FOM: The meaning of truth

[Joe Shipman]

Kanovei wrote:

> > Date: Wed, 01 Nov 2000 14:38:53 -0500
> > From: Joe Shipman <shipman at savera.com>
>
> > ... for mathematicians willing to commit to an ontology
> > containing infinite sets like {0,1,2,3,....} the problem with statements
> > like Goldbach's Conjecture (henceforth GC) is purely epistemological
>
> For anybody "willing to commit to an ontology containing" unicorns,
> and also willing to be a scientist, the problem is to present a
> unicorn to colleagues.
> So, are you going to present "infinite sets like {0,1,2,3,....}"
> in "ontology", that is, as I understand, empirically existing, of sort ?
> Are you also going to claim that those "sets" are determined enough to
> claim that any statement "like GC" is true of false in the same manner
> as 1=1 is true and 0=1 is false ?
> This means that you will finally let everybody to have a look at the
> notoriously invisible platonical "bats" flying between pages of
> philosophical books since centuries ago.
>

I am not going to do any such thing.  My statement above goes in one
direction:  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,...},+,*].  I do not claim that such an ontological commitment is
NECESSARY for saying what "true" means for arithmetical sentences, just that
it is SUFFICIENT.  The rest of my posting is addressed to mathematicians like
you and Sazonov, who don't make such a commitment and therefore must define
the meaning of "true" in some other way or regard it as an incoherent notion.

>
> > Professor Sazonov does not like to hear about "the" integers and would
> > not accept the structure {N,0,1,+,*} as a well-defined object.  He
> > therefore has trouble understanding what it could mean for an
> > arithmetical statement to be "just true" without reference to a theory,
>
> Why don't you explain to Professor Sazonov what does
> or "could" it mean, in order to relieve the poor guy from his
> troubles of understanding ?
>

I don't have to, I instead recapitulated your explanation, which is a
reasonable one that suggests three things it "does or could mean".  But I want
to clarify those three options some.

>
> > Professor Kanovei, while still unwilling to accept the infinite
> > structure {N,0,1,+,*} as a complete, well-defined object
>
> If in the ontological sense, yes, very much unwilling.
> It is since Renaissance if not earlier that anybody in the world
> of science claiming of having seen a unicorn was kindly asked to
> present an evidence. I only follow this pattern.
>
> > (a) "it has been correctly proved mathematically"
> > [Kanovei did not specify an axiom system
>
> I wrote "proved mathematically".
> Choice of axioms is not what was the subject of discussion.
>

But I want to know just what you mean by "correctly proved mathematically".
If I presented you with a proof that every even number >2 was the sum of two
primes, and the proof was rigourously formalizable in Peano arithmetic, I
imagine you would say that GC had been "correctly proved mathematically".  If,
on the other hand, I presented you with a proof of GC that used the assumption
of the consistency of n-Mahlo cardinals for all n, you might say that the
statement "Consistency of n-Mahlo cardinals for all n implies GC" had been
correctly proved mathematically, but I doubt you would say that GC itself had
been correctly proved mathematically.  Although I agree with you that it does
not affect this discussion where you draw the line, it does affect the
discussion that you draw the line SOMEWHERE because otherwise you cannot give
"correctly proved mathematically" a precise meaning.

>
> > I would hope his axiom system
> > includes PA but do not expect that it includes ZF since ZF allows a
> > truth definition for arithmetical statements like GC.]
>
> Has this anything to do with the ontological truth ?
>

If you accept the axioms of ZF, which include the Axiom of Infinity, then
there is a well-defined predicate for "Arithmetical truth", and one may make
sense of the statement "GC is true but not provable".   If you want to say
that your acceptance of the ZF axioms does not imply an ontological
commitment, that's OK, but you must still accept ZF's definition of
arithmetical truth and have no right to complain about the meaningfulness of
statements of the form "S is true but not provable" for arithmetical S.

>
> > (b) "it is true as a fact of nature"
> > [Here Kanovei refers to counting pebbles, but more generally to
> > statements about the physical universe.  In the case of facts that can
> > be established by computation,
>
> Computation is a method of mathematical proof
>
> > this is only different from case (a) as a
> > matter of scale.  For example, Appel and Haken proved (in a traditional,
> > humanly verifiable manner) that the 4-color conjecture 4CC was implied
> > by a certain logically simpler statement S that a particular computer
> > program had a particular output, and then verified S by a computation on
> > a real physical machine.
>
> What this has to do with the point of discussion ?
> The 4CC proof can be and is regarded as mathematical.
> Do you mean that it is not ?
>

OK.  I will admit that it is possible to print out the trace of the 4CC
computation in 100 massive volumes and point to that document and call it a
"mathematical proof".  But it is not a proof that has been verified in the way
mathematical proofs traditionally are!

> Of course we have to believe that the computers are not faking us
> and also, to that extent, that dosen or so of specialists who really and
> fully understand the recent proof of FLT are not just plotting in order to
> get fat grants, but I am pretty sure we are in these beliefs much less
> credulous than you who believe in
> "an ontology containing infinite sets like {0,1,2,3,....}".

I have not stated any such belief!  I have pointed out that such a belief
renders the question of "the meaning of truth" unproblematic, but am seriously
attempting to address this question without resorting to such a belief.


