FOM: Reply to Tennant on arithmetic, Einstein etc.

[Vladimir Sazonov]

Neil Tennant wrote:
>
> 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
> implicitly involved?"
>
> Sazonov claims:
>
> > 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
> ask this!"
>
> 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.

The only analogy here is the fruitfulness of asking the question
"What does it mean?" about *mysterious* concepts such as
non-relativistic absolute notion of simultaneousness of events
or "the unique standard model of PA". 

> What is this preposterous revisionist arithmetician going to do?

It is interesting if you would happen to live in the previous
century, would you ask: "What is this preposterous revisionist
geometrician going to do?"

What
> axioms or theorems of PA will he/she dispute? What fundamental aspects
> of a conceptual scheme with identity

What is this "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.

Unfortunately, I am not inclined to make such kind of 
experiments. (Probably this is the main point where we 
disagree.) Also, what does it mean the "disembodied Cartesian 
soul" and a "universe with no physical objects"?  This is again 
something mysterious. If there are no objects then there is 
nothing to count. I prefer experiments, say, with real physical 
objects or with measuring sum of angles of a triangle or with 
deducing theorems in a formal system to recognize whether this 
system responds to my expectations.

> The answer
> is: exactly Peano-Dedekind arithmetic.

as to geometry, the answer probably should be "exactly Euclidean 
one"?

> Now could someone please
> explain how the mere presence of some physical objects could show that
> Peano-Dedekind arithmetic is somehow incorrect?

Is the chess game correct or not? Instead of correctness I would 
say that Peano-Dedekind arithmetic is full of meaning. [Also, it 
has a standard model in *any* universe of ZFC. Also, it is 
likely that it is formally consistent.  Also, it is 
aesthetically very nice.] The meaning of some of its formulas 
may be related with physical objects (say, pebbles; essentially 
we are dealing here with feasible numbers) in the evident way.  
Therefore, as in the case of geometry, we may check them 
experimentally. Say, that (i) the successor operation for 
feasible numbers is total (i.e., formally speaking, there are 
infinitely many of feasible numbers) and (ii) logarithm function 
is bounded by a concrete(!) not very large number on all 
feasible arguments. Of course (i) and (ii) contradict to the 
axioms of PA. This means that PA is incorrect *on feasible 
numbers*.  Also it is interesting to include the "true" axioms 
(i) and (ii) into a formal system which would be formally 
consistent so that we could formally deduce anything interesting 
on feasible numbers.  It is important because we have almost no 
*rational* mental experience with feasible numbers despite only 
such numbers are involved in all our practical calculations. We 
need a help of a formal system. Constructing it is both a 
technical proof-theoretical problem and foundational one.  
Actually some reasonable compromise should be (and have been) 
found. As I wrote earlier, some simple, rather unexpected, but 
seemingly reasonable theorems are provable in this system. 
These considerations show that at least in principle there is 
possibility for some alternative *feasible* arithmetic.

> 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.

Taking into account the whole context of the discussion, you 
did not convince me. Note, that I appealed not to authority, 
but to a concrete example and to how it does work. It happens 
that good examples are usually related with great peoples. 
Therefore their authority will be inevitably involved.


Vladimir Sazonov