Do physics books butcher the math?

[micromass]

That's a very good point, Anti, but it's actually similar to the point I was making. Just as you said that most new fields of research are deemed to be too abstract to really be applied to anything, the same could be said about new fields of mathematics when they are first introduced. While subjects like calculus and differential equations saw use in physics and other disciplines as soon as they were created, I'd be willing to bet large sums of money that many of the more modern mathematics were of the style to see no real practical use at first, but then physics "caught up", if you will. And the same could be said about many of the highest features of theoretical mathematics today.

The point I was trying to make was that it's folly to discard a discipline because "there aren't any applications for it right now", especially if you're going to embrace things like high-energy particle physics.

Yes, there are plenty examples of things without applications in mathematics that ended up being useful. For example, functional analysis was of limited use. But then it suddenly saw applications in the fundamentals of QM and QFT, and even image processing. Number theory was long said to be completely useless (and the mathematician Hardy was extremely proud that he did such a useless thing!), but now it has found applications in cryptography. Mathematical logic was also of limited use outside of mathematics, but is now important in computer science.

One mathematicians whom I personally know did research on units in group rings. A completely useless topic. Until she got an email from an engineer designing telephone wires or something and was asked to explain some things about her research and perhaps even collaborate.

There are really myriads of example of math which eventually ends up being applied in something completely unexpected.

That said, many research in mathematics is done because it are fundamental questions which are good to know, not just because it is applicable. For example, Liouville's theorem answers a very natural question on which functions have elementary antiderivatives on which do not: http://en.wikipedia.org/wiki/Liouville%27s_theorem_(differential_algebra )
I would think this is good to know regardless of applications. It did eventually find applications in computer software which use it to evaluate many antiderivatives.

Many mathematics is also of historical interest. Things like squaring the circle are not immediately useful, but are historical questions which are good to know. Still, it ended up developing group theory (by Galois), which is now used in physics and in much more.

Differential geometry was completely useless when it was first invented. They used it to settle historical questions about the parallel axiom. Guess what? It is now used in relativity and even aerospace engineers use it.

 

[AnTiFreeze3]

That's a very good point, Anti, but it's actually similar to the point I was making. Just as you said that most new fields of research are deemed to be too abstract to really be applied to anything, the same could be said about new fields of mathematics when they are first introduced. While subjects like calculus and differential equations saw use in physics and other disciplines as soon as they were created, I'd be willing to bet large sums of money that many of the more modern mathematics were of the style to see no real practical use at first, but then physics "caught up", if you will. And the same could be said about many of the highest features of theoretical mathematics today.

The point I was trying to make was that it's folly to discard a discipline because "there aren't any applications for it right now", especially if you're going to embrace things like high-energy particle physics.

Gotcha, looks like I didn't really read your post too closely (my natural tendency is to disagree first). Glad to see that we agree on this.

 

[Arsenic&Lace]

I have not forgotten this thread, I am just currently pondering some of the mentioned applications of algebraic topology.

A few miscellaneous thoughts:
Particle physics does have useful applications in medicine, and accelerator physics has many industrial applications. Major discoveries in fundamental quantum field theory so far have no applications that I am aware of, but field theory itself is very helpful in condensed matter physics; ironically, then, theoretical physicists as mathematicians are helpful to other theoretical physicists.

Have I actually said that pure mathematics has no applications ever? If so that was a mistake and unintentional; I was rather attempting to argue that the assumptions that a). mathematical objects, ontologically speaking, can be distinguished meaningfully from their applied origins as a formal language and b). extremely high levels of rigor as an artifact of assumption a) significantly reduce the usefulness of the pure discipline. If I haven't mentioned it already, it is exceedingly clear that the mathematical culture of the 18th/19th century was incredibly productive and useful to others.

The latter point has been greeted with much derision and leads to the common analogy "math is to physics as physics is to engineering" as a counterargument, but this reaction is caused by assumption a). Physics is not merely a tool for engineering; it is a science. Mathematics is a formal language invented by humans to solve problems. The process of pure mathematics seems to be the process of studying extremely formalized arguments in this language in fastitidous detail. I would not say that they shouldn't do this, but I do not understand what the point is of inventing a set of premises and then studying highly formalized arguments which do not apparently pertain to anything. When I stated that one should look at the literature to see that this underlying machinery is useless to physicists, this is precisely what I meant; some good ideas have come from the math department, such as continuous groups, but the accompanying freight train of theorems and their meticulous proofs built in highly formal terms out of sets and stilted mathematical prose does not seem to come along for the ride.

Treating it as a science results in misleading statements such as "Physicists butcher the mathematics." No, they do not. The mathematics that they use is a formal language in which to express their ideas; if they are satisfied that these ideas have been clearly communicated then that should be the end of the story. In fact, there is a distinction between the ideas and the language used to express them. I can use an informal language to describe Newton's law "The rate of change of the rate of change of the position of an object is inversely proportional to its mass and directly proportional to the net force" or a much more efficient formal expression (a=F/m). Complaining that the precision of the language used is not high enough is irritating as it is, but even worse, mathematicians will state that the expressions do not communicate what physicists think they do. An integral to a physicist is not actually the exact same object as an integral to a mathematician, although they overlap and occasionally coincide.

The reason I dislike the notion of mathematicians as experts in mathematics is because it implies that these linguistic structures belong to the mathematicians and that the technical rules and formal arguments mathematicians concoct regarding them in lieu of application should bear some importance to people trying to use them. It is not apparent to me that the technical rules and formal arguments have any bearing whatsoever on people trying to use these constructions, and it is this aspect of the enterprise I see no evidence for having been consulted by individuals outside math departments. I don't need to open up an advanced calculus book when I'm playing with limits... ever. Mathematicians do engage in the process of abstraction though, which occasionally is useful.

