Why are Physicists so informal with mathematics?

[DrClaude]

There is a Millenium Prize for the Navier–Stokes existence and smoothness problem. Should all computational fluid dynamics stop until this problem is solved?

 

[martinbn]

There is a Millenium Prize for the Navier–Stokes existence and smoothness problem. Should all computational fluid dynamics stop until this problem is solved?

The complaint was about lack of rigour, not about the existence open problems. Otherwise he would be complaining about mathematics too.

 

[russ_watters]

The complaint was about lack of rigour, not about the existence open problems. Otherwise he would be complaining about mathematics too.

I believe @DrClaude 's point is that CFD is a cheap/ugly shortcut to avoid dealing with that unsolved problem. It's The Wrong Way To Do Things and maybe we should stop until someone figures out The Right Way.

I guarantee any stroke QM could give a mathematician, an engineer could make worse.

 

[DrClaude]

The complaint was about lack of rigour, not about the existence open problems. Otherwise he would be complaining about mathematics too.

I must say I have some trouble understanding the difference between the two in this case. If you can't prove existence and unicity, doesn't the use of the equation show a lack of rigor?In any case, physicists are only interested in whether it works or not (i.e., does it model nature), and leave the details to mathematicians (*cough* Dirac delta *cough*). While the work of mathematical physics is valuable, physics itself needs to satisfy itself with "it works" otherwise we would get nowhere, especially when learning the material.

 

[martinbn]

I must say I have some trouble understanding the difference between the two in this case. If you can't prove existence and unicity, doesn't the use of the equation show a lack of rigor?

No, why would it be lack of rigor? An unsolved problem can be rigorously and precisely formulated. (This is not important but the problem is about the long time behavior. Do smooth solutions exist for all ##t##, or do they develop some kind of singularity in finite time? Short time weak solutions are known to exist.)

In any case, physicists are only interested in whether it works or not (i.e., does it model nature), and leave the details to mathematicians (*cough* Dirac delta *cough*). While the work of mathematical physics is valuable, physics itself needs to satisfy itself with "it works" otherwise we would get nowhere, especially when learning the material.

This is a better example. Before distributions were introduced what Dirac did was not rigorous. An example of lack of rigor would be a physics text that uses Dirac-like "functions" with only a vague and intuitive explanation. (One of course, can argue that it may be better for the intended audience.)

My problem with the OP is that when he tries to present how things should be done, he is even less rigorous than any physics text or lecture.

 

[TurtleKrampus]

@TurtleKrampus There's a branch of physics called mathematical physics whose purpose is to formulate theories of physics by fully respecting the rigors of various branches of mathematics. It's not at all compulsory, or should I say it's not very common that universities have staff doing research and teaching in mathematical physics. Therefore, for lectures delivered to students, most of the staff use the "standard semi-rigor" of physics textbooks.

For example, none of the exposé here https://en.wikipedia.org/wiki/Ehrenfest_theorem uses the methods of mathematical physics. The theorem of Ehrenfest was proven within the realm of mathematical physics only in 2009 by Friesecke and Schmidt. arXiv:1003.3372v1.

Returning to your dismay, there are, if you want to learn theories of physics properly using mathematics, you can very well do so by opting to books instead of lecturers. For example, classical mechanics is neatly covered either by Abraham & Marsden or by V. Arnol'd, I don't think there's anything sketchy or less rigorous in their books.

Yeah, those books (and other similar material) will probably be how I end up studying physics this semester. Thanks

 

[TurtleKrampus]

There is a Millenium Prize for the Navier–Stokes existence and smoothness problem. Should all computational fluid dynamics stop until this problem is solved?

No? I don't see how this relates to what I wrote...

 

[PeroK]

Yeah, those books (and other similar material) will probably be how I end up studying physics this semester. Thanks

I think that the recommendation of the Abraham & Marsden book is shockingly bad advice. It's in no way suitable as an introduction to physics.

If you really can't accept physics as it is taught by physicists at your university, then you should find some way to drop the subject and stick to mathematics.

It's your decision, of course, but I think it's worth saying that you may be jeopardising your degree by a vain search for mathematical purity in physics.

