[FOM] Am I a Platonist?

[Harvey Friedman]

Reply to Davis. A very nicely written and thought provoking posting.

On 10/27/03 9:35 PM, "Martin Davis" <martin at eipye.com> wrote:

> In a telephone conversation, Harvey Friedman once accused me of being an
> "extreme Platonist". I did not recognize myself in that description, and
> have thought for a long time about how to express my actual views in a
> coherent manner.

I think that the sxxx hits the fan when one claims that there is "clearly no
difference in objectivity between an arbitrary first order sentence in
V(omega + omega) and an arbitrary first order sentence in V(omega)". I think
you like that position, and fall just shy of asserting

"clearly no difference in objectivity between an arbitrary first order
sentence in V and an arbitrary first order sentence in V(omega)".

As I said earlier on the FOM, I feel uncomfortable with ANY position. I feel
even more uncomfortable when I see people being comfortable. Nobody should
feel comfortable. 
 
> I take it for granted that our minds are the result of the activity of a
> brain that evolved to deal with the exigencies of life faced by our
> hunter-gatherer ancestors. I find it a source of wonder that as the result
> of a kind of logical "over-spill", our minds can obtain mathematical
> knowledge. I believe that mathematical knowledge is objective knowledge,
> that (as G\"odel has remarked), we have no ability to alter the properties
> of the natural numbers. We might decide that for every natural number to be
> the sum of three squares (rather than the four that Lagrange established to
> be the case) would be aesthetically pleasing, but we have no more ability
> to make this so than we have to alter the diameter of the earth. This
> contrasts with works of fiction where a writer can certainly produce a
> desired ending to order. I regard this objectivity as crucial, and regard
> questions concerning the actual "existence" of numbers or even measurable
> cardinals as having no clear content, and in any case, being beside the point.
> 
> I regard this objectivity as a hard-won result

This I find confusing. It would seem that, according to you, we also have no
ability to alter the objectivity of this all. So how can it be a "hard-won
result"?

>of mathematical practice and
> in no way given to some kind of a priori sense or intuition.

Do you mean then that this objectivity became only clear to us from
mathematical practice? Otherwise, although it certainly exists (the
objectivity), and we may never have discovered it without hard work??

>It is a source 
> of wonder and delight that with our finite brains, evolved for very
> different purposes, we are able to obtain objective knowledge about things
> that are infinite.

I would say the same for the large finite. We know quite a lot about
arbitrary subsets of {1,2,...,100}.

Actually, what are the most impressive things that we know about the subsets
of {1,2,...,100}? This is the kind of question that I wanted to address in
my future postings on formalism/Platonism.

>Surveying the history of mathematics, we see
> mathematicians as almost reluctantly being driven to consider the infinite.
> Aristotle provided a safe mathematician's playground with his notion of
> "potential" infinity, suggesting avoidance of  any dealings with the
> "actual" infinite. Gauss explicitly warned mathematicians in a similar vein.

However, it should also be mentioned that one can systematically pursue a
program of 

removal of the infinite.

With the advent of great computers, would it appear that we could rid the
whole of applied mathematics of consideration of the infinite for any
practical purpose???
 
> In the 17th century Torricelli stunned his contemporaries by flouting the
> Aristotelian doctrine by exhibiting a geometric body that was infinite in
> extent,  but had a finite volume. This discovery of what was OBJECTIVELY
> the case about the relation between the finite and the infinite was not the
> result of metaphysical speculation, but the direct result of the reach of
> our mathematical powers.

I do think that this can be attacked as an oversimplification. There is the
question of the meaning of volume - rather than any objective situation. It
could be said that Torricelli succeeded in showing merely that it was a good
idea to extend the notion of volume to more objects. Lebesgue also did that
later.

>When Cantor began his study of transfinite
> numbers, it was not at all clear that this could be carried out in a
> satisfactory manner, and of course many refused (and continue to refuse) to
> accept it. But the work of the set theorists continue to develop this
> domain, and even when assumptions that transcend ZFC are used, the
> objective nature of their results seems compelling.

This may seem compelling to you, but not to many people. I don't take sides.

>In a recent talk by
> John Steel, he spoke of the striking way in which bifurcations in the
> higher transfinite don't happen. As he put it, "There is only one road
> up."  To my mind this is clear evidence that one is dealing with objective
> knowledge.

