¿Quién puede nombrar el mayor número?
Jorge Alonso has kindly translated my essay “Who Can Name The Bigger Number?” into Spanish. You can read his translation here, or my original English version here.
Even though I wrote this piece seven years ago, as an undergraduate at Cornell, I still get more mail about it than about anything else I’ve written. Which depresses me, because I know I couldn’t write it today. I’d be too embarrassed to trot out the Ackermann and Busy Beaver numbers as if they were the awesomest things ever. This is standard, decades-old material! Doesn’t everyone know it by now?
They don’t, of course. But the price I paid for learning enough to do science is that I can no longer work up childlike wonder over, say, humankind’s inability to pin down BB(7). The things that are new to me are too hard to explain to a popular audience, and the things that are easy to explain are no longer new to me. How I long for the power to return, at will, to my intellectual adolescence.
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Comment #1 March 14th, 2006 at 8:27 am
i’ve encountered something slightly similar, whenever i find out about something i hate not being able to explain it to others. this is in the area of machine learning.
for instance: want to know the dirty secret of machine learning? a lot of it is just statistics and (especially) nonlinear optimization in drag, and the vague biological analogies that inspired neural nets are useless in practice. also, practical machine learning techniques are usually driven by heuristics and practical experience — educated guesses and voodoo.
Comment #2 March 14th, 2006 at 8:36 am
also, practical machine learning techniques are usually driven by heuristics and practical experience — educated guesses and voodoo.
I’m shocked — shocked!
Comment #3 March 14th, 2006 at 10:27 am
I hear you. I’m at the APS meeting in Baltimore right now. Six or eight years ago, I would be sitting in the front row of a session in each timeslot from 8am to 8pm, asking questions in every talk I attended. Now, I slink into the back row, and just as often slink out before the talks are done. In fact, I’m four blocks from the convention center as I write this right now.
Posting anonymously so my employer doesn’t find me…
Comment #4 March 14th, 2006 at 10:28 am
A lot of why Erdos is so popular is because his work was so approachable. A lot of it wasn’t, but he’s commendable for sticking with understandable conjectures.
Martin Gardner, the most popular recreational math writer of all time, could clearly not hack it as a mathematician. In some of his columns he conjectures things which it’s hard to see how he couldn’t realize how to prove them. It didn’t hurt his popularity though, in fact it probably helped, because it made his columns a lot more accessible. Recreational mathematics columns written by mathematicians tend to contain a lot of magic tricks.
Anonymous: other parts of AI are frequently even worse, with the best algorithms being simplistic things which were hand-tweaked by a human expert. No machine intelligence of any kind involved.
Comment #5 March 14th, 2006 at 10:31 am
Something about the gestalt of mathematics has changed a lot in the last hundred years. One gets the feeling that around 1900, or even 1950, you could sit down with a pencil and paper and start dreaming stuff up and be doing original mathematics. Now it seems like the bones have been picked clean, and any simple idea you could come up with has not only been thought of before, but been worked on a lot by real mathematicians, who have written very sophisticated published papers about their results.
Software is considerably more exciting, in that you can have an idea, slap together a web site, and in short order be providing a service which has never before been made available on planet earth.
Comment #6 March 14th, 2006 at 11:14 am
Something about the gestalt of mathematics has changed a lot in the last hundred years. One gets the feeling that around 1900, or even 1950, you could sit down with a pencil and paper and start dreaming stuff up and be doing original mathematics…
I tend to doubt that this was true of mathematics in 1900, but maybe it was true in the 1700s.
In any case, I would (almost) say the same about theoretical computer science in the 1950s-1980s vs. today, except that every once in a while one still sees some simple-yet-elegant results.
Comment #7 March 14th, 2006 at 11:26 am
(Bram) One gets the feeling that around 1900, or even 1950, you could sit down with a pencil and paper and start dreaming stuff up and be doing original mathematics…
(Anon. 11:14) I tend to doubt that this was true of mathematics in 1900, but maybe it was true in the 1700s.
I suppose whch year one pins on this depends on whether you think the most beautiful part of mathematics is number theory, group theory, real analysis, formal logic, etc — and correlates with whichever mathematician of yore you may secretly wish to have been.
The answer, I suppose, is to try to found the next interesting branch of mathematics and harvest the prettiest theorems, so that in your turn you may also frustrate mathematicians of future generations.
Comment #8 March 14th, 2006 at 12:24 pm
“The things that are new to me are too hard to explain to a popular audience….”
Hmm, why do I hear the voice of Marvin the Robot from Hitchhiker’s Guide: “Here I am, brain the size of a small planet….” ;-o
Don’t know whether you are disappointed purely at not being able to communicate the wonder you do feel, or partly because the sense of wonder has diminished. If the latter, Dave Baron recently posted about something similar – I imagine you saw it.