 

[TurtleKrampus]

My problem with the OP is that when he tries to present how things should be done, he is even less rigorous than any physics text or lecture.

I mean, all I want is to be presented with a model, which probably exists for the well known, well studied, topics that I will be studying in my physics class. I presented my ideas on a possible formulation of a model of Newtonian Mechanics, this isn't too say that the things I wrote have to work right off the bat, rather I believe that the overall ideas may possibly be expanded on.

Also I call bull on

less rigorous than any physics text or lecture.

Intuition only based arguments are not real arguments, they should either be acknowledged to be motivated assumptions, not used, or have some reference to it being true regardless of intuition. And I've heard several accounts of professors using Physical intuition only arguments to skip proving something in mathematics... Don't get me wrong though I don't have anything against saying something like "this turns out to be true AND agrees with what we believe should be morally right", even if a proof isn't presented.

 

[TurtleKrampus]

I think that the recommendation of the Abraham & Marsden book is shockingly bad advice. It's in no way suitable as an introduction to physics.

If you really can't accept physics as it is taught by physicists at your university, then you should find some way to drop the subject and stick to mathematics.

It's your decision, of course, but I think it's worth saying that you may be jeopardising your degree by a vain search for mathematical purity in physics.

Physics is mandatory here, so I unfortunately cannot drop out of it even if (very much so) wanted to, though thankfully it's only a semester.
I also don't believe that I'd be jeopardising my degree, I've skimmed the class notes / previous years exams, and the class seems to be one of the easiest ones this semester so hopefully I should be fine as long as I attend the classes.

 

[BWV]

Perhaps physics should learn from financial economics - the models there are mathematically rigorous and the discipline even has its own theorems. The failure of these models to either explain or predict real-world phenomena provides no impediment to plumbing their mathematical depths

 

[vela]

This thread reminds me of the joke where the punchline is that mathematicians take a long time to come up with an answer that is 100% correct but still utterly useless. I think in most contexts, insisting on mathematical rigor to the degree the OP wants is a waste of time. It doesn't really add much, if anything, and detracts from the goal of teaching the physics.

 

[symbolipoint]

This thread reminds me of the joke where the punchline is that mathematicians take a long time to come up with an answer that is 100% correct but still utterly useless. I think in most contexts, insisting on mathematical rigor to the degree the OP wants is a waste of time. It doesn't really add much, if anything, and detracts from the goal of teaching the physics.

Very interesting comment that is. Some situations make certain points important and other points unimportant. This seems to be the characterization going on in the topic. Different people have their own ways of trying to say the same thing.

 

[TurtleKrampus]

Perhaps physics should learn from financial economics - the models there are mathematically rigorous and the discipline even has its own theorems. The failure of these models to either explain or predict real-world phenomena provides no impediment to plumbing their mathematical depths

You didn't have to discredit financial economists... What approach do you suggest they should use?
Making a model gives a very good way to describe the assumptions the author used to study something

 

[andresB]

This thread reminds me of the joke where the punchline is that mathematicians take a long time to come up with an answer that is 100% correct but still utterly useless.

Like the very slow converging analytical solution for the three-body problem?

 

[AndreasC]

Given two inertial reference frames (both being isomorphic to ##\mathbb R^3##) ##R_1,R_2## there exists some ##A \in SO(3)## and ##v \in C^2(\mathbb R, \mathbb R^3)## with ##v'' = 0## such that the projection of the position of some point mass ##r## to ##R_2##, which I'll denote ##\pi_2(r)##, factors through ##R_1## from it's projection of its position in ##R_1##, which I'll denote ##\pi_1(r)##, in the following way:
$$\pi_2(r) = A\pi_1(r) + v$$
I believe that this enlarges into time being invariant across the reference frames, since our transformation is (in terms of ##\pi_1(r)##) time invariant.
There are some minor things that I wrote, for example we can consider the domain of the translation to be an interval, but I wrote it for the sake of simplicity. I don't know if it's possible for reference frames to rotate through time, but if so we just replace A with a continuous function onto SO(3). This is a first thought into how I'd go about writing things, there are probably many errors here..

