## Monday, January 25, 2010

### Leap to Infinity

I can't resist blogging on a recent article I read over at Republic of Mathematics. In it, the author wrestles with that perennial question of whether abstract mathematical concepts are real.

He talks about the set of natural numbers. We all know this set intuitively: {1, 2, 3, 4, 5, 6, ...} You know what the "..." means. It means the set goes on forever. If I have any natural number n, you know that n + 1 is another natural number, so the set of natural numbers can't have a biggest number.

You can't get this set from finite sets. If you work in the realm of the finite, you stay there forever. It takes a leap of faith to get to the infinite. This is what mathematicians call making an axiom. It's a leap of faith, but it's certainly not a leap in the dark. We have an intuitive idea that we just formalize. No big deal, until we think about what it is we've just done.

What have we done? Have we described something real? Depends on what you mean by real, I suppose. What got me interested in this subject again was the religious idea that came out in the article I read:

So at the heart of mathematics lies an act of reification, a taking as real some thing – the set of natural numbers – that is an abstraction from our human activity of counting.

The process of reification is explicitly addressed in Buddhist thought, where it is generally thought to be not a good thing because it leads to the delusion of permanence for mental constructions that are bound to decay:

All things and events, whether ‘material’, mental or even abstract concepts like time, are devoid of objective, independent existence. … things and events are ‘empty’ in that they can never possess any immutable essence, intrinsic reality or absolute ‘being’ that affords independence.” Dalai Lama (2005)
Contrast this with the Christian point of view, that the created world has real, independent existence, and that it is good, in the sense that it ought to be cared for and treated in the right way. As I blogged about earlier, Florensky had a lot to say about this.

Indeed, Florensky would say that it is precisely Trinitarian thought that leads us out of these labyrinths of epistemological despair. Human rationality, he says, is based on two things: the static and the dynamic. Clear thinking depends on things being static--A is not non-A--while proof, explanation, and learning depend on things being dynamic--A is also B. The combination of static and dynamic is found in Trinity--God is both Himself and not Himself, and by being not Himself He is found to be most fully Himself. But I digress.

The article ends with a rather nonchalant statement of pragmatism:

So in mathematics we treat the abstract construct of the set of natural numbers as a real object and – as if by magic – discover deep properties of this set.

On such myths mathematics and science thrive!

Just as modern science has reached the height of its self-confidence, believing itself to be the grand beacon of objective knowledge, mathematicians are here to cut the legs out from underneath that self-confidence!

Not that mathematicians think any differently about science than your typical modernist on a practical level. It's just that mathematicians seem to quite often be okay with accepting a world that isn't real and manipulating it anyway. Curiosity becomes its own reward.

Many of us will have a negative gut reaction to this attitude, and I think that's healthy. Curiosity is a wonderful thing, but is it really that wonderful if there's nothing real to be curious about?

Nevertheless, I do think that mathematics is mostly about mental abstractions created by us; they don't exist "out there" in some Platonic heaven. But they are real, because our brains our real! The concepts are as real as we are. Some craftsmen shape metal or wood; mathematicians shape the human mind. (And we are certainly not the only ones who do that.) That's about as real as it gets, if you ask me.

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