# FOM: The meaning of truth

Kanovei kanovei at wmwap1.math.uni-wuppertal.de
Fri Nov 3 11:42:53 EST 2000


> Date: Thu, 02 Nov 2000 16:46:44 -0500
> From: Joe Shipman <shipman at savera.com>

> My statement above goes in one
> direction:
(a)
> IF a mathematician accepts {0,1,2,3,...} (along with the
> operations + and *)  as a *well-defined* and *determinate* object (which is
> all I mean by ontological commitment), THEN he has no trouble making sense out
> of  "GC is true and GC is not provable" because "true" means true-in-the-model
> [{0,1,2,3,...},+,*].

This thesis is absolutely similar to the following:

(a')
IF a zoologist accepts unicorn as a *well-defined* and *determinate* object
THEN he has no trouble to admit that unicorns should habituate yet unknown
continent in the Earth,

in that 1) both express a relation of someone to something
rather than a fact of any kind, 2) both are unthinkable to
have anything like truth value in any sense, 3) both are,
perhaps, admissible and discutable in chat across the table,
4) the existential core of the IF part of both is, mildly
speaking, doubtful, and 5) which naturally makes the THEN part
pointless.

> But I want to know just what you mean by "correctly proved mathematically".

My idea was that "accordingly to current mathematical standard",
which nowadays means in ZFC, either prima facie or in its
caricaturic image called Category Theory. But this is not what
was the point of discussion.

> no right to complain about the meaningfulness of
> statements of the form "S is true but not provable" for arithmetical S.

No educated logician would deny such a statement, as well as
the Goedel theorem itself, as mathematical sentences which
may be provable mathematically, yet the point of
discussion was that they (i.e. some of them) do not yield
that ontological meaning which is often stuck on them.

> OK.  I will admit that it is possible to print out the trace of the 4CC
> computation in 100 massive volumes

Suppose you thumb a paper on analysis and find an argument like

now, because (*) e^\pi<\pi^e, the desired result is achieved.

What you will do is take a calculator, find the values with
enough precision, compare and conclude if the argument is true
or wrong. Nobody in good mood will start nowadays to spend ink
on manual verification. The difference with the 4-color is in
scale only.

> can you give a reference for this
> metaphysical misinterpretation?

The only sound ontological meaning of the Goedel theorem is
that there is no theory (of certain kind) which is both
complete and consistent. The misinterpretation of it claims
that there exist (ontologically) sentences which are true
(ontologically) but not provable (mathematically) -- this
misthesis was expressed by several contributors to this list.
To that extend, I do not care who (out of FOM) said that first,
be it even Goedel himself.

> A valid point -- to convince other scientists who are not theists, he will of
> course not use a theistic argument.  I'm just saying he COULD use it to
> convince other theists.

I simply do not see any religious mathematician who will buy
any reference do divine as a professional argument in a proof of
theorem.

> if GC is false a proof must exist.

Any mathematical statement qualifies as false if there is a
proof of the falsity. The proof may include, or take a
form, of computation
or some simple geometric demonstration, not necessarily with
any amount of "ink" actually spent.

> ...proof of 4CC, but that it is a mathematical proof is something
> that I know only with a "scientific" level of certainty and not a
> "mathematical" level of certainty.

My vision is that as soon as

(1) there is a program whose certain output leads to a certain
conclusion regarding 4-color,
(2) the program works in digital manner, deals only with entire
entities (no fractions that must be cut, etc.) and does not
appeal to random numbers,
(3) it does not exceed physical capacities of the computer,
for instance, the amount of data to be stored never exceeds
the storage memory, the estimated time of work is not more
than something, etc.
(4) the (1,2,3) above are established as mathematical facts,

then few runs of the program on different computers with one and
the same result mean that the fact has been mathematically established,
in principle ONE run is enough, few are invented to inghibit possible
misfunction of the hardware.

> There are certain extremely large integers which have been shown by a
> probabilistic argument to be almost certainly prime

I do not understand this. If the number in question is in fact a product
of two also very big primes then no probabilistic test can pretend to
establish the primeness with really huge probability, unless there is
also some mathematics behind which I do not know.
Anyway tests like this can only show that the primeness is a substantiated
conjecture.

> possibility of machine error
> in the 4CC computation (which involved both a more complex algorithm and more
> computational work).

If you are going to take into account this possibility as
something which distinguieshes mathematical proof from a
merely "scientific" one then this brings heavens with it
because then any numeric material in mathematical proofs
(see above) must undergo manual verification.

> This is following Godel, so if I am scientifically incorrect below at least I
> am in good company.

Following Goedel writing means to be in a
company with the written, not with Goedel.

> If you assume on faith the form of the Church-Turing thesis

This is perhaps an interesting topic but hardly connected with
the topic of discussion (mathematical and ontological truth).
Anyway, my understanding of CT is approximately as follows:

(CT) It seems to be an immanent property of human being that
everything which human mathematicians agree to consider as
"algorithm" can be formalized as a recursive function, and
there is little doubt that this will ever change.

V.Kanovei