I would expect that after spelling out this unwieldy thing, you would appreciate why physicists just write t=t'.

 

[AndreasC]

Yes, t = t' makes no sense to me

But you said it means time is absolute. So you understood it. So how does it make no sense? I don't even think this is a math vs physics rigor thing. Have you never seen in pure maths x being the coordinates in one system, x' being the coordinates in another?

 

[AndreasC]

No, why would it be lack of rigor?

Because we are using approximate solutions and an approximate solution is not rigorous unless you know an actual solution exists, and you have some kind of an error estimate for your approximate solution.

 

[TurtleKrampus]

Like the very slow converging analytical solution for the three-body problem?

That answers the question of whether or not we can find an analytical solution. It's not done to be useful for approximations

 

[TurtleKrampus]

I would expect that after spelling out this unwieldy thing, you would appreciate why physicists just write t=t'

Preferences I suppose

 

[AndreasC]

You didn't have to discredit financial economists... What approach do you suggest they should use?
Making a model gives a very good way to describe the assumptions the author used to study something

Coming up with detailed mathematical models is a waste of time when your model is completely off base. In many social sciences, it is recognized that mathematical models have very limited applicability so they don't use them. Economists want to pretend they are somehow different. They are not, they study social constructions, and social constructions are primarily controlled by social forces which are not well described by such models. That's not to say they have no applicability whatsoever, but it feels like a lot of it is just a thinly veiled game. To put it another way, Alexander the Great didn't study knot theory, he simply cut the Gordian knot.

Physics is not quite the same, because there quantitative mathematical models are very applicable. However, the full weight of MODERN mathematical formalism would only weigh down most areas of physics. That's not to say there are not areas in which it is profitable to work that way, but for others it is just a burden, either because the formalism clouds the intuitive ideas, because the formalism is unnecessarily complicated or even non-existent as of yet, or because the model is already kinda "wrong", and rigorous conclusions from an unsound model are still unsound.

In my opinion physics does need greater unity with mathematics, which does mean some linguistic changes should happen in physics. But that doesn't mean going to the other extreme, and it also works both ways, ie mathematics should steer a bit closer to physics as well. After all, that is how it developed historically, and still does to a certain extent. The modern standards of "rigor" would never have happened of there weren't the older, far less rigorous "classical" mathematics, and those mathematics would never have developed had there not arisen many questions from physics, questions which in turn could never have developed had physics waited for the final link in the chain! This is not just a historical artifact, it still happens today. Rigorous mathematical foundations of QFT is something that is important and interesting and close to my heart, but it is still a work in process which still has many gaping holes. Physicists could never have waited to use it until it somehow developed fully formed, skipping a bunch of steps in between.

 

[martinbn]

Because we are using approximate solutions and an approximate solution is not rigorous unless you know an actual solution exists, and you have some kind of an error estimate for your approximate solution.

May be people need to say what they mean by rigorous first. Here is an example: you have an equation, someone proves a uniqueness theorem, but noone has proven existence yet. Do you think that the uniqueness theorem is not rigorous?

 

[DrDu]

If you are complaining about physicists not doing rigorous math, what should I complain about, having to work everyday with biochemists? Its your job as a mathematician to bring physical ideas into Bourbaki style, not that of a physicist (they tend to fall immediately asleep with this task).

 

[martinbn]

If you are complaining about physicists not doing rigorous math, what should I complain about, having to work everyday with biochemists? Its your job as a mathematician to bring physical ideas into Bourbaki style, not that of a physicist (they tend to fall immediately asleep with this task).

Just to point out that not all mathematicians think the same way.
https://www.math.fsu.edu/~wxm/Arnol...thematics that is,ministers) and of the users.

 

[TurtleKrampus]

Coming up with detailed mathematical models is a waste of time when your model is completely off base.

A model is just a way to formalize your ideas.
Creating a model allows you to call things theorems, based on the axioms (i.e. assumptions).
There is no loss in generality by creating a model, unless you assume the existence of some type of object / property which hasn't been made rigorous in the language of mathematics, but I assume, with things like stochastic calculus there won't be many things like this.

Ultimately the main limitation with a financial model will be on the influence of real time events, which are in general hard to predict (something like an Elon Musk tweet can influence, and has influenced before, Tesla's stock prices).

