I really wish Posterous would allow Javascript customization, first because I want to use Disqus here and second because I want to make a post with a bunch of math equations and there's a nifty Javascript-only widget for that found at yourequations.com .
(What I write here includes a lot of confident-sounding guesses. Consider this more like a late night wine-soaked conversation than an exposition about things that I know. This was meant to be an exploration. It turned out to raise more questions than it answered.)
Last night as I was trying to fall asleep I was, for some reason, thinking about mathematics and innovation, and I was it occured to me that the point humanity passed a few decades ago, where a single mind [1] can no longer contain all our mathematics is a major and universal milestone for a civilization. What does it mean? Will the balkanization of math [2] slow progress? Are we approaching Peak Math? [3]
There is something else related to this: the more study required to reach the frontiers of a field, the fewer restless turbulent personalities will be in that field, because they won't have the patience to quietly learn at the feet of their predicessors for the many years necessary to reach the frontier. And without those sort of people, you won't get the sort of revolutionaries that shake the planet with their work. Thus as the frontier gets further from the starting point of pure ignorance, you should see more incremental advances and fewer revolutions. Do we see that in mathematics? If the revolutionaries are driven out of vast fields, where do they go? I have my suspicions. Is it useful to ask how they are driven out? Do they just find the culture and its practitioners no simpatico?
The distance to the frontier of a field should be related to how much of the field can be contained in a single brain, (it should increase as the square root of the amount of knowledge in the field if you assume the field grows in all directions) since you've got to have all the preceding stuff that is on the route to your speciality in mind before you can advance the field and by the time you get there you have less space for the new stuff. This implies that there will be a bigger chunk of your brain filled up with the common stuff shared between subfields than if you were in a smaller field.
So again, how many brains do you need to contain mathematics? Or physics? How many other fields have exceeded the capacity of a single brain? Economics? Certainly there are fields of Econ that have no contact with other fields - game theory has little in common with econometics - but is brain-filled-ness the reason for that? Geology - certainly soft rock geologists and hard rock geologists don't fully understand what each other are currrenly up to but is that because they can't or because they don't want to? [4] How about Computer Science - surely there is more than one brainfull there. I'm pretty sure that most of what's going on in machine learning is not understood at all by the people working in theory, and vice versa. Likewise the PL people can't be current on algorithims. I just don't know enough about these fields to know how many minds you need to encompass the field, just that its got to be more than one.
What fields aren't like that? New ones. [5] From what I understand, neuroscience is new enough that a neuroscientist can just be a neuroscientist.
So is this the difference between a field and a subfield? Two different fields can share some stuff, like math and physics (or math and anything) but from the beginning they are diverging - there are lots of things that a first year physics undergrad and a first year math undergrad learn together, but already there are things that they don't. Same is true for geologists and physicists - from the first year they are diverging. (Hmm... yet they all have the same pre-college experience.) But budding topologists and analysts are studying exactly the same things for years at college. So are the capabilities of the 18 year old college student the thing that makes electrical engineering and computer science different fields? Maybe. Honestly, academia politics probably has much to do with it. Hmmm... probably I'm seeing this thru my lens shaped by the current divisions in university departments. I mean, what the 18 year olds are studying isn't necessarily what they *should* be studying.
What I'd love to see is some way to measure the size of fields in units of brains. How? I was thinking that maybe a field branches into subfields every time it gets too big to be contained in a single head. Does this happen? Well, if you assume that the amount of knowledge in a leaf on the tree of fields is constant, the increase in fields and subfields should be proportionate to the increase in human knowledge.
How much have fields increased? It seems that there were no departments at universities 800 years ago. This page shows around 1500 of them today, but that list is under-branched. This shows many subfields not mentioned on the first page. Excluding repeats it makes at most a 5-fold increase. If it is underspecified and there are (or should be) similar pages for other topics we might get another order of magnitude. So four orders of magnatude increase in human knowledge in 800 years. For every one thing known 800 years ago do we now know only 1000 things? No way - that seems too low. Math has done far more than that, even excluding its offshoots (Computer Science, etc). On the other hand, theology was a major area of study at the time and we have probably not even doubled our knowledge of that since then. For example I think most Protestants would say that theological knowledge has been completely static for 2000 years. [6]
Here's something interesting: if your field is smaller than one brainfull, I don't think you can tell how much less. It seems to come as a surprise to all that the arts and sciences combined had gotten too big, and it was only by complaints by some mathematicians that they realized it was true in their field, too.
