Derivations vs Proofs in Physics Textbooks

[Monocles]

I've noticed in my physics textbooks that every time the author wishes to prove something, he'll go for a direct proof/derivation. Is there any particular reason for this? I think that I have yet to see any proofs by contradiction/proving the contrapositive/mathematical induction in any of my textbooks. I haven't taken anything beyond junior level classical mechanics/thermodynamics yet, though.

 

[alxm]

Well, physical derivations seldom qualify as mathematical proofs, although often assumptions/derivations made on physical grounds can later be proven mathematically.

Proofs by counter-example and such aren't usually terribly useful. E.g. proving that the solution does not have certain properties doesn't tell you much about what way it is.

For instance, the Hohenberg-Kohn theorem is a very important reductio ad absurdum proof in atomic/molecular physics, which proves that the ground-state energy of a system of electrons is a functional of the electronic density. (and that this functional obeys the variational principle)

Problem is, by the nature of the proof, we get no indication whatsoever as to what that functional actually is, only that it exists. Nobody's found it yet.

 

[turin]

Physicists tend to avoid real rigorous proofs, because then we would have to come face-to-face with the aweful truth ...

However, there are a few examples in which, not only are the proofs rigorous, they are by contradiction. When/if you take EM, you will probably see a proof by contradiction for the uniqueness theorem and Green's theorem. If you go into QFT, you may encounter proofs by contradiction in various group theory topics. Basically, though, the proofs are to justify some mathematical property that we want to interpret as a physical property, so the proofs are really only mathematical, not physical.

Science itself doesn't really have formal proofs because one of the most important criteria of science is that every fact is subject to experimentation, and the link between experimentation and theory is subject to interpretation.

 

[gmax137]

Science itself doesn't really have formal proofs because one of the most important criteria of science is that every fact is subject to experimentation...

I think that's the answer. Experiment is the difference between the 'natural philosophers' of antiquity (Aristotle) and 'modern' science. Even theories that can be 'explained' entirely by thought (like special relativity) have an experimental basis, and would be mere curiosities if not confirmed by observation (meson decay times, accelerator design, etc. are often offered to answer the questions "yes, but is time dilation / mass increase with velocity*real*")

 

[AUMathTutor]

For a physicist, it's a proof if you write QED below it. Try it sometime. You can also add an optional footnote at the bottom of the page acknowledging that what you just said isn't true, but it's good enough for now (true story... check Griffiths QM / E&M).

 

[turin]

Mere experimentation is inadequate. The experiments must satisfy their own criteria, and these are subject to interpretation. Even pseudoscientists (e.g. those damn ghost hunters) do "experiments". I claim that they are not practicing science because they seem to merely exploit the accepted authority of experimentation as an exoticism to buffer themselves from the scrutiny of their "clientele", who seem to have no concept of, e.g. double-blind practice.

 

[AUMathTutor]

Don't be so hard on paranormal research. People used to say Chemistry was crazy, too.

You don't want to wake up one day and see "ghosts proven real" on CNN. Then you'll lose face on the PhysicsForums and have to buy us all a coke.

 

[AUMathTutor]

Seriously, though, their proofs aren't any worse than physicists'.

 

[turin]

People used to say Chemistry was crazy, too.

And so it was, but that's irrelevant. That sounds exactly like a pseudoscientist's justification for pseudoscience. The issue is not what people say, nor used to say, about the discipline.

You don't want to wake up one day and see "ghosts proven real" on CNN.

I certainly don't equate CNN, nor any other entertainment medium, with e.g. Phys. Rev. D or something. I don't see how that is relevant. In fact, not CNN, but various other networks basically claim (or at least broadcast the claim) that ghosts are real on a daily basis.

Then you'll lose face on the PhysicsForums and have to buy us all a coke.

That's a lot of coke.

Seriously, though, their proofs aren't any worse than physicists'.

Then I guess that's where we simply disagree.

 

[n!kofeyn]

I've noticed in my physics textbooks that every time the author wishes to prove something, he'll go for a direct proof/derivation. Is there any particular reason for this? I think that I have yet to see any proofs by contradiction/proving the contrapositive/mathematical induction in any of my textbooks. I haven't taken anything beyond junior level classical mechanics/thermodynamics yet, though.

What physics are you taking or what textbooks are you reading?

Often times, the derivations in physics textbooks use mathematics that we already know to be true, and the derivations are just physical interpretations of those facts or laws. For example, one of the reasons quantum mechanics is so successful is because it is strongly based in probability and algebra, two areas in mathematics that have both been extensively and rigorously developed.

If you have access to a good library and have had a certain amount of math and physics, on p. 110-111 in the book Quantum Mechanics by David Griffiths, you will see that he actually gives a very rigorous mathematical proof of the generalized uncertainty principle. We then use this very mathematical principle in physics by giving it a physical interpretation. The first form usually seen by physics students is the relationship between the standard deviations of position and momentum, but there are other interpretations which depend on what you are wanting to observe.

 

[gmax137]

Seriously, though, their proofs aren't any worse than physicists'.