> > (c) "That it is true is given in a sacred script" [I suspect Kanovei is
> > joking here, thinking that nobody believes in the truth of a
> > mathematical statement because of a religious text.
>
> Read carefully, I wrote about "sacred scripts", not about any religy
> nor any deity nor a someone's confession, of course.
> By those I mean, to be concrete, the known metaphysical thesis of
> Goedel's misinterpretators, which is nothing but a sacred script of
> a branch of modern pseudophilosophy  of mathematics.
>

I understand the distinction here, but can you give a reference for this
metaphysical misinterpretation?

>
> > A committed traditional theist will probably believe not only in
> > the existence of infinite sets (since infinity is a traditional
> > attribute of God) but also in arithmetical corollaries like Con(PA) and
> > maybe even Con(ZF).]
>
> A "committed traditional theist", if also a scientist, first of
> all separates scientific matters from matters of faith.
>

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.

>
> > In the following, I will use the Twin Prime Conjecture (TPC) rather than
> > GC as an example, since it is Pi^0_2 and neither it NOR its negation
> > can, if true, be finitely verified so far as we know.  This will avoid
> > some confusion.
>
> Both of the two can be verified only by mathematical proof, which,
> with respect to not-GC, can be just a computation provided
> an unexpected counterexample is found.
>

Yes, a computation is a kind of proof; but I want to discuss the possibility
that no proof exists, which can lead to confusion when discussing GC because
if GC is false a proof must exist.

> > By the Principle of Parsimony, we should therefore not introduce a
> > notion of "truth" that is distinct from provability because it is not
> > needed and accomplishes nothing for us
>
> The notion of "truth" in the system mathematician vs. mathematics
> (ie, ontological truth, so to speak) is well known: true is what
> has been mathematically proved.
> If somebody wants to give any other notion this must be done in
> scientifically correct manner, at first. Is it needed and how it
> will be useful is another matter.

I regard the 4CC as having been shown to be scientifically true because we
understand how computers work well enough and their historical performance has
been good enough.  There exists a very large document which is alleged to be a
mathematical 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.

There are certain extremely large integers which have been shown by a
probabilistic argument to be almost certainly prime (10,000 numerical
witnesses were picked at random using a physically generated table of random
numbers, and a test was applied which always succeeds for primes but fails for
3/4 or more of the potential witnesses for composites, and it succeeded for
all 10,000 witnesses chosen).  I regard the primality of such an integer as
having been established with as much or more "scientific certainty" than the
4CC has been established, because the probability of bad luck (< 0.25 to the
10000th power) is far outweighed by the greater possibility of machine error
in the 4CC computation (which involved both a more complex algorithm and more
computational work).  But of course there is not even an alleged mathematical
proof in this case.  (In my opinion, such a probabilistic calculation can
always be replaced with a mathematical proof, but this is just an opinion
until someone proves the conjecture "BPP=P").

>
> > But I disagree with this
>
> You disagree because you do not argue in scientifically correct manner as
> the following habbracadabbra shows.
>

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

>
> > I would also allow the possibility of empirical discovery
> > of a mathematical-sentence-generating oracle which had never been known
> > to emit a sentence that was known to be either false or inconsistent
> > with its previous utterances.
>
> So you add a like of medieval folktales to "sacred scripts", here we go...
>

This is not a folktale -- various ways in which noncomputable functions might
still be experimentally derivable have been proposed.  If you assume on faith
the form of the Church-Turing thesis which says no such experiment can be
built, then of course it can never be more than a folktale, but our current
physical theories do NOT allow us to derive this assumption.

> > The degree of faith we would have in the
> > truth of the sentences it provides would depend on how good a physical
> > model we had for it; but for sufficiently plausible physical models and
> > sufficiently impressive oracular performance the epistemological status
> > of an utterance, while not attaining "mathematically proven", might
> > still qualify as "scientifically known".
>
> A mathematical statement qualifies as "scientifically known" if
> it is mathematically proved. The proof can include computation,
> in particular, made by a computer, provided the circumstances of
> the computation are mathematically verifiable and reproducible.
> In addition, another program can convert the computation in a
> formal proof in ZFC (in the 4-color case, probably, in PA), and yet
> another formally check the correctness, so I see no problems to
> qualify proofs that include human-uncomputable computation in this way.
>
> > We "know" the 4-color map
> > theorem to be true although no human has verified the proof
>
> This is wrong, the proof is verified because experts verified
> the program, observed the repetitious runs of computation on
> independent computers, and have sufficient knowledge as how those
> computers work.
>

Yes, but although the "experts verified the program" part is mathematical, the
"observed the repetitious runs of computation on independent computers" and
"have sufficient knowledge as how those computers work" are merely scientific.

> > The only difference here
> > is that the physical experiment we run will not necessarily be
> > algorithmically representable so that a human cannot "in principle"
> > verify it as he could for the proof of 4CC.
>
> Any kind of experiment, which cannot, in principle, be
> transformed into a mathematical proof (the 4-color proof can)
> cannot qualify as a proof (of a mathematical statement)
> to a sound mathematician.

It cannot qualify as a *mathematical* proof, but it can certainly qualify as a
*scientific* proof.  As for the 4-color proof, your grounds for asserting its
correctness are scientific but not strictly mathematical.

>
> V.Kanovei

-- Joe Shipman