaware of a hole (here I say most and not many). I mean the notion of capacitance matrix. When I read once elementary books in EM, I immediately felt the hole (I call that my mathematician intuition). Then I asked several physicists (even in this forum if my memory is good), and they reply to me that for them, it is evident that the charges of the conductors depend linearly on the potentials, invoking the "linearity of the equations of EM". After all, that's what is written in Jackson, Landau and Lipschitz etc. etc.
Then I tried and succeeded to answer by myself to the question for conductors in free space, which involves a uniqueness theorem by no way trivial, that I found in some lectures by Dr Konoplisky. I couldn't believe that this is not dealt somewhere, and I continued to seek in books. I finally found that Griffith was aware of the problem and of its solution (but he writes nowhere the solution), and that Jefimenko book is perhaps the only book that contains an (almost) valid proof by induction. Regarding Maxwell, I think his proof is incomplete (he proves only the multiplicative linearity, which is trivial, but not the additive linearity).
And all of that concerns only conductors in free space, but what about conductors in a mixture of dielectrics whose properties are described a tensor?
Only recently were two articles published on the question. Unfortunately, in some aspect, they are misleading (I am currently writing an article where I develop the theory in a general framework, even if it is an extremely modest contribution to physics.)
The point I wanted to illustrate is that holes exist even at such an elementary level. How is it possible that elementary physics has not been duly formalized, akin to what the Bourbaki school did in mathematics ?
I would like to cite the problem of microscopic to macroscopic EM field, where I feel that something big has been missed as well; but I am currently trying to study the books of Robinson and De Groot who have dealt extensively with the subject. So, I will abstain.

Edit: actually, Jefimenko does provide a valid proof for the elastance matrix, but assert without proof that it is invertible (its inverse is the capacitance matrix). Nevertheless, the tools he provides are sufficient to establish the capacitance matrix as well.

 

[malawi_glenn]

Thank your for this contribution @coquelicot
Though the notion of a capacitance matrix is not that important for physics as a whole.
What are the "real" problems in physics? Can these problems we solved solely by mathematicians? What are your thoughts on that?

 

[coquelicot]

Thank your for this contribution @coquelicot
Though the notion of a capacitance matrix is not that important for physics as a whole.
What are the "real" problems in physics? Can these problems we solved solely by mathematicians? What are your thoughts on that?

Well, many high level problems are well known (see Vanhee71, and all the insight articles of this forum). I don't think these problems can solely be solved by mathematicians, because, as I said above, physicists are actually mathematicians as well, sometimes at a very high level. I say that it is possible that one day, a pure mathematician propose an original idea that proves to be excellent and fruitful for the development of physics.
Obviously, the unification of GR with QFT is one of the most fundamental problem in physics. But I disagree with you about the fact that "smaller" problems (and even very small ones like the capacitance matrix) should be neglected.

 

[malawi_glenn]

Still this "sloppy" treatmeant of operator-valued distrubtions have given us some of the most accurately known physical models which are tested to extreme high precision. Is it then a "problem" that such mathematical operation is not rigoursly defined? Or perhaps physicsts were just lucky... ?

the unification of GR with QFT is one of the most fundamental problem in physics.

We might be working with the wrong premisse: that it IS possible to unify these two frameworks at all. This might be wrong.

 

[coquelicot]

In addition to what I said above, I would like to point two possible problems of the physics today. I'm not sure I am right, and I am not sufficiently good to defend these ideas. So, please, just consider them as "feelings" from a non professional physicist.

• The use of Lagrangian and the action principle as a "magical tool" to produce theories. As a person who try to read a lot of articles, I feel that there are as many physical theories as there are Lagrangians. Everyone can "tinker" a Lagrangian to produce (almost :-) ) all what he wants, and propose a new theory. Again, I may well be wrong, but from the exterior side, this looks like a jungle of Lagrangian derived theories, almost magical, hiding the real physical concepts.
• The tendency to present a jungle of equations as a well established theory. The problem is real: you have to spend two years to understand the equation jungle, just to realize that there is no clear directing idea behind them. And the worse is that you can even not criticize the theory, because you will be considered as an idiot for not understanding the (alleged) idea behind the equations. The book of de Groot I am trying to study may be an example of that (not sure, I'm just beginning). That's bad for the physics, because some problems may be considered to be solved while they are actually not

 

[coquelicot]

Is it then a "problem" that such mathematical operation is not rigoursly defined? Or perhaps physicsts were just lucky... ?

