antinegationism reblogged your post and added:

Unfortunately, in a truly physical context, your…

(Replying like this because the post chain is becoming very large)

About the weights: that was supposed to be a hypothetical setup in Newtonian mechanics, not a thought experiment about something you could actually do.  Sorry if I didn’t make that properly clear.

My point is that there are cases in physical theories when an infinity really should appear, either corresponding to some process that can continue without bound, or to some breakdown in one of the postulates of the theory, or both.

The point is that that infinity is telling us something accurate, and shouldn’t be renormalized away or otherwise removed.  Here’s another example – this is a cool one because it’s an actual infinity rather than just a potential one:

In inviscid gas dynamics, solutions with discontinuities can form in finite time from smooth initial data.  In some sense, these discontinuities are unphysical, because the idea of a bulk quantity like density having a true discontinuity is hard to square with the idea that such quantities are only defined as averages over large sets of particles.  This is an example of the theory going beyond its own strictly defined range of validity, like piling together more and more weights until Newtonian mechanics stops being accurate.

However, these discontinuities (shock waves) can be dealt with in a self-consistent and useful way.  They are a limit of the large-but-finite slopes you’d get in systems with very small viscosity, and often it’s simpler and better to just use inviscid theory and shocks than add a tiny viscosity to get rid of the infinities.

The point of this example is that the infinite slopes in gas dynamics, though they technically correspond to something finite in the physical world (there are no “density discontinuities” in reality), should really be infinite if the viscosity is truly zero.  (This is a widely accepted concept, not something I’m making up.)

Now, can we connect that example to the sum of the natural numbers?  Well, it’ll be pretty contrived, but really we can connect anything infinite to the sum of the natural numbers.  Write the slope as a function of time, rescale time so the divergence happens only as t –> inf rather than at finite time, take some function of the slope so that it grows like t^2 or slower (if it didn’t before), then say that that function is bounded above by (some constant times) the quadratic staircase thing defined by f(x) = (sum of natural numbers up to ceil(x)).  But that f is the partial sums of the natural numbers, and the natural numbers sum to something finite while the slope goes to infinity, so we have a contradiction.

You’re undoubtedly groaning right now because that was a hugely convoluted and belabored way of saying that if you use the usual definition for the sum of a series (which says the sum diverges) together with one that says it doesn’t, you can derive a contradiction.  Duh.

But that’s the whole problem here: we need to know which is valid in which case.  You could say “it’s obvious we can’t do the -1/12 thing here because we’re talking about something that really diverges.”  But when we’re talking about the sum of the natural numbers, we’re talking about something that always “really diverges,” in the ordinary sense.  When are we allowed to treat the sum as finite, and what licenses us to do so?  (That’s a non-rhetorical question, and if you know the answer I would love to hear it.)

As for the complex numbers issue, I think we may not disagree so much.  I am speaking from my historical moment, which is one in which ideas like “complex eigenvalues” and “complex exponentials” (and even maybe “contour integrals”) are things the average physical scientist gets comfortable with in their undergrad days and doesn’t lose any sleep over thereafter, while “renormalization” is this mysterious, creepy thing which still hasn’t been put on a fully rigorous foundation.

It’s true that there are cases where complex numbers work but are mysterious, but the paradigm cases are all unambiguous.  I use complex numbers in my day-to-day work and don’t even think of them as stranger than real numbers anymore, because they’re built into the theories I use in a well-defined and self-consistent way that I’ve become comfortable with.  No one thinks of the -1/12 thing that way because no one is sure how to do the same thing with it.

What you’re saying is that you’re really confident that one day we will view the two the same way, that one day we’ll understand how to wield the -1/12 thing rigorously.  And that we should be, as it were, “mathematical progressives” and speak as though we lived in that more enlightened age to come.  Which is fine if you are actually 100% sure of that prediction.  I’m not.  Some old mysteries have turned into unmysterious tools of the trade; that doesn’t entail that every current mystery will become an unmysterious tool of the trade.