Re: Cantorian pseudomathematics

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 
.

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