We all know that they are not "just lucky". I think that yes, there is a problem because mathematical operations must be rigorously defined.
Fortunately, historically, mathematicians have always succeeded in establishing on rigorous bases the ideas of their physicist colleagues.

Examples are:

• distribution theory, that was used by Heaviside and Dirac and established rigorously by Schartz.
• path integral of Feynman that was established by specialists of integration theory.
• General relativity where Einstein was helped by Levy-civita, Grossman and in some sense by Minkowski, Hilbert and Noether.

 

[malawi_glenn]

, there is a problem because mathematical operations must be rigorously defined.

Can you give an example what consequences of this problem give, other than an itch in mathematicians brains? E.g. practical results

path integral of Feynman that was established by specialists of integration theory.

This is off topic, but do know of any good path integrals for mathematician style textbook?

 

[BillTre]

To me, not a physicist, physics seems to be doing OK.

A quote from member @coquelicot (now deleted) was suggesting that most physicsts are unaware of the "real" problems with physics and that they [the physicsts] choose to ignore them.

This is why I am entering this tread.

This seems to be a common complaint in more fields than just physics.

As a biologist, I could say the same thing about origin of life issues. While basic to biology, many ignore these unresolved issues "at the base of biology" and carry on with what ever they are studying.
It is a common sense approach to dealing with issues in a complex field.

If this weren't done, generations of biologists would have wasted their careers on trying to figure this kind of thing out before the field was in a state where the subject could be realistically approached.
These are individual decisions each person makes for themselves.

 

[coquelicot]

Can you give an example what consequences of this problem give, other than an itch in mathematicians brains? E.g. practical results
This is off topic, but do know of any good path integrals for mathematician style textbook?

q1: Obviously, distribution theory allows rigorous thinking in physics. I am not sufficiently good to give you examples, but I am pretty sure that it is possible to find articles with minimal effort.

q2: Yes, for example Henstock book "A general theory of integration" (not sure this is the exact name but this is close to).

 

[coquelicot]

To me, not a physicist, physics seems to be doing OK.This is why I am entering this tread.

This seems to be a common complaint in more fields than just physics.

As a biologist, I could say the same thing about origin of life issues. While basic to biology, many ignore these unresolved issues "at the base of biology" and carry on with what ever they are studying.
It is a common sense approach to dealing with issues in a complex field.

If this weren't done, generations of biologists would have wasted their careers on trying to figure this kind of thing out before the field was in a state where the subject could be realistically approached.
These are individual decisions each person makes for themselves.

A good point.
I'm not sure that my sentence quoted by Malawi-Glenn reflects exactly what I think (it was after all a short sentence inside parentheses, written quickly and carelessly to reply to the underlying claim that physics books by mathematician are bad). My opinion is that mathematicians may be very qualified to discover flaws and holes inside physics, and even to propose new and original theories.

 

[malawi_glenn]

q1: Obviously, distribution theory allows rigorous thinking in physics. I am not sufficiently good to give you examples, but I am pretty sure that it is possible to find articles with minimal effort.

That is ok, but don't you think that physics has "enough" rigorous thinking considering all the success physics has made so far? Not sure why there is an urgent need for making sense of operator-valued distributions will lead to any practical result. Let's see if someone can chime in.

I'm not sure that my sentence quoted by Malawi-Glenn reflects exactly what I think (it was after all a short sentence inside parentheses, written quickly and carelessly to reply to the underlying claim that physics books by mathematician are bad)

Which is why I created this thread, to have a better place to discuss these matters.

