Do physics books butcher the math?

[atyy]

c). that the "inconsistencies" of the delta function resulted in spurious results or prevented physicists from actually advancing physics.

http://arxiv.org/abs/quant-ph/0303094

 

[ZombieFeynman]

Fair enough. But I do think to some respects that an undergraduate ought to understand that those with more research experience at the graduate levels and beyond likely know what they're talking about, and rather than ignoring what they say and pursuing vapid points, he ought to take it as evidence that he may be wrong.

We in the sciences should be encouraged to question authority. However, it's not always the most productive rout; unless one is a genius it may lead to a lot of headaches and wasted time.

 

[micromass]

The argument I was making regarding Hilbert spaces is that completeness is indeed one of their properties, but that it is a useless property to learn about as a physicist and utterly irrelevant to physical theory.

So something like $\sum |\psi><\psi| = I$ is seen as useless and utterly irrelevant nowadays?

 

[ZombieFeynman]

So something like $\sum |\psi><\psi| = I$ is seen as useless and utterly irrelevant nowadays?

I can answer that! NO!

I use resolutions to identity with great regularity.

 

[rubi]

I can answer that! NO!

I use resolutions to identity with great regularity.

But Arsenic&Lance doesn't use them, so they are useless.

 

[atyy]

I guess you don't know what a Hilbert space is. It's complete by definition. And its completeness is used in QM all the time, although it is usually just swept under the carpet.

The argument I was making regarding Hilbert spaces is that completeness is indeed one of their properties, but that it is a useless property to learn about as a physicist and utterly irrelevant to physical theory.

Isn't this $\Sigma|n \rangle \langle n| = 1?$

 

[micromass]

But Arsenic&Lance doesn't use them, so they are useless.

This is the best reply of this thread

Isn't this $\Sigma|n \rangle \langle n| = 1?$

Could very well be. I'm not really good in braket notation. Thanks for the correction.

 

[ZombieFeynman]

This is the best reply of this thread



Could very well be. I'm not really good in braket notation. Thanks for the correction.

As long as $\ket{n}$ or $\ket{\psi}$ is summed over a complete set of states it doesn't matter!

I don't know how the forum latex thing works = (

 

[AnTiFreeze3]

We in the sciences should be encouraged to question authority. However, it's not always the most productive rout; unless one is a genius it may lead to a lot of headaches and wasted time.

Well I certainly have a headache from this thread

 

[Arsenic&Lace]

Software like ANSYS just implements algorithms that have been discussed by mathematicians. Of course, they rely on rigorous results proved by mathematicians. They even employ mathematicians. You have to be blind to not see this. Additionally, of course they need to benchmark their software. Software development consists of more than just implementing algorithms. The greatest performance gain is due to the use of efficient algorithms, however. If you use an algorithm of complexity $O(n^2)$ instead of $O(\log(n))$, then you can optimize as much as you want, it will always be inferior.


You couldn't even come up with the obvious harmonic oscillator counterexample on your own. I still think you have absolutely no clue what you are talking about.

Here's what I remember of the discussion, correct me if I'm wrong:
rubi: I challenge you to determine the speed at which an algorithm converges without pure mathematics.
Arsenic: This question is irrelevant, in the real world we only use benchmarks.
rubi: Corporations use theoretical methods to determine speed because it is too costly to benchmark.
Arsenic: My industrial/academic experience is that it is impossible to develop theoretical methods in most cases so benchmarking is used instead (I'd add after the fact that it really isn't that hard to benchmark multiple packages).
rubi: The corporation you cited uses theoretical methods to determine the speed of the algorithms.

This is now an empirical question.

All I ask of you, now, is to reiterate what exactly it is you're arguing against. Because I feel you know it's a lost cause, yet find your only redemption in asking more and more obscured questions,

I'm arguing against many things, but I'll pick a couple and briefly list the conditions under which my views will change so that people can decide if they are completely unreasonable or not.

1. The levels of mathematical rigor employed by mathematicians serve no useful purpose for the practitioners of mathematics.

My mind would change if someone could provide an empirical example of where rigorous proofs actually aided the development of applied disciplines.

2. That it is pointless to divorce mathematics from its applications.

It seems to me that extremely general reasoning about say, PDE's, has produced nothing of use. There are numerous grand theorems, but these are ignored outside the math department because people actually studying real PDE's realize that there is very little separating the symbolic expression of the problem from the underlying physics/real world rules.

