FOM: The meaning of truth

[Kanovei]

> 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