 

[fresh_42]

Physics is about nature, mathematics is about philosophy. Nevertheless, they developed pretty much in parallel, especially after the physicist Newton (Or was he a mathematician?) and the mathematician Leibniz (Or was he a philosopher?) discovered differential calculus. When I read about the history of these sciences between the seventeenth century and the great eruptions in both fields at the beginning of the twentieth century, then nobody ever really made a difference: Gauß, the Bernoullis, the French mathematicians, they never made a distinction. Calculus and linear algebra seemed to be just the language to describe and solve physical problems.

Things changed a bit in the early decades of the twentieth century. Physics had its determinism crisis, mathematics its logical crisis. Whereas the issue in physics isn't really settled until today, as you can see in our QM interpretation forum, mathematicians shrugged over Russell and Gödel and proceeded as usual. The real switch came with Bourbaki when mathematics was all of a sudden written in patterns [: definition - example - lemma - proposition - theorem - corollary - example :|]) instead of prose. This makes a real difference today. I call it the Bourbakian transition. It also led to the fact that mathematics nowadays follows the route of inventing some crude environment by making up some technical definitions, often looking as if they were artificial and at random, and then playing around with them like a boy does with his toy trains.

This decoupling of physics and mathematics didn't serve either in my opinion. Maybe Noether was the last big result of both fields together.

 

[coquelicot]

The real switch came with Bourbaki when mathematics was all of a sudden written in patterns [: definition - example - lemma - proposition - theorem - corollary - example :|]) instead of prose. This makes a real difference today. I call it the Bourbakian transition. It also led to the fact that mathematics nowadays follows the route of inventing some crude environment by making up some technical definitions, often looking as if they were artificial and at random, and then playing around with them like a boy does with his toy trains.

I agree about everything you said but this claim. Of course, the pattern "definition, theorem , problem" was not invented by Bourbaki. This is the merit of the ancient Greeks to invent it, and they come to it by the way of philosophy: the idea of the philosopher Thales was that in order to reach the knowledge, one has to start with the clearest principles, and to deduce from them less clear, or totally unclear, principles, in a deductive way. This method set by the ancient Greeks, and marvelously illustrated in Euclid's elements, is one of the greatest progress, if not the greatest, for the mankind.
On the other hand, the merit of the Bourbaki school is that they have thought and established mathematics on deep and solid bases.
I would like to see such a revolution in physics.

 

[Haborix]

I would like to see such a revolution in physics.

Let's not. There are a few mathematical physicists and applied mathematicians running around. That's enough. Do you want to Bourbaki-ize chemistry? How about biology? It seems like an incredible waste of time in an experimental science.

 

[coquelicot]

That is ok, but don't you think that physics has "enough" rigorous thinking considering all the success physics has made so far? Not sure why there is an urgent need for making sense of operator-valued distributions will lead to any practical result. Let's see if someone can chime in.

The situation can be compared to the situation of infinitesimal calculus in mathematics before the development of "limits" by Cauchy. Mathematicians had reached a high level in infinitesimal calculus, but everthing was rather intuitive and not rigorous. Near this time, Abel complained that most reckonings in this domain are not valid, and that Cauchy is the only man that knows how to write mathematics (he himself invented several series convergence methods to improve the situation). Because of that, many results were erroneous. Thanks to the revolution of Cauchy, everything became clear and rigorous, and mistakes were eliminated from the domain. The same can be said about everything in physics that is based on intuitive principles whose limits have still not be determined. Eventually, this will necessarily produce erroneous statements and theoretical mistakes, if this is not already the case. Not everything can be easily checked by experiments.

 

[coquelicot]

Let's not. There are a few mathematical physicists and applied mathematicians running around. That's enough. Do you want to Bourbaki-ize chemistry? How about biology? It seems like an incredible waste of time in an experimental science.

Bourbaki a waste of time? are you serious?

 

[Haborix]

Bourbaki a waste of time? are you serious?

For physics, yes.

 

[coquelicot]

For physics, yes.

This precisely shows that this revolution has to come, in order for people
like you realize how necessary it was.

 

[coquelicot]

For physics, yes.

Let try a test: please, explain to me what is the macroscopic E.M field, and why a surface charge appears when a dielectric is exposed to an exterior field, and why a volume charge appears inside the dielectric when there is a div field. Elementary question isn't it? you have probably learnt that at your beginning. But think well, think very well.

