Bruno Marchal wrote:
> Le 16-août-06, à 18:36, Tom Caylor a écrit :
> > I noticed that you slipped in "infinity" ("infinite collection of
> > computations") into your roadmap (even the short roadmap).  In the
> > "technical" posts, if I remember right, you said that at some point we
> > were leaving the constructionist realm.  But are you really talking
> > about infinity?  It is easy to slip into invoking infinity and get away
> > with it without being noticed.  I think this is because we are used to
> > it in mathematics.  In fact, I want to point out that David Nyman
> > skipped over it, perhaps a case in point.  But then you brought it up
> > again here with aleph_zero, and 2^aleph_zero, so it seems you are
> > really serious about it.  I thought that infinities and singularities
> > are things that physicists have dedicated their lives to trying to
> > purge from the system (so far unsuccessfully ?) in order to approach a
> > "true" theory of everything.  Here you are invoking it from the start.
> > No wonder you talk about faith.
> >
> > Even in the realm of pure mathematics, there are those of course who
> > think it is invalid to invoke infinity.  Not to try to complicate
> > things, but I'm trying to make a point about how serious a matter this
> > is.  Have you heard about the feasible numbers of V. Sazanov, as
> > discussed on the FOM (Foundations Of Mathematics) list?  Why couldn't
> > we just have 2^N instantiations or computations, where N is a very
> > large number?
> I would say infinity is all what mathematics is about. Take any theorem
> in arithmetic, like any number is the sum of four square, or there is
> no pair of number having a ratio which squared gives two, etc.
> And I am not talking about analysis, or the use of complex analysis in
> number theory (cf zeta), or category theory (which relies on very high
> infinite) without posing any conceptual problem (no more than
> elsewhere).

When you say infinity is what math is all about, I think this is the
same thing as I mean when I say that invariance is what math is all
about.  But in actuality we find only local invariance, because of our
finiteness.  You have said a similar thing recently about comp.  But
here you seem to be talking about induction, concluding something about
*all* numbers.  Why is this needed in comp?  Is not your argument based
on Robinson's Q without induction?

> Even constructivist and intuitionist accept infinity, although
> sometimes under the form of potential infinity (which is all we need
> for G and G* and all third person point of view, but is not enough for
> having mathematical semantics, and then the first person (by UDA) is
> really linked to an actual infinity. But those, since axiomatic set
> theory does no more pose any interpretative problem.
> True, I heard about some ultrafinitist would would like to avoid
> infinity, but until now, they do have conceptual problem (like the fact
> that they need notion of fuzzy high numbers to avoid the fact that for
> each number has a successor. Imo, this is just philosophical play
> having no relation with both theory and practice in math.
> > The UDA is not precise enough for me, maybe because I'm a
> > mathematician?
> > I'm waiting for the interview, via the roadmap.
> UDA is a problem for mathematicians, sometimes indeed. The reason is
> that although it is a "proof", it is not a mathematical proof. And some
> mathematician have a problem with non mathematical proof. But UDA *is*
> the complete proof. I have already explain on this list (years ago)
> that although informal, it is rigorous. The first version of it were
> much more complex and technical, and it has taken years to suppress
> eventually any non strictly needed difficulties.
> I have even coined an expression "the 1004 fallacy" (alluding to Lewis
> Carroll), to describe argument using unnecessary rigor or abnormally
> precise term with respect to the reasoning.
> So please, don't hesitate to tell me what is not precise enough for
> you. Just recall UDA is not part of math. It is part of cognitive
> science and physics, and computer science.
> The lobian interview does not add one atom of rigor to the UDA, albeit
> it adds constructive features so as to make possible an explicit
> derivation of the "physical laws" (and more because it attached the
> quanta to extended qualia). Now I extract only the logic of the certain
> propositions and I show that it has already it has a strong quantum
> perfume, enough to get an "arithmetical quantum logic, and then the
> rest gives mathematical conjectures. (One has been recently solved by a
> young mathematician).
> Bruno

What is the non-mathematical part of UDA?  The part that uses Church
Thesis?  When I hear "non-mathematical" I hear "non-rigor".  Define
rigor that is non-mathematical.  I guess if you do then you've been
mathematical about it.  I don't understand.


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