What's the meaning of "random" in Mathematics?

[fbs7]

The mathematical criterion did not change. An algorithm "exists" whether it is known or not.

Ah, I see! Good point!

Now, how about this question; say that P is the set of all possible programs for some given machine; P is certainly infinite. Then for a certain string s we want to consider algorithms p ∈ P whose size(p) < size(s); we don't want to consider algorithms that are too big. Even so, the number of algorithms will be immense; if that string is 1,000,000 bytes, then the programs cannot be more than say 500,000 bytes, therefore, that will be up to 256^500,000 programs, or 10¹⁰⁶. Then how to prove that among that gazillion of programs there isn't any program that will compress that string to 500,000 bytes.

This criteria seems very interesting, but I suspect the application of it may be challenging at best! I'm so happy I didn't become an Information Theory guy, because my poor old brain would be fried many times in the first week!

 

[jbriggs444]

Then how to prove that among that gazillion of programs there isn't any program that will compress that string to 500,000 bytes.

In general, it is not a solvable problem. Are you familiar with the halting problem? There is no general way to tell which of those programs compute anything at all.

 

[fbs7]

Right; I wonder what's the benefit of the compressibility criteria thingie, other than being an interesting statement.

Man, this random string stuff is a tough cookie to crack!

 

[FactChecker]

The compressibility criteria is a rather profound insight that allows one to immediately say that pseudorandom generated numbers are not random no matter what statistical tests they pass. So, although proving a sequence is random may still be challenging, proving that some sequences are not random can be done formally. It also allows one to say that replaying a recorded sequence of random numbers does not make the sequence nonrandom, even though the sequence is determined and repeatable.

However, that definition of random does not address the statistical properties of the sequence. We are almost always more interested in the distribution and certain statistical properties than we are in pure randomness. (There are some exceptions.)

 

[Hornbein]

Having a masters degree in probability and statistics, I can answer this question with authority. I can't be bothered to read this whole thread, so forgive me if I'm redundant.

Probability was axiomatized by Kolmogorov in 1950 or so. Probability is the study of normed measure spaces. There is nothing random about the math that is used to predict probabilities. One can do the math with no concept of probability at all if one so wishes. It can of course be applied to random processes, but like all math the responsibility for the validity of such an application lies with the applier, not the mathematician.

It doesn't help that "random" often implies a uniform distribution. Things that have a 1% chance of going wrong aren't considered random. You just have to keep that in mind.

In my opinion for the most part "random" means "unpredicatable." It is subjective.

However it seems possible to prove more, that some things can never be predicted, and are hence essentially or truly random. Sometimes but not always this is what physicists mean when they say "random."

If you want to prove that something will never be predictable, then you have to show that if you can predict it then some contraction arises. Perhaps Conway and Kochen succeeded in doing that with quantum spin in their Free Will Theorem. Arguments that the spin of entangled quantum particles cannot be used to communicate are of this sort. If you could communicate in that way then the concept of causality falls apart.

I would say that presidential elections are random. But this is not how the word is commonly used. We think -- rather dubiously, say I -- that there is some reason that so-and-so won. So this sort of thing is excluded. More often "random" is something that is done for incomprehensible reasons, or a mechanical system that is not well understood. But maybe tomorrow such reasons or systems will be better understood. Then they will no longer be random to a person who understands it. But it will be random to someone who doesn't. A good example is the magnetic field reversals of Earth. Today they are random. But sooner or later I believe they will become at least somewhat predictable. Or the orbit of the planets. They were random until Ptolemy came up with his epicycles.

I have personal experience with this. I have some videos I made on Youtube. The viewership is quite random day to day. But recently I looked at some graphs of views over the last two years. One was a linear, another exponential, and a third logarithmic. It was amazing how close the curves fit what I had thought were isolated blips. I feel I can confidently predict that these trends will continue. Then as well some of the vids had viewership graph lines that wiggled around. Random.

Is the CMB random or not? Hard to say, hard to say. So I don't worry about this issue much. I just use my simple definition, that "random" and "unpredictable" are the same, and let others discuss definitions.

 

[fbs7]

Wow, what an impressive post! Thank you so much for it - I'll definitely re-read it several times, as it refers to several concepts I'm not familiar with! Thank you!

Let me ask a related question... say that Civilization 9 From Outer Space decides to announce themselves by sending a radio signal that is beyond any possibility of confusion with any natural process. So they send say 100 billion trillion bits of the number pi in binary. Their thought is... "the chance of any natural process to generate pulses that are the same as the number pi with 100 billion trillion bits is nil, and any Civilization 9 child with IQ of 1 billion will recognize pi in a blink!".

We know that the bits of pi in binary are not random at all; but it passes our random tests as random, and as we don't have an IQ of 1 billion we can't really distinguish that signal from random noise. So, in that case we may not be able to distinguish an intelligent, designed, procedural, calculated and predictable sequence from random noise.

The question is... where's the mathematical gap in scenarios like this? Once you know the context of a sequence (say the algorithm or a "meaning") it's easy to know if something is not random. But context is not mathematical, so short of having some additional information about a sequence or being lucky in compressing it, in general it seems to be an impossible problem of separating if something is truly random or is just generated by some mechanism that we don't understand - that would be my guess.

ps: Now extend the scenario to say alpha decay. We can get timings of alpha decay, and they may seem completely random to us... but maybe there is an underlying deterministic process for alpha decay, but it's too complicated for our low IQ and "random" there just means we don't really know the rules that control it?

 

[FactChecker]

@Hornbein , Good post. I have a couple of thoughts.

It doesn't help that "random" often implies a uniform distribution. Things that have a 1% chance of going wrong aren't considered random.

One field where very small probabilities are studied is in reliability engineering with tools like fault trees. They often study rare events that have catastrophic consequences.

In my opinion for the most part "random" means "unpredicatable." It is subjective.

I am not sure that I would call this subjective. I agree that the interpretation of "random" is often dependent of the amount of information available. But, given a certain set of information, either there are more than one possibility to guess about or the result is predictable. That would make it objective, but dependent on the information available.

There are common examples where the lack of information is key to treating something as random. Suppose a coin is tossed, caught, and the result is covered. After that, there is nothing physically random about the result -- it has already happened and the result is determined but unknown. If we try to guess the result, we still treat this as a random process.

 

[sysprog]

@Hornbein , Good post. I have a couple of thoughts.
One field where very small probabilities are studied is in reliability engineering with tools like fault trees. They often study rare events that have catastrophic consequences.I am not sure that I would call this subjective. I agree that the interpretation of "random" is often dependent of the amount of information available. But, given a certain set of information, either there are more than one possibility to guess about or the result is predictable. That would make it objective, but dependent on the information available.

There are common examples where the lack of information is key to treating something as random. Suppose a coin is tossed, caught, and the result is covered. After that, there is nothing physically random about the result -- it has already happened and the result is determined but unknown. If we try to guess the result, we still treat this as a random process.

And even if it's an unfair coin, if I don't know which way it's biased, my odds of guessing correctly or incorrectly which way up it is are still 50% each.