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.
<snip>
> 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".
>

I agree; this is what Jesse seemed to be getting at when he said that
no finite set of observations of, for example, rolls of a die,can act
as "proof"that that die is random, because the assumption "this die is
random" is consistent with every finite set of observations (well, as
long as it never comes up "7":) ).

The property that I defined as "random" is a property of some infinite
sequence; in particular, one that is not computable. It's my way of
formalizing this observation.

>
> >> 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".

"Diehard", "Die Harder", "Die Hard with a Vegeance", is a sequence of
random movies :).

>
> >> 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.
>

While we can't say for sure that "obeys a probability distribution" is
what we "really" mean by random (philosophically), we /can/ say "if it
obeys a probability distribution, then it has these properties (say,
Kolgomorov axioms, or frequency limits, etc.)".

To me, this is as satisfying (or unsatisfying) as saying "by random, I
mean some object (probability distribution) which obeys certain
manipulations". It's very like "6 is a perfect number".

> >> 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 :-)

Or collection, or class, or species. The point being: it can be
formalized.

> 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.

"The integers under addition are an infinite cyclic group" is a
mathematical statement; because I can define what those terms mean in a
mathematical fashion. "The integers are really big" is not, until I
define what "really big" means.

Why would we apply a different standard to the term "random natural"
than we do for any other mathematical term?

>
> > 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 !).
>

Maybe that's why defining it is so hard - how does one define the
undefinable? (Shades of Zen!).

> > 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.
>

Sidebar: My personal opinion is that the whole "potential infinity
versus actual infinity" debate comes from an unfortunate history where
the term "infinite", in a non-mathematical context, was related to the
God of the Christians. So it was not possible philosophically to reject
"infinity" - it was equivalent to rejecting God.

Now we have a way of saying "there are no infinite sets": state it as
an axiom. Thus to me, the distinction between "potential" and "actual"
is just confusing word play.

> 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 :-)

I certainly don't assert that /only/ set theory (in the sense of ZFC)
is rigorous enough; there are plenty of "equally" rigorous constructs.

>
> > <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 ? ;-)
>

Nah; it's just normal mathematical neurosis. Some days I'm a Platonist;
other days I'm a Formalist.

Cheers - Chas

.

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