Of course, if one could show that the powerful theorems learned in a pure PDE's course are actually helpful to applied mathematicians, I would change my mind.

So something like $\sum |\psi><\psi| = I$ is seen as useless and utterly irrelevant nowadays?

The notion of completeness carries much more baggage than this. One can understand the value of this expression simply by analogy to orthonormal vector spaces. I had in mind more mathematical notions such as the fact that every Cauchy sequence in a complete metric space converges to a value in that metric space.

Most of your posts seem to read "I haven't had to use this and don't think I will have to, therefore no one does!"

You should consider reading them more carefully then.

 

[micromass]

The notion of completeness carries much more baggage than this. One can understand the value of this expression simply by analogy to orthonormal vector spaces. I had in mind more mathematical notions such as the fact that every Cauchy sequence in a complete metric space converges to a value in that metric space.

Saying that a Hilbert space is complete is exactly the same as saying that $\sum |\psi><\psi| = I$. So it doesn't carry any more baggage.

 

[ZombieFeynman]

The notion of completeness carries much more baggage than this. One can understand the value of this expression simply by analogy to orthonormal vector spaces. I had in mind more mathematical notions such as the fact that every Cauchy sequence in a complete metric space converges to a value in that metric space.

Do all notions from finite dimensional vector spaces carry over to the infinite dimensional case? (Hint: no) How do you know which ones do and don't without rigorous mathematics? Cantor showed the intrinsic non-intuitiveness of sets with infinite and (moreso!) with uncountable cardinalities. I'd be seriously careful here.

I challenge you to prove that you can have two canonically conjugate matrices A and B in finite dimensional space (akin to momentum and position).

ie AB - BA is the identity, up to a constant.

Before you waste too much of your night on it, it's impossible. I double dog dare you to convince me of that without being...rigorous.

 

[micromass]

My mind would change if someone could provide an empirical example of where rigorous proofs actually aided the development of applied disciplines.

It's hard to change your mind if you ignore most of the examples we give. But again, consider wavelets.

 

[rubi]

Here's what I remember of the discussion, correct me if I'm wrong:
...
This is now an empirical question.

I just argued that mathematical rigour is essential for the development of numerical PDE methods and this is undeniable. It is unthinkable that a software package like ANSYS would yield reliable results if it didn't depend heavily on rigorous results. Anyone who has the slightest idea of how these packages work, will agree with this. If you don't believe it (which would be totally ridiculous), go ahead and check out some of the open source FEM packages. There are plenty. I won't help you though, because it is a waste of my time.

 

[jergens]

My mind would change if someone could provide an empirical example of where rigorous proofs actually aided the development of applied disciplines.

These examples have been provided before (and there are many, many more examples still to be named as well), but these are all pretty concrete and you can verify each of them by pretty much asking anyone working in these fields.

1. If you work in finance or data analysis or do computer science focusing on machine learning (three very important industries these days) you are going to be needing techniques from stochastic calculus that were only possible because of rigorous foundations. Most of the important results were not "intuited" first, as they were in ordinary calculus, but only arose as the relevant proofs came with them.
2. If you work in economics or again finance there is good chance you will be needing fixed-point theorems whose development required rigorous proofs. Things like the Brouwer and Kakutani fixed-point theorems were utilized to establish various equilibria phenomenon (like Nash equilibrium) that have since become staples in the industry.
3. If you work in some of the cutting-edge data analyst groups you will be needing lots of tools from algebraic topology for topological data analysis. Here you actually need quite a bit of machinery like knowledge of various (co)homology theories, their connections with cobordism and Morse theory, spectral sequences, etc.

If you pay attention to all of the examples people have been giving, rather than honing in one or two (like the applications of algebraic topology in condensed matter theory) you would see there is a big real world market for results from pure mathematics.

Edit: Just to list a few more applications off the top of my head, you should check out: stochastic calculus in statistical physics; number theory and algebraic geometry in cryptography with stuff like elliptic curves; algebraic topology in biology for modeling protein structure and interactions; category theory in computer science.

 

[WannabeNewton]

I would ask for this thread to be closed because at this point it is akin to a cowering cat cornered by a gang of dogs closing in for the kill but I feel like too many people are getting entertainment value out of it.