An analogy would be like somebody saying "I'm feeling good today.", which makes perfect sense to most people, and having someone else respond by telling them that this statement is meaningless; according to their technical rules composed in a bubble somewhere at a university, yes, this statement is meaningless. In the real world, it isn't.

 

[Fredrik]

...misleading statements such as "Physicists butcher the mathematics." No, they do not. The mathematics that they use is a formal language in which to express their ideas; if they are satisfied that these ideas have been clearly communicated then that should be the end of the story.

No one is saying that every non-rigorous argument is "butchering" the math, or that every physics book is doing it. The objection is against those who dumb things down to the point where the ideas aren't even close to being clearly communicated.

 

[1MileCrash]

You make it sound as if mathematical rigor is some sort of arbitrary convention.

Sure, you may not need an advanced calculus book to use limits. That's the difference between using math and doing math. The people who would read an advanced calculus book are not doing it so they can use anything. The "tool" that you call mathematics wasn't handed to Moses on a stone tablet. It was developed by people, the same type of people that would read an advanced calculus book. New mathematics can never be developed if we are only concerned with "using" it.

 

[rubi]

I don't need to open up an advanced calculus book when I'm playing with limits... ever.

And why don't you need to do that? Because you have learned the rules for limits like for example $\lim a_n \cdot b_n = a \cdot b$ whenever $a_n\rightarrow a$ and $b_n\rightarrow b$. And why can you be sure that this rule always works? Because some mathematician (probably Weierstrass or Cauchy) has proved it using the $\epsilon$ definition of a limit. A rigorous proof is absolutely necessary to establish the truth of that limit rule. You have been taught this rule, because it has been proved to work, not the other way around.

 

[dkotschessaa]

Another thing that is rarely mentioned is that some math has "applications" to other math, which in turn may have "real world" applications. A lot of math is building bridges, finding connections, seeing how a problem in one place can be solved in another.

-Dave K

 

[WWGD]

As a mathematician in training, I wish I knew enough physics to be able to butcher it. Because that would imply I knew something beyond basic undergrad level.

WWGD:
What Would Gauss Do?

 

[WannabeNewton]

Honestly this is all a matter of taste. I don't see much else to it. There are some "physics for mathematicians" books that completely butcher physics, like "General Relativity for Mathematicians"-Sachs and Wu, but that's ok because the goal of the book isn't to do physics justice. Indeed often times they don't. It is to present the underlying formalism of the physical theory in a rigorous way accessible to mathematicians. The way the aforementioned book presents GR is almost comical because of its high level of rigor-that just isn't the way physics is done. But regardless I still managed to learn a lot from it because it presents ideas in a very clear, rigorous, and logical fashion that quite literally all true GR books with the exception of Wald fail to do. A more extreme example is Spivak's book on physics which is just absolutely laughably terrible. I don't even know why it was written.

In that same light, physics books don't cater to mathematicians. They try to present things in a way that is accessible to physicists and for such a purpose rigorous mathematics is absolutely useless. But that is not to say they shouldn't present mathematics clearly. Fredrik gave one example of this but there are countless in the physics literature. There's a difference between physics books not presenting math rigorously, which is fine, and presenting it in an unclear fashion, which is not fine because this has the terrible effect of making the underlying physics less coherent and more confusing to understand. Nowhere is this more pronounced than in typical QM books and in typical QFT books, the latter with regards to representation theory in particular. In this respect I almost always find myself turning to math books for clearer expositions and with good consistency find the physics much more understandable after getting a more lucid understanding of the math from proper pure math texts.

To say pure math is not useful to a physics student is just patently false unless you're of the persuasion who just takes mathematical results for granted and doesn't get frustrated by the cryptic explanations of math found in many physics books. The first chapter of Maggiore's QFT book was so frustrating for me because its development of the representation theory of the Poincare group was so unclear and laden with such poor physics terminology to the point of making me cry.

 

[atyy]

But regardless I still managed to learn a lot from it because it presents ideas in a very clear, rigorous, and logical fashion that quite literally all true GR books with the exception of Wald fail to do.

How about Hawking and Ellis, or Straumann? Or even Weinberg?

OK, maybe not Weinberg, but he's clear and logical, and has his eye on the quantum theory.

Incidentally, I would be interested to know when differential geometry books became mathematical. Did Riemann or Levi-Civita use the modern definitions, or did they use the physics "a tensor is a thing that transforms as ..." ?

A more extreme example is Spivak's book on physics which is just absolutely laughably terrible. I don't even know why it was written.

To show that it can be done

 

[atyy]

Arsenic&Lace is not qualified to defend his point, because he is a sophisticated mathematician in disguise, claiming that imaginary time and Grassmann numbers are intuitive. He is probably trying to bring a Trojan horse into the domain of physics.

Nonetheless, he is correct.

Typical condensed-matter physicist’s opinion:
“C’mon, there always is a speed of sound!”

Typical mathematical physicist’s opinion:
“Hmmm, interesting, but very difficult”

http://profs.if.uff.br/paraty07/paraty09/palestras/Eisert_2.pdf (p49, but their best slide is p57)

 

[WannabeNewton]

How about Hawking and Ellis, or Straumann? Or even Weinberg?

Hawking and Ellis is so dense I honestly can't tell if it is logical and clear or just pretentious. Straumann is unequivocally logical, clear, and rigorous: it's a brilliant physics book. I don't know why I forgot to mention it. And Weinberg is...well Weinberg. His books are about as clear as the air in China.

Did Riemann or Levi-Civita use the modern definitions, or did they use the physics "a tensor is a thing that transforms as ..." ?

I have no idea to be honest.