 

[fresh_42]

I agree about everything you said but this claim. Of course, the pattern "definition, theorem , problem" was not invented by Bourbaki.

That does not change the fact that mathematics was written in prose before Bourbaki, cp. Kurosh, van der Waerden, Courant, et al., and written in the pattern above after Bourbaki, cp. Lang and basically every modern textbook. Their explicit goal was to deduce and present mathematics on an axiomatical base. That wasn't the case before, possibly except for the ancient Greeks. Well, you could note that the idea was based on Hilbert's program, but Bourbaki turned the idea into books.

 

[coquelicot]

That does not change the fact that mathematics was written in prose before Bourbaki, cp. Kurosh, van der Waerden, Courant, et al., and written in the pattern above after Bourbaki, cp. Lang and basically every modern textbook. Their explicit goal was to deduce and present mathematics on an axiomatical base. That wasn't the case before, possibly except for the ancient Greeks. Well, you could note that the idea was based on Hilbert's program, but Bourbaki turned the idea into books.

But the fact that it was sometimes written in prose did not reduce the rigor of the exposition, nor its beauty. Have you read the "Disquisitiones" of Gauss?

 

[coquelicot]

That does not change the fact that mathematics was written in prose before Bourbaki, cp. Kurosh, van der Waerden, Courant, et al., and written in the pattern above after Bourbaki, cp. Lang and basically every modern textbook. Their explicit goal was to deduce and present mathematics on an axiomatical base. That wasn't the case before, possibly except for the ancient Greeks. Well, you could note that the idea was based on Hilbert's program, but Bourbaki turned the idea into books.

And here is a page of Van der Waerden book you cite as an example, demonstrating that you are wrong in this case.

 

[fresh_42]

Have you read the "Disquisitiones" of Gauss?

Only took a look (after download).

 

[fresh_42]

And here is a page of Van der Waerden book you cite as an example, demonstrating that you are wrong in this case.

View attachment 323706

I have both, modern algebra volumes 1 and 2, and both are prose. That does not mean van der Waerden wrote without formulas or structure, au contraire, only without Bourbaki's strict corset.

 

[coquelicot]

I have both, modern algebra volumes 1 and 2, and both are prose. That does not mean van der Waerden wrote without formulas or structure, au contraire, only without Bourbaki's strict corset.

I think that Bourbaki did two very great things:

• First, the work was thought by a group of great mathematicians that discussed and decided how mathematics should be established and written.
• second, regarding the form: they always specify all the assumptions in every theorem. Believe it or not, this is great. I have had troubles with many, many modern books in mathematics, just because you have to read the whole book to know what is the meaning of their notations and what is really assumed in theorems. Of course, this would probably be useless in physics. Actually, I don't want to see an imitation of Bourbaki's book in physics. But I think that great physicists should discuss and decide how physics should be established and written, and provide the deal in the form of a book collection.

 

[fresh_42]

I think that Bourbaki did two very great things:

• First, the work was thought by a group of great mathematicians that discussed and decided how mathematics should be established and written.
• second, regarding the form: they always specify all the assumptions in every theorem. Believe it or not, this is great. I have had troubles with many, many modern books in mathematics, just because you have to read the whole book to know what is the meaning of their notations and what is really assumed in theorems. Of course, this would probably be useless in physics. Actually, I don't want to see an imitation of Bourbaki's book in physics. But I think that great physicists should discuss and decide how physics should be established and written, and provide the deal in the form of a book collection.

I am a fan of Bourbakian style, don't get me wrong. I'm just saying that
a) there is a huge difference in style between Kurosh and Kargapolov/Merzljakov.
and
b) physics never made this transition. They still say generators if they mean a vector, they still speak of infinitesimal quantities as if Leibniz were still in the room, and they are trapped in a world of coordinates.

 

[coquelicot]

I am a fan of Bourbakian style, don't get me wrong.

Of course not. This was just an addition.

b) physics never made this transition. They still say generators if they mean a vector, they still speak of infinitesimal quantities as if Leibniz were still in the room, and they are trapped in a world of coordinates.

