Do math like a jellyfish

It occurred to me this afternoon as I sat down to blog that my blog can basically be distilled down to a few recurring themes:

1. Pro-life issues
2. Miscellaneous political issues
3. Science and religion
4. Nerdy math stuff

Aside from the random things I find on the Internet and just post to say, "That's cool!" I think this really sums up my blog. I didn't necessarily mean for this to happen (my blog is still what my title says it is) but I guess these are things I think about a lot.

Anyway this afternoon I was thinking about topic 4. Nerdy math stuff and I decided my blog could use a little diversion from the more serious (read: relevant) topics normally discussed.

So I started reading this book called Is God a Mathematician? which is about the question, "Does math describe something that actually exists, or is it all just made up in our minds?" In other words, do circles and lines and numbers actually exist in some Platonic universe, or are they just a convenient fiction?

Personally, my intuition says that yes, there is some Platonic universe in which all these concepts actually exist. I guess. I mean, there's nothing to force an intelligent creature to come up with the concepts of "1, 2, 3, 4, ..." and so on, but if they did, whatever theorems about those concepts they came up with, they would match up with ours. There's an actual logical structure to the universe that exists, whether we try to imagine our way around it or not.

But there was one motivating example that I found striking. It comes in the book as a quote from Sir Michael Atiyah, whom I had not heard of before. Here's his idea:

"[L]et us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jelly-fish, buried deep in the depths of the Pacific Ocean. It would have no experience of individual objects, only with the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count."

As a math grad student who is interested in analysis (for all of you non-math majors out there, think Calculus... on steroids) I am very much interested in the study of continuous things. So it strikes me as a good thought experiment to try to develop some sort of mathematics starting with the continuum, rather than with discrete objects.

This is hard. We human beings, perhaps thanks to God's blessing, started doing math by counting things. So all of our theorems from back in the day (before we invented weird things like Calculus) have to do with finite numbers, like 1, 2, 3, 4, and so on. What Atiyah is saying is, okay, now imagine there's nothing to count. You can't count one cow, two cows, three cows because there are no cows. It's really hard for us to imagine, because we're trained from the time we're little to count things.

So when we did finally start to study continuous things, we were able through a series of brilliant strokes to describe continuous things in terms of discrete things. (Or if you are currently unable to do this, that is why you never got Calculus. Sorry.) That is, if you want to understand a nice curve, just chop it up into little tiny pieces and "take the limit" as the number of pieces you chop goes to infinity. It's a brilliant strategy for moving out from the finite numbers you can count to the continuum.

I wonder if it's possible to work the other way around, or simply to bypass discrete math altogether, by starting with the continuum. What if we could do math like a jellyfish? What kind of mathematics would we come up with if all we had to experience was a set of continuous phenomena? Personally I can't quite imagine it at this point, but I'm going to try and think about it to see what I come up with.

This is what we mathematicians think about as we're walking home from lunch...

Yeah...