Re: Cantorian pseudomathematics

Jean-Claude Arbaut wrote:
> Chas Brown wrote:
> > Jean-Claude Arbaut wrote:
> >
> >> cbrown@xxxxxxxxxxxxxxxxx wrote:
> >>
> >
> > <snip>
> >
> >>> It's not enough to simply specify what intuitions one hopes to capture
> >>> (although this is always nice); one must also specify what one actually
> >>> /means/ by the terms, in the mathematical sense.
> >>
> >>
> >>
> >> I'm nut sure... I mean, usually, to select something at random, one
> >> relies on a "true random generator" (or considered so, like a Geiger
> >> counter),
>
> > Is this a magical device, or does it actually have certain mathematical
> > relevant properties?
>
> Well, I don't know what kind of random you can expect from a Geiger
> counter. Take a die, it will suffice: *experimentaly* you see no reason
> why a face appear at some time, and you can check all appear equally
> often by computing the frequencies. You can even check other facts told
> by probability theory, but I cannot see a *mathematical* reason why a
> die should give a random value: I cannot even define random better than
> checking frequencies or that kind of property, and that's obviously not
> enough. At some point, it seems necessary to *admit* a die gives random,
> period. Maybe I'm wrong, but then give me your definition of "random".
>
>
> >> or a pseudo random generator, which is deterministic, but "looks like
> >> what is expected as true random".
>
> > And what would that "look like", mathematically speaking?
>
> See above. And statistical tests such as diehard help find pseudorandom
> numbers "sufficiently random".
>
> >> Then one manipulates these
> >> random values to get an object as needed. So, ultimately, you need to
> >> know what random is, or is expected to be, and I don't know that.
>
> > It seems like your operational definition is "some thing which is
> > consistent with a set of mainpulations being valid"; e.g., countable
> > additivity on countable disjoint unions, or whatever.
>
> Errr... maybe. I don't see the relation between countable additivity and
> my statement.
>
> >> The only thing I could tell, I suppose, is it's possible (or
> >> impossible), to
> >> get some kind of random object, based on some random value (usually
> >> uniform on {1,2,...,n} or on [0,1]). So at one step, I'll assume I
> >> already have something random.
>
> > Hence, you have some object or set of objects for which a set of valid
> > operations have been defined.
>
> You can call that a set if you like :-) But it's not necessary to use
> set theory to think about all that; and mathematicians didn't wait for
> Cantor or Kolmogorov to compute probabilities, with or without a good
> definition of probabilityor random, btw: in his "Calcul des
> probabilités" in 1912, Pincaré, after a 20 pages introduction about what
> random is, starts its first chapter by (more or less) "One cannot find a
> satisfactory definition of probability", and goes on with the "old"
> definition (k/N, not sets, of course).
>
> >> Hence, I can't define completely
> >> mathematically what "select a natural at random" means. Nevertheless,
> >> using assumptions made by basic probability theory (i.e. that true
> >> random exists and that this theory is able to model it is some way,
> >> dealing with frequencies of appearance of random events), one can infer
> >> what a uniform random integer would need to be, and to conclude it is
> >> not possible - almost all you need to admit is that an event with
> >> probability 0 cannot happen, well, at least in "real life", because
> >> measure theory won't agree...
>
> > Right. To assert that something like "a random natural" /should/ exist
> > is fine.
>
>
>
> > But to make a mathematical statement about a random natural
> > requires us to make a definition of what that means.
>
> I suppose.
>
> > Our general failure to do so is not a mathematical one; it a
> > philosophical one.
>
> This failure is a lack of knowledge. In fact, random itself is nothing
> more than a lack of knowledge (maybe !).
>
> > For each of our reasonably intuitive mathematical
> > definitions of "random", it seems impossible to find one that applies
> > equally "intuitively" well to finite, countably infinite, and
> > uncountable sets.
>
> Hence it seems reasonnable to wonder whether set theory his intuitive
> enough or not (but I don't want to discard it for this kind of reason).
> Well, problems only arise with infinity (and even uncountable) sets, and
> they are not easily found in nature, so the difficulty is not a major
> drawback. And when these sets are found (say, pick a random point on a
> line), it can usually be avoided by a limiting process. That's probably
> why potential infinite was considered enough, before Cantor.
>
> There is one funny point with set theory: the idea of rigor has
> completely changed. Today, in all parts of mathematics, a rigorous
> definition or proof, is necessarily based on sets. Before,
> formalism was not considered as important - it's the impression I get,
> at least. Who is right ? It's too easy to assert only set theory is
> rigorous enough :-)
>
> > <snip>
> >
> >>>
> >>> It's not like I don't have different definitions for use in different
> >>> situations; I just try to be clear about which definition I'm using at
> >>> any given point in time.
> >>
> >>
> >>
> >> On that topic, I fear I'll be quite dumb: I've no definition of random,
> >> of selecting at random, and still, I claim I can study probability,
> >> which deals essentially with random events. It must be a nightmare :-)
> >
> >
> > WAKE UP!!
>
> I like set theory, but I dislike set theory... Is that schizophrenia ? ;-)
>
> > :)
> >
> > Cheers - Chas

Arbaut,

You are a fascinating intellect. On the one hand, you are unknown, and
pretty much everyone here has more volubility in this volume than you.
On the other hand, you're voluble, and full of dense prose for those
who care to consider. Comp.unix.misc, statistics, and demographics?

Yeah, so what.

Infinite sets are equivalent. Would you care to argue that with me?
If so, why?

You might want to be prepared to correctly argue that.

There is one universe, and it is infinite. Infinite sets are
equivalent.

I argue, for example, in the extension of Cantor's first that there is
a maximal element, which there can not be, in ZF, Zermelo-Fraenkel set
theory. There can not be that, it's axiomatized away from possibility
of existence.

So, there is a minmal and maximal element of the continuum. There is.
There is not in ZF, from which you might be able to conclude, ..., ZF
is inconsistent.

There is ONLY one theory, that with no axioms. The rest: are
inconsistent, and incomplete.

That's the Finlayson Null Axiom Theory.

Would you care to argue otherwise? Goedel then damns you.

Ross

.

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