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

Bill Taylor W.Taylor at math.canterbury.ac.nz
Thu Jun 5 23:14:12 EDT 2003

->>Do you admit the reality of 10^10^10^10 in the same sense (whatever that is)
->>that you admit the reality of 3 and 7 ?   Would you put these three numbers
->>on the same footing, ontologically.
->In a sense yeas, in a sense not.

Aha.  So you can give me a definite maybe on that one, then?   ;-)

I was expecting an answer of that type, though I was hoping for a better.
I'm not interested in this system or that system, as in your reply.
My query was quite specific.  I specified "ontologically".  If you
insist that this has no meaning in the context, fine, but you could
have at least said so.

If you will not answer the question as asked, then there is no point
going any further.   I should have known an ultrafinitist would
be canny enough to avoid a definite answer.   ;-)

->What (as you say) "disinformed" me? Some deeper that in the school
->things like Goedel's theorems, especially on incompleteness and
->Goedel/Cohen proof on independence of CH, what demonstrated (to me)
->that both N and continuum are vague concepts,

WOW!  That's a mighty leap from a standing start!   I have sympathy with
the view that CH-independence and the like could make you newly unsure
of exactly what sets were; indeed this is my own case to some extent.
I am less sympathetic to it making one uncertain about the reals, though
even there I can see some vague concern.  But N !?  I quite fail to discern
the psychological processes that could take you from CH-independence
to unsureness about N.

But everyone is different; and if you say it happened to you, then I'm
sure we all believe you.

->Yes, yes! What are we ever talking about when discuss about
->this N? I know - this is some illusion. This is, I believe,
->a more honest position than to assert that there exists
->some idealized unique N. By which way unique? Illusion or
->idealization are something about which uniqueness makes
->no real sense. This is rather from religion-like views
->on which science should not rely.

First of all math is not science, though I know it is usually included
so in continental thought.  (I have heard this is mostly a linguistic
problem, that the German word for "science" means more like what we Anglos
would call "studies".)

Secondly, though it is mere mudslinging to call it religion, there is
undoubtedly an aspect of metaphysical thought in FOM.  However one might
like to remove it as much as possible, it seems to me (and most of us)
that metaphysical objections to N are so absurd as to amount to throwing
out the baby with the bathwater.

->I feel the difference, but in a sense they are equally ullusive,
->i.e. ALL of them are illusive.

Then you might as well say, so are small deductions from simple premises
illusive, so are small numbers, so are any ideas, so even are material
objects like tables and chairs and stones.  This cheap undergraduate
radical skepticism is ultimately a dead end.  Declaring something
to be illusive is no 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.
->Not all (I do not know what means `all' here), but only those
->we can really do.

Which changes with time, understanding and technology.   So you want to
make math reality dependent on biology and technology?   This is possibly
a tenable philosophical view, but is far beyond what most of us would
consider to be true mathematical thought.   You will soon be saying,
may have already said,   "Before computers were invented  1893....21
wasn't prime, but now it is".   What a strange and unpalatable conclusion!

->> The latter [ Th(PA) ] is more complicated than the former  [N] ,
->> and just as big, so why regard it as more fundamental?
->Not so complicated. Say, PA consists of several axioms
->and one axiom schema. It is based on FOL based on several
->(schematic) rules. We easily understand how to use these rules.
->All of this is quite concrete, unlike N,

Wrong.   You are comparing apples and oranges.  You should compare
the operations above with the simple operation of adding one to a
(written down) number and writing down the answer.   I'm sure even you
will agree that the latter is a LOT simpler and more fundamental than
the former!

->I am not sure that I will convince many participants of FOM
->as the previous experience shows.

I'm sure everyone will agree with that, at least.

               Bill Taylor       W.Taylor at math.canterbury.ac.nz
               Philosophical clarity would have the same effect
               on mathematics as sunlight does on potato shoots

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