Bruno Marchal wrote:
> 
> At 13:36 09/01/04 +0100, Georges Quenot wrote:
> >Bruno Marchal wrote:
> >
> > > It seems, but it isn't. Well, actually I have known *one* mathematician,
> > > (a russian logician) who indeed makes a serious try to develop
> > > some mathematics without that infinite act of faith (I don't recall
> > > its name for the moment). Such attempt are known as "ultrafinitism".
> > > Of course a lot of people (especially during the week-end) *pretend*
> > > not doing that infinite act of faith, but do it all the time implicitly.
> >
> >This is not what I meant. I did not refer to people not willing
> >to accept that natural numbers exist at all but to people not
> >wlling to accept that natural numbers exist *by themselves*.
> >Rather, they want to see them either as only a production of
> >human (or human-like) people or only a production of a God.
> 
> What I mean is that their arithmetical property are independent
> of us.

I don't think this is very different. I could argue that even if
natural numbers were not out there, as soon as anybody consider
them, their properties automatically come with and impose themselves.
Even this seemingly weaker statement can be contested and it is not
actually weaker but equivalent since there might be no other way than
this one for natural numbers to be out there.

Some people do argue that there is no arithmetical property
independent of us because there is no thing on which they would
apply independentkly of us. What we would call their arithmetical
properties is simply a set of tautologies that do come with them
when they are considered but exist no more than them when they
are not considered.

> Do you think those people believe that the proposition
> "17 is prime" is meaningless without a human in the neighborhood?

" "17 is prime" is meaningless without a human in the neighborhood"
is exactly the kind of claim these people make (possibly generalizing
the concept of human to aliens and Gods). After discussing with some
of them I think they actually believe what they claim. I am not sure
however that we always fully understand each other and that you or I
would exactly understand such a claim in the same way as they do.

> Giving that I hope getting some understanding of the complex human
> from something simpler (number property) the approach of those
> people will never work, for me.

And certainly vice versa. Though it is difficult to have them saying
it explicitely I have the feeling that the reason why they do not
want the natural numbers to be out there and even as not possibly
being considered as out there is that they do not accept that the
complex human be understood from something simpler (number property).
They do not even accept the idea being considered, were it as a mere
conjecture or working hypothesis. Their more official argument is
that such a view would prevent the foundation of human dignity.

> Also, I would take (without added explanations) an expression
> like "numbers are a production of God" as equivalent to
> arithmetical realism.

Yes and there are several ways to understand this.

> >And I said "unfortunately" because some not only do not want to
> >see natural numbers as existing by themselves but they do not
> >want the idea to be simply presented as logically possible and
> >even see/designate evil in people working at popularizing it.
> 
> OK, but then some want you being dead because of the color of the skin,
> or the length of your nose, ... I am not sure it is not premature wanting
> to enlighten everyone at once ...
> I guess you were only talking about those hard-aristotelians who
> like to dismiss Plato's questions as childish. Evil ? Perhaps could you be
> more precise on those people. I have not met people seeing evil
> in arithmetical platonism, have you?

I have not met any of them physically but I had discussion with
some of them via Internet. There might not be so many of them but
there are. You will find, at least in the US, a lot of people
considering the views of evolution and/or of the big-bang as evil.
If they finally have to abandon these positions due to the amount
of evidence in favor of it, the last line of defence for their
conception of a personal God and for a significant role for Him
could be at the level of artihmetical realism. Artihmetical
realism by itself (not from a distinct personal God) is therefore
seen as evil by them. As I mentionned, they usually do not put it
that way. Rather they argue that such a view would prevent the
foundation of human dignity and the like.

Georges Quénot.

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