Re: randomness

Look, I'm really sorry for being a pest now, but there really is just one
more thing which must be mentioned.


We know that it is problematic to prove anything about a random number r,
because there is no way to distinguish between a random number r and a
nonrandom r', where r = r'. So, any proof which concerns "the randoms"
will be indeterminate at best.

We can see that there would be some huge problems if we allowed "the
randoms" to be considered a legitimate number system. Yet, probability
theory and all of statistics works just fine. So, you have this very uneasy
situation where it makes sense that the randoms are indeed like a legitimate
number system, and also are not.

It seems almost silly to say that there is no such thing as a random number.
That such a number cannot exist. It seems ridiculous. You can try to
comprimise by saying that there is no such thing as a random number, only
random variables. Perhaps you dont need random numbers anyway if you have
random variables, who cares about numbers, probability theory is rescued
and the world can go on living happily knowing that alll is well. I dont buy
it.

Probability theory and statitics will perform exactly as advertised on data
which is random or not - as long as the nonrandom data looks suffciently
disordered. Everybody knows that.

I can give you a string of 1's and 0's and tell you that they were generated
"at random" and it will be impossible for you to verify this or disprove it.
Impossible. Everybody knows that.

So the question becomes, :"Is probability theory _really_ a theory which
deals with randoms, or just things which _act_ like randoms?"

Do you have a theory which truly analyzes uncertainty, or is this simply a
theory which tells you what uncertainty would look like if it actually
existed ?

Questions like this can go on forever. There are some standard excuses which
attempt to alleviate the apparent discrepancy, such as the usage of "random
variables" instead of actual "random numbers". But, in my opinion, none of
these rationalizations is as sensible as simply acknowledging existential
indeterminacy.

If you have existential indeterminacy, then you can say things like this ;
"We dont know if the randoms exist or not, their existence is indeterminate.
But if they exist, then you have probaility theory." Or, "If randoms dont
exist then probability theory is still valid, but it is merely an
interesting subset of mathematical locial structure which would have had
additional signifagance had those randoms really existed but too bad they
dont".

All you have to do is think about this for a moment and you will realize the
potential implications.

If there are other number systems which are "existentially indeterminate",
like the randoms, then this would imply that there may be other theories
buried just beneath the surface - just like probability theory.

All of this is highly speculative. But I think that it is a very interesting
lineage of speculations to think about.

Could there be other entire theories of disorder - just waiting to be
uncovered ? Or, is probability and statistics the final word ?

One such possible validation of these overt hallucinations of a mathematical
madman - the "partially random numbers" which you alluded to in the original
question of this thread. Where a random number A added to a nonrandom B
yields a third number C.

There could in fact be an entirely new theory of probability out there
regarding such numbers like C, a whole new theory which just needs to be
discovered..

Perhaps.













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