 

[ZombieFeynman]

I would ask for this thread to be closed because at this point it is akin to a cowering cat cornered by a gang of dogs closing in for the kill but I feel like too many people are getting entertainment value out of it.

I would ask for it to stay open. I think it is (somewhat) intellectually dishonest to declare victory and close up shop. As long as no forum rules are being broken, I don't see why it should not remain opened.

 

[Jorriss]

I am also in disagreement with A&L, however, I don't think an argument from authority is a good way to proceed.

I read it more like, 'everyone disagrees with you, perhaps you should reconsider your view.'

 

[Char. Limit]

One more thing: Why does pure mathematics need applications? Would you say someone who studies art for 50 years isn't an expert on art because "there's no applications for their work"?

 

[Arsenic&Lace]

I would ask for this thread to be closed because at this point it is akin to a cowering cat cornered by a gang of dogs closing in for the kill but I feel like too many people are getting entertainment value out of it.

Well I'm enjoying this thread so I hope it continues.

Firstly my apologies to jergen, micromass, and others for not examining each and every application in detail. I promise I'm not trying to cherry pick applications which are easiest for me to argue with. However, it is much easier for me to choose applications I'm familiar with, and if the applications I was familiar with did not conform to my point, I wouldn't believe it (although I may merely be misinterpreting them). More importantly, if you merely mention an application or post a textbook, it puts the ball in my court to construct the argument for you. I'm not saying you're lazy, but I am saying this thread would likely progress much more rapidly if you were to construct more thorough arguments around your evidence.

Secondly, I have been accused of waving my hands and not really providing concrete arguments. This is duly noted and I have consistently attempted to increase the rigor (...ha!) of my arguments with each post. However, I have not observed many concrete arguments from my (admittedly numerous) foes. Mostly I am told "if you only read this textbook" or "surely this must be the case", which may very well be true, but it is extremely challenging for me to read every paper, extract the argument you imply with said paper, and then respond to it.

Finally, I think we should concentrate on one of these topics at a time. Either it will constitute evidence that the mathematician's theories are very helpful and the thread will die a peaceful death, or it will not, and we will proceed onto the next application.

I would prefer we begin with algebraic topology as applied to protein structure since I presently work in a computational biophysics lab and have been pondering more theoretical approaches to the problem of protein conformational change for several years now. The problem space appears to admit itself very well to a geometric or topological approach, yet protein conformational change prediction or first principles predictions of protein folds are extremely challenging unsolved physics problems (some colleagues of mine are currently engaged in CASP, a refinement/prediction challenge and advanced mathematical trickery which gave them an edge would certainly be interesting ;)). I have repeatedly explored more esoteric approaches and have been unimpressed.

The laboratory in which I work (surprise surprise) relies heavily on brute force, running molecular dynamics simulations on protein systems where the trajectories for every atom are simulated, although I work on algorithmic/more theoretical approaches. What is interesting to me is just how far out of our reach conformational change actually is; just obtaining a microsecond of simulation, significantly below the timescales for full conformational change, can take several months.

So what I would like to know is, what are these approaches, what pure mathematics do they rely upon,and how do they perform?

 

[Arsenic&Lace]

One more thing: Why does pure mathematics need applications? Would you say someone who studies art for 50 years isn't an expert on art because "there's no applications for their work"?

It doesn't. I'm fine with it being art. But much in the same way I find it baffling when a fine artist tells me that a Pollack painting is a work of genius, I find it baffling when a mathematician bends over backwards to handle a delta function.

In either case the explanation is that they are wrong or that I'm just a tasteless rube. I'll accept the latter if evidence can be presented, but I'm as optimistic for the mathematicians as I would be for Pollack.

 

[Jorriss]

The laboratory in which I work (surprise surprise) relies heavily on brute force, running molecular dynamics simulations on protein systems where the trajectories for every atom are simulated, although I work on algorithmic/more theoretical approaches. What is interesting to me is just how far out of our reach conformational change actually is; just obtaining a microsecond of simulation, significantly below the timescales for full conformational change, can take several months.