Well, I think that tensor calculus in physics is as useful as matrix calculus in linear algebra. There too, you can do almost everything without matrices; that's just... much less intuitive. I'm not sure physicists are trapped in a world of coordinates. They perfectly know other tools like differential forms with hodge star operator etc., or even geometric algebra. That's just a matter of choice.

 

[coquelicot]

b) physics never made this transition. They still say generators if they mean a vector, they still speak of infinitesimal quantities as if Leibniz were still in the room, and they are trapped in a world of coordinates.

I'm just curious: how do you make physics without infinitesimal quantities? do you have another way?

 

[fresh_42]

I'm just curious: how do you make physics without infinitesimal quantities? do you have another way?

No. I just do not behave as if they were tiny little creatures. In most cases, they are real numbers $\left. \dfrac{dy}{dx}\right|_{x=x_0}$ and in the rest of the cases they are differential forms or basis vectors.

I love Weierstraß's formula for derivatives: $f(x+v)=f(x)+D_x(f)\cdot v + o(v).$ No tiny little creastures necessary except for the remainder function.

 

[coquelicot]

No. I just do not behave as if they were tiny little creatures. In most cases, they are real numbers $\left. \dfrac{dy}{dx}\right|_{x=x_0}$ and in the rest of the cases they are differential forms or basis vectors.

I love Weierstraß's formula for derivatives: $f(x+v)=f(x)+D_x(f)\cdot v + o(v).$ No tiny little creastures necessary except for the remainder function.

So, how should the derivation of the propagation of heat be written according to you?

 

[fresh_42]

So, how should the derivation of the propagation of heat be written according to you?

I do not suggest another concept, just another wording. Do not use the word "infinitesimal". It creates more confusion than it solves. There is no need. The only case I can think of that has a bit of a justification is the chain rule, but even this is cheating. There is no problem in using $dx$. There is a problem with treating it as a small quantity.

 

[coquelicot]

I do not suggest another concept, just another wording. Do not use the word "infinitesimal". It creates more confusion than it solves. There is no need. The only case I can think of that has a bit of a justification is the chain rule, but even this is cheating. There is no problem in using $dx$. There is a problem with treating it as a small quantity.

I don't see what you mean. $dx$ is actually supposed to represent the limit of $\Delta x$, so what is the problem to name it "infinitesimal quantity"?

 

[fresh_42]

I don't see what you mean. $dx$ is actually supposed to represent the limit of $\Delta x$, so what is the problem to name it "infinitesimal quantity"?

$dx$ is a differential form. $dy/dx$ is a limit, and it is not "something small", especially not for undergraduates. How would you rigorously define the word infinitesimal without falling to the seventeenth century? And, of course, without using hyperreals or surreals. Wikipedia writes

In mathematics, a positive infinitesimal is an object which is greater than zero in the order of the real numbers but less than any positive real number, however small.

I consider this pure nonsense as long as we teach calculus traditionally (meanwhile) with epsilontic. This definition is a crime.

 

[coquelicot]

$dx$ is a differential form. $dy/dx$ is a limit, and it is not "something small", especially not for undergraduates. How would you rigorously define the word infinitesimal without falling to the seventeenth century? And, of course, without using hyperreals or surreals. Wikipedia writes

I consider this pure nonsense as long as we teach calculus traditionally (meanwhile) with epsilontic. This definition is a crime.

yes I agree with you that such a definition in Wikipedia is rather ridiculous, except in the context of non standard analysis. But the use of the wording "infinitesimal" is valid in my opinion, as an intuitive and very useful abbreviation for limit processes. Once you have specified what is boiling down mathematically, it's perfectly OK to use such a jargon, which has a natural and intuitive meaning. Everyone with sufficient maturity in calculus is then able to translate, if he wishes, a demonstration exposed this way into a demonstration with epsilons (even thought he would never do that as this complicates the matter uselessly).

 

[malawi_glenn]

Speaking of wikipedia, you are always welcome to edit it...

 

[fresh_42]

Speaking of wikipedia, you are always welcome to edit it...

I would delete it.

 

[malawi_glenn]

I would delete it.

Well that is a subset of editing :D

 

[romsofia]

I think this is a very interesting question and depends on which field of physics you're in.