 

[atyy]

Hawking and Ellis is so dense I honestly can't tell if it is logical and clear or just pretentious. Straumann is unequivocally logical, clear, and rigorous: it's a brilliant physics book. I don't know why I forgot to mention it. And Weinberg is...well Weinberg. His books are about as clear as the air in China.

I've always wanted to visit.

 

[Matterwave]

Hawking and Ellis is so dense I honestly can't tell if it is logical and clear or just pretentious. Straumann is unequivocally logical, clear, and rigorous: it's a brilliant physics book. I don't know why I forgot to mention it. And Weinberg is...well Weinberg. His books are about as clear as the air in China.

Hey...hey hey... the air in Beijing and Shanghai and other major cities in China is polluted...but not through all of China! Especially not true in the forest and dessert areas.

 

[atyy]

Hey...hey hey... the air in Beijing and Shanghai and other major cities in China is polluted...but not through all of China! Especially not true in the forest and dessert areas.

I'd especially love to visit the dessert areas! http://en.wikipedia.org/wiki/Dessert

 

[Matterwave]

 

[WannabeNewton]

Hey...hey hey... the air in Beijing and Shanghai and other major cities in China is polluted...but not through all of China! Especially not true in the forest and dessert areas.

Mmm dessert.

 

[Matterwave]

Desert*...dang...

 

[micromass]

The reason I dislike the notion of mathematicians as experts in mathematics is because it implies that these linguistic structures belong to the mathematicians and that the technical rules and formal arguments mathematicians concoct regarding them in lieu of application should bear some importance to people trying to use them. It is not apparent to me that the technical rules and formal arguments have any bearing whatsoever on people trying to use these constructions, and it is this aspect of the enterprise I see no evidence for having been consulted by individuals outside math departments. I don't need to open up an advanced calculus book when I'm playing with limits... ever. Mathematicians do engage in the process of abstraction though, which occasionally is useful.

OK, then who are the experts in mathematics? Are you an expert in mathematics?

An analogy would be like somebody saying "I'm feeling good today.", which makes perfect sense to most people, and having someone else respond by telling them that this statement is meaningless; according to their technical rules composed in a bubble somewhere at a university, yes, this statement is meaningless. In the real world, it isn't.

Funny, I always thought that physicists were as bad as mathematicians in this respect. Look at some threads started in the physics section of PF, many start out by saying that their question is ill-posed.

Or see this video of a poor reporter asking what you feel when you put two magnets together.

https://www.youtube.com/watch?v=36GT2zI8lVA

 

[Arsenic&Lace]

OK, then who are the experts in mathematics? Are you an expert in mathematics?



Funny, I always thought that physicists were as bad as mathematicians in this respect. Look at some threads started in the physics section of PF, many start out by saying that their question is ill-posed.

Or see this video of a poor reporter asking what you feel when you put two magnets together.

https://www.youtube.com/watch?v=36GT2zI8lVA

Experts in differential and integral calculus can be found in math departments, engineering departments, physics departments and elsewhere. Once you've taken the first 3 semesters of calculus (and gotten A's, well maybe a B... at least if you've solved lots of elementary calc problems correctly!), you're an expert in elementary calculus. A math professor with a specialty in analysis who's taken two semesters of advanced calculus, four semesters of undergraduate and graduate real analysis, and publishes papers in I don't know PDE's or something, doesn't really know anything more than you do about calculus, because the knowledge s/he possesses cannot be meaningfully be described as "more calculus" since calculus is a tool for solving engineering/physics/other types of problems and the vast majority of what s/he uses cannot be used for this purpose and never will be.

Earlier in the thread somebody stated, essentially, that my usage of elementary calculus is contingent upon all of those very precise and rigorous proofs of theorems using epsilon's and deltas, Cauchy sequences and what not. It isn't. The rules of calculus, from the product rule to L'Hospital's rule, were figured out and used as much as a thousand years (MVT/Rolle's theorem) before these proofs were written. It is completely unclear what purpose this rigor actually serves.

Of course this thread is actually about how authors can explain things poorly or well. I've read very lucid explanations of mathematics by mathematicians and physicists, and very poor explanations of mathematics by mathematicians and physicists. I am attempting to define what butchering actually means. The best explanations of mathematics, pedagogically speaking, are among the least rigorous. If every engineering calculus student began in a bog of Dedekind cuts, set theory, and basic topology, it would do nothing for them as far as actually performing calculus except to confuse them; likewise, they thankfully spend hardly any time at all with epsilons and deltas.

I will say this though: I actually have a huge preference for learning mathematics over physics, generally speaking, and for solving mathematics problems. My favorite sections of physics textbooks are those which discuss theory; when matters turn to actually computing things, this can be somewhat interesting but is nevertheless often tedious , depending upon the problems you have to solve. I've been doing physics research in computation and theory for several years now, and there is a great deal of joy to be had before implementation (in my case through programming usually). I don't completely hate implementation and calculation and often enjoy it as well, but I have a vast preference for theory. In other words, if it had not occurred to me that pure mathematics was useless, I would have chosen to become a mathematician. I personally find a bog of Dedekind cuts, set theory, and basic topology to be a nice place to hang out.

Of course people have come up with cases where algebraic topology is not "useless"; I'm still exploring these, and if I'm convinced that they are right I might just consider trying to become a topologist.

 

[WWGD]

Experts in differential and integral calculus can be found in math departments, engineering departments, physics departments and elsewhere. Once you've taken the first 3 semesters of calculus (and gotten A's, well maybe a B... at least if you've solved lots of elementary calc problems correctly!), you're an expert in elementary calculus. A math professor with a specialty in analysis who's taken two semesters of advanced calculus, four semesters of undergraduate and graduate real analysis, and publishes papers in I don't know PDE's or something, doesn't really know anything more than you do about calculus, because the knowledge s/he possesses cannot be meaningfully be described as "more calculus" since calculus is a tool for solving engineering/physics/other types of problems and the vast majority of what s/he uses cannot be used for this purpose and never will be.

