What exactly are mathematical objects like π and the unit circle? In what sense do they exist?
In the last post I proposes neofictionalism, the idea that the mathematical universe is a collective fiction (like Lord of the Rings, Star Trek or GOT) but about which statements can be objectively true or false. Thus the ordinal ω is comparable to the Iron Throne and the statement “ω is the set of all natural numbers” is objectively true in the same way that “the Iron Throne is made of swords” is objectively true.
Why neo-fictionalism? Because there is already a philosophy called “fictionalism” which presents a similar account but which declares that mathematical statements are all false because they refer to nonexistent objects. (Presumably statements about the Iron Throne are also all false).
What I’m saying is that ω and the Iron Throne are products of the human imagination. Humans are part of the real world and so is our imagination. So the products of our imagination surely exist in some sense. And if they exist, we must be able to make assertions about them.
All the members of the mathematical universe are like characters in a play, inventions of the human mind. This is true even if they are patterned after objects in the real world.
The perfect circle, an early fiction
Take circles, for example. As one comment on the Hacker News post pointed out, there are lots of circles in the world – the shape of the moon, the waves when a stone is thrown in a lake, the iris in your friend’s eye. But real world circles don’t hold up to close inspection.
Some are not quite circular. And all of them, under high magnification, are not made of perfect arcs – quantum mechanics, if nothing else – rules this out.
Mathematical circles, on the other hand, are idealizations or refinements of real world circles. The arcs are perfect (uniform curvature) at every level of magnification and have zero width. There are no such circles in the real world but we can imagine them. And, more importantly, we can come to a consensus about their properties so we can talk to each other about them.
Mathematics advances as we imagine new objects and work out a consensus about their properties. This has taken a long time – centuries in some cases. Think how long it took for a consensus to emerge to not use infinitesimals.
The role of infinity
Notice that already in the discussion of circles the word “infinite” popped up – perfect circles are infinitely accurate. Not an accident. The simple counting numbers are infinite in number, The ordinal ω is an infinite object. We mentioned infinitesimals, numbers which are infinitely small. Pure (mathematical) line segments are infinitely thin, have infinitely many points, and can be extended indefinitely.
If mathematical objects are imagined refinements of real world objects, then it seems that the refinement process involves proceeding from the finite to the infinite. Does the infinite exist in the real world? This is a difficult question but not one we have to answer. The important point is that humans can (consistently and collectively) imagine infinity.
What is our inspiration?
And where do we get the inspiration for these imaginary objects? From our experience solving technical problems in the real world.
The connection is not always straight forward but we can see the outline. Negative numbers record debts. Rational numbers record the results of dividing quantities into equal parts. Algebraic numbers (like the square root of two) allow measurements of geometric constructions and roots of equations. The real numbers and the continuum were created to describe continuous (smooth) change. The imaginary numbers allowed any polynomial to have roots.
The fact that mathematics is useful for describing the real world should hardly be a surprise. Mathematical objects were designed for this purpose. Saying math is unreasonably effective is like saying knives are unreasonably effective for cutting meat or hammers are unreasonably effective for hitting nails.
Not so unreasonable effectiveness
But perhaps that’s going a bit too far. It regularly happens that mathematical entities introduced for one purpose turn out, often decades later, to be perfect for another use that couldn’t possibly have been foreseen. My favorite example is Weierstrass functions, that are continuous but nowhere differentiable. Discovered in 1872, they were considered to be rare and pathological. Several decades later physicists realized they modeled ideal Brownian motion.
This was possible only thanks to centuries of struggle to find the right definition of a function. At first functions were considered as a defining expression, then piecewise by several expressions, then piecewise by infinitely many expressions. Finally the consensus emerged to allow arbitrary correspondences.
Nor should unforeseen applications surprise us. All really good technology is useful for purposes for which it was not designed. Take the world wide web: it was launched on networked personal computers, which had been designed for word processing, gaming, and exchanging recipes on bulletin boards. All good technology has redundant power which sometimes makes its effectiveness seem unreasonable.
The wolf in the woods
Neofictionalism also sheds light on issues of constructivism. A consensus has emerged in set theory, which is to use Zermelo – Fraenkl axioms plus the axiom of choice (ZFC). ZFC has been used for almost a century without a contradiction emerging and without finding a mathematical idea that can’t be formalized.
The axiom of choice is a perfect example of proceeding to the infinite. For a finite number of sets, our real world experience tells us it’s possible to make the choices. Visit each (say) jar and take one candy from each. AC says thus can be done even when there are infinitely many jars – and we can certainly imagine doing this even if in the real world it doesn’t make sense.
The problem is, infinite choice allows the construction of surprising objects.
In particular (AC) implies the existence of a nonmeasurable set – e.g. a surface that has no area. It implies the existence of an infinite game for which neither player has a winning strategy (and the game has no ties). It implies the existence of a voting system for denumerably infinite constituencies.
What does a nonmeasurable set look like? Or the game? How does the voting system work? Don’t ask. There are a whole group of results that say, basically, if you can give some sort of of concrete description of a set, it is measurable. Give a description of a game, there’s a winner. Of a voting system, it doesn’t work.
This is the explanation from a neofictionalist perspective: we imagine that there is a nonmeasurable set without there being a particular set that we imagine is nonmeasurable. There’s nothing impossible about this situation. It’s like writing a novel in which you say there’s a wolf in the woods without saying anything more about the wolf.
Constructivists reject the axiom of choice on principle but I don’t think there’s any principle at stake. It’s an engineering decision whether to include (AC) – we can imagine either possibility. And AC greatly simplifies the universe of set theory; it implies, for example, that the cardinals are linearly ordered.
Is ZFC the end of the story? I don’t think so. Mathematicians have been searching for alternatives to AC without the unpleasant side effects. So far without success, though the axiom of determinateness (every infinite game has a winner) looks promising.
Wherever this goes, I’m sure the last word has not been written. We’re free to choose our mathematical universe.