I just finished Mario Livio's Is God a Mathematician? a book about whether mathematics is invented or discovered. It was a nice read, filled with wonderful stories of mathematics throughout the ages--you know, just the thing that gets nerds like me excited.

The essential conclusion was nice and pragmatic. Livio thinks that the question of whether mathematics is invented or discovered is an ill-posed question, because it is both.

My feeling is that to most modern mathematicians this feels like common sense. We invent the mathematical concepts that we're going to study, and then we discover all the properties of, and relationships between, those concepts.

It's a little deeper than just inventing concepts at random, of course. These concepts typically find both motivation and application in the real world, whether we're talking about physics, economics, or whatever.

So that's great, but then, where did this question come from in the first place? One thing that's nice about Livio's book is how he traces the history of how this question was answered. There have been some really interesting answers.

The Platonists would say that math studies forms or objects that exist in some sort of reality beyond the world of the senses--a Platonic heaven, you might say. This idea has largely died out in the modern era, as people have discovered just how culturally unique certain mathematical concepts are.

For instance, the Babylonians, the Egyptians, and far Eastern cultures developed sophisticated mathematics, but only the Greeks thought of the concept of prime numbers. The kinds of questions you ask shape the kinds of concepts you formulate. There are other reasons why Platonism has lost credibility, but this is a pretty good one.

Then there are the formalists, who say that basically math is all just a construction of the human mind that happens to be self-consistent, nothing more. GĂ¶del's incompleteness theorems created some problems with that idea, but I won't get into the details.

My own view accords well with the kind of realism that Livio seems to espouse in the last chapter of his book--the actual definitions and axioms with which we create mathematics are technically invented, but nevertheless there is objective content to mathematical theorems.

However, I don't really like the rabbit holes he goes down to try to explain this. He cites many different people who give pseudo-scientific answers that somehow get linked to the theory of evolution. They try to make the mathematics we now have out to be some "chosen species" weeded out from a number of potential candidates.

But evolution works on very long time scales, whereas the time it took for modern mathematics to develop was, comparatively, not that long. Besides, often the development of mathematical theories has a lot more to do with beauty than with fitness.

A "beautiful" mathematical concept is a purely subjective idea, but it's no less real because of that. What's so surprising to me is how beautiful mathematics can be--and still truly describe the world we live in. Some will say, "Oh, well, that's just because the world we live in happened to have these certain symmetries." Well, then, why not marvel at how beautiful the world is!

As I see it, mathematics has to develop as something of a human invention--but always, of course, seeking something that is objectively real. In fact, all knowledge must progress this way. We can gain no knowledge of the world in prepackaged form--it all comes in the form of changes in ourselves.

I have every theological reason to believe this as I contemplate the Trinity. Truth is not found in one person but in three. Just the same, truth is not found in nature alone, but it is found in nature, in ourselves, and in the interaction between the two. It is all about the interplay.

Consider an ancient mathematician charged with developing some accounting scheme for commerce. He starts by observing people trade their goods. He realizes that he can quantify these transactions using whole numbers. This is because the things traded always behave according to the rules of his number system. I have two cows and three sheep; you trade me a cow for my sheep; now I have 2 + 1 = 3 cows. Whether its cows or sheep or whatever, my goods always seem to follow these basic properties of arithmetic.

So this mathematician has immersed himself in the nature of the world around him, and he has come up with a way to describe it, the whole numbers. These are defined by simple rules: basically, that each number can be obtained by adding 1 to itself enough times, and given any number, adding 1 to that number gets you another number. But from here it is possible to define the operations of subtraction, multiplication, and division, where applicable, and it is possible to start proving all sorts of theorems in arithmetic.

This description of nature has a life of its own, apart from nature, but always intimately linked to nature. So long as the goods in this mathematician's world continue to behave like cows and sheep (which add together just like whole numbers) all the theorems about whole numbers will also be applicable to trading real goods.

But suppose the ancient mathematician's task is different. Perhaps it is to record the yearly cycles of water levels of the local river. If he has never done any arithmetic beforehand, who is to say that he will start by developing the whole numbers? Maybe he will not choose to create a system of definite quantities at all, but merely record comparisons throughout the year. His mathematics will be all inequalities, rather than definite quantities.

So we see that there is always a third party involved, one to whom we are accountable to and who influences how we see the world.

In this way the road to truth is not a linear motion, but rather a dance between three parties, each of whom must influence and be influenced by the other two. Just as God is one in three, so each of us must learn to be one of three, and we must learn that knowledge only comes from making these three one.

I am a bit mystified by this myself, but there's something compelling about it.

What I mean really is that I don't think a mathematician should not be discouraged by the fact that mathematical concepts are an invention. As part of this triad consisting of human, nature, and whatever this third party may be (perhaps "context" will suffice), the mathematician must simply work on his relationship with the other two. He must both establish his own real identity and work to seek the other two with sincerity. His own identity comes from his invention of mathematical systems. His seeking the other two comes from testing whether these systems help him describe something real in nature, and in seeing whether they are relevant in the context he is given.

It would be an unhealthy relationship if one part of this triad was favored over the others, so a mathematician should feel free to be creative, but never so free that he is not grounded in some reality. While this may be common sense to many, I like how there is this possibility of real theological grounding for it. If God is Trinity, why should that not affect all that we do?

Is God a mathematician? If by mathematician you mean one who studies the mathematical constructions that he invents, then probably the answer is "no." (Does God need to invent such things and study them?)

But on the other hand, I do believe that the study of mathematics is so linked with His character that I would have to say that God seems to have constructed this universe with mathematicians in mind. Somehow, God loves a quantifiable universe.

In any case, I know this will all sound like a lot of esoteric nonsense to most people out there, but after all my purpose in all this is written at the top of the page. I thought, therefore I blogged.

I like your blog! Mine blog isn't half as intellectual, but I appreciate a good theological, philosophical or scientific discussion...

ReplyDeleteThanks! I like your blog; I wish I had more of a talent for creative writing. Math has trained me to iron out details--a different kind of cleverness from creativity or wit. My brother is a good writer, though; you might want to check out "i am probably overthinking this."

ReplyDeleteI'll check out your brother's blog. It sounds great. You are an excellent writer though, and thinker. I wish I had a more scientific mind. I always shied away from math in college. I took physics my senior year, however, because they forced me, and I ended up with the best grade in the class. It made me regret not doing a science major in the first place: hence, my blog url being "ishouldhavebeenadoctor."

ReplyDeleteA blog doesn't have to be "funny" to be interesting though. I feel guilty for my posts after reading yours. I'm afraid I'm not being the best witness to the world! I will have to update my bible journal blog more often: lure them in with the funny stuff and hope they wander on over to the Christian stuff! Ha ha!