Earlier in the thread somebody stated, essentially, that my usage of elementary calculus is contingent upon all of those very precise and rigorous proofs of theorems using epsilon's and deltas, Cauchy sequences and what not. It isn't. The rules of calculus, from the product rule to L'Hospital's rule, were figured out and used as much as a thousand years (MVT/Rolle's theorem) before these proofs were written. It is completely unclear what purpose this rigor actually serves.

Of course this thread is actually about how authors can explain things poorly or well. I've read very lucid explanations of mathematics by mathematicians and physicists, and very poor explanations of mathematics by mathematicians and physicists. I am attempting to define what butchering actually means. The best explanations of mathematics, pedagogically speaking, are among the least rigorous. If every engineering calculus student began in a bog of Dedekind cuts, set theory, and basic topology, it would do nothing for them as far as actually performing calculus except to confuse them; likewise, they thankfully spend hardly any time at all with epsilons and deltas.

This is (at best possibly- ) true for calc at a basic level, i.e., for calc I,II, maybe for calc III , and this knowledge can take you pretty far -- that is evidence of the power of Calculus. But if you want/need to go further into advanced Calculus, a deeper knowledge of analysis is almost necessary. At the end of the day ,yours are little more than strong opinions, with little, if any rigor behind them, and , if you don't think Mathematics beyond the level of Calc. is necessary , just don't do it, and you're free to hold it and express your views. But don't claim yours is anything other than a strong opinion until you provide something that looks like actual evidence to support it. And I don't know of anyone who has advocated teaching Dedekind cuts to Engineering Calc. students; do you?

 

[Arsenic&Lace]

Ok WWGD, I'm all ears; as I've stated, I'd like an excuse to become a mathematician since I have a preference for the subject.

However, I simply do not believe that advanced analysis underpins anything other than the production of more academic mathematics; I cannot think of a single example where important theorems and extremely rigorous arguments born in the math department actually influenced applications in engineering and physics.

The jury is still out on some of this algebraic topology stuff people have mentioned in condensed matter physics and other disciplines, since I am coming up to speed on it. So far it is very difficult to keep track of just how much of the pure mathematics is necessary for these applications; if it appears to be very little (which it does), is this just because that little piece stands on the shoulders of countless sophisticated theorems? Or is the rigor just a meaningless illusion? I suspect the latter at present. My mind is subject to change, however.

 

[micromass]

Experts in differential and integral calculus can be found in math departments, engineering departments, physics departments and elsewhere. Once you've taken the first 3 semesters of calculus (and gotten A's, well maybe a B... at least if you've solved lots of elementary calc problems correctly!), you're an expert in elementary calculus. A math professor with a specialty in analysis who's taken two semesters of advanced calculus, four semesters of undergraduate and graduate real analysis, and publishes papers in I don't know PDE's or something, doesn't really know anything more than you do about calculus, because the knowledge s/he possesses cannot be meaningfully be described as "more calculus" since calculus is a tool for solving engineering/physics/other types of problems and the vast majority of what s/he uses cannot be used for this purpose and never will be.

So basically you're saying that you are as much of an expert in mathematics as somebody like Terrence Tao? OK... Well, since you're going to resort to completely insane positions such as this, I think I'll stop arguing with you.

Earlier in the thread somebody stated, essentially, that my usage of elementary calculus is contingent upon all of those very precise and rigorous proofs of theorems using epsilon's and deltas, Cauchy sequences and what not. It isn't. The rules of calculus, from the product rule to L'Hospital's rule, were figured out and used as much as a thousand years (MVT/Rolle's theorem) before these proofs were written. It is completely unclear what purpose this rigor actually serves.

Thousand years? I think your timeline is a bit messed up, but I'll let it slide since it's completely off-topic.

 

[micromass]

Ok WWGD, I'm all ears; as I've stated, I'd like an excuse to become a mathematician since I have a preference for the subject.

However, I simply do not believe that advanced analysis underpins anything other than the production of more academic mathematics; I cannot think of a single example where important theorems and extremely rigorous arguments born in the math department actually influenced applications in engineering and physics.

OK, what about wavelets? This is a book by one of the very founders of the theory: http://books.google.be/books?id=Nxn...ce=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
It looks rigorous enough, no? It is currently used in image processing.

Also, whether it is useful or not is a red herring. A lot of math is philosophical too. It is philosophically pleasing to base an argument on axioms and sound reasoning. It is what humans do. In that sense, math is also very important.

 

[Arsenic&Lace]

So basically you're saying that you are as much of an expert in mathematics as somebody like Terrence Tao? OK... Well, since you're going to resort to completely insane positions such as this, I think I'll stop arguing with you.



Thousand years? I think your timeline is a bit messed up, but I'll let it slide since it's completely off-topic.

In the first instance, beyond a shadow of a doubt, Terrence Tao knows nothing more about calculus than a good physicist does, because I would argue that modern analysis does not even constitute "more knowledge about calculus". If it really is more knowledge about calculus, then one could perhaps conceive of a use for something like Cauchy sequences outside of a pure math class. It is impossible to conceive of a usage for such a structure except to develop more philosophical academic "mathematics." It has nothing to do with calculus; it is a misnomer to describe such a topic as having anything to do with calculus, and utterly unclear how real analysis bears any relation, aside from philosophical, to something like calculus.

In the second case, Bhāskara II used and knew about Rolle's theorem in the 12th century. So a thousand years was a few hundred years off the mark, but it is nevertheless impressive that he was able to use it without having a professional, expert mathematician prove it first.