Physics is not quite the same, because there quantitative mathematical models are very applicable. However, the full weight of MODERN mathematical formalism would only weigh down most areas of physics.

I get that for the development of physics creating a model with rapidly changing beliefs / assumptions is not a good thing. What I argue is that for topics in physics which have been explored really well, and that do in fact have faithful models, I argue that introducing one of those models can be a good thing, specially when the people you're exposing the material to study math.

ie mathematics should steer a bit closer to physics as well. After all, that is how it developed historically, and still does to a certain extent. The modern standards of "rigor" would never have happened of there weren't the older, far less rigorous "classical" mathematics, and those mathematics would never have developed had there not arisen many questions from physics, questions which in turn could never have developed had physics waited for the final link in the chain!

Mathematics has become independent of Physics, topics that are closest to my heart, i.e. something like abstract algebra, Galois theory, representation theory, and to some extent category theory, wouldn't really gain anything by steering into Physics specifically (there are a things that can be motivated by Physics like studying the Heisenberg groups, but the theory itself doesn't really evolve by steering into Physics. In fact Physics isn't special in this aspect, a lot of branches of mathematics are influenced by other Sciences like game theory & Biology and PDEs & Chemistry).

The modern mathematician is probably less restricted to rigor than you might believe. Somewhat counterintuitively for new topics, you often look at a property some things satisfy and then write definitions based on constraints you needed to arrive at that, which ultimately become theorems. That is to say, the theorem in new areas often predates the definition. Or if you'd like, the definitions are motivated by the theorems, and often evolved over time.

When the theory becomes sufficiently well studied the first textbooks are written about it.
There are also experimental mathematical magazines, which expose patterns & possibly corresponding conjectures to "fit" those patterns.

 

[AndreasC]

May be people need to say what they mean by rigorous first. Here is an example: you have an equation, someone proves a uniqueness theorem, but noone has proven existence yet. Do you think that the uniqueness theorem is not rigorous?

Uniqueness means if it exists, it is unique. That's rigorous even if existence is not known. But an approximation has to approximate something that exists.

 

[TurtleKrampus]

May be people need to say what they mean by rigorous first. Here is an example: you have an equation, someone proves a uniqueness theorem, but noone has proven existence yet. Do you think that the uniqueness theorem is not rigorous?

Uniqueness and existence are unrelated properties, though they are often paired up in differential equations.
Where you may expect something to be a solution of a differential equations, so first you may ask "does it make sense, do there exist solutions to this?" and secondly you may ask "ok, let's suppose that there are solutions, are they unique? so that I can use the unique solution to estimate future behaviour?".

Often times, asking if there's uniqueness is basically asking if we have specified enough initial / boundary conditions.

 

[TurtleKrampus]

Uniqueness means if it exists, it is unique. That's rigorous even if existence is not known. But an approximation has to approximate something that exists.

You can make approximations to problems without solutions, but yeah.

 

[AndreasC]

A model is just a way to formalize your ideas.

And some ideas gain nothing by being formalized, or are way too hard to formalize. When you only look at things that you can formalize, then you inevitably lose the big picture. And that is a huge part of modern economics, there is tons of literature of economists coming up with pristine mathematical analyses of a silly idealized scenaria that have nothing to do with the real world.

What I argue is that for topics in physics which have been explored really well, and that do in fact have faithful models, I argue that introducing one of those models can be a good thing, specially when the people you're exposing the material to study math.

On the one hand, yeah. On the other hand, in the particular example you brought up, there is no discernible reason why you should bury the simple intuitive idea of different observers counting time the same way under a mountain of unnecessary formalism. Why would you demand that people know about Lie groups and projections and whatnot before you explain this simple fact? You need these things AFTER you get this point through, if you want to generalize.

Physics, topics that are closest to my heart, i.e. something like abstract algebra, Galois theory, representation theory, and to some extent category theory, wouldn't really gain anything by steering into Physics specifically (there are a things that can be motivated by Physics like studying the Heisenberg groups, but the theory itself doesn't really evolve by steering into Physics. In fact to believe Physics isn't special in this aspect, a lot of branches of mathematics are influenced by other Sciences like game theory & Biology and PDEs & Chemistry).

