Do physics books butcher the math?

[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.