 

[disregardthat]

Arsenic, you are not giving proper credit to those devoted to the rigorous treatment of mathematics for what you have today. Historically, the lack of rigor was indeed a major problem, and mathematicians as well as physicists ran into trouble with inconsistencies. Once upon a time it was assumed that all functions were differentiable. Without the invention of epsilon/delta-definitions and proofs (or something with equal rigorous force), mathematics would have halted. We would not have seen much of the theory which is of extreme importance today. Without sufficient rigor, any attempt to come up with the necessary theory of today's physics will likely result in a useless mess of a theory with huge errors.

It may, in one sense, seem like much rigor indeed was not necessary for calculus. But this is because the type of non-rigorous arguments which survived are those which more or less could be made rigorous. We do not see today the erroneous arguments of yesterday, because it became apparent that they did not work and were simply expelled. A calculus student today is playing on a platform built on rigor, and if he can't see the edge he may (falsely) conclude that he's an expert. But, he is not, and he has only seen and understood a tiny part of mathematics, and does not realize the body on which it is based.

Quite frankly, your point of view reeks of ignorance.

 

[Arsenic&Lace]

Arsenic, you are not giving proper credit to those devoted to the rigorous treatment of mathematics for what you have today. Historically, the lack of rigor was indeed a major problem, and mathematicians as well as physicists ran into trouble with inconsistencies. Once upon a time it was assumed that all functions were differentiable. Without the invention of epsilon/delta-definitions and proofs (or something with equal rigorous force), mathematics would have halted. We would not have seen much of the theory which is of extreme importance today. Without sufficient rigor, any attempt to come up with the necessary theory of today's physics will likely result in a useless mess of a theory with huge errors.

I sincerely can think of no instance on the history of physics where these concerns slowed progress or resulted in lousy theories. Of course my knowledge of the topic is not exhaustive; can you think of one?

Quite frankly, your point of view reeks of ignorance.

Perhaps, but wouldn't it be more potent to just exhibit an example of where this rigor is valuable rather than just declaring me ignorant?

Does anybody have more expertise on algebraic topology and condensed matter physics? So far the only papers I have seen which employ it for experimentally realized results use an apparently minute amount of the subject, but I won't have taken a course in algebraic topology until after next semester and am not equipped to judge just how much is being employed. There are turgid, esoteric treatises written by the likes of (*wretch*) Ed Witten, but these appear to have no relation to reality.

EDIT: I forgot to finish my thoughts. The Ed Witten style pieces use very sophisticated maths but talk about things like anyons in 4+1 dimensions, so I don't consider them evidence that this stuff is actually helpful.

 

[micromass]

Not sure why you ignored my wavelet example?

Also, what about complex analysis? This certainly is useful in physics and engineering, no? It were the mathematicians who first made sense of complex numbers without really an application in mind. Without this, physics would certainly have been set back.

Or differential geometry on manifolds? This was invented as a rather useless generalization of curves and surfaces. It served as a counterexample to the Parallel postulate, but nothing really more. But now it is being used in general relativity and even aerospace engineering.

All of these things are very rigorous mathematics which are now being used. There would be a rather huge setback without these tools.

 

[WannabeNewton]

The point being made is that rigor is necessary in order to know the boundary between sensical and nonsensical. If you do calculations in physics haphazardly with no regard to rigorously proven results underlying the calculations and you end up with nonsense then you obviously present an example of how rigor is important for its own sake. There are quite a few examples of this in QM. They are academic but illustrative nonetheless.

Physics isn't just about getting computational results, be it haphazardly or not. It is also about gaining a deep conceptual understanding of physical theories and their structures. How would you do this if mathematicians didn't already prove rigorous results pertaining to these structures? It is ridiculously hilarious to assume rigorous mathematics has no usefulness in physics. Modern general relativity relies head to toe on rigorous results from differential topology to the point where relativists can't even investigate the theory conceptually or computationally without knowledge of these results which mathematicians had proven to exist earlier.

 

[disregardthat]

I sincerely can think of no instance on the history of physics where these concerns slowed progress or resulted in lousy theories. Of course my knowledge of the topic is not exhaustive; can you think of one?


Perhaps, but wouldn't it be more potent to just exhibit an example of where this rigor is valuable rather than just declaring me ignorant?

The very invention of calculus did not only forward physics to extreme lengths, but was also a huge leap in mathematical rigor. At the time, a lot of physicists were advocating an aristotelian view (and had been for hundreds of years)... You can draw the conclusions yourself.

 

[rubi]

Earlier in the thread somebody stated, essentially, that my usage of elementary calculus is contingent upon all of those very precise and rigorous proofs of theorems using epsilon's and deltas, Cauchy sequences and what not. It isn't.

You're just wrong. If you honestly think that rigorous proofs aren't required to establish the truth of mathematical statements, it just shows that you're not intelectually mature enough to understand it, yet. This isn't supposed to be an insult. Many students fail to understand this, when they first learn about it, so you're in good company. You're problem is rather that you have a strong opinion on things that you don't understand and instead of trying to understand them, you're just stubborn. We won't be able to convince you, since it takes years of study to develop the intellectual maturity that it takes to understand the requirement for the rigour in mathematics. Even people, who have been doing mathematics for a long time, get back to their analysis books after years, because they suddenly feel that they have acquired enough mathematical maturity to read them again and new learn things that they hadn't realized when they first read them. So if you claim that you're entitled to judge the necessity of rigour in mathematics, then this is highly questionable, to say the least.

It is completely unclear what purpose this rigor actually serves.

It is unclear to you. But your opinion isn't the measure of all things.

 

[gopher_p]

Much like science without methodical naturalism isn't science, mathematics without rigor is not mathematics.

