Meta-metamathematics

[Vaughan Pratt]

In FOM, Vol 229, Issue 6, Joseph Shipman proposes, "There is no physical
experiment which could ever provide persuasive evidence for or against
statements of set theory that are not absolute," and asks "What is your
opinion of this proposition? I especially want to hear from physicists."

My original training was in theoretical physics.  My understanding at the
time was that future progress in physics would require crazy thinking, and
so by way of preparation I took a year dedicated to Pure Maths, eschewing
the alternative of a year in Applied Maths (a department then headed by
professors Keith Bullen and Bruce Bolt) as hopelessly pedestrian, before
taking a year of Physics.  (And now that much of my work in retirement is
in geophysics I'm a bit regretful that I made that choice in 1964.)

I believe you'll find large cardinals beyond crazy for most theoretical
physicists and pretty much all experimental physicists.  While the
continuum is a nice approximation to reality, Democritus is well respected
these days even if he wasn't thinking as small as the Planck scale, let
alone the vibrating strings and loops of string theory which for now are
considered far beyond the reach of any experiment.

And cosmologists seem to view the Big Bang in some sense as a compact space
that remained compact as it cooled to its present state, acquiring its
topology in the process.  One intriguing candidate for its topology is the
three-torus model of the universe (google it) proposed in 1984 by Alexei
Starobinsky and Yakov Zel'dovich at the Landau Institute in Moscow, which
could have either positive or negative curvature throughout, or a random
mix, or be completely flat.  (Yes, the customary embedding of the 2-torus
in R^3 has negative curvature on the inside and positive curvature on the
outside, but (i) the cosmos is 3D and (ii) need not be embedded in R^4.)

But even if the universe turned out to be infinite in extent, the
observable part is not infinite, that is, *surely* finite if we treat that
double negative intuitionistically and interpret "not-not" as "surely".
Any experiment performable on the universe that brought its result back to
Earth would, by definition I think, be an experiment on the observable and
surely finite universe.

Crazy thinking to be sure, but physical crazy.  Interpreting "father" as
per the Mathematics Genealogy Project, we might paraphrase John 14:2 as,
"In my advisor's house are many mansions."  If you and your advisor are
mathematicians you can imagine many universes containing large cardinals.

The difference between mathematicians and theoretical physicists is that
the latter are funded on the coattails of the experimental physicists and
find themselves competing for that funding with those theoretical
physicists who stick to physical crazy like Caltech's Sean Carroll.

Vaughan Pratt
-------------- next part --------------
An HTML attachment was scrubbed...
URL: </pipermail/fom/attachments/20220107/d1c4d07b/attachment-0001.html>