The "one way up" with regard to INTERPRETABILITY is even more general and
striking. So far, it appears that despite work in myriad directions in math.
logic, for any two natural S,T,

S is interpretable in T, or
T is interpretable in S

provided S,T contain a tiny amount of stuff. More accurately,

S is interpretable in T', or
T is interpretable in S'

where S',T' are the extensions of S,T by truth definitions.

If we use any appropriate sort of equiconsistency, we have

S proves T is consistent; or
T proves S is consistent; or
S,T are equiconsistent.

Of course, the one road up idea is wrong if not stated carefully. E.g., the
existence of a probability measure on all subsets of [0,1] does not follow
from even an elementary embedding from a rank into a rank. And some people
are more attracted to "the existence of a probability measure on all subsets
of [0,1]" as an axiom than they are to the existence of elementary
embeddings, or even the existence of a measurable cardinal, or even the
existence of an inaccessible cardinal.

Of course, it follows from your Platonism/realism, that

the existence of a probability measure on all subsets of [0,1]

has a "determinate" truth value.
 
> In a recent post, Harvey Friedman has raised the question of the
> possibility of the coherence of a belief that every sentence in some
> language has a determinate truth value while accepting that this truth
> value will never be found by the human race. He mentions in particular,
> sentences of enormous length.

I was proposing that it is hard to convince people that "every sentence has
a determinate truth value" if truth values cannot be found, or if truth
values are known to be non findable, or there is absolutely no evidence that
truth values can be found, or there is no plan or idea for finding truth
values, etc. Also, it is even harder to convince people of this, or what it
means, if the sentences involved are of length 10^100.

>But the fact is that there are many perfectly
> ordinary sentences of which it is perfectly coherent to believe they have a
> definite truth value which we are unable to determine. Does a planet
> recently discovered about some very distant star contain life? Or more
> mundanely, is the number of leaves on trees on my property at this moment
> even?

The trouble with this line of argument is that in the examples you cite, we
know how we could have become convinced of its truth value. We have a story
to tell, that cannot be assailed easily.

Only very special sentences in mathematics have the property that we can
even give any sort of plan for determining its truth value unassailably.

If we are talking about truly verifiable sentence of mathematics, I know of
no case YET where we can get our hands on a consequence of a logically high
powered axiom that can be confirmed in this way.

> The name I suggest for my point of view is Empirical Objectivism:
> "empirical" because our understanding of the entities occurring in
> mathematics is not the result of any transcendent intuition, but rather of
> direct experience as practicing mathematicians, and "objectivism" to
> express the belief in the objective character of mathematical truth.

When you say "as practicing mathematicians", you know that, historically,
only a very few want to seriously consider the type of statements that are
needed to be considered in order to even begin getting any meaningful input
from mathematicians - other than those who have a large stake in the
promotion of certain axioms.

Granted - we both know somebody who wishes to change this situation, to draw
a significant number of mathematicians into the arena...

I would not second guess how such things will play out. My esteemed
correspondent is not easily moved from his very formalist position - despite
his full knowledge of what logicians have done, and are doing.

> G\"odel was certainly not an empirical objectivist, but he approached that
> stance in the following statement from his 1951 Gibbs lecture: "If
> mathematics describes an objective world just like physics, there is no
> reason why inductive methods should not be applied in mathematics just the
> same as in physics. The fact is that in mathematics we still have the same
> attitude today that in former times one had toward all science, namely we
> try to derive everything by cogent proofs from the definitions (that is, 

> from the essences of things). Perhaps this method, if it claims monopoly,
> is as wrong in mathematics as it was in physics."  Of course in mathematics
> we don't have the check that experiment provides in physics. But that
> comparison belongs in another post.

The big difference between physics and mathematics that has to be confronted
is as follows.

When Einstein's GR is confirmed over and over again, in various settings, it
is clear that it could have been horribly disconfirmed - to the point of
refutation, by these experiments.

How do we set up an experiment for the correctness of ZFC - or even the
arithmetical correctness of ZFC - which is on a par with the Einstein GR
confirmation? 

COMMENT.

The alternative explanation for why it is that we can know so much about
infinite sets is that it turns out that for lots of infinite contexts, a
little bit goes a long way - i.e., very very few principles suffice to
formally derive so much.

A formalist reads this just as it is. The formalist notes that this
situation - a little bit goes a long way - starts to break apart as we move
more remotely from the integers.

The formalist refuses to believe that this is any evidence of any kind of
"objective reality" or "determinate truth values".

Harvey Friedman