Make attention to use the entropy . The entopy is based on the idea that the amount of information in binary string S is Log(S) , the number of bit to directly code the string . This is wrong , the correct information is the Kolmogorov complexity of S . Nowaday the scientific literature don’t focus on this difference and very often use Log(S) instead of K(S) ( K is the Kolmogorov complexity function ) and justify this becouse K(X) is uncomputable and becouse for the major part of K(X) value it is approximable by Log(X) in a mathematical context . This wrong assumption is done becouse people think that every object , every binary string can happen . This is wrong , only few bit string with lenght 1000000 can happen becouse a system that produce 2^1000000 object , string can not exist. What this mean is that the object are always small ! The object stay in that value smaller than Log(X) in the function K(X) .
A._Denis
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“So these people are frauds?”
To answer your question , I think the card can be one of the better random generator existing, so it is absolutely not a frauds ( and pheraps I will buy one … thank you for the link ) . But there are many definition of random . The only one theoretical random definition I accept as true is that a bit string S is random only if this string has K(S) >=Len(S) . My strong deterministic opinion is that in the real world these random objects exist only if we take short object , also for quantum field ( like Wolfram think ). I read articles where people reasoning on the quantum field and the relation on exponential or polynomial world , I don’t know what is the answer but I don’t think that quantum filed open the door for an exponential world. This opinion come to me from many discrepance I find in the mathematical description and what happen in practical . For example for the K function ( Kolmogorov complexity ) there are proof say that for major part of value we have K(X)>=Log(X) and if you watch on the function you can say it is absolutely correct , not only but for very very few case we have K(X)<Log(X) and also from a statistical point of view all is coherent. The probability to compress X using the optimal K function with a dimension of N bits into a Y with a dimension of N/2 bits is (2^(N/2))/(2^N) . This probability is exponential low! So for increasing value X the probability to have a small K(X) is very low. What this mean for practical point of view? This mean that if I take a file from my pc and then I try to compress this file using a very very bad compressor ( becouse the K function is an idealized compressor with an infinite power ) it is crazy to hope to compress it . But I absolutely sure that I am able to compress it and with high probability of 50% ! . Why? What happen ? There are many explanation we can give to this phenomena but the follow is my opinion. When we get an object , when we receive an input this object is only a representation of the information of the object computed by a program , every program has limited power , we can define this power as a limit on the size of the bitstring that this program can do . I call this limit M and using this limit something change , for examples the number of available bit string of lenght N in a mathematical view are 2^N , now become min( 2^N , M/N ) . The compression probability change … The universal distribution change … But the interesting property of this vision is that it make pactical aspecative coherent with theretical function . In the standard mathematical/statistical view is more easy to compress small string and become more difficult to compress large string in absolutely opposition in what happen in the real world! When I have a small file I think will be difficult to compress it and when I have a big file I think that I will get a big compression ratio.
If you watch this function min( 2^N , M/N ) what happen is this! for small string we have exponential behaviour and this is coherent with mathematical classical view but after a limit M what happen change and the probability to compress for example increase! .
Another important characteristic is that big variation on the parameter M make small variation on the behaviour of the function , so is more important to assume the existence of this behaviour also if we don’t know M also if it is impossible to compute M !.
This cause a discrepance in the entropy , in the assumption of Log as function of information measurement , etc … becouse this theories suppose an exponential world ! a very big world! ( exponential functions are very big and we can not underrate them ! )
There are many consequence of this simple observation and many to be investigate .
I don’t know if quantum field will open the exponential door but the world behaviour seem to me polynomial.
Denis.