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).
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
http://iridia.ulb.ac.be/~marchal/
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