When the physicists tell you they have found the mass of the Earth to be 6 * 10 to the whatever power kilograms, *you* can go check for yourself. When the ghostbusters tell you dear old mum is watching over your shoulder, how do you check? And why is it that whenever anyone does check, they don't see her?

 

[turin]

... one of the reasons quantum mechanics is so successful is because it is strongly based in probability and algebra, ...

So is economics. I don't think economics is particularly successful.

 

[n!kofeyn]

So is economics. I don't think economics is particularly successful.

Well to be honest, I do not know much about economics, but I mentioned that was one of the reasons, not the sole reason. From what I know, the physical theory of quantum mechanics meshes very well with the mathematics. Also, quantum mechanics is a tool we use to describe nature. For me, what economics tries to describe is more man-made than particles are. Physicists are able to actually make accurate predictions, and these predictions have been verified through experiments. Again from what I know, which may very well be very little, economists do not make such strong and accurate experiments, which doesn't mean the math is wrong. It most likely means that their interpretation of and use of the mathematics is off.

 

[turin]

From what I know, the physical theory of quantum mechanics meshes very well with the mathematics.

I don't know what that means. Perhaps I can better understand what you mean if you give an example of a scientific theory that does NOT "mesh very well with mathematics".

Also, quantum mechanics is a tool we use to describe nature.

Again, can you give a counterexample of a scientific theory that is not "a tool we use to describe nature"?

Physicists are able to actually make accurate predictions, and these predictions have been verified through experiments.

Pseudoscientists also boast accuracy and experimental verification. In fact, the excessive accuracy claimed by a pseudoscientist is one of the tells of pseudoscience. They achieve this accuracy by ignoring a large majority (usually all) of their null results (yet they will happily use the null result of a scientific inquiry in order to further support their claims). Embarrassingly, this practice is sometimes adopted in science, because it is so tempting. I will hand it to the pseudoscientists, though, for being able to continue to pull this trick. The world is becoming exceedingly less obliging to this practice as information can be transmitted across the world for fractions of a cent at fractions of a second at the touch of a virtual button.

... economists do not make such strong and accurate experiments, which doesn't mean the math is wrong. It most likely means that their interpretation of and use of the mathematics is off.

Economics is not my field, either. You may be correct. However, the mathematics themselves are perfectly accurate. In fact, that was my point. It is not the mathematics that makes the theory accurate, it is the development of the understanding of the mechnisms envolved, and what can and cannot be described and predicted by the theory.

 

[Hurkyl]

E.g. proving that the solution does not have certain properties doesn't tell you much about what way it is.

I have to disagree with this; knowing what properties something doesn't have is equally important to knowing what properties it does have.

 

[n!kofeyn]

I don't know what that means. Perhaps I can better understand what you mean if you give an example of a scientific theory that does NOT "mesh very well with mathematics".

Again, can you give a counterexample of a scientific theory that is not "a tool we use to describe nature"?

Well the example the comes to mind is psychology. While this scientific field uses statistical analysis heavily, it does not use mathematics similar to how quantum mechanics uses mathematics. It does not ,on whole, use math to build a theory and make predictions.

I think the point I was trying to make is that economics is just a huge man made mess. It has basic properties that come not from nature, but rather from man. Again, just my non-professional opinion.

Pseudoscientists also boast accuracy and experimental verification. In fact, the excessive accuracy claimed by a pseudoscientist is one of the tells of pseudoscience. They achieve this accuracy by ignoring a large majority (usually all) of their null results (yet they will happily use the null result of a scientific inquiry in order to further support their claims). Embarrassingly, this practice is sometimes adopted in science, because it is so tempting. I will hand it to the pseudoscientists, though, for being able to continue to pull this trick. The world is becoming exceedingly less obliging to this practice as information can be transmitted across the world for fractions of a cent at fractions of a second at the touch of a virtual button.

I honestly don't know what you are talking about.

In fact, that was my point. It is not the mathematics that makes the theory accurate, it is the development of the understanding of the mechanisms envolved, and what can and cannot be described and predicted by the theory.

Well that was my point as well, because quantum mechanics uses mathematics in a useful way, which is one reason it is successful. It has physical interpretations of rigorous theory that works.

I think we're getting a little off topic from the concerns of the original post, and I'm honestly not well versed enough to continue this sort of vague discussion.

 

[turin]

Well the example the comes to mind is psychology. While this scientific field uses statistical analysis heavily, it does not use mathematics similar to how quantum mechanics uses mathematics. It does not ,on whole, use math to build a theory and make predictions.

OK, I see what you mean. Unfortunately, I don't know psychology at all, so I have to take your word for it. Actually, after having some discussions with psychology students to try to understand what it is that they study, I'm not entirely convinced that psychology is a science, but that is for precisely the reason that you have given here for psychology being an example of a science that does not mesh well with mathematics - i.e. psychology does not make quantitative predictions.

I honestly don't know what you are talking about.

The ambiguity of accuracy, and the logical fallacy of confirmation bias.

I think we're getting a little off topic from the concerns of the original post, and I'm honestly not well versed enough to continue this sort of vague discussion.

OK.