Metadynamics, multiscale coarse graining, symplectic integrators, relative entropy methods, monte carlo methods, etc. These are all techniques which were (primarily) developed by chemists and physicists but whose development was assisted by having firm, rigorous mathematical foundations to build off of or were rigorously developed themselves. For the latter, see http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.112.240602
From the abstract.
"Metadynamics is a versatile and capable enhanced sampling method for the computational study of soft matter materials and biomolecular systems. However, over a decade of application and several attempts to give this adaptive umbrella sampling method a firm theoretical grounding prove that a rigorous convergence analysis is elusive. This Letter describes such an analysis, demonstrating that well-tempered metadynamics converges to the final state it was designed to reach and, therefore, that the simple formulas currently used to interpret the final converged state of tempered metadynamics are correct and exact. "

 

[ZombieFeynman]

A&L, it seems that you missed my previous post on the difficulties in generalizing finite dimensional vector spaces to infinite dimensional ones.

 

[Matterwave]

So something like $\sum |\psi><\psi| = I$ is seen as useless and utterly irrelevant nowadays?

Ewwwwwwwwww micro...seriously? Ewwwwwww...use the left and right commands to make this look not so disgusting.

$$\sum_\psi \left|\psi\right>\left<\psi\right|=I$$

Also, in physics, one rarely uses $\left|\psi\right>$ to denote a complete set of basis states, but rather one particular state vector. Much more common is $\left|n\right>$ for energy eigenstates.

 

[Jorriss]

However, it is much easier for me to choose applications I'm familiar with, and if the applications I was familiar with did not conform to my point, I wouldn't believe it (although I may merely be misinterpreting them).

You aren't really qualified to argue your point if you are unfamiliar with the amount of fundamental topics people have mentioned here. You can't really make your point if you only know brute force MD (which apparently you're missing a lot of if you don't know where any rigor is used there).

 

[micromass]

Ewwwwwwwwww micro...seriously? Ewwwwwww...use the left and right commands to make this look not so disgusting.

$$\sum_\psi \left|\psi\right>\left<\psi\right|=I$$

Also, in physics, one rarely uses $\left|\psi\right>$ to denote a complete set of basis states, but rather one particular state vector. Much more common is $\left|n\right>$ for energy eigenstates.

Thanks a lot for the correction!

 

[rubi]

Here's a job offer by ANSYS: http://www.linkedin.com/jobs2/view/10302122

Minimum Requirements/Qualifications:
- A Master’s degree in Computer Science, Applied Mathematics, Engineering or related discipline
- 2+ years of proven accomplishments in the mesh generation community via academic, research or industry experience.
- Deep knowledge of various meshing data structures and techniques are critical

Surely, they are looking for someone who benchmarks random algorithms on their supercomputers.

--

Here's an excerpt from the LAPACK user manual: http://www.netlib.org/lapack/lug/node81.html
It clearly shows that the developers have taken rigorous results on the accuracy of the algorithms into account. If you don't know LAPACK: It's a widely used linear algebra library that is used in many FEM packages for solving sparse matrix equations. If this still doesn't convince you, look at the Bibliography section.

 

[Arsenic&Lace]

Do all notions from finite dimensional vector spaces carry over to the infinite dimensional case? (Hint: no) How do you know which ones do and don't without rigorous mathematics? Cantor showed the intrinsic non-intuitiveness of sets with infinite and (moreso!) with uncountable cardinalities. I'd be seriously careful here.

I challenge you to prove that you can have two canonically conjugate matrices A and B in finite dimensional space (akin to momentum and position).

ie AB - BA is the identity, up to a constant.

Before you waste too much of your night on it, it's impossible. I double dog dare you to convince me of that without being...rigorous.

What physics does this model? Physics on a lattice? I remember reading an interesting paper on lattice models of spacetime, where strange things happened to the uncertainty principle because of the lattice. I will post the paper if you are interested.

But I contend that unless discrete position/momentum operators actually model something interesting, this problem would never cross my desk. If it did, it would probably go the way of the paper in reference and work out just fine, but not in the framework you described, because we must make different physical assumptions when working on a lattice.

