Re: Cantorian pseudomathematics

Jean-Claude Arbaut wrote:
> cbrown@xxxxxxxxxxxxxxxxx wrote:
> > Jean-Claude Arbaut wrote:

> >>* taking another point of view, the impossibility comes from the
> >>infinite number of naturals: you couldn't in real life compute
> >>frequencies on an infinite number of objects.
> >
> >
> > The precise definition doesn't make a statement about being able, in
> > real life, to compute anything.
>
> Then probabilities have nothing to do with random, random games, or
> anything real ?

That's not what I'm saying.

I'm saying that /if/ you formalize what is meant by "uniform
distribution on the naturals" as "a function from the naturals, having
certain properties: A, B, C, etc.", then there are mathematical
implications I can draw from that formalization. These inplications
have nothing to do with the real world, and everything to do with the
mathematical model.

On the other hand, if you /don't/ formalize what you mean by "uniform
distribution on the naturals", then there are no mathematical
implications I can draw at all; you haven't described a mathematical
model.

> But then, why compute probabilities ? I thought the
> basic idea (of Pascal and others) was to make the real world more
> understandable, not to create abstract mathematics unrelated to anything.
>

I agree with your statement; so I think you are misreading my intent.

We consider our intuitions about some real world problem. We propose
that it can be described by a mathematical model.

We describe the model in precise mathematical terms. We demonstrate
that the model is logical and internally consistent, from a
mathematical viewpoint. We assume that to the degree that the model is
"ggod", logical implications about mathematical statements in the model
should correspond to phemnomena observed in the real world.

We may then test the model by making experiments in the real world; but
the results of those experiments don't determine the validty the logic
of the mathematical model (which already follows from the model's being
mathematical); they relate to the applicability of the model to the
original real-world problem.

Conversely, if you claim that, in your mathematical model, a random
number is a number with property of being random; then I can neither
validate nor invalidate the logic of your mathematical model until you
state, mathematically, what it means for a number to have the property
of being random.

Only when you have done so, can I then apply the results of real world
experiments to your model, in an effort to determine how well the model
represents the real world occurrences.

> > The proof of nonexistence relies on
> > properties of the reals, inherent in their definition, to show that
> > P({k}) must be 0 for all singleton {k}; and that the sum of a countable
> > number of 0's is 0, which can be derived from, say, standard
> > delta-epsilon definitions of limit of a sum.
>
> Yeah, inherent to the definition of real numbers. I won't argue on that
> point, I know some modern probability theory too :-)
>
> > If you don't accept that a function can have as its domain the set of
> > naturals, because the set of naturals does not exist; then the whole
> > question is meaningless to you.
>
> Actually, what I find natural about function is slighlty more complex:
> "computable functions" are natural to me, and "computable numbers" as
> well. Just a funny example to show the necessity to use (some) real
> numbers; it's a game with a coin (I'll call A and B the two faces)
> * 1st step, throw the coin. If you get an A, you continue at step 2
> otherwise, you get a B, and you stop. Then I say you win if the first
> bit of the dyadic expansion of e-2 is a 1, and you loose if it's a 0.
> * 2nd step, you throw the coin again, you continue if you get an A, and
> you stop with the same result as in 1st step, except you look at 2nd
> bit.
> * etc.
>
> Obviously, you need not know what "e-2" really is, all you need is a
> dyadic decomposition, or equivalently, a string of bits.
> What's the probability to win ?
>

I understand exactly what you mean, as a thought experiment.

But this thought experiment relies on some mathematical assumptions.

What is a "fair" coin? Most people would say "one that is equally
likely to come up A or B on any given throw".

But people are notoriously horrible at simulating the outcomes of coin
flips; for example, there is the often overwhelmingly strong intuitive
expectation that, after tossing 4 heads in a row, that tails are "due"
to come up.

To me this indicates that we must mean something more specific than "it
seems equally likely"; because not everyone agrees on what "equally
likely" really means.

We can say "comes up A or B, with equal probability". But what does
/that/ mean, as a mathematical statement? (I know what the usual
definition is).

