The story actually goes back to Leibniz, who "dreamt of building a machine that could manipulate symbols in order to determine the truth values of mathematical statements."

The problem got its modern form as the Entscheidungsproblem, which is German for "decision problem." According to Wiki,

The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" according to whether the statement is universally valid, i.e., valid in every structure satisfying the axioms.

By the completeness theorem of first-order logic, a statement is universally valid if and only if it can be deduced from the axioms, so the Entscheidungsproblem can also be viewed as asking for an algorithm to decide whether a given statement is provable from the axioms using the rules of logic.
— Wikipedia

In 1936 Alan Turing published his famous paper, On Computable Numbers, With an Application to the Entscheidungsproblem. Turing was played by Benedict Cumberbatch in the highly fictionalized but definitely worthwhile movie, The Imitation Game.

In this paper Turing showed that there could be no mechanical procedure to decide the truth of mathematical propositions.

As part of his proof, he invented a formalized model of computation called the Turing machine (TM). A TM can be programmed to solve a given mathematical problem. He showed that there are many real numbers whose digits can be cranked out by a TM. He defined such a number as computable. For example any rational number is computable, since given two integers n and m we can just apply the grade school division algorithm to compute as many digits as we like of the quotient n/m.

Surprisingly, even many irrational numbers are computable. For example pi can be computed via the famous Leibniz series:

pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - ...

which can easily be turned into a computer program or TM.

However there are many noncomputable real numbers, whose decimal representation can never be computed by any TM or any computer program.

Now, how many real numbers are computable? Since each TM program consists of finitely many finite-length instructions, there are only countably many such programs: finitely many of length 1, finitely many of length 2, and so forth.

On the other hand we know from Cantor's diagonal argument and also from Cantor's theorem, that there are uncountably many real numbers. So there are "too many" real numbers to compute them all with TMs. These are the noncomputable real numbers. They are deserving of the name random, because there is no mechanical or deterministic process to generate their decimal representations.

Now when we say that "almost all" real numbers are noncomputable, we mean that a countable set is vastly smaller than an uncountable one. So "most" real numbers are noncomputable.

We can formalize this idea using measure theory. Any countable set has measure zero. So for example in the unit interval consisting of all the real numbers between 0 and 1, the set of computable reals has measure zero, and the noncomputable reals has measure 1.

Intuitively this means that if you put all the real numbers in a paper bag, closed your eyes, and reached into the bag and pulled out a real number, the probability is zero that it's computable, and probability one that it's not.

Another way to think of it is that if you flip a fair coin infinitely many times, regard heads as 1 and tails as 0, and put a binary point in front of the sequence, you have the binary representation of some real number. The probability that a computer program could crank out that exact sequence is zero; and the probability that no possible computer program or mechanized procedure could do so is one.

This is actually plausible. If we randomly flip infinitely many coins to generate a sequence of 0's and 1's, it's very unlikely that there will be any kind of pattern or mechanical procedure that predicts the sequence. It's far more likely that the sequence will be patternless, or random.

The noncomputable real numbers are sort of the "dark matter" of the real numbers. They don't have names. They don't have individual qualities or characterizations that uniquely describe any of them. They are simply there, holding the real line together.

So my thesis is that even beyond the mysteries of quantum mechanics, if anything in the world can legitimately be called random, it's the noncomputable real numbers.

Constructivist mathematicians don't believe in noncomputable numbers. They try to do all of math using only numbers that can be specified via algorithms or procedures. They're gaining some mindshare these days due to the influence of computers in math.

Well I wrote a lot but I left out a lot, so please ask if anything's not clear.
18 minutes ago — fishfry