[FOM] On Existence of Mathematical Objects

[Harvey Friedman]

Reply to Sazonov.

On 10/9/03 3:42 PM, "Vladimir Sazonov" <V.Sazonov at csc.liv.ac.uk> wrote:

> More precisely, if you will take my posting as the whole, you will
> see that I even agree that all these fictions, as well as electrons
> and mathematical objects has some kind of existence, but quite
> different. When we say "exists" we should be sufficiently precise,
> in which sense. Many of them exists only in your, my and any other
> person's imagination and have no immediate relation to the real
> world. 
> 
>...

Perhaps Sazonov thinks that almost everybody else is talking

*NOT about the natural number system*

but rather talking about the

**imaginary natural number system**.

Then Sazonov may agree that the axiom scheme of induction for

**imaginary natural numbers**

is highly compelling, if not evident?

>... I do not say here about imagination.
> This is quite different thing. We can imagine anything
> without any belief in what we imagined. All mathematics
> can be understood and practically developed in this way.
> 
>... The great Cantor also had some very nice idea. It proved to be
> contradictory. 

The current view of the history of set theory is that Cantor's ideas were
NOT (as far as we know) contradictory. The current view is that the
large/small distinction is implicit in Cantor's writings (i.e., class/set
distinction). 

> I do not see 
> any essential difference between this idea and the idea of natural
> numbers. 

This does not seem to be defensible. The imaginary natural number system
enjoys some very fundamental properties that the imaginary set system does
not.

>May be PA will be once shown to be contradictory, too.

Are you suggesting a fundamental defect in the imagination of working
mathematicians? 
 
> It would be disaster [inconsistency in Peano Arithmetic] only for those who
have superbeliefs. 
> Just as it was in the case with the old paradoxes in set theory.

Wouldn't various theorems in real analysis and other areas of mathematics
have to be rewritten and retaught with negations in front of them?

Taranovsky:

> >most fields of
>> knowledge and practical constructions depend on arithmetic (although
>> most practical applications can actually be carried out in weaker
>> systems). 

Sazonov: 
> 
> Yes, of course. This or other way the most important mathematical
> considerations will survive. We will probably get a very good
> lesson on the general philosophy of mathematics.

Why do you believe this? From your point of view, why shouldn't the
contradiction appear in our imagining, say, all permutations of the first
100 natural numbers? What imaginations do you accept and what imaginations
are you skeptical of?

Taranovsky:

>> Fortunately, we know a priori that every axiom of PA is true,
>> and hence PA is consistent.

Savonov:
> 
> ??????????????????????
> ----------------------
> 
Is every axiom of PA true in the imaginary natural number system?

Harvey Friedman