FOM: Reply to Sazonov on arithmetic, Einstein etc.
neilt at mercutio.cohums.ohio-state.edu
Sun Oct 11 00:37:01 EDT 1998
Vladimir Sazonov complains that I provide
> a typical example of how mathematical and especially
> philosophical education may sometimes prevent an "intellectual
> elitist" to see even the trivial point on some (let imaginary)
> physical experiment.
Then he trots out the tired old reference to Einstein as the thinker
who 'broke out of the box':
> The meaningfulness of the ABSOLUTE notion of simultaneous events
> in spatially different places (the God sees what is simultaneous
> and what isn't!) was considered dogmatically as non-questionable
> Einstein ... dared to ask a "silly" question "WHAT
> DOES IT MEAN?"
Let's be a little more attentive to the details here. What Einstein
asked was "What does it mean to say that two events are simultaneous?
What is the operational definition of time-measurement implicitly
involved? What is the operational definition of distance-measurement
> In the case of arithmetic we may analogously ask: WHAT DOES IT
> MEAN the unique up to isomorphism standard model and absolute
> truth in it? I think this should be a legal question of f.o.m.
> But it seems that our education does not allow us to ask it.
I shall not go over the same ground as Charles Silver did in reply to
this. I just want to add, emphatically, "No, we may NOT analogously
Sazonov's analogy is completely broken-backed. Einstein was dealing
with the structure of spacetime, and was trying to accomodate the
constancy of the speed of light in all inertial frames, while
preserving the principle that no inertial frame is privileged. One
cannot do anything remotely similar or analogous with the natural
numbers. Einstein would in all likelihood have scratched his head and
exclaimed "Waaaas? Spinnst Du?" (or words to the effect) if anyone had
suggested revising *arithmetic* in a way (vacuously) pronounced
"analogous" to the way in which Einstein himself had revised mechanics
and physical geometry.
What is this preposterous revisionist arithmetician going to do? What
axioms or theorems of PA will he/she dispute? What fundamental aspects
of a conceptual scheme with identity are to be jettisoned or mangled?
There is a simple thought experiment that ought to put an end to the
suggestion that one might revise arithmetic. Just ask yourself what
theory of arithmetic would be excogitable by a disembodied Cartesian
soul in a universe with no physical objects in it at all. The answer
is: exactly Peano-Dedekind arithmetic. Now could someone please
explain how the mere presence of some physical objects could show that
Peano-Dedekind arithmetic is somehow incorrect?
Sazonov wrote further:
> Instead of direct answering, desirably in simple and clear terms
> (as Einstein did in his case), we are going around the question
> by appealing just to the great authority of Goedel. (Cf. posting
> of Tennant from Fri, 9 Oct 1998 08:55 EDT.)
First, I did not appeal to the great authority of Goedel for
anything. I simply referred to him in the context of my reply to
Silver, because everyone know that Go"del himself was an unrelenting
platonist about mathematical objects such as numbers and sets. It was
Go"del's philosophical convictions to which I was *referring*, not to
which I was *appealing* for the sake of argument.
Secondly, Sazonov appears not to realize that his own analogy
involving Einstein comes across as a thinly-disguised appeal to
authority, this time in an attempt to lend credence and dignity to
what might otherwise be regarded as acts of wanton intellectual
vandalism. We'd all love to pose as intellectual iconoclasts able to
effect the same sea-changes as Einstein did. But people who have
thought through these matters aren't impressed by the sort of
misdirected handwaving in which Sazonov has indulged.
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