 

[atyy]

Does anybody have more expertise on algebraic topology and condensed matter physics? So far the only papers I have seen which employ it for experimentally realized results use an apparently minute amount of the subject, but I won't have taken a course in algebraic topology until after next semester and am not equipped to judge just how much is being employed. There are turgid, esoteric treatises written by the likes of (*wretch*) Ed Witten, but these appear to have no relation to reality.

I'm hardly an expert, since I am the true non-rigourous guy here, not you. But roughly, there are two different sorts of topology in condensed matter physics.

(1) There is the topology of the integer quantum hall effect, involving Chern numbers. Topological insulators are generalizations of this idea.
http://www.physics.upenn.edu/~kane/pedagogical/WindsorLec2.pdf
http://physics.princeton.edu/~haldane/talks/dirac.pdf
http://www.bioee.ee.columbia.edu/downloads/2013/nature12186.pdf

(2) Then there is the topology of the fractional quantum hall effect, one sign of which is that the ground state degeneracy depends on the topology on which the Hamiltonian is placed. A proposed use of this sort of topology is in Kitaev's topological quantum computation.
http://stationq.cnsi.ucsb.edu/~freedman/publications/96.pdf
http://www.simonsfoundation.org/quanta/20140515-forging-a-qubit-to-rule-them-all/

From the Haldane slides above:
"The moral of this long story: suggests three distinct ingredients for success.
• Profound, correct, but perhaps opaque formal topological results (Invariants, braid group, etc)
• Profound, simple and transparent “toy models” that can be explicitly treated (The honeycomb Chern Insulator, the Kitaev Majorana chain, etc)
• Understanding the real materials needed for “realistic” (but more complex) experimentally achievable systems that can bring “toy model results” to life in the hands of experimentalist colleagues."

 

[Matterwave]

I have a question for Arsenic&Lace (or anyone else really): would you find the reasoning in this video: sufficiently rigorous to conclude

$$\sum_{n=1}^\infty n = -\frac{1}{12}$$

?

This video is interesting to me because the physicist at the end explicitly gives his justification for why he believes this result: because we use the result in a physical theory.

 

[atyy]

I have a question for Arsenic&Lace (or anyone else really): would you find the reasoning in this video: sufficiently rigorous to conclude

$$\sum_{n=1}^\infty n = -\frac{1}{12}$$

?

This video is interesting to me because the physicist at the end explicitly gives his justification for why he believes this result: because we use the result in a physical theory.

We discussed this back in posts #3,7,9,11. Physics and experiments are important for rigour. If something is non-trivially true in a physical theory, that suggests there is a way to make it rigourous. The calculus is an example of this. In fact, micromass's link to the Arnold article goes even further and claims mathematics is an experimental science!

 

[atyy]

Landau damping is another example of non-rigour in physics later being made rigourous (or at least the mathematicians claim, I tried reading the proof and it's gobbledygook to me).
http://en.wikipedia.org/wiki/Landau_damping
"Mathematical theory of Landau damping is somewhat involved—see the section below. However, there is a simple physical interpretation which, though not strictly correct, helps to visualize this phenomenon."
http://www.icm2010.in/prize-winners-2010/fields-medal-cedric-villani

 

[Jorriss]

In the first instance, beyond a shadow of a doubt, Terrence Tao knows nothing more about calculus than a good physicist does, because I would argue that modern analysis does not even constitute "more knowledge about calculus". If it really is more knowledge about calculus, then one could perhaps conceive of a use for something like Cauchy sequences outside of a pure math class. It is impossible to conceive of a usage for such a structure except to develop more philosophical academic "mathematics." It has nothing to do with calculus; it is a misnomer to describe such a topic as having anything to do with calculus, and utterly unclear how real analysis bears any relation, aside from philosophical, to something like calculus. .

People have conceived of uses of Cauchy sequences outside of academic math, the banach fixed point theorem for example. Don't get me wrong though, you're wrong for much more fundamental reasons.

 

[WannabeNewton]

In fact, micromass's link to the Arnold article goes even further and claims mathematics is an experimental science!

Well I think it's safe to say Arnold is wrong in that regard unless one interprets "experimental science" in the loosest fashion imaginable. Personally I can only think of one math professor I've met who would even come close to sharing Arnold's view but this math professor also thought topology was useless so that doesn't mean much.

 

[micromass]

Well I think it's safe to say Arnold is wrong in that regard unless one interprets "experimental science" in the loosest fashion imaginable. Personally I can only think of one math professor I've met who would even come close to sharing Arnold's view but this math professor also thought topology was useless so that doesn't mean much.

Arnold may have exaggerated a great deal when he made that statement, but he has a point. But first, to understand Arnold's point, you must realize that he was totally against mathematics as a discipline that "makes up some axioms and then derives consequences". Rather, he thought of mathematics as describing nature in one way or another. This is a very unconventional point of view with mathematicians, but it did allow him to give a great deal of intuition in his works.

When doing research in mathematics, we sure do experiments, but of course not in the sense that physics does experiments. For example, when developing a theory, we will always look at some special cases first and make some simple computations. Then we can gradually build up an abstract theory from these special cases. We don't just take a few axioms and start deriving things from those, we always have some specific phenomena in mind that we wish to describe. In that sense, we do experiments and in that sense we describe reality.

 

[dkotschessaa]

Arnold may have exaggerated a great deal when he made that statement, but he has a point. But first, to understand Arnold's point, you must realize that he was totally against mathematics as a discipline that "makes up some axioms and then derives consequences". Rather, he thought of mathematics as describing nature in one way or another. This is a very unconventional point of view with mathematicians, but it did allow him to give a great deal of intuition in his works.

When doing research in mathematics, we sure do experiments, but of course not in the sense that physics does experiments. For example, when developing a theory, we will always look at some special cases first and make some simple computations. Then we can gradually build up an abstract theory from these special cases. We don't just take a few axioms and start deriving things from those, we always have some specific phenomena in mind that we wish to describe. In that sense, we do experiments and in that sense we describe reality.