Related: The Last Days of the Polymath.
FOOTNOTES:
[1] I should clarfify that the amount of knowledge that you can fit into one brain is going to be very hard to differentiate from the amount you can learn before you start going senile. Perhaps, in fact, that is the limit. Maybe the brain has no limit of knowledge, but instead is limited by rate of intake. Then that rate times the lifespan of a person equals the size of a brain.
[2] Just how balkanized is math? Do they in fact know, in absolute terms, less about other fields than they used to? Of course they know a smaller percentage of the whole field than they used to, but do they also just know less? Would a topology mathematician of 30 years ago know things about analysis that a topologist today wouldn't? Probably if someone wanted to simply know as much as possible about math and did not intend to contribute, he or she could know much more than the typical mathematician. I.e. it might be just that no one ambitious wants to 'waste' any time keeping up with the rest of their field when their subfield is so demanding. But that's OK - couldn't I just redefine my definition of "too big for a brain to encompass" to "too big for an ambitious brain to bother learning all of"?
[3] I don't think this would hurt us for some time. Our current known reserves of unapplied math should last centuries.
[4] Here's what wikipedia says are the 'subdisciplines' of geology: Economic geology, Mining geology, Petroleum geology, Engineering geology, Environmental geology, Geochemistry, Geological modelling, Geomorphology, Historical geology, Hydrogeology, Mineralogy, Paleontology, Petrology, Sedimentology, Stratigraphy, and Structural geology. Certainly no ambitous geologist is atop the current trends of all of these. (I have the advantage of knowing some.)
[5] Here's an ancient one that can be pretty well understood by one mind: music. I'm pretty sure one person could understand everything that's currently happening and everything that is known about previous music of humanity. (It's really interesting. You can start here: http://en.wikipedia.org/wiki/Musical_scale .) That's really weird. People have been innovating music forever and yet the field isn't too big to know? Why? Is our innate musical sense just that much better than our ability to comprehend math or science? Probably.
[6] You could compare the religions of the world based on how they see the advance of theological or spirtual knowledge over time. Christians would see it as a starting in Eden with miniscule knowledge, then a spike when Eve bites the apple, a long period of occasional increases by the odd prophet here and there until a big jump at Moses, and eventually a giant leap by Jesus, and finally a small increase with the writing of Revelations. After that Catholics and Orthodox would suppose continued small increased through the present day while Protestants would draw a flat line. Islam's line wouldn't look too different except with a giant leap around the year 800 followed by a flat line. I don't really know enough about other religions to say further, even by the loose standard I'm holding myself to here.
I'm in Mexico right now working around people in the mining industry and I just got an invitation from one of the Mexican geologists to join a Dropbox folder to share files about a mining project. Now, I've been using and telling people about Dropbox since they starting beta testing. Hearing about a startup from a source unrelated to the people building is always really cool and feels like a big milestone. That I'm far from Silicon Valley makes it all the more impressive. Plus these are corporate users who have told me they'd be perfectly happy to upgrade to a paying account when they need to. ¡Dropbox les gusta a todos aca!
I'm writing this in a store that sells laptops, on one of the tackiest computers I've ever seen. I think it's called a Gateway FX or something - imagine a Pontiac Trans Am with a touchpad. Yet I'm envious and irritated because it has something that I can't have in my wonderfully designed MacBook Pro: fantastic keyboard feel. My MBP has the mushiest, least responsive keyboard I've ever used, at least among those keyboards I've tolerated. It's crap. I'm going to be ordering up a nice IBM Model M keyboard to use while I've seated at a table, which will be great since that must account for, gosh, at least 5% of my usage. I wonder if I can get a laptop case that will hold a seven pound keyboard?
My sister and two of my best friends just joined Facebook after years of grouchy protest. Now they love it. Why wouldn't they? All their friends are already there - it's like a personal theme park that's been waiting for them. It's funny, because their early adopter buddies had a much harder time, yet talked the experience up.
So the sticks-in-the-mud have more fun. Is this true of other experiences? I was going to say that that's true of anything with a heavy network effect. But that's not true - I was the first of my circle of friends to get involved with the internet and it was a fantastic experience. The early Twitter had that feel, and blogging. CB radio is said to have been like that too. Why? Simple: that's where the cool kids hang out. Your friends may not be there, but the friends you wish you had are.
So, there you have it. If you want a new crew of hip friends, jump aboard the newest network. They're waiting for you there.
