FOM: Effective Bounds in Core Mathematics

[Fred Richman]

Harvey Friedman wrote:

> By the way, are you a constructive mathematician?

This was the last question but I'll answer it first so that I can
write in the first person for the others. I consider myself a
constructive mathematician. I satisfy the definition I gave earlier:
"a mathematician who systematically proves theorems without appeal to
the law of excluded middle".

> Richman writes:

> "I think that it is more accurate to say that constructive
> mathematicians and classical mathematicians are talking about
> the same things."

> Do you really want to say this? Isn't the constructive
> mathematician tempted to accept or regard as plausible, a
> statement like:

> * if x is a real number then there is a recursive sequence
> of rationals q1,q2,... such that for all i, |qi - x| < 1/i?

> the classical mathematician regards this as absurd, [so] how
> can you maintain that "they are talking about the same things"?

My reasons for rejecting * are the same as those for rejecting the
weaker, negative versions ** and ***, given later by Friedman, so I
will not address ** and *** as such. Indeed, I prefer * to the others
because it at least says something interesting. (Rejection does not
mean denial: I am not claiming that * is absurd.)

The short answer is that I am not tempted to accept *, or even to
regard it as plausible. Classical refutations play a role in
constructive mathematics similar to that of Brouwerian
counterexamples: classically refuted statements are rejected. This is
not as controversial as Brouwerian counterexamples, hence not so
talked about, because classical mathematicians also have no trouble
rejecting those statements. Such rejection is essential if we want the
theorems of constructive mathematics to be theorems of classical
mathematics. I doubt that I would pursue constructive mathematics were
this not the case.

Now I certainly can't be less likely to regard * as plausible than the
classical mathematician, so I suppose I must be more likely to. It is
consistent with my view of the universe and I am free to investigate
its consequences, which the classical mathematician cannot (in any
interesting way). But I don't think that precludes that we are talking
about the same things.

Before continuing I want to quote the question Friedman poses after
introducing ***.

> doesn't the constructive mathematician think this is
> extremely plausible as a constructive assertion?

I think a key point here is that the constructive mathematician does
not think in terms of "constructive assertions". That is a distinction
made by classical mathematicians to explain the behavior of
constructive mathematicians, or it may just reflect their own view
toward constructivity. The question is whether the constructive
mathematician views *** as plausible. The answer is no, within the
general framework of constructive mathematics, because *** is
classically refutable.

I'll continue by giving an analogy, at the risk of going beyond my
depth. Suppose the vast majority of mathematicians believed so much in
the continuum hypothesis that their work was most accurately described
as taking place in ZFC + CH. Yet some small minority rejected the
continuum hypothesis and worked within ZFC. 

You could say that the two factions must have different concepts of
what a set is, that they are talking about different things. That is
certainly a defensible position, but it doesn't seem very attractive
to me from the minority point of view. Would it be more accurate to
say that the two factions are talking about different things, or that
they have different views of the same thing, the minority simply not
being committed to the continuum hypothesis?

The majority, in trying to understand the minority view, or for
reasons of their own, might build a model within ZFC + CH where ZFC
holds but CH fails. I assume this can be done. Am I wrong? They might
then think that the minority had that model in mind, and so were
talking about different things. To confirm that, they could ask of the
minority whether some particular sentence, true in the model but
refutable in ZFC + CH, was plausible to them. Given that this sentence
doesn't open their eyes so that they would now be inclined to affirm
or to deny CH, how should the minority respond? And if they thought it
was plausible, just plausible, would that confirm that they were
talking about different things?

--Fred