A lot of math grad school is reading books and papers and trying to
understand what's going on. The difficulty is that reading math is not
like reading a mystery thriller, and it's not even like reading a
history book or a New York Times article.
The main issue is that, by the time you get to the frontiers of math,
the words to describe the concepts don't really exist yet.
Communicating these ideas is a bit like trying to explain a vacuum
cleaner to someone who has never seen one, except you're only allowed to
use words that are four letters long or shorter.
What can you say?
"It is a tool that does suck up dust to make what you walk on in a home tidy."
That's certainly better than nothing, but it doesn't tell you
everything you might want to know about a vacuum cleaner. Can you use a
vacuum cleaner to clean bookshelves? Can you use a vacuum cleaner to
clean a cat? Can you use a vacuum cleaner to clean the outdoors?
The authors of the papers and books are trying to communicate what
they've understood as best they can under these restrictions, and it's
certainly better than nothing, but if you're going to have to work with
vacuum cleaners, you need to know much more.
Fortunately, math has an incredibly powerful tool that helps bridge
the gap. Namely, when we come up with concepts, we also come up with
very explicit symbols and notation, along with logical rules for
manipulating them. It's a bit like being handed the technical
specifications and diagrams for building a vacuum cleaner out of parts.
The upside is that now you (in theory) can know 100% unambiguously
what a vacuum cleaner can or cannot do. The downside is that you still
have no clue what the pieces are for or why they are arranged the way
they are, except for the cryptic sentence, "It is a tool that does suck
up dust to make what you walk on in a home tidy."
OK, so now you're a grad student, and your advisor gives you an
important paper in the field to read: "A Tool that does Suck Dust." The
introduction tells you that "It is a tool that does suck up dust to make
what you walk on in a home tidy," and a bunch of other reasonable but
vague things. The bulk of the paper is technical diagrams and
descriptions of a vacuum cleaner. Then there are some references:
"How to use air flow to suck up dust."
"How to use many a coil of wire to make a fan spin very fast."
"What you get from the hole in the wall that has wire in it."
So, what do you do? Technically, you sit at your desk and think. But
it's not that simple. First, you're like, lol, that title almost sounds
like it could be sexual innuendo. Then you read the introduction, which
pleasantly tells you what things are generally about, but is completely
vague about the important details.
Then you get to the technical diagrams and are totally confused, but
you work through them piece by piece. You redo many of the calculations
on your own just to double check that you've really understood what's
going on. Sometimes, the calculations that you redo come up with
something stupid, and then you have to figure out what you've understood
incorrectly, and then reread that part of the technical manual to
figure things out. Except sometimes there was a typo in the paper, so
that's what screwed things up for you.
After a while, things finally click, and you finally understand what a
vacuum cleaner is. In fact, you actually know much more: You've now
become one of the experts on vacuum cleaners, or at least on this
particular kind of vacuum cleaner, and you know a good fraction of the
details on how it works.
You're feeling pretty proud of yourself, even
though you're still a far shot from your advisor: They understand all
sorts of other kinds of vacuum cleaners, even Roombas, and, in addition
to their work on vacuum cleaners, they're also working on a related but
completely different project about air conditioning systems.
You are filled with joy that you can finally talk on par with your
advisor, at least on this topic, but there is a looming dark cloud on
the horizon: You still need to write a thesis.
So, you think about new things that you can do with vacuum cleaners.
So, first, you're like: I can use a vacuum cleaner to clean bookshelves!
That'd be super-useful! But then you do a Google Scholar search and it
turns out that someone else did that like ten years ago.
OK, your next idea: I can use a vacuum cleaner to clean cats! That'd
also be super-useful. But, alas, a bit more searching in the literature
reveals that someone tried that, too, but they didn't get good results.
You're a confident young grad student, so you decide that, armed with
some additional techniques that you happen to know, you might fix the
problems that the other researcher had and get vacuuming cats to work.
You spend several months on it, but, alas, it doesn't get you any
further.
OK, so then, after more thinking and doing some research on extension
cords, you think it would be feasible to use a vacuum cleaner to clean
the outdoors. You look in the literature, and it turns out that nobody's
ever thought of doing that! You proudly tell this idea to your advisor,
but they do some back of the envelope calculations that you don't
really understand and tell you that vacuuming the outdoors is unlikely
to be very useful. Something about how a vacuum cleaner is too small to
handle the outdoors and that we already know about other tools that are
much better equipped for cleaning streets and such.
This goes on for several years, and finally you write a thesis about
how if you turn a vacuum cleaner upside-down and submerge the top end in
water, you can make bubbles!
Your thesis committee is unsure of how this could ever be useful, but
it seems pretty cool and bubbles are pretty, so they think that maybe
something useful could come out of it eventually. Maybe.
And, indeed, you are lucky! After a hundred years or so, your idea
(along with a bunch of other ideas) leads to the development of aquarium
air pumps, an essential tool in the rapidly growing field of research
on artificial goldfish habitats. Yay!
Visto aquí:
https://gist.github.com/stoutbeard/4158578
