[FOM] Query for Martin Davis. was:truth and consistency

Bill Taylor W.Taylor at math.canterbury.ac.nz
Sun Jun 15 03:06:29 EDT 2003

Vladimir Sazonov wrote:

> I should also note that I am not an ultrafinitist,

Here I definitely think we can say you are wrong.  It seems to me to be
the most helpful approach that in such debates, if a person is to be
described by some such label, it should be what OTHERS think is most
appropriate, not by what he wants to call himself.  A politician who
thinks he is a leftist but who comes to espouse rightist views and
enact rightist legislation will be called a rightist by others, correctly,
much as he still wants to think of himself as a leftist.     And we have
all heard the whining semi-redneck complaining "I'm not a racist BUT..." .

So; as I think most here would call you an ultrafinitist, so you are one.
But it is not necessarily a term of opprobrium!

> I am rather a formalist

Certainly that too!  Formalist and ultrafinitists do neither exclude
nor include one another, they overlap either way, but there is certainly
a very strong correlation between the two; as instanced in the article
by Henle in "Math Intelligencer" a few years ago, wherein he wrote
a formalist apology, but also espoused (in the Q & A) very ultra views.

As a quick attempt at definition, I think we could maybe say:

(a) An ultrafinitist denies any meaning to numbers as great as
    (some fixed number in orthodox math, typically a power tower).

(b) A formalist is one who denies meaning to any mathematical idea
    unless and until it has been precisley formalised.

I think you are clearly both.      But this is linguistics, not math.

>> that metaphysical objections to N are so absurd as to amount to throwing
>> out the baby with the bathwater.

> Which baby? Formalist view throws nothing, except things like
> the doubtful "standard" N, etc.

That's the baby!!   N is the baby!      YOU may regard it as doubtful,
but you must know you are in a tiny minority, we might say an "ultratiny"
minority, of mathies, so your doubt can have very little significance.

>> As I see it, the real problem with formalism, is that you deny the reality
>> of all of N, but somehow re-admit the reality of all derivations from
>> a system of logic.

OK, I withdraw this; I now understand your point here, you admit only
"as few" derivations as you admit natural numbers.  Fair enough.


The remainder of my post is not really for Vladimir; and I neither seek
nor desire a response from him on my final point, as I'm pretty sure I
know how it would go.  But I add this coda, merely lest anyone else
should imagine I am somehow conceding any point by silence.
(Not that I really think anyone else is interested in my musings.)

Vladimir wrote:

>> "Before computers were invented  1893....21 wasn't prime, but now it is".
>>  What a strange and unpalatable conclusion!

> My formulation would be: "Before computers were invented  1893....21
> wasn't proved to be prime, but now it is proved".

Of course we will all agree on this trite formulation.  And I guess VS
means to say there is nothing else much to be added to it.  But I guess
we others almost all disagree.  We want to say something about its TRUTH.

On previous occasions VS has specifically denied any meaning to truth,
other than provability.  Or even more stringently, I think he must say,
than "historical has-been-proved-ness"!

Let me re-form the above excerpt.  Let us imagine that someone, around 1900,
first proved that 1893....21 was prime.    He used only methods known for
centuries, just took somewhat more time about it than anyone else ever did.

I can ask:  "Before he proved it, was it TRUE?"

VS can only interpret this as "was it provable?"  But no, not even that,
because I doubt he'd want to admit the Platonic meaning of "provable but
not yet proved" any more than he would of "truth".     So he can only
interpret the question as "before that had it been proved?", which we all
agree is trivially false.   So he is virtually reducing math to history;
and actually only to attested history, not even events themselves.

Could we not ask him, "I have rolled this dice under a hat,
now I look and see it is a 4; so was it a 4 just before I looked?"

I think, with his views, he will have to say "no", or "I dont know",
or "it is meaningless", or some other such response; where we would all
unhesitatingly say "yes of course it was!"

IMHO, his view of the primes is very like his view of the dice,
and IMHO it is most un-mathematical at root.

            Bill Taylor           W.Taylor at math.canterbury.ac.nz
      MATH:      the discovery, clarification and rigorous study of
            precise relationships in number, pattern, and structure.

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