About the other sciences, yes, I agree, and to rephrase, I meant mathematics should steer closer to the physical sciences in general, because that is the most fertile ground for inspiration, intuition, and new problems. Incidentally almost all of the topics you mentioned are VERY closely intertwined with physics, and even today there is cross pollination. For instance, representation theory is incredibly central to quantum mechanics (furthermore, a lot of techniques developed motivated specifically by its application in physics), and there are even very advanced, very abstract and pretty recent concepts (such as quantum groups) that are directly inspired by problems of physics. After all, 2 of the most famous open problems in mathematics today are explicitly related to physics.

That is to say, the theorem in new areas often predates the definition

Exactly. So why would you want to insist that getting into physics starts the other way round?

 

[AndreasC]

You can make approximations to problems without solutions, but yeah.

With the standard of rigor of physics, yeah. But how are you supposed to do that in "rigorous", "pure" mathematics? To make an approximation rigorous, you need a rigorous error estimate. To get a rigorous error estimate, it has to err against something. If that something does not exist, then it's very unlikely you would be able to come up with an error estimate. Maybe one that is conditional to the solution's existence. But then it would still be suspect.

 

[TurtleKrampus]

With the standard of rigor of physics, yeah. But how are you supposed to do that in "rigorous", "pure" mathematics? To make an approximation rigorous, you need a rigorous error estimate. To get a rigorous error estimate, it has to err against something. If that something does not exist, then it's very unlikely you would be able to come up with an error estimate. Maybe one that is conditional to the solution's existence. But then it would still be suspect.

Im on my phone, but for example x² -2 has no rational roots, but we can approximate it with rationals. We just to talk about something similar to a error estimate of the difference of the squares, |x² -y²|, and substitute y² with 2.

Usually (? Not sure, not my area of interest but I assume so) this type of problem is solved using a type of completion. In this case a Pythagoric completion works to find all square roots.

Either way, I can't really talk, as I said I dunno. Maybe it does really only make sense if there exists some extension where you have existence

 

[AndreasC]

Im on my phone, but for example x² -2 has no rational roots, but we can approximate it with rationals. We just to talk about something similar to a error estimate of the difference of the squares, |x² -y²|, and substitute y² with 2.

Usually (? Not sure, not my area of interest but I assume so) this type of problem is solved using a type of completion. In this case a Pythagoric completion works to find all square roots.

Either way, I can't really talk, as I said I dunno. Maybe it does really only make sense if there exists some extension where you have existence

Yeah you are right there. But we were talking more about Navier-Stokes etc. Generally in physics there are various physically motivated simplifications and approximations, and nobody bothers to check what the error is, or whether there even exists a solution, as in Navier-Stokes. Ideally, we should have an error estimate, and we should know there is a solution. But sometimes that is not available and we have to do physics anyways. For instance existence has not been established for most realistic QFTs (Yang Mills being a famous open problem in mathematics) but physicists proceed with perturbation theory and other approximations regardless. Despite the fact that we don't know what the Hilbert space of these theories is, we write down state vectors and perform operations with them. It's not ideal but it's what we've got so far.

 

[TurtleKrampus]

And some ideas gain nothing by being formalized, or are way too hard to formalize. When you only look at things that you can formalize, then you inevitably lose the big picture. And that is a huge part of modern economics, there is tons of literature of economists coming up with pristine mathematical analyses of a silly idealized scenaria that have nothing to do with the real world.

I don't know financial economists so I really can't speak on that. Perhaps you should talk to one yourself if you believe that they're hindered, or revolutionize the field yourself.

On the one hand, yeah. On the other hand, in the particular example you brought up, there is no discernible reason why you should bury the simple intuitive idea of different observers counting time the same way under a mountain of unnecessary formalism. Why would you demand that people know about Lie groups and projections and whatnot before you explain this simple fact? You need these things AFTER you get this point through, if you want to generalize.

