I don't understand why you said that random numbers are non-sense, what about Chaitin's Omega number? The approach here is incompresibility. You don't even need probability to define randomness, just Turing Machines. What exactly means that randomness implies probability in math terms? Have you read Kolmogorov axioms on probability? One of its success -not failure- is that it doesn't need a definition of randomness to build its theory.
> You don't even need probability to define randomness, just Turing Machines.
You don't need the integral of 1/x with respect to x to define e, but you can do it that way. Mathematical definitions are bidirectional in many cases, in that we can use A to define B or B to define A without any loss in power for defining C in terms of B and/or A. But that doesn't mean that this works for any inversion of the definitions, as in the case I outlined above.
> I don't understand why you said that random numbers are non-sense, what about Chaitin's Omega number?
As you can see, the definitions we use to construct these make all the difference! The definition of the Chaitin constant that I'm familiar with is a probability, and the probability is not random; rather, we assign a probability to a random event (or, more precisely, the outcome of a random function). If * probabilities* were themselves random, they wouldn't be very useful, would they!
> Have you read Kolmogorov axioms on probability? One of its success -not failure- is that it doesn't need a definition of randomness to build its theory
I think you misunderstood my point, which is pretty much orthogonal to Kolmogorov. I didn't say that probability requires an assumption of randomness; I said that randomness (as used by the author in this post) implicitly invokes a notion of probability ('likelihood', in the casual use of the word). And certainly as used in the 'Is 14 a random number' example.
