Mathematics has the most demanding standards of truth of any field of study. For a fact to be considered mathematically true, it must follow from previously-known facts using precise rules of inference. In turn, those facts are justified by inference from previous facts, and so-on.
But this process has to terminate at some point, or nothing could ever be justified. You need some facts which don’t need to be justified by something else. They’re called axioms — they are the sources of truth.
Because axioms are assumed true without justification, choosing axioms to believe in is more subjective than maths usually is.
Today, most mathematicians agree to work under the Zermelo-Fraenkel axioms. These axioms assert that there is only one kind of thing in existence: a set, i.e. a collection of things. Just as our understanding of people could be reduced to the movements of atoms, many beautiful structures in mathematics – numbers, space, geometric objects — can be reduced to sets.
The Zermelo-Fraenkel axioms include reasonable assumptions like “Two sets are the same set if they contain the same elements,” and “If you have two sets, then you can create a new set containing everything from both those sets.” Not especially controversial.
But there was one axiom whose proposed inclusion in ZF caused a stir: The Axiom of Choice, also known simply as Choice. (You know a mathematical statement is famous when it has a mononym like Prince.)
Choice can be written so innocuously as to believe its controversy:
Given a collection of nonempty sets, it is possible to pick one element from each set.
You could also call it the Axiom of Decision: if you have a bunch of decisions to make, you can always make every decision. Still sounds totally obvious and noncontroversial.
But, viewed from the right angle, the obvious can be troubling.
Choice asserts, without reservation, that you can always make any collection of decisions, no matter how many decisions that may involve. This is a little troubling: how do we make infinitely many decisions?
Actually, algorithms do that all the time. If you want to program an autonomous vehicle – say a rover – to travel for an indefinite length of time, it must be prepared to make infinitely many decisions. Every millisecond – or however short a length of time it takes to adjust the vehicle’s course – it has to decide on a direction to move in. But you don’t actually have to make infinitely many choices. You could program the vehicle to move in a straight line, or turn 90 degrees every 300 metres, or list 1° to the right every two minutes. Making decisions infinitely is not so difficult if you can write an algorithm or procedure that makes the decisions for you.
The trouble with Choice is that it asserts, even if you can’t write down an algorithm, that there is always a way to make infinitely many decisions. Choice is nonconstructive — it guarantees the existence of a decision process, but doesn’t tell you how to describe it.
And, by invoking Choice, one can argue for the existence of strange and unintuitive things that can’t be explicitly described. Choice implies that you can cut a set of points out of a finite line so that the set can’t be assigned a meaningful length. Even more amazingly, we have the Banach-Tarski paradox: a solid three-dimensional ball can be split into five parts, which can be moved around by rigid motions, and reassembled into two balls of the same size as the original.
The existence of indescribable objects (let alone indescribable objects that violated beliefs about measurement) was objectionable to some mathematicians when Choice was first introduced by Ernst Zermelo in 1904. It was a key reason for some mathematicians to reject Choice.
But there were also mathematicians prepared to support it, because of its astounding usefulness. Any areas of maths with incredibly useful applications, would be impoverished without Choice. Also, just as Choice has many strange consequences, rejecting Choice does too. For example, rejecting Choice implies that there exists a set that can be split into more parts than it contains objects.
So Choice sat in the middle of a crossfire. Who was right? How could we determine whether Choice is true or not?
In 1938, Kurt Gödel showed that Choice is consistent with the Zermelo-Fraenkel axioms. That means the ZF axioms, broadly accepted by mathematicians as a foundation of truth, couldn’t prove Choice wrong. That means Choice had to be true, right?
Not exactly. In 1963, Paul Cohen proved that ZF axioms couldn’t prove Choice true, either.
Choice is in the strange position of being totally independent from the ZF axioms. There are “mathematical universes” where the ZF is true and Choice is true, and there are also universes where ZF is true and Choice is false. Accepting Choice is equally valid as rejecting it. How, then, did mathematicians decide what to do?
Eventually, Choice became widely accepted. Except for mathematicians who study axioms themselves, most other mathematicians accept Choice as an axiom. Choice plus ZF is called ZFC – it’s the foundation for most mathematics done today.
Should that worry us? Choice has consequences that are physically absurd: a ball can be broken apart and reassembled into two balls. Should we be concerned that modern maths has no bearing on reality, because it relies on Choice?
No, not really.
Choice brings about some strange, unintuitive things, but they can’t be explicitly described, and can’t really have any impact on the real world. Choice is certainly not at odds with the success that mathematical models have enjoyed at describing our world.
Originally published in Woroni Vol. 72 Issue 2 ‘To Be Confirmed’
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