Bruno Marchal wrote:
> Le 06-sept.-06, à 21:23, Tom Caylor a écrit :
>
> >
> > 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?
>
>
> Robinson Arithmetic (Q or RA) is just the ontic theory. The
> epistemology is given by Q + the induction axioms, i.e. Peano
> Arithmetic.
> This fix the things. The SK combinators (cf my older post on this
> subject) gives a more informative ontology, but in the long run none of
> the ontic theories play a special role. With regard to the TOE search
> they are equivalent. Now, RA is not interviewed. It defines the UD if
> you want (RA is turing equivalent). But RA cannot generalize enough. We
> need PA for having the machinery to extract physics from the ontic RA.
>
So you are saying we need induction for epistemology. I will wait to
see more of the roadmap.
>
>
>
>
> >
> >> 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/
> >
> > What is the non-mathematical part of UDA?
>
> If only the "yes doctor". UDA is applied math.
>
>
> > The part that uses Church
> > Thesis?
>
> Also, but in a lesser measure. Church Thesis is at the intersection of
> math and philosophy. But is, nevertheless, 100% Popper-scientific by
> being clearly refutable.
>
>
> > When I hear "non-mathematical" I hear "non-rigor".
>
>
> Gosh! "Mathematical" can be non rigorous (cf Euler). Rigorous can be
> non mathematical (take any good text in any field, or just take the
> following non mathematical reasoning:
> Socrate (a non math concept) is a human (also)
> All human are mortal
> Thus Socrate is mortal.
>
OK. I guess we are just assigning different scopes to math. The above
uses set theory, a part of mathematics; and it uses logic. Purely
mathematical proofs make use of set theory and logic also, but that
doesn't make them non-mathematical. Also, saying 2 apples + 2 apples =
4 apples is not non-mathematical simply by adding "apples" (a "non
math" concept). But is this what you mean by "applied math", when you
add the "apples"?
>
>
> > Define
> > rigor that is non-mathematical.
>
> Clear definition, clear postulate, valid reasoning, etc.
>
Again, I would call this math. If there is nothing outside of this in
the UDA, then I am content with continuing to listen.
>
> > I guess if you do then you've been
> > mathematical about it. I don't understand.
>
> Not necessarily. Or with "math" in a large sense: any applied math I
> would say.
>
> OK?
>
> Bruno
>
> PS Must go now, I will comment Russell and Stathis asap.
>
>
>
>
>
>
> http://iridia.ulb.ac.be/~marchal/
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Received on Thu Sep 07 2006 - 12:29:21 PDT