Re: Cantorian pseudomathematics

cbrown@xxxxxxxxxxxxxxxxx wrote: 
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.

Ok, I understand. We agree that a model is needed, but we don't
necessarily agree on the exact constitution of this model.
Moreover, I must say I don't like the way some "models" (like probabilities) are now introduced only from pure theories, I mean,
it looks like some mathematicians erase all remaining traces of
links with "real life", and concentrate on deductions from axioms.
I get that impression when reading Rudin, for example. Therefore,
that's more the way mathematics are taught I dislike ;-)

We describe the model in precise mathematical terms. We demonstrate
that the model is  logical and internally consistent,


We try :-) Consistency is relative to the consistency of a stronger system.

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.


Yes.

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);


I agree here.

they relate to the applicability of the model to the
original real-world problem.

"Not applicable", or invalid from a practical view point. Actually, it's not very important, all mathematical models agree quite well with reality. For instance,when set theory doesn't agree, that's because the result is not to be applied to real life.

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.

Ok. But you'll probably agree that there might be different models of random, leading to different and incompatible results. Hopefully, experiments would help to choose the best, from a practical view point.
The other would be mathematically correct, but probably unusable in practice. Well, we must put that into perspective, when considering Newton's gravitation for example.

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.


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).

The old definition, with frequencies, could be a basis.
And I think it's possible to use even an unfair coin to get random bits:
throw it twice, and keep only AB and BA, which should "look" equiprobable enough for most people.


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. 

Yes.

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".


But that's the truly interesting part, in probability theory. Without
that assumption, it's just a sequence of deduction, leading to nowhere.
I admit it's only an assumption, though.

I claim that the statement "uniform distribution of the naturals" has
no /mathematical/ meaning, until we give it one.

Obviously :-) I have no such definition, for the moment, and since I belive the usual (old) definition of probabilities is mostly, correct,
I don't think there can be one either. But I'll let Han try to provide
one. At least, he knows what he has to show: using traditionnal maths to
prove his statement about uniform random distribution of the naturals, is meaningless, and we cannot look for something else, for him.

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.


Yes.

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.

Yes. Therefore no mathematical argument may dismiss any theory, as long as it's logically coherent. Since Han has not provided such a theory, nor an experiment showing he may be right, the conclusion is rather obvious :-)

Actually, the same argument shows that set theory is very good. It's funny, at first sight I intended to show it's not necessarily the best possible: it's true, but it's not the worst either :-)

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.

Then (the intuitive concept of) random has no mathematical definition or meaning. And probability theory won't prove otherwise: the only (or amlost the only) appearance of the word "random", in "random variable", is a very bad choice, btw.

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.

It's not a purely real statement either. :-) It relies on some idea of what gravity may be, and even that is not very clear, except the law of gravitation, that can be observed. Actually, I'd say your quote is indeed a mathematical statement, about the law of motion: if I remember well, Newton starts his Principia by a series of *axioms*, from which all results are deductions.


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.

Yes, but a statement involving gravity has to give some definition of it, and the usual one uses Newton's law (or maybe Eistein's theory, but
I prefer the simpler one, at least for this discussion).

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".

I agree. The mathematical model is still true, but some applications are not. In fact, all applications are wrong, but in most cases nobody can see it - that's why Newton's law are still taught and used.

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.


:-)

You are right. Intuition says it should be possible, but some experiment
with basic probability theory shows it is not. Since we can observe the
same poor intuition in cases that can be checked by experiments, we ought to agree with the theory more than with our intuition. QED ;-)

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.


Err, yes. I apologize. :-)

My statements are only intended to indicate what a
mathematical model is; not how to determine that model's validity.

Still, one question remains: are "computable objects" enough ?
[I didn't give a definition of computable, but I think it should
be rather intuitive at first sight. I may define it by <<computable
by a finite program in a "computer language" with which you can use as much memory as you want, and as much time as you want, but both must also be finite, and which can express integers, variables, usual flow
of control and recursion>>.]

When I tried to find the least necessary part of set theory (necessary for most uses, well, I know it's not mathematically defined :-) ), I had the impression it may be based on computability.
There is an example of limitation I find interesting: programs are countable, therefore real numbers are not all computable. What would change in theorems (I have mainly analysis in mind), if we only care about computable reals ? Is there a completeness definition resembling
"a computable Cauchy series of computable numbers, converges to a computable number" ? Actually, even the diagonal argument of Cantor may
be declared wrong: it states, if a sequence of all real numbers is written, you get another one by writing a number with at least a difference in a figure, with any number in the sequence. That's right,
but this number wouldn't be computable: the sequence of all computable
numbers is not computable either.

So, the best I can tell so far: naturals and fractions are not enough - or you must always manage with approximations, which I find awful, and not intuitive: in history, even when irrationnal numbers where used (by
approximation), it was rather implicit in notations, that they were also true numbers, for example roots or logarithms. Traditionnal real numbers
are enough, but they introduce infinitely more numbers than rationnals, which nobody can show. It's rather amazing: among this continuum of
real numbers, nobody can show one that wouldn't be computable. Hence, I
think a more restrictive definition wouldn't hurt (i.e. computable numbers), but I don't know how to look further in that direction.
.

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