FOM: reply to Holmes (about set theory and genuine Platonism)

Joseph Vidal-Rosset jvrosset at
Sat Sep 2 13:45:27 EDT 2000

Dear Randall,

First thanks for your mail. I apologize for being so late to reply. I 
felt free to post the correspondence to these lists because I thought 
that these philosophical issues could be interesting for colleagues. 
I don't want that one believes that I avoid the discussion on the 
lists after having started. But I will try to reply to you rather 
shortly and, I hope, more clearly than I wrote first. The best is to 
avoid useless polemics and that is why I want to insist only on the 
main points of my position.

First I agree with you when you wrote that "there is no philosophical
reason why a Platonist could not be committed to a different set
theory (for example, NF) and there is nothing about strong set
theoretical realism that dictates the adoption of any particular set
I reply that I did not hold that is impossible to base Platonism on 
NF. I think only that there are strong reasons in set theory in 
general which are very puzzling for genuine Platonism, and the most 
famous theorems about NF belong to these reasons. I see a 
philosophical importance in Specker's theorem and I remember that you 
are reluctant to accept it, or, more exactly, you do not give to this 
theorem the same philosophical interpretation.

At 11:48 -0600 9/08/2000, holmes at wrote:
>This is not to say that I don't think there are
>philosophical arguments which might lead a realist to adopt one set
>theory rather than another: I think that the Platonist who believed in
>NF would have more explaining to do than the one who believed in ZFC

I reply that it would have been interesting to know from you why 
there is more explaining to do for the Platonist NF fan, because I 
wonder if it is not already a move toward an agreement with me.

But let me explain me better what I am thinking. My argument concerns 
only realism in the strong meaning, i.e. Gödel's realism for example, 
" "realism" for the view that there is one objectively-determinate 
set theoretic universe, so that CH (and the rest) must be either true 
or false there." (P. Maddy, Naturalism in Mathematics, p. 87) More 
precisely again my goal is to show the difficulties of genuine 
Platonism with set theory pluralism and especially with NF. By 
"genuine Platonism" I mean also an anti-empiricist philosophy, like 
Plato's was. I am afraid that such a goal does not seem very 
interesting in a time and in a country where most philosophers are 
positivist and empiricist. But because I respect sincerely the 
genuine Platonistic position, I think that it is not philosophically 
trivial to show why set theory can produce skepticism about it (what 
Quine's realism, very different from Gödel's, does in fact).

The Platonist philosopher interprets the pluralism in set theory as a 
sign of transcendence and independence of the mathematical universe 
that mathematicians are trying to see. At the exact opposite, Logical 
Positivism interprets such a plurality as the result of different 
conventions on the ground of pragmatism (Carnapian Principle of 
Tolerance): Syntax gets rid off Metaphysics. I think that there is no 
way to prove that one of these philosophical positions, correctly 
understood and honestly exposed, is wrong. But I claim that there 
exist logical arguments which weaken the former. I come to Bernay's 
paper and Specker's theorem. You wrote:

>For a Platonist, for whom mathematical theorems are _facts_,
>there really should be no philosophical difference between an axiom
>and a theorem _as mathematical facts_ (of course an axiom plays a
>distinguished role in his derivation and organization of the facts,
>but for the Platonist this is quite independent of the facts

If your meaning of "Platonist" is the same than mine, I deny that 
what you say is right from a strict philosophical point of view. When 
Russell tried to show that the actual infinite can be deduced from 
mathematical induction (Introduction to Mathematical Philosophy, Ch. 
3, 1921), it is well known that he moved towards Nominalistic 
positions (1921). To get an infinite set as a theorem and not as an 
axiom is not without philosophical consequences. If you have in mind 
the history of theological proof the actual infinite of the concept 
of God was given as an existential axiom, and never given as a truth 
which can be proved from finite ways and finite elements (that is 
obvious in Descartes's and Spinoza's works, but I have a doubt about 
Leibniz, maybe someone will help me). Maybe you will think that is 
nothing to do with our discussion about Platonism, but it does if you 
remember that I am concerned with genuine Platonism: if, in our 
knowledge, there is at least one a priori idea which is considered as 
a truth and which cannot be deduced from finite elements, thus 
empiricism becomes impossible  because  every empirical knowledge is 
finite or indefinite, and there is no empirical knowledge of an 
infinite set (if you pay attention to the Kantian distinction between 
"to think" an "to know"). I recognize that if you do not care to 
empiricism, there is not a great philosophical interest to the 
alternative of being committed to an infinite set via an existential 
axiom or via a theorem, that is not important from the Quinean point 
of view of ontological commitment. But to not pay attention to this 
basic philosophical point (to be or not to be empiricist) is a 
betrayal of the systematic spirit of philosophy where all questions 
are linked together. (Analytic Philosophy has great merits, but some 
of its development are sometimes so technically specialized that some 
philosophers believe wrongly doing science and forget that 
philosophical problems are larger than they see...). When Bernays 
wrote his paper and talked about the presupposition of the infinite 
set of integers, it is obvious that the mathematical context was the 
classical set theory with an existential axiom of infinite and 
without theorem about it, so the word "presupposition" that he uses 
denotes the axiom of infinity in a Zermeloian set theory, as shows 
the following of his paper (written in French). I do not understand 
what you say when you write:

>The totality of integers (as defined in NF) is presupposed if one
>supposes that the axioms of NF are true.  This has nothing to do with
>the fact that Infinity is a theorem rather than an axiom.

In the ordinary meaning of "presupposition" it seems to me that the 
truth of axioms is presupposed (or supposed) in the deduction and 
that is why it is interesting to prove new theorems. If the infinity 
is seen as "presupposed" that begs the question. (Maybe you have in 
mind another meaning of this word or I do not understand something, 
but that is only a parenthesis in my argumentation.)

Last, when Specker gave his astonishing result about NF, he 
demonstrated that infinity in set theory can be formally deduced from 
the syntactical trick of stratification + strong extensionality, and 
he demonstrated also that the Axiom of Choice could be interpreted as 
false for sets like V and true in the field of Cantorian sets. Even 
if this demonstration is not constructive enough because of its use 
of excluded middle, it is satisfying, as far as I can see, for a 
Quinean or a Carnapean philosophy of mathematics, and very bad for a 
genuine Platonism à la Gödel.
I am afraid that my mail is already too long but I hope that you 
understand now better my philosophical reading of this interesting 
story. If there remain only details in this discussion, do not feel 
forced to reply to the lists, that is up to you, and I apologize if 
you felt uncomfortable because of my ccing to the lists. (I hope that 
my English is not too terrible.)

Best wishes,

Joseph Vidal-Rosset

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