Robert Helling is giving a sensible argument to why infinities in a physics theory are some kind of big warning sign:
Most of the time, when in a physics calculation you encounter an infinity that should not be there (of course, often “infinity” is just the correct result, questions like how much energy I have to put into the acceleration of an electron to bring it up to the speed of light? come to my mind), you are actually asking the wrong question. This could for example be because you made an idealisation that is not physically justified.
He gives a number of examples, some pretty simple, for instance:
A similar thing can be seen in fluid dynamics: The Navier-Stokes equation has singular solutions much like Einstein’s equations lead to singularities. So what shall we do with for example infinite pressure? Well, the answer is simple: The Navier-Stokes equation applies to a fluid. But the fluid equations are only an approximation valid at macroscopic scales. If you look at small scales you find individual water molecules and this discreteness is what saves you actually encountering infinite values.
Fighting the belief in “magic” in physics
I find his explanations somewhat more convincing than the Lubos typical babbling that prompted this reaction:
A typical example of a mathematical fact that the anti-talents in theoretical physics can’t ever swallow are the identities that appear in various regularizations
The problem with Lubos is that he believes in magic (which he calls “Nature” below with a capital N, but he could have written God or Knuth or Grand Schtroumpf):
When a physicist writes an integral, she usually doesn’t care whether you use the Lebesgue integral or the Riemann integral. For a physicist, these two and other definitions of an integral are just man-made caricatures to calculate some expressions in practice and to give them a rigorous meaning in a particular system of conventions.
That’s not exactly what a physicist means by the integral. A physicist always means nothing else than Nature’s integral that coincides with the Riemann and Lebesgue integral in most well-behaved situations. But whenever there is something unusual about the integral, we must leave it up to Nature – not Riemann or Lebesgue – to decide what is the right thing to do with the integral.
And he wants you to believe too:
I say “handwaved” because only experts are capable to understand how these arguments work and decide whether they’re correct. Believe me or not.
Sorry, Lubos, but I won’t gratify this with your favorite “not even wrong” here. This is simply wrong. Why? Because the problem is not that Lebesgue or Riemann are “anti-talents who can’t swallow that 1+2+3+...=-1/12“. The problem is that physicists like you take pride in writing incorrect mathematical expressions for the problem at hand, like putting an infinity where it physically does not make sense. A mathematical expression that diverges only appears to converge in physics because the physics situation is not understood well enough, most often simply because there is some cutoff that was blissfully ignored.
Feeding the world, the Lubos way
Therefore, it is simply wrong to claim that the correct result (to use Lubos’ misguided expression) for 1+2+3+.. is always -1/12. Let’s consider a different physical situation to show how vacuous this “correct result” is. Imagine a tribe where one new person is born each day. How much food do you need to feed them forever? Well, the first day, they eat one food ration. The second day, they eat two food rations, 1+2. The third day, three food rations, 1+2+3, and so on. Therefore, the “correct result”, according to Lubos, is that you only need -1/12th of a food ration to feed them forever.
“Je suis chargée de vous le dire, je ne suis pas chargée de vous le faire croire.”