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guslThe other day, I wrote the following on my PDA:
Why I am no longer a mathematician:
· Tired of working hard just to be clever. Life is short. The real world is more interesting.
· Phenomenology, introspection drove me towards cogsci.
· it's more productive to do meta work: computers will eventually do math much more cheaply than me. (see Zeilberger)
----
Here's something of an academic autobiography, of my time at Bucknell. It says nothing about my ideas, or what I read. I tell the story of how undergraduate curricula shaped my choice of majors:
Mathematics
The last time I did serious mathematical research was my junior year of college... and even that was very much empirically-aided: it was about counting the number of roots of polynomials over finite fields... my discoveries were made with the aid of a C++ compiler.
Since then, I have proven things about cute games (Nim, thanks to![[livejournal.com profile]](https://www.dreamwidth.org/img/external/lj-userinfo.gif)
agnosticessence), toy theorems (prove that number_of_divisors_of(n) is always even except for when n is a perfect square), and created neat correspondences (e.g. if you represent natural numbers as multisets, GCD is intersection, LCM is union), but nothing that could count as serious mathematics.
Already my senior year, in topology class, I no longer saw the point of doing pure math. The only way I could interpret infinite products of topological spaces was as a game of symbols: it had no real meaning to me.
Not only was I starting to get a formalistic view of mathematics, but I was increasingly bothered by the normal approach to mathematics, the standard mathematical language and the paper medium. This was made much worse by the fact that I had grown intolerant of confusing notation/language and informal proofs. Thankfully, I didn't stay in mathematics. Advanced mathematics requires a lot of effort and things are not always beautiful. The real world has many more interesting things to understand. During this time, I considered going for a PhD in Applied Math, but became disappointed with that idea too. It was still too much like other math.
By my senior year, mathematics was no longer fun. Still not "hard", but I no had motivation left. I had become enthusiastic about statistical modelling... even if I got labelled a Bayesian by our frequentistics department (I think it was meant as a compliment). And it was my interest in AI, by far, that dominated my intellect.
Physics
The reason I had liked mathematics before that was that it had been, for me, easy and fun. And its formal structures were much more satisfactory and easier for me to understand than the things people did in physics, my original major. My physics teachers never seemed to explain things clearly, and never gave me good logical reasons for why they were doing what they were doing. It was often unclear which model and assumptions were being used. And even after pressing them, I still had foundational questions that went unanswered. Quantum Mechanics class was extremely frustrating: while "nobody understands quantum mechanics", the theory still has a reason to be, but they didn't give us a chance to try to make sense of the experimental results that motivated the theory, or convince me that the theory was the best we could do.
Although I started out with bad grades in physics, they were steadily improving. Still, my professors saw promise in me, and wanted me to stay. Despite liking and doing well on my last class on Thermodynamics & Statistical Mechanics, I decided that I was going to focus on math: I was just too different from the physicists, and talking to them took too much effort. Now I want Patrick Suppes to be my next physics teacher. Among the physicists, I was definitely a philosopher.
Computer Science
I had to overcome my initial prejudice against CS. I only started it because of my father's argument that it would be a good idea if I wanted to make money. As a freshman, I had thought that it was just going to be about programming techniques, and similar boring-sounding things. The sort of person who did CS at my school was not far from the "typical management major": financially ambitious, if not particularly mathematically-talented. When I joined the group, I learned that there were exceptions... so now, I realized that there were also "computer geeks", as well as the former type. I was never a "computer geek". Programming geek, yes, for a long time... but one who couldn't get Linux installed, and who would call a technician to troubleshoot my network. Among them, I was solidly seen as a math geek. It bothered me that their AI class assumed neither knowledge of basic probability or basic logic, and that the computer graphics class couldn't do a simple linear projection.
But I really liked ProgLan. Also, designing algorithms was fun. Algorithmic reductions even more. And I learned some useful programming techniques.
Philosophy
I've always been a philosopher. But I did not like the prospect of reading shelffuls of philosophy books, learning the ins and outs of useless arguments (for instance, about metaphysics), and rereading & struggling to understand what exactly writers mean. Philosophy is great for breaking people out of their epistemological vices: questioning their prejudices, intuitions, etc., but some things are just overanalyzed. I think this is because they talk past each other. Case in point: the Monty Hall problem. Why are they still writing papers about it?? I think that philosophers should benefit the most from computational aids to reasoning, argumentation maps and such. At least, they already know logic.