I like how fresh_42 put it where math is more akin to philosophy, and physics is more akin to nature. In the 60s-80s, some fields saw those who could philosophize nature succeed the most. Wheeler, DeWitt, Penrose, Choquet-Bruhat, Unruh, etc

In the fields where people like this thrive, I would think a mathematician (although, they better be a good one!) could find the supposed holes of the field. But the closer we get to actual experimental data, say designing the simulation of a binary black hole merger, or calculating the velocity of a black hole kick, it's less likely that a mathematician (even a great one) could be of that much help, even less that they'd be able to find the supposed holes in it.

You need actual physics for that, you can't just "abstractize" the experimental set up. Nor should we put experimental design in a "definition: corollary: proposition: remark: " fashion. Now, it was talked about back in the 80s (one of my favorite essays of all time "What have we learned from Quantum field Theory in curved space-time?" by Fulling talks a little about this at the start), and should be emphasized: the foundational community *does* need to transition to that point. There is way too much overlap and usage of common words, and each subfield of this overall community has their own definitions, and it gets confusing fast. Heck, I'd wager that at least 40% of the posts on the interpretation forum is about someone's usage of a word in a way they don't use it. Imagine how fast a Bourbakian style would clean up that field and the potential progress we would see if those great minds talked to each other properly! But does that mean a mathematician in the current state could find the problems the physicists aren't seeing? Maybe.

To put it more bluntly: If we're talking about topological censorship, then yes, I could see how a mathematician could see some conceptual junk the physicist is unaware of, and be able to break the problem down to a more abstract view which could lead to some connections, and that will patch some holes in the framework. If we're talking about experimental design, simulations of events, and analyzing real world data, I don't see why a mathematician would be any better than the common human in moving the field forward. You need actual scientific protocol, and training.

 

[fresh_42]

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If we're talking about experimental design, simulations of events, and analyzing real world data, I don't see why a mathematician would be any better than the common human in moving the field forward. You need actual scientific protocol, and training.

I think this is a bit one-dimensional. As long as SUSYs and Branes are on the table, as long there are serious mathematical aspects to be considered. E.g., mathematicians could find theorems that would allow actual experiments to test those theories. I guess that would help a lot! And I can't get these slime molds out of my mind. If they really represent a valid model for the distribution of DM, then mathematics could well be a first step to connect the two, possibly easier than the very different physical entities can.

I like the, admittedly nostalgic view of scientists who do not distinguish so sharply as we do these days and consider physics and mathematics as two aspects of the same thing. I mean, a circle is a circle:

• It does not exist. (Nature)
• Except as an idea of Plato. (Philosophy)
• $re^{it}$ however, is good enough to be useful. (Physics)
• $re^{i\varphi }$ however, is good enough to be useful. (Mathematics)

The only difference is time (t) versus angle ($\varphi$). And everyone far enough from QM does not really care about this different perspective.

 

[apostolosdt]

We, who studied physics, are not in a war with mathematicians—-for goodness sake! But please, PLEASE, let us remember that physics is an experimental science, and yes it’s full of errors, unanswered questions… perhaps the worst is yet to come!

Speaking of experiments, an “experiment” is going on for more than forty years now, superstring theory. Mathematicians’ contribution is unquestionable; let’s await to witness progress.

 

[Astronuc]

What are the "real" problems in physics? Can these problems we solved solely by mathematicians? What are your thoughts on that?

Knowing that one does not know what one does not know. It applies to a current situation of some physics (a natural phenomenon) that has been overlooked, probably because it was/is unexpected.

The tendency to present a jungle of equations as a well established theory. The problem is real: you have to spend two years to understand the equation jungle, just to realize that there is no clear directing idea behind them. And the worse is that you can even not criticize the theory, because you will be considered as an idiot for not understanding the (alleged) idea behind the equations. The book of de Groot I am trying to study may be an example of that (not sure, I'm just beginning). That's bad for the physics, because some problems may be considered to be solved while they are actually not

Not so much 'well-establish', but perhaps established enough to solve a problem (behavior of a physical system in which a multitude of phenomena are occurring, and one would like to describe different aspects in an economical way). Economical can mean solving a problem on a workstation (multicore), or a small cluster or work stations, and not a supercomputer with 100,000 cores, and I want an answer within 24 hour (or less ideally) and not a week, or weeks, or longer. And the solution shouldn't take MWh (MJ) of energy, but maybe many kWh (KJ).