I mean, I don't think anyone cares about the physical / philosophical implications of time invariance (on my class that is), we will only use it for calculations at which point we'll have to write what it means in some sense that's compatible with what we'll be trying to do (like convert from a reference frame to another).
Maybe you don't see this, because you already have an understanding of how it works, but a sentence ##t=t'## can be ambiguous by itself. What does it mean mathematically, in the sense of how do I use it when doing algebra? (rhetorical questions)
But answering that questions already gives you a way to write what you mean with ##t = t'## which is more rigorous.

About the other sciences, yes, I agree, and to rephrase, I meant mathematics should steer closer to the physical sciences in general, because that is the most fertile ground for inspiration, intuition, and new problems. Incidentally almost all of the topics you mentioned are VERY closely intertwined with physics, and even today there is cross pollination. For instance, representation theory is incredibly central to quantum mechanics (furthermore, a lot of techniques developed motivated specifically by its application in physics), and there are even very advanced, very abstract and pretty recent concepts (such as quantum groups) that are directly inspired by problems of physics. After all, 2 of the most famous open problems in mathematics today are explicitly related to physics.

I think that you're mixing a few things, there are very few areas in mathematics where sciences give intuition on how to solve problems. What I meant by motivation is that most mathematicians don't want to study some object that has no known interest (also who'd want to fund that?).
A group appearing in something like Chemistry gives mathematicians motivation to study that specific group*, and some times these groups end up having interesting properties (and would overall just be easier to fund).
(* contrary to popular (?) belief most mathematicians don't want to do something completely without uses, though with uses I'm also counting uses within mathematics itself, the motivation of study is not just motivated by science).
Though to be honest I'm kinda interested in the Physics inspired techniques you're referring to

 

[TurtleKrampus]

Yeah you are right there. But we were talking more about Navier-Stokes etc. Generally in physics there are various physically motivated simplifications and approximations, and nobody bothers to check what the error is, or whether there even exists a solution, as in Navier-Stokes. Ideally, we should have an error estimate, and we should know there is a solution. But sometimes that is not available and we have to do physics anyways. For instance existence has not been established for most realistic QFTs (Yang Mills being a famous open problem in mathematics) but physicists proceed with perturbation theory and other approximations regardless. Despite the fact that we don't know what the Hilbert space of these theories is, we write down state vectors and perform operations with them. It's not ideal but it's what we've got so far.

I am completely out of my water here lmao

 

[AndreasC]

I mean, I don't think anyone cares about the physical / philosophical implications of time invariance (on my class that is), we will only use it for calculations at which point we'll have to write what it means in some sense that's compatible with what we'll be trying to do (like convert from a reference frame to another).

Well then this notation is definitely fine! In classical mechanics the position of a point particle in space is given by coordinates x,y,z, which are continuous functions of time t. In other words, its path through space is simply a curve. In another reference system, these are x',y',z', parameterized by time t'. But what you learn here is that you can use the same parameter for both. If you want to convert between reference frames in classical mechanics, you won't generally need all that stuff you mentioned, you will just need to make coordinate changes.

This gets more difficult when you get to special and general relativity. In that case you have to work with spacetime, in other words you have to add the time t as a coordinate, instead of just using it as a parameter, and it changes from frame to frame. At that point you are dealing with curves in non-Euclidean spaces, and to change reference frames you need differential geometry etc.

Though to be honest I'm kinda interested in the Physics inspired techniques you're referring to

What I meant was not so much that problems were solved using physics intuitions, but rather that people who worked on the problems were often physicists or mathematicians very close to physics, who came up with techniques tailored towards physics applications, which later also found other uses. For representation theory, Wigner was a physicist who worked a lot on it for example, von Neumann was another influential one, and then you have people like Cartan who were very motivated by physics problems.

However, there is a bit of the other thing (physics inspired techniques used to solve math problems) as well. Ed Witten has done some of that (and he has a Fields medal for it). There is also various things in mathematics where their name betrays their physics origins (entropy, quantum groups, "sources" in differential equations, etc).

Since you are learning classical mechanics and want a mathematical treatment, you would like V.I. Arnold's book. By the way, he was one of the mathematicians arguing for mathematics to lean a bit closer to physics. But of course his language in the book is mathematical, and very rigorous throughout.