FOM: The meaning of truth

[Joe Shipman]

Kanovei wrote:

> > 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,

By determinate I mean that ALL the properties of the object are determined.  In the
case of the integers, 1st-order arithmetical sentences is what I mean by "property",
but for unicorns (a') is vacuous because only if unicorns really existed physically
would we say that all their properties were determined.  Because there are no real
unicorns, only a few properties of unicorns are determined (such as are "hoofed
quadruped with a single horn in the middle of the forehead"), which suffice to
distinguish unicorns, were any to be discovered, from all the other creatures we
know about.  This is not the same thing as ALL properties!

I think we can agree that "X exists" implies "All the properties of X are
determined".  You seem to think that I am reversing this, saying that even if all
the properties of X are determined then X does not necessarily "exist".  I am
willing to grant that there is a sense for the word "exist"  which goes beyond "all
properties of X are determined", but this is irrelevant because I was only trying to
argue for the other direction!

This thread is about "the meaning of truth".  "Truth" is a property that sentences
may have.  For arithmetical sentences, a question we are discussing is whether
"truth" means anything other than provability.  Basic results in model theory allow
us to define "truth" for arithmetical sentences in a way that goes far BEYOND
provability, IF we have accepted the "existence" of the set of integers (in ZF, once
you have accepted the Axiom of Infinity you can get the set /omega={0,1,2,3...} and
define the operations + and *).

> > 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.

I don't understand, if you accept ZFC, why you don't accept the standard ZFC
definition of truth for arithmetical sentences.  For any given arithmetical sentence
S, ZFC proves that "exactly one of S and ~S is true", but WHICH of the pair is true
can only be established by ZFC in a recursively enumerable set of cases.  (Actually,
ZF is good enough because ZFC and ZF prove the same arithmetical sentences.) You're
denying the set {0,1,2,3...} "existence" in a certain sense, but are you also
denying that for every pair of arithmetical sentences {S,~S} exactly one of them is
true?  I know that Professor Sazonov would deny this because he does not think the
notion of "the set of integers" is sufficiently specific, but I don't think he
accepts ZF either.

> > 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.
>

Yes.  For 4CC, it is only a difference in scale -- and we have "scientific proof",
not only of 4CC, but ALSO of "a mathematical proof of 4CC exists".  The distinction
I am making is that there may be OTHER mathematical statements S for which we can
have scientific proof of S but NOT scientific proof of "a mathematical proof of S
exists".

>
> > 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.
>

Since you call this a misinterpretation, you must think that only sentences which
are provable mathematically are true "ontologically".  But I can prove (in ZF) that
EVERY pair of arithmetical sentences {S, ~S} contains exactly one true sentence, and
that there exists a pair of arithmetical sentences {G,~G} which contains either zero
or two provable sentences.  It's still not clear to me whether you are rejecting the
concept of arithmetical truth or rejecting a concept of existence that is stronger
than "all arithmetical sentences have truth values".

> > 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.
>

Of course not, if you are referring to the mathematical and not the theological
profession.

> > ...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.

I say that the existence of a mathematical proof of 4CC has been scientifically
established, and that 4CC has been scientifically established.  I am willing to
adopt your language and say that "If the existence of a mathematical proof of X has
been scientifically established, then X has been mathematically established".  What
I would like to know is why you think that "X has been scientifically established"
entails "The existence of a mathematical proof of X has been scientifically
established".


>
> > 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.

There is apparently mathematics you do not know.  Given an odd number n, choose
using physical methods (such as a table of random numbers generated by radioactive
decay processes) K integers w1, w2, ..., w_K between 1 and n.  Let m be the largest
odd divisor of n-1 ((n-1)=m*(2^r) for some r >= 1).  For each w_i, calculate the
sequence (w_i^m mod n, w_i^(2m) mod n, w_i^(4m) mod n, ...., w_i^(n-1) mod n).  This
sequence has r+1 elements.

For prime n, either the first item in the sequence will be +1 or the sequence will
contain the value (n-1) followed by the value 1.  For composite n, this will be the
case for at most 25% or the possible values of w.  (Miller and Rabin.)  Therefore,
if you are given a number which passes the test for a few thousand RANDOMLY chosen
w, you may be sure that either it is prime or you have been extraordinarily unlucky,
so unlucky that "machine" error is a MUCH more likely explanation even if you ran
the calculation on 100 different independent machines.  If you are willing to only
call this a "substantiated conjecture", why are you willing to call 4CC "proven"
when the chance of machine error in the 4CC proof is much larger than the chance of
"machine error OR bad luck" in the other case?

>
> > 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.
>

Why "must"?  As I said, I am willing to accept your usage that we call
"mathematically proven" a statement X such that we can scientifically prove that a
mathematical proof of X exists.  This is a STRONGER property of X than "We can
scientifically prove X".

>
> > 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.
>

Your form of CT is weaker than the one I stated.  The one I stated is not a
statement about human psychology but about the physical universe.  It says that any
sequence we can generate by well-defined physical experiments is recursive.  Now, if
this were false and the nonrecursive experimentally generated sequence were
mathematically DEFINABLE (i.e. in the context of some super-grand-unified physical
theory), then we would have a mathematical fact which could be scientifically
established as true but not mathematically proven.

> V.Kanovei
>

-- Joe Shipman