Is the argument about how physicists use mathematics or describe the mathematics (or mathematical tools) employed in solving a problem, or describing a problem.

When I began studying physics in earnest, in high school then university, I quickly realized that I needed a firm background in mathematics, at least calculus, and then advanced calculus, but I didn't seen the need for subjects like group theory.

Consider some 'small' problems, or pieces of physics, involving photon interactions in matter, and simultaneously, interactions of electrons (and positrons) in the same matter, and the response of that matter to the population of photons, electrons and positrons moving through it (I haven't introduced neutrons yet). I have dozens of elements (atoms) jiggling about. Ideally they stay near there initial location before the photons, electrons, positrons (and neutrons) starting flying about. And there's more physics to add - transmutation of atoms (one element to another) and cations wanting to meet up with anions.

Consider the mathematics in this text describing the theory applied in the PENELOPE code.
https://www.oecd-nea.org/upload/doc...mulation_of_electron_and_photon_transport.pdf

It only covers part of the physics of photons, electrons, positrons (and doesn't involve neutrons, atomic displacements/diffusion, nuclear transmutations, nuclear fission, chemical reactions). The output is generally the energy deposition (dose) in some structure of matter, which could be simply a volume of water, air, human tissue, medical plastics, a detector material (NaI, GeLi, . . . ), semiconductors, ceramics (e.g., SiC) stainless steels, Zr-alloys, nuclear fuel (UO₂). When high Z materials are involved, the problem takes hours to run on a small PC (Intel I7-6600U CPU @ 2.60 GHz).

PENELOPE has been incorporated into GEANT4.
https://twiki.cern.ch/twiki/bin/genpdf/Geant4/LowePenelope
https://twiki.cern.ch/twiki/bin/view/Geant4/LowePenelope

Another code similar to PENELOPE is EGS5 (5th version of EGS, Electron-Gamma Shower).
https://rcwww.kek.jp/research/egs/egs5_manual/slac730-160113.pdf

Just one example of application - https://www.sciencedirect.com/science/article/pii/S027288422030883X

I'm interested in the effects of radiation on the evolution of structural materials and their performance in rather aggressive environments, and in particularly, how high energy gamma rays (and coincident/consequential electrons/positrons) change the atomic microstructure (crystal lattices and grain structure) of structural materials. The above codes PENELOPE does not yet consider photoneutron or photofission reactions, which only become significant in the presence of very light atoms (isotopes of Z=1 through Z=8) and heavy atoms (Z≥92), nor have the codes. EGS5 does appear to do photoneutron reactions.

Nuclear reactor core simulation codes do not consider photonuclear reactions necessarily, and in some cases, the cross section data libraries are incomplete.

Now if we turn to simulating the evolution of all the stars (do we consider the influence/perturbations of planets?) in all the galaxies in the universe over say a billion years - that's a much bigger problem. Or we could pick a small problem and just do all the stars in the galaxy in which we find ourselves. Or we pick a piece of a large problem and see if our mathematical models are sufficiently rigorous and robust to predict the behavior and outcome (describe the physics) we observe now and in the future.

 

[Astronuc]

So, how should the derivation of the propagation of heat be written according to you?

As simply as possible without sacrificing too much accuracy. Note that there are competing effects: conduction, convection and radiation.

 

[Astronuc]

How about a simpler problem looking to be solved.

EXISTENCE AND SMOOTHNESS OF THE NAVIER–STOKES EQUATION
https://www.claymath.org/sites/default/files/navierstokes.pdf

from
https://www.claymath.org/millennium-problems/navier–stokes-equation

It's not so simply give that turbulence is a bit chaotic.

A lot has been written about the challenge problem. Yet - not much progress in a solution.

https://www.quantamagazine.org/mathematicians-find-wrinkle-in-famed-fluid-equations-20171221/