An attempt to clarify the notion of randomness.

DISCLAIMER
These views are controversial.


Definitions
- Randomness will not be defined because it is here assumed to be a trivial property, simply defining it yields an immediate paradox.

- Randomness is currently treated as an isolated state throughout mathematics, and this author suspects that order/disorder exist on a continuum. Pure order and pure disorder (randomness), are suspected to be trivial.


- Order and Disorder can be defined tentatively as being complementary properties which characterize various organizational states of being. This treatment seeks to quantify and qualify the level of organization of various states with respect to each other.
- Further, I dont think that there is an all-encompassing definition of order. I think that it is contextual, and this treatment will attempt to look at various contexts where order and disorder can be further understood.


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RANDOM NUMBERS



Question 1

Consider a number A = a1 a2 a3 ........ a_n where each of the a_n are unknown or uncertain.

If none of the a_n are determined then is A really a number ? Is it a trivial number ? If such an A is the best example of a truly random number, one which each digit remains completely uncertain, and if this A is not a number, then can we use this to prove that randomness is indeed trivial ?


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Question 2

Consider a number A = a1 a2 a3 ........ a_n where each of the a_n are unknown or uncertain, to be selected by some "random" process.

Then consider a number A' = 7 2 5 a3 a4 a5 ........ a_n where some of the digits are known and the rest are uncertain.

I'd like to know if A' is more or less disordered than A, and by how much, or if the question even makes sense in the first place. Is A' an example of partially disordered number ?

If A' is finite, then how do we quantify this disorder ?
If A' is infinite, then how do we quantify the disorder ?
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Question 3

Consider a number A = a1 a2 a3 ........ a_n where each of the a_n are unknown or uncertain.

Let U represent absolute uncertainty. Then A can be characterized as

A = a1 a2 a3 ........ a_n = Sigma a1*U + a2*U + a3*U +......a_n*U

So, what is a1*U ? Is it really a number at all ?

Can we construct Sigma1 and Sigma2 where Sigma1 =/= Sigma2 but U*Sigma1 = U*Sigma2 ?
And, does this not appear quite similar to 0/1 = 0/2 = ..... = 0/n ?

Might be bullshit - will come back to that.

But we do have that

Sigma a1*U + a2*U + a3*U +......a_n*U = Sigma U + U/10 + U/100......U/(1*10^n)

or alternatively, = Sigma 1*U + 10*U + 100*U + .....(1*10^n)*U

more on that later -
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Question 4

Let A = a1 a2 a3 ........ a_n where each of the a_n are unknown or uncertain.

For any given digit a_n, a_n is in a state where uniqueness has been violated. In this uncertain state, a_n may be considered to be 1,2,3,4,5,6,7,8,9, or 0. The choice is arbitrary, hence uniqueness has been violated. Hence, a_n cannot be considered a number as long as it remains in a state of uncertainty.

Question : What's wrong with that argument ? Anything ?

If it is uncertain whether a given a_n has an assigned value, then a_n is not a number. It is an uncertainty. You could argue that the uniqueness of a given number establishes it as a certainty. If a6 = 4, then you are "certain" that it cant be anything else because 4 is unique. 4 is not 1 or 2 or 9, nor is it 4_1, 4_2, or anything else. 4 is 4 and 4 is unique.
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Speculation

This work is attempting to show that order and disorder exists on a continuum. To say that absolute order and absolute disorder are trivial, basically this is the same as saying that this order-disorder continuum does not contain it's endpoints.
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In the photo above, it is clear that area becomes trivial as you take the union of A and B. Consider this question - which points are made trivial and which points remain after one has taken the union ? Did points disappear from A or B ? Clearly, this is arbitrary, and somehow it has much to do with the rest of these arguments. This is a very fascinating question which goes to the heart of existentialism and math.

Then consider that the union of A and B does equal A+B when the intersection of A and B is empty. This is very profound from an existential standpoint.



An attack on poker
1) A condition of not knowing an outcome in advance is considered acceptable impetus to assume maximal disorder of a shuffle.
2) While this assumption of maximal disorder is an acceptable abstracted model of a real world event, the basic methodology and dogma of probability theory has become so engrained that most people's understanding of disorder is now based on such models and the notion of randomness has become more a matter of faith than substance.



CA

pictures go here