Psychology
It was fascinating. But it wasn't rigorous enough for me. If they had offered cognitive science, I probably would have taken lots of it.
Economics & Linguistics
I also flirted with economics, although never for credit. It was interesting, but they were too slow on the math. Like CS, only worse. I also took a class in linguistics (the only one offered!), but as I wasn't about to start doing NLP, it remained a mere curiosity.
Why I am no longer a mathematician:
· Tired of working hard just to be clever. Life is short. The real world is more interesting.
· Phenomenology, introspection drove me towards cogsci.
· it's more productive to do meta work: computers will eventually do math much more cheaply than me. (see Zeilberger)
----
Here's something of an academic autobiography, of my time at Bucknell. It says nothing about my ideas, or what I read. I tell the story of how undergraduate curricula shaped my choice of majors:
Mathematics
The last time I did serious mathematical research was my junior year of college... and even that was very much empirically-aided: it was about counting the number of roots of polynomials over finite fields... my discoveries were made with the aid of a C++ compiler.
Since then, I have proven things about cute games (Nim, thanks to
agnosticessence), toy theorems (prove that number_of_divisors_of(n) is always even except for when n is a perfect square), and created neat correspondences (e.g. if you represent natural numbers as multisets, GCD is intersection, LCM is union), but nothing that could count as serious mathematics.Already my senior year, in topology class, I no longer saw the point of doing pure math. The only way I could interpret infinite products of topological spaces was as a game of symbols: it had no real meaning to me.
Not only was I starting to get a formalistic view of mathematics, but I was increasingly bothered by the normal approach to mathematics, the standard mathematical language and the paper medium. This was made much worse by the fact that I had grown intolerant of confusing notation/language and informal proofs. Thankfully, I didn't stay in mathematics. Advanced mathematics requires a lot of effort and things are not always beautiful. The real world has many more interesting things to understand. During this time, I considered going for a PhD in Applied Math, but became disappointed with that idea too. It was still too much like other math.
By my senior year, mathematics was no longer fun. Still not "hard", but I no had motivation left. I had become enthusiastic about statistical modelling... even if I got labelled a Bayesian by our frequentistics department (I think it was meant as a compliment). And it was my interest in AI, by far, that dominated my intellect.
Physics
The reason I had liked mathematics before that was that it had been, for me, easy and fun. And its formal structures were much more satisfactory and easier for me to understand than the things people did in physics, my original major. My physics teachers never seemed to explain things clearly, and never gave me good logical reasons for why they were doing what they were doing. It was often unclear which model and assumptions were being used. And even after pressing them, I still had foundational questions that went unanswered. Quantum Mechanics class was extremely frustrating: while "nobody understands quantum mechanics", the theory still has a reason to be, but they didn't give us a chance to try to make sense of the experimental results that motivated the theory, or convince me that the theory was the best we could do.
Although I started out with bad grades in physics, they were steadily improving. Still, my professors saw promise in me, and wanted me to stay. Despite liking and doing well on my last class on Thermodynamics & Statistical Mechanics, I decided that I was going to focus on math: I was just too different from the physicists, and talking to them took too much effort. Now I want Patrick Suppes to be my next physics teacher. Among the physicists, I was definitely a philosopher.
Computer Science
I had to overcome my initial prejudice against CS. I only started it because of my father's argument that it would be a good idea if I wanted to make money. As a freshman, I had thought that it was just going to be about programming techniques, and similar boring-sounding things. The sort of person who did CS at my school was not far from the "typical management major": financially ambitious, if not particularly mathematically-talented. When I joined the group, I learned that there were exceptions... so now, I realized that there were also "computer geeks", as well as the former type. I was never a "computer geek". Programming geek, yes, for a long time... but one who couldn't get Linux installed, and who would call a technician to troubleshoot my network. Among them, I was solidly seen as a math geek. It bothered me that their AI class assumed neither knowledge of basic probability or basic logic, and that the computer graphics class couldn't do a simple linear projection.
But I really liked ProgLan. Also, designing algorithms was fun. Algorithmic reductions even more. And I learned some useful programming techniques.
Philosophy
I've always been a philosopher. But I did not like the prospect of reading shelffuls of philosophy books, learning the ins and outs of useless arguments (for instance, about metaphysics), and rereading & struggling to understand what exactly writers mean. Philosophy is great for breaking people out of their epistemological vices: questioning their prejudices, intuitions, etc., but some things are just overanalyzed. I think this is because they talk past each other. Case in point: the Monty Hall problem. Why are they still writing papers about it?? I think that philosophers should benefit the most from computational aids to reasoning, argumentation maps and such. At least, they already know logic.
