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Peter Webb wrote:
> Here is the original question, and my claim:
>
> > Can be any math theory INVENTED in a sense its basic consepts (like
> > member of set, element, etc.) in no way are related to relaity? Would
> > such "invented" theory have any sense at all? [Another poster]
> >
>
> Many people (FWIW, not including me) believe that set theory itself is such
> a theory. Where is the physical model of the Axiom of Infinity? Even if you
> accept a physically infinite universe, Set Theory (ZF) proves the existence
> of objects which cannot exist in a physical universe, such as uncomputable
> numbers, and different types of infinity. Some people believe that set
> theory cannot "have any sense at all" because it allows the construction of
> entities which cannot exist in the physical universe.
Okay, I understand that as your report of certain kinds of objections
to set theory that some people have.
My initial response was to the notion (if I'm paraphrasing it
correctly) that some mathematical objects "correspond to" (if that's
the term) physical objects and some don't. Whether that reflects your
own view, I don't know, but it's a view I have not (at least not yet)
been able to find much sense in or explanatory benefit from.
> > Would you give an example of such a modeling of a physical system as
> > you have in mind? [MoeBlee]
> Flight of a cannon ball.
That's a particular problem. I had more in mind an entire theory. But
this doesn't seem very important now, as the whole subject now seems to
me to be vague.
> > Getting back to the earlier remarks you made, how do you give a
> > mathematical formulization of a physical space without the continuum?
> > As I understand your latest reply, you recognize that the continuum is
> > needed for such things. But the continuum includes real numbers that
> > you don't seem to think have any role in describing a physical world.
> > Isn't their role at least that of being members of the continuum that
> > we rely upon to describe the physical world in such descriptions as
> > provided even by ordinary calculus?
>
> I have two objections to that statement. Firstly, the OP wanted to know
> whether there were mathematical objects that have no bearing on "reality". I
> don't think that you can simply define the continuum as "reality", at least
> not in the sense the OP seemed to mean.
Again, I'm starting to see this ('bearing on reality", etc.) as too
vague for me to have much stake in it. Nevertheless, I'm not claiming
that the continuum is itself a physical reality, only that it is a
mathematical object (and made of mathematical objects) that has as a
"bearing upon" an explanation of physical phenomena as do positive
natural numbers, at least in the sense of such applications of
explaining physical phenomena with ordinary calculus.
> Secondly, I am not even sure it is true. I can define R' as being the
> numbers defined as the limits of computable Cauchy sequences. I can then (I
> think) build calculus entirely within R', and nothing breaks.
That might be so; I don't know. I need to learn more about alternatives
to classical analysis. But it strikes me that some people (not
necessarily you) who are not very familiar with axiomatics are quick to
say that we don't need the classical continuum; but these people don't
suggest an axiomatization that is an alternative. That there are
alternatives, I'm not disputing, for lack of my knowledge of how truly
rigorous such alternatives are or are not; so I'm only pointing to a
certain kind of bluster I find in certain people who say things like
"We only need computable..." but without saying just how that can be
rigorously accomplished and what the heuristic drawbacks may be.
MoeBlee
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