> This example shows that rationnal numbers are not always enough, even
> when dealing with basic games with coins. But when trying to see what
> is really necessary, I'd say: an infinite sequence of bits, that is
> *computable*, i.e. for which you can provide a mecanical way to compute
> terms. The "set of programs" is necessarily countable, hence not all
> real numbers, and not all sequences of integers, are computable.
>
> For me, that's the limit for "usable" mathematics. But that doesn't mean
> uncountable sets or such unreachable objects are not interesting. I just
> wonder whether "computable objects" are enough to describe the real life.
>
> > However, the fact that it is meaningless to you /still/ does not
> > constitute a /logical/ disproof of "if we accept that the set of
> > natural exists, then it follows that there is no such thing as a
> > uniform distribution on the naturals".
>
> You are right. What is interesting to ask is: are natural numbers,
> or real numbers, or whatever, the most useful objects to describe
> *real* random. If they aren't, then probabilities as they are known,
> are useless.

Well, not quite useless; just not as useful as some other description.

>No _definition_ can answer this question.
>

I didn't claim that a definition of "uniform distribution of the
naturals" answers the question "what is the nature of 'randomness' in
nature".

I claim that the statement "uniform distribution of the naturals" has
no /mathematical/ meaning, until we give it one. Thus, to use that
phrase (or the equally ambiguous terms "random", or "probability") as
part of a mathematical argument without first defining it results in a
non-mathematical argument.

> >>Hence, the concept of
> >>probability based on frequency cannot help, and the previous argument,
> >>though mathematically correct, may be void. Let's make an analogy: for
> >>the same reason, it's impossible to select a random real in [0,1[.
> >>Really ? Well, yes, but it's possible to get an approximate random
> >>number, and to refine the approximation without loosing anything about
> >>an intuitive "equidistribution": just compute each bit with a coin.
> >>The resulting number, though not really a random real number, would
> >>qualify as the object we need.
> >
> >
> > Mathematically speaking, I can only really accept that when you tell me
> > precisely what you mean by "the object you need".
>
> Mathematically speaking, I don't care. Sorry.

Er, Apology accepted :).

> I take care of mathematics
> when they provide what I /think/ is useful. You ask for a definition in
> the mathematical world, of a fact mathematics cannot discuss at all:
> whether they describe something real or not. The answer cannot come
> from mathematics, but from confrontation of mathematical results, with
> real life results. Do you want a mathematical definition of that ?
>

We agree: no such definition is possible, because this is not the
domain of mathematics.

> > The definition of a uniform distribution on the naturals is not "an
> > object required to allow me to say that I can select a natural at
> > random". It's much more precise than that.
>
> If you an define "much more precisely that that", what random is,
> mathematically, then you are very smart.

Not at all; just very insistent that terms have a mathematical meaning
if they are part of a a mathematical argument.

If you say "gravity is the thing that keeps us from floating off the
earth", I will say that that is not a mathematical statement.

If you say "if we model gravity as a force that obeys the inverse
square law, then it follows that a distance 2*d, an object will
experience 1/4 the force that it does at distance d", then that is a
mathematical statement.

However, the latter statement does not become false if we discover that
gravity does not obey the inverse square law in the real world. What
becomes false is the statement that "this mathematical model of gravity
is supported by experiments in the real world".

> But I fear the definition
> will be of no use to explain (real) random events, or what they might
> be. Mathematics are not the only way to talk about random. I mean, it's
> rather a natural question, to ask whether one can get a random integer,
> such that no integer is more "probable" than another. You don't even
> need a numerical definition of "probable".

See my earlier comments in re a coin being "due" to be heads. People's
poor intuitions about probability is what keeps the casinos in
business.

> The current mathematical
> framework says no, but to which degree does that help ? Let's suppose
> someone comes up with a formal system where the answer is true, how in
> hell *mathematics* will help to tell which is right, in real life?

You are confusing a clearly stated mathematical model with the
assertion that that model is "really" a model of some real world
phenomenon. My statements are only intended to indicate what a
mathematical model is; not how to determine that model's validity.

Cheers - Chas

.

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