[FOM] Query for Martin Davis. was:truth and consistency
Vladimir Sazonov
V.Sazonov at csc.liv.ac.uk
Mon Jun 23 16:59:49 EDT 2003
Sorry, by some reasons I reply rather late.
Bill Taylor wrote:
>
> 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.
I wrote that my views are RELATED with ultrafinitism. I tried to be
as precise as possible. I hope you understand the word "related".
But I cannot tell precisely what ultrafinitism is. And OTHERS do
not know well enough what do I mean by ultrafinitism. The discussion
demonstrates that some of the FOMers, including yourself, misrepresent
my views. Thus, branding me in this way leads to even more
misunderstanding, because OTHERS may have their own views on what
ultrafinitism is, strongly different from my views. Any branding is
offensive, whatewer you are writing below, especially after I wrote
that I am not an ultrafinitist and also because of the strength by
wich you insist on this branding. You know very well, at least from
my words (I also described this in some details), that my general
position is different. My finitist views describe better what I mean
(if you really want to understand me).
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).
I will not say for those who ARE ultrafinitists. As to me (I am
forced to repeat what I already wrote), I am interested in
formalizing the (vague) concept of feasible numbers. I do not
deny ANY meaning to large numbers considered in mathematics.
But I strongly distinguish between the meanings of 3, 7 and,
for examle, 2^1000. Distinguishing does not mean denying.
What I deny is the ABSOLUTE meaning. I understand what is 2^1000
in a sutable context.
> (b) A formalist is one who denies meaning to any mathematical idea
> unless and until it has been precisley formalised.
Replace "has been" by "will be eventually". I consider ANY vague
adea as (pre)mathematical if it is subject to formalizing. If
it proves to be that there is no way of formalization, then, of
course, the initial idea (intuition, imagination) does not deserve
to be called even premathematical.
> 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,
Your "standard", quasi-religious N is daubtful. My imaginary and,
of course, vague N (if this is the baby) is quite helthy. Nothing
bad was happened with it.
> 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.
Reference to minority/majority is not a scientific argument.
>
> >> 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.
As I wrote above I admit various numbers in corresponding context,
even exponential and superexponential towers. However, derivations
should be feasible/physical objects (if it is not a metamathematics).
For example, the last posting of Friedman are based just on this
considerations. Give me any example of mathematical proof of
non-feasible lenght! It is extremely strange for me that I should
tell these trivial things.
>
> .............
>
> 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.
Although this is addressed not for me, my opinions here are
predicted. I am forced to comment.
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.
I just replaced a stupid formulation YOU wanted assign to me
by something related and meaningful. I am not responsible
that stupid became trite.
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.
Again, voting instead of argumenting.
> We want to say something about its TRUTH.
I want to be a Chinese emperor! What is TRUTH if it is not
something based on physical experiments or it is not a truth
on imaginary objects governed by a formalism and discovered by
a formal/rigorous/mathematical derivation? (I hope it is clear
that I do not mean here the model theoretical definition of
the truth by Tarski which is done, say, within ZFC.)
See also an addition below.
>
> On previous occasions VS has specifically denied any meaning to truth,
> other than provability.
Not exactly so. In addition to the above note, I have a (vague) idea
of truth helping me to work in a formal system which has in my mind
some imaginary universe (like non-Euclidean geometry which was
called be Lobachevsky Imaginary one - a good example to recall;
there was no precise model, only axioms, intuition and ability
of rigorous reasoning.)
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?"
I would say only: "is it provable?" For me there is no reason
and interest to go in these "deep" considerations.
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.
I would never go into these considerations. They are highly not
interesting to me, as well as the consideration on the dice below.
It is really trivial that mathematical formalisms and proofs
in these formalisms appear during the time. There is nothing
to discuss here. The goal of Bill Taylor is to assert that
there is an ABSOLUTE TRUTH, but without giving ANY explanation
what does it mean. Instead of some constructive explanation, the
opposite point of view is presented as an absurd. On the other
hand, I tried to present my views by explaning each essential
step.
>
> 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.
Of course, Bill Taylor, as anybody else, may have any opinions.
The problem whether they are "non-religiously" convincing, that
is based on arguments, instead of beliefs or references to
majority. Actually this is the crucial ponit of the whole
discussion. Beliefs and majority may be changed. I presented
a rather simple sequential point of view where no "Holy Cow"
is taken as a foundation.
------------------------------------------------------------------------------
> Bill Taylor W.Taylor at math.canterbury.ac.nz
> --------------------------------------------------------------------------
Vladimir Sazonov
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