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Sometimes in physics, the mathematics leads to "un-physical solutions or terms", that are readily tossed by the physicist. For example, when deriving absorption and emission rates for via quantized light-atom interactions in quantum optics class, we toss out 2 terms from the Hamiltonian, on the grounds that they don't correspond to any observed physical process:

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How should we understand what is going on here from an epistemological point of view? It feels a bit convenient to toss these 2 terms from the point of view of mathematics - well maybe even wrong, in the sense that the resulting Hamiltonian is no longer the result of a sound mathematical derivation? Would it be desirable for the physicist to work with maths that would not yield unphysical solutions? Isn't maths in fact unreasonably ineffective in physics?

  • The goal of science is to make theories that lead to testable predictions. Epistemically, science's goal is to have a system that allows you to correctly predict the outcome of experiments. Science isn't the same thing as doing pure math, and if there is a mathematical result that is "un-physical" and tossing it aside doesn't alter the accuracy of the predictions, then there isn't anything epistemically wrong in doing so, as far as science goes. The enterprise of physics isn't "we need to adhere strictly to all aspects of mathematical rigor" it's "we need to correctly prediction outcomes." – Not_Here Apr 5 '17 at 19:03
  • The word is "physicist." A physician is a medical doctor. And as Heisenberg noted, you should always ask your physicist for a second opinion. – user4894 Apr 5 '17 at 19:07
  • I could see a few things "wrong": 1. maths is backed by surefire deduction, so if those terms arose in the derivation, then the axioms are wrong, meaning, the maths used is not the maths nature follows, if it follows any maths? ; 2. in other situations, maybe what we toss today will be observed tomorrow ; 3. the resulting is mathematically wrong, should we proceed on a mathematically incorrect result? Or - would a maths basis free of such unphysical terms not be more desirable to the physicist? – Frank Apr 5 '17 at 19:07
  • (it also kind of puts a dent in the "unreasonable effectiveness of maths in physics" - not so impressive anymore haha ;-) – Frank Apr 5 '17 at 19:14
  • See en.m.wikipedia.org/wiki/Philosophy_of_mathematics#Fictionalism and Field's book in regards to science without math. The goal of science is to create systems that lead to correct predictions. If you are asking about the epistemic aspects of science in regards to math, science cares more about accurate predictions than mathematical rigor. Math may be necessary for physics but tricks like renormalization lead to better predictions so they are perfectly valid tools of physics. – Not_Here Apr 5 '17 at 19:19
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I think this sort of thing happens all the time. For example, in some calculation or other for the height of say a telegraph pole I may need to extract a square root.

Now, there are two square roots, one the negative of the other; for example, the square root of 25 is 5 or -5.

For the problem at hand, -5 makes no obvious physical sense; in what way is a telegraph pole -5m in height? So I throw out that solution, and keep only the 5m solution.

The moral of this little story, is that the epistemology here is that much abused term, physical intuition.

  • Of course, it happens all the time. The philosophical point of this thread is to question and think about why it is so, and/or if it would be desirable to have maths that more closely adheres to physical situations. Note that for the pole example, you should work with positive numbers, which are the only adequate ones for measuring lengths, and if you stay within R+, you do not get negative roots. So we can readily find a "maths" that better works for this physical situation, in fact. – Frank Apr 7 '17 at 0:28
  • But isn't that how positrons were discovered? because Dirac's equation had both positive and negative solutions? Based on your calculation, -5 m telegraph poles exist, we just need to look for them. – Alexander S King Apr 7 '17 at 0:36
  • @AlexanderSKing - which is the troubling part: sometimes it seems useful to consider the seemingly "unphysical" solutions, but sometimes you have to cross them out for good! That doesn't seem very "reliable", or at least it violates maths left and right. This is ok since observation takes precedence in physics, but it ought to give at least a little pause to the philosopher, no? – Frank Apr 7 '17 at 0:40
  • @alexander king: if we go looking for them, we will find them buried underground! Sure (about positrons), actually t'Hooft or was it Tomanaga, I think the latter, called Diracs thinking 'acrobatic'; it's how many things are found, even in mathematics; should we have thrown away the square root of -1, well there goes complex analysis because we couldn't find a nice physical interpretation for it; and how about 0/0? It's not often realised that calculus, as dx/dy provides the solution for this. – Mozibur Ullah Apr 7 '17 at 0:49
  • By the way, I chose simple mathematics to illustrate the same conceptual problem you're attempting to think through; I tend to think, that when illustrating a problem, chose the simplest examples that one can get away with, because this way people aren't scared by the formalism thrown at them (even if it is in nice pretty colours) and can concentrate on the conceptual problem at hand. Of course if it is a matter of affirming ones credentials as a mathematician or physicist, then fire away; but one ought to notice, this is not a maths/physics site. – Mozibur Ullah Apr 7 '17 at 0:57

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