Author's note: not that anything is ever 100% original, but this was written shortly after reading Elie Ayache's The Medium of Contingency and this vignette is greatly indebted to his ideas.
Randomness is prosthetic.
We have a lack of knowledge about a coin tossed in the air before it lands, but we know its properties well enough to feel confident that no matter which side you bet on you'll break even.
In mathematics, we write proofs that show a property holds for any relevant value. The material world obviously doesn't work like this, so we do the next best thing and create a game of chance: you bet that a rule holds regardless of what's out of your control. The double-blind randomized trial is the die, it's built to prevent you from betting on what you have not explicitly stated.
Science is evolutionary in this sense: nothing is ever proven, or even falsified (since unlike mathematics there's no such thing as an unequivocal counterexample); there's only constant competition in which you put your money where your mouth is.
But there's no "true" randomness necessarily at play: there's simply degrees of subjective uncertainty that we end up expressing with our actions and different games we play that define our relationship to the roll of the dice.
The idea that this sense of uncertainty can be expressed as a number between 0 and 1 is technologically contingent on the proliferation of money.
Money makes everything fungible, you can sell something and buy something else, and when enough people are using money with enough frequency you can use any fine-grained amount of money at virtually any time.
And with that you have the ability to make a profit off of a loaded die: if the probability of something is even infinitesimally asymmetrical, there's a percentage of your money you can repeatedly bet that will guarantee a profit over a long enough timespan, and that percentage is directly calculable from the probability.
Picking stocks is an exercise in futility by design: any reliable asymmetry will be exploited, which makes them reliably random for those playing a game one level up (derivatives).
But the movements of a stock are obviously not the roll of a die: there's no limit to the values it can take on and the pattern of movement in the long run isn't an even distribution of outcomes.
Yet at the same time there's no obvious pattern, which is where that thing we call "finance" comes in: it no longer suffices to make some simple bet, financial derivatives are created in order to bet on patterns with more dimensions, which themselves have a price that allows the same to be done to them.
The market never settles down to equilibrium because there's always room for more games to play.
At every step of this process there is a logical leak that allows something new to be constructed: a kind of uncertainty still bound to a number by definition but completely devoid of any kind of reliable expectation (unlike a die, where we can be sure of its average value.)
But at no point is any of this divorced from causality: the very uncertainty itself is a logical consequence of the fact that there was something certain and people took advantage of it; a process equivalent to making a die. There's no break from logical necessity by which things miraculously wiggle.
But this doesn't mean that the future is already written...
The answers then have to come from "outside", and this is what happens in the market all the time. The reason why real financial trading isn't gambling is because this is obviously an act of creative understanding, it's not the same thing as some jackass watching a line go up and down.
And so for any example I give I can only tell you that causality within that example is insufficient, not that there's no causality elsewhere to dictate what gets plugged in from outside.
But if no non-trivial logical system can be complete, there's always an outside, never a view from nowhere; causality remains the backbone of reasoning but we can never fully articulate causality without the help of a substrate indifferent to it.
There is always contingency, incomprehensible by logical necessity, and probably a good definition of God. At no point does it contradict causality: causality is a local property that acts so long as it's capable of giving an answer.
"Free will" by necessity exists but not in the form of some homunculus piloting a human; it's something more diffuse, a kind of creative and evolutionary process that could lead to a better definition of what we call "life".
Randomness is uncertainty with a name: a set of outcomes that are only significant in relation to a game, their meanings defined by convention; a materially constructed opacity made in the hopes that the ruler measures the table and not the other way around.
That's the gist of it but that last part could use an explanation:
The more you know about the conditions of your experiment that aren't explicitly stated as part of the hypothesis, the more your hypothesis can become a predictor of your conditions instead of the other way around.
Making your test conditions opaque to yourself prevents this: in other words, the game is what gives language stable meaning. But this is entirely artificial; it requires both a way to make it truly independent and an agreement by everyone about what it represents.
None of this is to denigrate it: like all technology, it's what gives us ground to stand on.
Financial markets are a nice example because the supreme uncertainty of an asset's price represents a gap in the universe where cause and effect isn't fully present to dictate what comes next: a trader constructs a new game that defines a new set of relationships, a new language, a new material. The newly created derivative is a real thing that gets traded, and it's not purely a product of what came before.
Yes, there are forces acting on the trader that influenced or dictated what he made, but within the logic of the market there was no definite answer.
The formalization of this idea is Godel's Incompleteness Theorem: the inevitability in any sufficiently complex logical framework of relevant questions that can't be answered within the logic of the system.
The important thing to understand here is that this is not a "random" answer: the very fact of an undecidable problem necessitates new axioms, things that cannot in any way be defined in relation to any other part of the system (because otherwise they would have been derived).
The constructed object that fills the void inherently can't be something that was previously deemed "possible", it's the defining of something entirely new, not a value but a predicate.