Psychology
It was fascinating. But it wasn't rigorous enough for me. If they had offered cognitive science, I probably would have taken lots of it.
Economics & Linguistics
I also flirted with economics, although never for credit. It was interesting, but they were too slow on the math. Like CS, only worse. I also took a class in linguistics (the only one offered!), but as I wasn't about to start doing NLP, it remained a mere curiosity.
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(no subject)
Date: 2005-09-12 05:33 pm (UTC)"A year spent in artificial intelligence is enough to make one believe in God"
- Perlis
I guess just expressing my doubts that computers will ever be able to develop new mathematics effectively as humans.
prove that Euler-Phi(n) is always even except for when n is a perfect square
Um, Phi(n) is even for all n except 2 (which isn't a square).
(no subject)
Date: 2005-09-12 06:04 pm (UTC)Well, I believe that AI is coming, possibly in our lifetimes.
(no subject)
Date: 2005-09-12 06:07 pm (UTC)(no subject)
Date: 2005-09-13 01:07 am (UTC)(no subject)
Date: 2005-09-13 06:27 am (UTC)Mathematics, yup... I avoid it for the same reason. Really fun and really cool, but I just feel like it's not worth spending all that time just to be clever.
Philosophy, yup. I hate reading history, I want to learn about the ideas without caring that much who had them or in what order. This is why I do a lot of reading of philsophy encylopedias and such. Actually, I just bought the Cambridge dictionary of Philosophy yesterday and I've been really enjoying it. An entry for all the major schools of thought, and entry for each jargon term that's often used, and an entry for each major philosopher and what their views were. Pretty cool way to learn, I think. Better than reading huge volumes of pages by each person.
CS, yup. Psychology yup. Economics, yup.
Physics, I had a different experience with. Yes, a lot of the teachers were unclear. But I think what happened in my case was that I read enough of the history of quantum mechanics and other things before taking the classes that I already had a solid foundational understanding of why we were doing it. In particular, one of the best books I read in high school was Quantum Mechanics and Experience by David Z. Albert. He's a philosopher who has a degree in physics, and he explains the issues pretty well without using any equations. Once I got into the classes, it was mostly just about learning the math behind it. I've always sort of taken the approach that whatever isn't covered in class I go learn on my own. And it's worked very well for me. Fortunately, I've had some really incredible teachers in grad school (along with some bad ones, but extremely good ones for the most important classes). So this has helped continue my appreciation for physics. There may be a difference in how we approach it, though, because I don't usually care for it to be any more formal... I feel like it's good enough to use without ambiguity and that's what counts. But there could be new things we learn by formalizing it further, so I suppose it couldn't hurt.
Other areas I sometimes wish I had gone into are AI, neuroscience, and philosophy (of language, mind, mathematics, or science). I also wish I understood certain parts of mathematics a lot better like logic, godel's incompleteness theorem, category theory, or anything relating to the foundations of mathematics.
(no subject)
Date: 2005-09-13 07:53 am (UTC)(no subject)
Date: 2005-09-18 12:40 am (UTC)Not that I know much, but I find studying the foundations of mathematics very unsatisfying. It doesn't really answer any interesting philosophical questions, and makes me even more of a coherentist: there are no ultimate foundations, and in fact no foundational language is more fundamental than the others. It's all just about being coherent.
Modeling mathematical cognition seems much more interesting... e.g. modeling (and automating) mathematical reasoning, based on intuitions, diagrams, etc. Foundationalists can explain why "math is right" (or rather, consistent with other math), but the interesting question for me is "how is math done?".
For Goedel's incompleteness stuff, I would suggest you look at uncomputability first. They are the same thing afterall, and computational problems feel much more concrete than the Platonic mathematical world. Uncomputability is also related to problems of induction in the philosophy of science (see Kevin Kelly)... and these parallels are much more intuitive than provability in Peano arithmetic.
(no subject)
Date: 2005-09-18 09:47 pm (UTC)Of note: I had the please of studying CS at Carnegie Mellon, where the people who can't do at least competent math either aren't accepted or get weeded out fairly quickly. (Or I should say that's the way things were a few years ago.) I think partly I also gravitated towards topics that math-incompetent people tended to shy away from. (I think one of the biggest turnoffs for AI for me was just that most of the students in the AI class I took seemed dumber than the average students in many of my other CS classes.)