Mathematics as Hermeneutics

[elfs]

One of the more common themes among the mathematically inclined writers like Greg Egan and Ted Chiang is that of mathematics and hermeneutics: the notion that much of what we think of as advanced mathematics, the kinds of stuff done by people with Erdos numbers (if you know what that is, you're really a math geek), is just the bottom floor, and that the stuff we'll be interested in next is too complicated, too big, and just too damned hard for the human brain. Mathematics will become hermeneutics ("The study of the methodological principles of interpretation.") as we try to grasp exactly what it was our computers are telling us about the world, as the proofs for the various interesting parts of mathematics become more than can fit into a single human consciousness.

Now, it seems mathematicians have started to understand that that's what it may come down to. Many people are familiar with the quote: "The universe is not only queerer than we think, it's queerer than we can think." I would move the emphasis now: "The universe is not only queerer than we think, it's queerer than we can think." But not our descendents.

Steven Strogatz addresses the question of mathematics as hermeneutics in his essay, The End of Insight.

Comments

I very much doubt we're going to hit the wall anytime soon.

[(Anonymous)]

There are still plenty of very cool things to be discovered at the human-accessible level in mathematics. What mathematicians are doing know is synthesizing the known fields of mathematics into a coherent and unified whole.

This started with algebraic geometry (synthesis of algebra [not what you're used to calling algebra, something else] and geometry). Then algebraic topology, and things like knot theory, which combine topology, combinatorics, and whole bunch of other interesting things. Category theory, which is a theory about theoretical frameworks and how to boil them down to their most abstract (and hence most general) underpinnings.

The biggest problem that mathematicians will be working on in the next couple of decades is something called the Langlands Program. To give you an idea of how big it is, it contains both Fermat's Last Theorem and the Riemann Zeta Hypothesis as minor spinoffs. The problem is, this one is very close to the wall. To even understand the statement of the Langlands Program requires a Ph.D. in math (I have a Ph.D. in theoretical physics, and I can barely comprehend the beginning of the statements in this program).

If we crack Langlands, I suspect there will be vast vistas opening up to us -- not just a wall.

The interesting thing is that while some of these things are extremely abstruse _now_, mathematical history shows us that people tend to find explanations to put them on more accessible levels.

-Malthus

Re: I very much doubt we're going to hit the wall anytime soon.

[shunra.livejournal.com]

I second Malthus' opinion, here.

What we'll probably continue to see, though, is a divergence of professional mathematicians from the rest of the population - it'll be much harder to be a dilettente. But covering more territory in order to get to the starting point won't disable the race.

Moreover, mathematics has the property of being entirely a creature of the mind(s): people invent it and go off on tangents, creating infinite spaces inwards as well as outwards, with infinite pockets in infinite dimensions. Fractals ain't got nuttin' on the way math works! And then other people come along and apply these new constructs to the world around us, and then the math becomes pertinent.

In other words, as long as people don't run out of ideas, math will have the potential to grow new appendages and pull us up to new levls of insight and understanding.