Random Questions
We can’t tell with certainty if something is random. Is this sequence random [1, 2, 3, 4]? Did I get these number by adding 1 to subsequent steps starting at 1 or by just randomly selecting numbers 1 to 10?
We can’t even say with probability if something is random. Another sequence: [1, 2, 7, 4]. What’s the probability that this is random?
We have no idea. Without knowing the underlying generating process, we can’t say anything about the probability of randomness. Sequences that appear random might have a deep underlying pattern that we can’t detect.
Or maybe we can? We do have mathematical techniques to attempt to determine this. One can look at things like the frequency of integers and integer sequences across multiple bases. Equally-frequent 1-10 (in base 10) and 0/1 (in base 2), (1, 2), (2, 3), (3, 2, 9), etc. etc. etc. over a large number of observations would seem to indicate that the outputs of the process are in fact random.
What about determining if a process is not random? Using similar reasoning: is the above sequence definitely not random? Impossible to say just using the observations.
There is one saving grace, one way in which outputs can be useful. They can tell us if some process is probably not random.
Sequence in base 2: [1, 0, 1, 0, 1, 0, 1, 0, 1, 0]
Sequence in base 10: [1, 2, 3, 4, 5, 6, 7, 8, 9]
Selections from a category (animals): [lion, lion, lion, lion, lion, lion]
In all these examples, there seems to be an explainable pattern that can be used to predict future observations. This explanation is better (in probability AKA more likely) than simply referring to these sequences as random, and the observations from these sequences are very unlikely without this non-randomness:
[1, 2, 3, 4] –> [5, 6, 7, 8]
I can explain this process by stating that to get the next observation one must add 1 to the previous observation, and use this explanation to predict the next 4 values. If these next 4 observations occur, it seems more likely that my explanation of the process is correct, rather than the purely random explanation.
It’s difficult to quantify/define this exactly, but it is real.
What’s less clear is semi-random sequences.
Random processes: [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
Here the best explanation seems to be that this sequence only contains 1s. But we could just as easily say that this is a stochastic process, with 1s being drawn from a distribution in which the probability of a 1 is very high. You might say that the former is a better explanation invoking some Occamic argument. But this can get us into trouble.
Trouble: [2, 4, 1, 3, 2, 3, 2, 1, 3, 4]
The best-seeming explanation here could be that these observations come from a process that equi-probably spits out numbers 1-4. But what if larger numbers are possible just with low probability? We have no way of knowing if that’s a possibility, and can’t be certain that it’s not, even with a huge number of observations. This is why understanding the underlying mechanism is vital, and relying purely on observations is troublesome.
But do we always need to do that? Contrast the above example with the following: [2, 4, 6, 8, 10, 12]
What are the next 5 observations? In some sense we don’t need an explanation (other than the sequence will continue). We just extrapolate purely from past observations (as in literally draw a line through them and extend it).
And sort of by definition if something is predictable, it’s not random. Although note again the asymmetry: if something is non-predictable, it might not be random.
An example is any process that Steven Wolfram has dubbed computationally irreducible. To figure out the output at time t, you need to run the program until time t. There’s no other way of knowing/predicting what the output will be otherwise. Sounds like randomness, right? But it’s not - these processes are often based on simple deterministic rules.
But then even though the process itself isn’t random, could the outputs be? Yes. They can be explained, but not predicted. Another example of this is pseudo-random numbers: they are generated as the result of a deterministic mechanism but seem to have lots of attributes of random numbers. Wait so are they really random, or not?
Another way of assessing randomness is compressibility: any sequence that can’t be compressed - represented in a smaller (like Kolmogorov complexity smaller) way - is random. Relating to the above: computationally irreducible processes can be generated via simple deterministic mechanisms, hence represented in a highly compressed way. The same is true of pseudo-random numbers. But the observation at time t can’t be (compressed), you still have to run the process until t. Hence, incompressible. So…random?
What about irrational numbers? Something like Pi or sqrt(2) can be represented in a compressed way (by using symbols)…but that’s not really the same thing as compression. Pi’s calculation can be expressed in a simple way (see Euler, Ramanujan, etc.) And yet its digits seem random (i.e. it’s normal).
This could be in some way related to intelligence. Intelligence is the ability to understand, abstract, explain, and apply across a variety of domains (my definition). Random processes are those which we can’t understand/explain, can’t create lossless compressions/abstractions/representations for, and can’t use to apply derived lessons to other domains. Randomness is in some ways the antithesis of intelligence.
How about things in the physical world? Do CO_2 molecules move randomly in our atmosphere? Could their movement be calculated if only we had enough compute and accurate-enough measuring devices?
Maybe. Laplace, extrapolating from Newton, certainly seemed to think so. But then Heisenberg and Schrödinger cast doubt on that belief with the uncertainty principle and Schrödinger equation, respectively.
Complexity science has shown how processes are at the same time intractably complex and yet at some level, in some way, to some people, compositionally and behaviourally explainable and, even, sometimes, gasp, predictable.
Does true randomness exist? Is it a fundamental feature of our universe like the finite maximum speed at which we can travel?
In some sense it doesn’t matter. There is no practical difference between a random process and one for which we don’t understand the generating mechanism. And the randomness “washes out” a lot of the time too, due to statistical phenomena like the law of large numbers. Things can also be “random” at one scale or dimension or level of accuracy, and yet deterministic in/at another.
But in a deeper sense, it does. Matter. Randomness is something we can’t predict, and have difficulty explaining. But this is not true of non-random processes: we can understand them if we work at the problem long enough. And we will, eventually.
From a mathematical perspective, randomness exists. A random sequence can be generated quite easily from a deterministic process, and they seem to exist everywhere in physics, nature, society, and even mathematics itself.
We can’t prove this apparent randomness, but we can test for it, build arguments for it, and say things about it in probability.
But, we must be careful. This is self-referential: randomness exists in mathematics, where we define randomness…mathematically. There might be a deep pattern to the digits of Pi that we just haven’t discovered yet.
Was the mathematics of randomness only invented as a way to deal with processes we don’t understand? Or does its invention/discovery reveal the presence of true randomness in our universe?