FOM: Determinacy of statements -- reply to Richman

Fred Richman richman at
Mon Jun 26 15:43:49 EDT 2000

"V. Sazonov" wrote:
> JoeShipman at wrote:

> > Richman:
> > >However, when constructivists do
> > >number theory, I would think that they have the same model in mind
> > >that every other mathematician does.

> > Professor Sazonov would disagree (if I have understood previous
> > posts of his correctly).
> More precisely, I do not understand what does it mean "the same model"
> in this context. However, I understand if we mean by a model here
> just a formal system or its preliminary semiformal version
> (describing "this" model).

The question is whether we are talking about the same things. This may
be a bad question, but it is an intriguing one. It would be difficult
to establish that we had the same model in mind; as Professor Sazonov
points out, it's not clear what that even means. Nevertheless, I feel
that I have some idea of what I'm talking about when I state the twin
prime conjecture, and I believe, from the communications I have had
with other mathematicians, that they have the same thing in mind.

I don't think that we have a formal system in mind. However, our
communications would appear to be in the form of a "preliminary
semiformal version". Because of that, one could view the common model
to be that preliminary semiformal system, or something closely related
to it. That is to say, what we are talking about could be identified
more or less with the system we use to talk about it. I don't see any
way to refute this view, nor am I interested in so doing. What I am
suggesting is that, whatever the nature of the model, the constructive
mathematician and the classical mathematician are thinking of the same


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