It feels like an experimental science in much the same way mathematical objects feel like they are ontologically existing entities. Though I think mathematicians know they aren't really experimenting and that they are dealing with abstractions (Unless they are a super platonist).

-Dave K

 

[ZombieFeynman]

I'm hardly an expert, since I am the true non-rigourous guy here, not you. But roughly, there are two different sorts of topology in condensed matter physics.

(1) There is the topology of the integer quantum hall effect, involving Chern numbers. Topological insulators are generalizations of this idea.
http://www.physics.upenn.edu/~kane/pedagogical/WindsorLec2.pdf
http://physics.princeton.edu/~haldane/talks/dirac.pdf
http://www.bioee.ee.columbia.edu/downloads/2013/nature12186.pdf

(2) Then there is the topology of the fractional quantum hall effect, one sign of which is that the ground state degeneracy depends on the topology on which the Hamiltonian is placed. A proposed use of this sort of topology is in Kitaev's topological quantum computation.
http://stationq.cnsi.ucsb.edu/~freedman/publications/96.pdf
http://www.simonsfoundation.org/quanta/20140515-forging-a-qubit-to-rule-them-all/

From the Haldane slides above:
"The moral of this long story: suggests three distinct ingredients for success.
• Profound, correct, but perhaps opaque formal topological results (Invariants, braid group, etc)
• Profound, simple and transparent “toy models” that can be explicitly treated (The honeycomb Chern Insulator, the Kitaev Majorana chain, etc)
• Understanding the real materials needed for “realistic” (but more complex) experimentally achievable systems that can bring “toy model results” to life in the hands of experimentalist colleagues."

This is strictly not all of the topology that lies in modern Condensed Matter Physics. The QH effect and its Chern number is rather different than Topological Insulators and the Z₂ topological QSH effect in 2D. This is generalized to a whole family of 3D Topological Materials. See the excellent review by Qi and Zhang.

I truly don't mean to quibble but this is an extremely exciting area of physics to me!

The points made by Haldane above are brought together in a very harmonious way in the original BHZ paper I cited above.

EDIT: Perhaps at a very rough approximation, I agree with your division.

 

[atyy]

This is strictly not all of the topology that lies in modern Condensed Matter Physics. The QH effect and its Chern number is rather different than Topological Insulators and the Z₂ topological QSH effect in 2D. This is generalized to a whole family of 3D Topological Materials. See the excellent review by Qi and Zhang.

I truly don't mean to quibble but this is an extremely exciting area of physics to me!

The points made by Haldane above are brought together in a very harmonious way in the original BHZ paper I cited above.

EDIT: Perhaps at a very rough approximation, I agree with your division.

Glad to hear your quibbling! I'd love to learn more from someone who's working on it:)

Yeah, the division is very rough, something that Hasan and Kane http://arxiv.org/abs/1002.3895 mentioned.

 

[Arsenic&Lace]

This is strictly not all of the topology that lies in modern Condensed Matter Physics. The QH effect and its Chern number is rather different than Topological Insulators and the Z₂ topological QSH effect in 2D. This is generalized to a whole family of 3D Topological Materials. See the excellent review by Qi and Zhang.

I truly don't mean to quibble but this is an extremely exciting area of physics to me!

The points made by Haldane above are brought together in a very harmonious way in the original BHZ paper I cited above.

EDIT: Perhaps at a very rough approximation, I agree with your division.

Intriguing, my own opinion of the field wasn't based upon experience, I had simply heard from a peer working in quantum computing at IBM that the theorists/experimentalists there generally felt that it was purely academic and impractical. Has anyone attempted to recast it in a more physical light, rather than in terms of formal, obtuse topology? Or is this inefficient/impossible? It was quite a while ago but Feynman's contributions to our understanding of supercooled helium were due to taking a very mathematically convoluted theory from the condensed matter group and trying to make it as simple as possible, in so doing obtaining everything they had and more. But that might not be the case here.

It certainly seems to be the case in modern particle physics, if you look at supersymmetry or string theory.

People have conceived of uses of Cauchy sequences outside of academic math, the banach fixed point theorem for example. Don't get me wrong though, you're wrong for much more fundamental reasons.

How is the Banach fixed point theorem used outside of pure math?

You're just wrong. If you honestly think that rigorous proofs aren't required to establish the truth of mathematical statements, it just shows that you're not intelectually mature enough to understand it, yet. This isn't supposed to be an insult. Many students fail to understand this, when they first learn about it, so you're in good company. You're problem is rather that you have a strong opinion on things that you don't understand and instead of trying to understand them, you're just stubborn. We won't be able to convince you, since it takes years of study to develop the intellectual maturity that it takes to understand the requirement for the rigour in mathematics. Even people, who have been doing mathematics for a long time, get back to their analysis books after years, because they suddenly feel that they have acquired enough mathematical maturity to read them again and new learn things that they hadn't realized when they first read them. So if you claim that you're entitled to judge the necessity of rigour in mathematics, then this is highly questionable, to say the least.

One can make a very suggestive argument for the chain rule by ignoring the mathematician's warning that differentials are not real numbers. Mathematicians claim that this reasoning is wrong because it does not cover pathological cases and does not handle differentials properly. Yet Leibniz was reported to have employed the chain rule long before rigorous proofs could be forumlated.

I am not saying that this demonstrates that rigor is always useless, but I think this debate would end extremely quickly if somebody could find a specific example of where, had it not been for formal mathematical rigor, progress in science or engineering would grind to a halt or follow false paths. Grand claims have been made that theories in physics would be a mess without rigor, but no actual evidence has been presented that this is the case. Indeed, I can even provide evidence to the contrary, given that QFT is still not that mathematically rigorous of a theory (to my knowledge).