Comment #9 March 14th, 2006 at 1:28 pm
Bram:
Now it seems like the bones have been picked clean, and any simple idea you could come up with has not only been thought of before, but been worked on a lot by real mathematicians, who have written very sophisticated published papers about their results.
To expand on what Niel said: yes, the bones in plain sight have been picked pretty clean. But that was probably how things looked to most people in 1900 and 1950 as well. So we keep on picking, hoping that we’ll eventually find a forkful of meat or even an entire leg that hasn’t yet been touched. One reason for optimism is that the chicken is infinite.
Comment #10 March 14th, 2006 at 3:07 pm
There’s no reason why sustainable development is incompatible with exponential growth … as long as the availability of resources grows at an Ackermannic rate.
BTW, the energy necessary to split a mere six billion tons of carbon dioxide hits the Earth in just a few minutes.
Comment #11 March 15th, 2006 at 10:42 am
I would think that Aggrawal, Kayal, Saxena’s “PRIMES is in P” paper on its own would be a good antidote to worries that the bones have been picked clean.
Comment #12 March 15th, 2006 at 2:31 pm
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Comment #13 March 15th, 2006 at 2:39 pm
The feeling you described has I think, some interesting relation, with a comment I posted on
Monday, February 20, 2006 12:01:30 AM to your post on The Fable of Chessmaster, that indicates why scientists need to start learning to profess or atleast delegate the job cleverly.
Comment #14 March 15th, 2006 at 6:47 pm
“ One reason for optimism is that the chicken is infinite ”
The forks, however, have to start digging from a similar spot.
Comment #15 March 16th, 2006 at 1:11 pm
I submitted a link to the number article to metafilter, and people there loved it.
Comment #16 March 17th, 2006 at 6:28 pm
In response to everyone’s comments, the mathematician I wish I could have been is Erdos (not that I wish I could have lived his life, I just wish I could have come up with some of his results). This puts my birth regret squarely in the post-1900 range.
I didn’t really mean that mathematics as a whole has been picked clean, but that the sort of simple mathematics which one can explain to anyone in high school feels quite played out. Perhaps this will all seem ridiculous in hindsight though. It certainly is the case that theoretical CS has emerged as a field of innumerable readily understandable conjectures which everyone knows to be true but nobody can prove, and where steady progress is in fact being made, without having to use opaque technical pyrotechnics.
I think there really has been a shift in the approachability of mathematics proper though. Among hilbert’s problems I can easily understand most of them. Of the millenium prize problems I can understand two of them, and pretend I understand a third, but the remaining majority are totally arcane.
CS is clearly the new frontier (and software is even more approachable, at least from my perspective, because I’ve made real contributions there), and there may be some new insight which involves, for example, simplifying the techniques used to prove fermat’s last theorem enough that high school students can use them to prove original theorem’s, but I for one don’t see one on the horizon.
So all of this leads to the question: What parts of mathematics do people feel have the property that the cutting edge research is understandable by a typical gifted high school student, both in terms of the theorems proved and the proofs themselves?
Comment #17 March 18th, 2006 at 1:08 am
What parts of mathematics do people feel have the property that the cutting edge research is understandable by a typical gifted high school student, both in terms of the theorems proved and the proofs themselves?
I like this question. I’ve always gravitated toward areas where the theorems and open problems — but not necessarily the proofs — can be understood by a high-school student. Indeed, I can get myself interested in a highfalutin theory if, and only if, I can see what HSSU (High-School-Student Understandable) problems it helps me answer. That’s why, for example, I’ve tried to study approximation theory, representation theory, and Galois theory, but never category theory or sheaf theory.
I share your assessment that theoretical computer science is the richest source of HSSU problems today — that’s why I work on it!
Combinatorics, discrete and combinatorial geometry, number theory (of course), stochastic processes, quantum information, and knot theory all come to mind as good sources of HSSU problems as well. But I suspect one could find them in many (or even most) areas of math if one knew how to look for them.
Comment #18 March 18th, 2006 at 1:22 am
As for areas where the proofs are HSSU, that’s harder — since even if you ask an HSSU question, you don’t know in advance whether it’ll have an HSSU answer. At this point it largely depends on how smart the HSS is, and how much time he or she can spend learning real stuff without flunking the courses he or she needs to get into college.
Comment #19 March 19th, 2006 at 11:07 am
Scott, you forgot to mention the Collatz conjecture. We’ll always have that one 🙂
Comment #20 December 21st, 2006 at 7:57 pm
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