Joriss: I've implemented numerous Monte Carlo simulations (I'm working on one right now), and it's a very intuitive, heuristic process. Indeed Monte Carlo methods are so enormously varied that there is little consistent theory and no strict forms for them to take, just various prescriptions for how one should design, generally speaking, the rejection/acceptance step. Symplectic integrator is an overly convoluted way of saying "obeys Hamilton's equations." Apart from a nice picture of how the phase space has no sinks or sources, to implement something like a Verlet integrator (something I recently used actually in a GPU driven simulation) you need absolutely no knowledge of differential geometry. Metadynamics using umbrella sampling is also a heuristic process, although I am less familiar with it (a colleague in the lab employs it for free energy calculations I believe); the paper you attached formally confirms physically motivated guidelines which have been empirically supported for decades. This is an example of one of the more amusing phenomena where pure mathematicians develop rigorous proofs ages after the methods are developed, casting doubt on the notion that the proofs are necessary at all. Of course Parinello is not a mathematician but a brilliant physicist who's made some great contributions, and I found his argument in the paper to be quite clever and delightful; it was a pleasure to read, and it's satisfying to see it formally proven. But it's minor at best. The only benefit the paper cites is that the conditions might be made more permissive, but the authors couldn't determine how this would effect the convergence rate.

I've heard of multi-scale coarsegraining. It's a very neat technique. The paper establishing it makes no reference to pure mathematics like functional analysis or algebraic topology. They use the variational principle, which is kosher in my book since it was first explored somewhere in the 18th-19th century by Euler before standards of rigor got utterly extreme. It seems you may have confused my attitude towards mathematics; I have no bone to pick with applied mathematicians, I just feel extremely skeptical about pure, artsy maths. I'm not a Luddite about theory in general, I just have a lot of doubts about pure mathematics.

Finally I've heard of the Kullback Leibler divergence, but sadly I'm not very clever and it's too hard for me to understand in an evening. The jury is out on that one.

rubi: So ANSYS hires Applied mathematicians! Great. Applied mathematics departments seem to not always require their students to take pure math courses, at least in the random sample I looked in, where some schools had no pure math requirements (that I could see), some schools required 1/8 courses be pure math, and other schools required more. As I said I don't really have a beef with applied mathematicians, but notice how they don't want an MS in PURE mathematics! Also notice how a Computer Science or Engineering major would be completely acceptable; individuals who have probably never seen the inside of a real analysis textbook.

As for the LAPACK bibliography, well, all of the citations are from computational or applied mathematics journals, or applied books. Maybe there's one which sneaked past me when I skimmed it, but I didn't see, for instance, a citation from the AMS or a journal on pure PDE theory. Perhaps you share Joriss' confusion about my stance. Applied/computational mathematics departments generally seem productive and don't get my goat. There are varying levels of rigor in comp/applied departments so the jury is still out as to whether or not it is prevalent or important in FEA.

 

[Jorriss]

Symplectic integrator is an overly convoluted way of saying "obeys Hamilton's equations." Apart from a nice picture of how the phase space has no sinks or sources, to implement something like a Verlet integrator (something I recently used actually in a GPU driven simulation) you need absolutely no knowledge of differential geometry. theory in general, I just have a lot of doubts about pure mathematics.

Well obviously you don't need pure math knowledge to actually implement velocity verlet, that's not what I'm arguing.

This is an example of one of the more amusing phenomena where pure mathematicians develop rigorous proofs ages after the methods are developed, casting doubt on the notion that the proofs are necessary at all.

No it doesn't, read the abstract. The simulations ran over years were not satisfactory with regards to questions of whether it will converge.

Of course Parinello is not a mathematician but a brilliant physicist who's made some great contributions, and I found his argument in the paper to be quite clever and delightful; it was a pleasure to read, and it's satisfying to see it formally proven.

It was primarily James Dama's work in this paper.

I've heard of multi-scale coarsegraining. It's a very neat technique. The paper establishing it makes no reference to pure mathematics like functional analysis or algebraic topology.

Yeah, they didn't cite pure math papers because they used well understood results that were grounded by rigorous math. You know people don't actually publish papers at the level they think about the material at right?

By the way, you are aware applied math is rigorous right? They do theorem-proof too.

 

[jostpuur]

One of the biggest problems with physicists' bad math is that it attracts wrong kind of people.

When a scientific community insists that explanations and claims must be logical, it serves as a sieve that filters out those who are capable of only babbling nonsense. The policy of physicists to allow nonsensical pseudomathematical carbage under the pretense of intuition has had the consequence that the sieve isn't working. Wrong kind of people get into the community and corrupt it from inside.

Some people defend the bad math with argument that it hasn't caused any harm. They might demand evidence that some harm has been done. Well it is the job of future historians to study what harm the modern pseudomathematical culture has produced. I wouldn't be surprised if the mandkind could already have achieved warm superconductors and fusion energy if only physicists had not declared war on mathematics.