1
$\begingroup$

There is a philosophic debate about whether there could be infinite quantities in nature.

Definitely we cannot measure infinite quantities with measurement instruments.

But we know the regularized sums for some infinite series, such as $$\operatorname{reg}\sum_{k=1}^\infty k=-1/12.$$ And such sums appear in the measurements, for instance in Casimir effect.

Can we conclude from this fact that infinite values in fact underlyingly exist but we only can observe them indirectly as finite parts of those infinite values?

$\endgroup$
1

2 Answers 2

2
$\begingroup$

Regularisation is a very precise way to provide some sense to apparently meaningful results. It is possible to prove that the analytical continuation methods are equivalent to subtraction or point splitting procedures where physics is evident. Those are very well known results and I cannot see any reason to literally consider these “identities”. There is nothing like actual infinity here, just a bad understanding of a physical reasoning. In my honest view, insisting on these issues, from a philosophical perspective, is just wasted time.

$\endgroup$
4
  • 1
    $\begingroup$ No. It seems you misunderstood. I did not ask if regularized sums are literal sums (it is evident they are not). What I asked is whether the emergence of regularized sums in physics an indication of existing underlying infinite quantities. $\endgroup$
    – Anixx
    Commented Oct 29, 2023 at 11:21
  • 1
    $\begingroup$ Sorry, indeed I misunderstood. However the answer, in my view, is negative as well. Infinities are just a symptom of a bad use of math, not the problem itself. There is a procedure, for instance, to produce UV renormalization terms in QFT without passing through infinity. $\endgroup$ Commented Oct 29, 2023 at 11:25
  • 1
    $\begingroup$ @Anixx I believe another way to state this answer (Valter can correct me if I am wrong) is that the regularized sum in your question appears in one particular way of calculating the Casimir energy. However, there are other calculational methods (including some that are more physically transparent) where this sum never appears at all. Since whether or not this regularized sum appears depends on how you do the calculation, it isn't particularly fundamental, and probably not worth spending a lot of time worrying about philosophical implications. $\endgroup$
    – Andrew
    Commented Nov 6, 2023 at 17:42
  • $\begingroup$ @Anderw I completely agree with your comment. $\endgroup$ Commented Nov 6, 2023 at 18:42
1
$\begingroup$

Mathematical objects are products of human imagination. They are not physical, although they sometimes are adequate models of physical things. Your question confuses the mathematical with the physical. No physical object is "actually" a mathematical object.

$\endgroup$
4
  • $\begingroup$ This is like to claim that there are no physical quantities at all. $\endgroup$
    – Anixx
    Commented Oct 28, 2023 at 14:41
  • $\begingroup$ @Anixx Not at all. When you measure something, there's physics behind it. But the mathematical objects we use to represent the measurement are not physical. It's not any different from understanding that the word "dog" is not an actual dog. $\endgroup$
    – John Doty
    Commented Oct 28, 2023 at 14:59
  • $\begingroup$ Are there physical quantities? If yes, are they only real? Or complex as well? $\endgroup$
    – Anixx
    Commented Oct 28, 2023 at 16:31
  • $\begingroup$ @Anixx There are physical phenomena. We construct models of these from mathematical objects. We use various mathematical objects, including "real" and "complex" numbers as needed. Again, you're confusing the description (mathematics) with the reality (phenomena). $\endgroup$
    – John Doty
    Commented Oct 28, 2023 at 16:40

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.