This thread is so active that I have missed numerous replies (I think micromass complained that I didn't notice his link on wavelets, which I had to dig to spot), so my apologies if I do not comprehensively reply to everything that is mentioned.

 

[rubi]

One can make a very suggestive argument for the chain rule by ignoring the mathematician's warning that differentials are not real numbers. Mathematicians claim that this reasoning is wrong because it does not cover pathological cases and does not handle differentials properly. Yet Leibniz was reported to have employed the chain rule long before rigorous proofs could be forumlated.

This is how math works. Of course, we have conjectures, based on heuristics, before we prove them rigorously. No theorem has ever been proven before it had been conjectured. There surely had been many more proposals for theorems in the history of calculus, but only those remained that could be proven. It's an evolutionary process. (By the way.. The use of differentials isn't wrong in general. Today we understand precisely why they work. See non-standard analysis. We just teach it the $\epsilon-\delta$ way today, because it's easier.)

I am not saying that this demonstrates that rigor is always useless, but I think this debate would end extremely quickly if somebody could find a specific example of where, had it not been for formal mathematical rigor, progress in science or engineering would grind to a halt or follow false paths. Grand claims have been made that theories in physics would be a mess without rigor, but no actual evidence has been presented that this is the case. Indeed, I can even provide evidence to the contrary, given that QFT is still not that mathematically rigorous of a theory (to my knowledge).

There are literally millions of practical methods that could only be developed using rigorous mathematics. Just think about numerical methods for solving partial differential equations for example. It is easy to come up with more examples, but I won't do it, because this is not the point I want to make. (Also, it is very arrogant to think that none such examples exist, only because you aren't aware of them.) Your problem is that you don't acknowledge the fact that even though some formulas might seem to be true heuristically, it's necessary to be able to rely on those formulas. And a formula can only be relied on, if we can be sure that it works and if we know under what circumstances it may fail. Such results can only be established using rigorous mathematics.

 

[Arsenic&Lace]

This is how math works. Of course, we have conjectures, based on heuristics, before we prove them rigorously. No theorem has ever been proven before it had been conjectured. There surely had been many more proposals for theorems in the history of calculus, but only those remained that could be proven. It's an evolutionary process. (By the way.. The use of differentials isn't wrong in general. Today we understand precisely why they work. See non-standard analysis. We just teach it the $\epsilon-\delta$ way today, because it's easier.)


There are literally millions of practical methods that could only be developed using rigorous mathematics. Just think about numerical methods for solving partial differential equations for example. It is easy to come up with more examples, but I won't do it, because this is not the point I want to make. (Also, it is very arrogant to think that none such examples exist, only because you aren't aware of them.) Your problem is that you don't acknowledge the fact that even though some formulas might seem to be true heuristically, it's necessary to be able to rely on those formulas. And a formula can only be relied on, if we can be sure that it works and if we know under what circumstances it may fail. Such results can only be established using rigorous mathematics.

I am not assuming that they do not exist! I simply cannot find any. Please, give an example where powerful applied methods clearly rely on rigorous proofs that might be taught in a theory course.

 

[Arsenic&Lace]

Let's maintain the productivity of this discussion in the following manner: Wikipedia has a list of numerical methods for PDE's here:
http://en.wikipedia.org/wiki/Numerical_partial_differential_equations

How about we look through them and exhibit where it would not be possible or would be dangerous to use them without extremely rigorous pure mathematics? If I learn nothing else I'll learn lots of numerical methods for solving PDE's :P

 

[disregardthat]

Let's maintain the productivity of this discussion in the following manner: Wikipedia has a list of numerical methods for PDE's here:
http://en.wikipedia.org/wiki/Numerical_partial_differential_equations

How about we look through them and exhibit where it would not be possible or would be dangerous to use them without extremely rigorous pure mathematics? If I learn nothing else I'll learn lots of numerical methods for solving PDE's :P

Are you just trolling at this point? What you're suggesting is about as ridiculous as saying that no evidence is ever necessary in physics, because our current theories seem to work pretty well without it. How on Earth would one ever arrive at the intricacies of the current mathematics used in physics without continually backing the process up rigorously? And that's not mentioning how you would even get started without the growing mathematical generalizations which inspire and allows for new ideas to form.

 

[rubi]

I am not assuming that they do not exist! I simply cannot find any. Please, give an example where powerful applied methods clearly rely on rigorous proofs that might be taught in a theory course.

I just gave you an example: Numerics of PDE's. Finite element analysis for example. Micromass has also given you an example, which you ignored. But as I said, that's not the point I want to make. The point I want to make is:

Your problem is that you don't acknowledge the fact that even though some formulas might seem to be true heuristically, it's necessary to be able to rely on those formulas. And a formula can only be relied on, if we can be sure that it works and if we know under what circumstances it may fail. Such results can only be established using rigorous mathematics.

 

[rubi]

Let's maintain the productivity of this discussion in the following manner: Wikipedia has a list of numerical methods for PDE's here:
http://en.wikipedia.org/wiki/Numerical_partial_differential_equations

How about we look through them and exhibit where it would not be possible or would be dangerous to use them without extremely rigorous pure mathematics? If I learn nothing else I'll learn lots of numerical methods for solving PDE's :P

How do you think those methods have been developed in the first place? Has there been some genius who just wrote them down? They have been developed by mathematicians over many years, using rigorous mathematics. Distribution theory, $L^p$ spaces and so on.

 

[disregardthat]

I would also point out that asking for examples of where lack of mathematical rigor halting physics is beginning from the wrong end. You should rather ask for how rigorous mathematics helped physical breakthroughs to come by. The classical examples are many, most notably the differential geometry which allowed for general relativity to be expressed and understood., which I believe is already mentioned.