The great math mystery

Last night Nova aired The great math mystery, a documentary that describes mathematicians’ ideas about whether math is discovered or invented, whether it is “out there” or “in our head”. It was well-done. Things were explained clearly using images and metaphors, although they did show Maxwell’s equations as algebra (without explaining it). The visual illustrations of connections between Maxwell’s equations and music and electromagnetic waves was one of the best parts of the documentary.

In my opinion they made good choices of mathematical ideas to cover, but I imagine a lot of research mathematicians will have a hissy that they didn’t cover XXX (their subject).

The applications to physics dominated the show (that is not a complaint), but someone did mention the remarkable depth of number theory. Number theory is deep pure math that has indeed had some applications, but that’s not why some of the greatest mathematicians in the world have spent their lives on the subject. I believe logic and proof was never mentioned, and that is completely appropriate for a video made for the general public. Some mathematicians will disagree with that last sentence.

Where does math live?

The question,

Does math live

• In an ideal world separate from the physical world,
• in the physical world, or
• in our brains?

has a perfectly clear answer: It exists in our brains.

Ideal world

The notion that math lives in an ideal world, as Plato supposedly believed, has no evidence for it at all.

I suppose you could say that Plato’s ideal world does exist — in our brains. But that wouldn’t be quite correct: We have a mental image of Plato’s ideal world in our brains, but that image is not the whole ideal world: If we know about triangles, we can imagine the Ideal Triangle to be in his world, but we have to know about the zeta function or the monster group to visualize them to be in his world. Even then, the monster group in our brain is just a collection of neurons connected to concepts such as “largest sporadic simple group” or “contains\[2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71\]elements” — but there is not a neuron for each element! We don’t have that many neurons.

The size of the monster group does not live in my brain. I copied it from Wikipedia.

Real world

Our collective experience is that math is extraordinarily useful for modeling many aspects of the real world. But in what sense does that mean it exists in the real world?

There is a sense in which a model of the real world exists in our brains. If we know some of the math that explains certain aspects of the real world, our brains have neuron connections that make that math live in our brain and in some sense in the model of the real world that is in our brain. But does that mean the math is “out there”? I don’t see why.

Math is a social endeavor

One point that usually gets left out of discussions of Platonism is this: Some math exists in any individual person’s brain. But math also exists in society. The math floating around in the individual brains of people is subject to frequent amendments to those people’s understanding because they interact with the real world and in particular with other people.

In particular, theoretical math exists in the society of mathematicians. It is constantly fluctuating because mathematicians talk to each other. They also explain it to non-mathematicians, which as everyone know can bring new insights into the brain of the person doing the explaining.

So I think that the best answer to the question, where does math live? is that math is a bunch of memes that live in our social brain.

References

I have written about these issues before:

• Constructivism and Platonism — a third way
• Math and the modules of the mind
• Conceptual blending
• Shared mental objects
• Modules for mathematical objects
• Thinking about thought

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