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Thursday, January 12, 2006

Mathematical seed bank

The Norwegians are building a seed bank to safeguard the world's crops in case of various catastrophes - nuclear war, climatic chaos, pandemic, etc. Rather than wasting all that effort of farmers over the millennia, two million carefully stored varieties of crop seed will allow future survivors the opportunity to reconstruct agriculture by recovering the hoard from Spitzbergen. Reading about this put me in mind of Michael Polanyi's warning:
The transmission of mathematics has today been rendered more precarious than ever by the fact that no single mathematician can fully understand any longer more than a tiny fraction of mathematics. Modern mathematics can be kept alive only by a large number of mathematicians cultivating different parts of the same system of values: a community which can be kept coherent only by the passionate vigilance of universities, journals and meetings, fostering these values and imposing the same respect for them on all mathematicians. Such a far-flung structure is highly vulnerable and, once broken, impossible to restore. Its ruins would bury modern mathematics in an oblivion more complete and lasting than that which enveloped Greek mathematics twenty-two centuries ago. (Personal Knowledge, 1958: 192-3)
As we now know, 'oblivion' is rather too strong a word to use for the fate of Greek mathematics. Although there were some shaky moments in the passing on of the baton, a continuous line can be traced. "Medieval Europe learned a lot of Greek science by reading Latin translations of Arab translations of Syriac translations of second-hand copies of the original Greek texts!" was John Baez's summary.

But what if we had a cultural meltdown? Polanyi was worrying here about a less catastrophic onset of an intellectual Dark Ages, but whatever the cause of the breakdown, perhaps we should start planning for the storage of our intellectual products. Storage on hitech devices would clearly be risky, but presumably acid-free paper would be safe enough, although the Babylonian clay tablet has its advantages. But more to the point, what should we store of our mathematics? Would you deposit works in the form of Bourbaki's Elements? Surely what would be more useful would be crates full of commentary, informal exposition, and the history of conceptual development. This is precisely what would be suggested by taking mathematics as a tradition of enquiry, and as such a socially-embodied argument seeking to further human understanding. Perhaps different seed banks would have to be set up to reflect different views of the best ways of organising mathematical priorities.

Someone I imagine who would have a distinctive idea about how to do this is Doron Zeilberger. I met Doron at the 'Mathematics and Narrative' conference in Mykonos last summer. He's a thoroughly likable mathematician, based at Rutgers, and famous for his opinions. I agree with the spirit of many of them, but feel most distance between us when he takes what humans have achieved to date as being of a trivial level of complexity compared to what computers will be doing in the future (e.g., opinion 69). Certainly, humans with computers are capable of more than humans alone. But I can't agree that, catastrophe permitting, "In fifty years (at most) human mathematicians will be like lamp-lighters and ice-delivery men. All serious math will be done by computers." No, unless their activity is embedded within the ongoing histories written by the mathematical community, it is not mathematics.

Doron will no doubt think this a glorification of the weakness of our minds, but I end with a quotation, copied from Alissa Cran's webpage:
Many people who have never had occasion to learn what mathematics is confuse it with arithmetic and consider it a dry and arid science. In actual fact it is the science which demands the utmost imagination. One of the foremost mathematicians of our century says very justly that it is impossible to be a mathematician without also being a poet in spirit... It seems to me that the poet must see what others do not see, must see more deeply than other people. And the mathematician must do the same.-Sofya Kovalevskaya, 1890


Doron Zeilberger said...

Let's hope that David is right, and that human mathematics will survive. But it would be a completely different, very high-level "meta-math". 99% of the articles written today
would be done faster and better by computers, but it is possible that humans will find a niche in what they are good at, namely philosophizing.
Then people like David Corfield (whose book I read twice cover-to-cover) would be main-stream mathematicians.
Think what mathematics is all about rather than do it.

January 16, 2006 7:12 PM  
david said...

I can see large amounts of the kind of work which is really just calculating within a fixed system becoming largely automated. But how about the activity Grothendieck described as "bringing new concepts out of the dark"? Is there any sign that computers could devise the notion of a scheme or a motive or a quantum group? Or, do you think that this is merely human style mathematics which will become swamped by more powerful computer activity?

January 17, 2006 1:52 PM  
John Baez said...

David Corfield asks:

"But how about the activity Grothendieck described as "bringing new concepts out of the dark"? Is there any sign that computers could devise the notion of a scheme or a motive or a quantum group?"

There's clearly no sign of it happening now, and bringing new concepts out of the dark is the most interesting form of mathematics to me, so I feel utterly unthreatened by competition from computers at present. Right now computers are mainly just wonderful tools for doing stuff that you tell them precisely how to do... which is great when you want to do something like check an identity involving binomial coefficients, or generate data whose mysterious patterns suggest interesting conjectures.

I don't see any a priori reason why computers couldn't eventually get good mathematics, or music, or other forms of art... and if they do, then they might get a lot better than us - for example by being faster, or being able to handle of data structures of greater complexity.

For example, I can tell that the levels of abstraction we've reached in category theory are only the tiny foothills of some very tall mountains. It's scary to imagine what we might think of if our minds didn't collapse under the strain of abstraction so easily.

But, James Dolan has pointed out - and I agree - that some of the very best mathematics arises from our limitations, especially our limited memory. Our inability to remember heaps of small facts pushes us to subsume them under big generalizations. People whose memory is too good tend not to do well in math: they get stuck in piles of detail instead of reaching for simplicity. It's possible that computer mathematicians will merely produce huge piles of boring crap unless we constantly force them to summarize and compress.

So, it will be interesting to see what happens as computers get better at math... I don't think we can predict this.

January 28, 2006 12:18 AM  
mathpoet said...

An amazing quote at the end of that post. Thank you for sharing.

February 01, 2006 2:23 PM  

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