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goldfishlaser comments on Open Thread: August 2009 - Less Wrong
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Mathematics
Its the foundation for everything else I want to learn. I don't know why I didn't major in it- other than concerns for a paycheck.
The nice thing about mathematics is that you can easily do it outside of school and independently, and when you do it as a mature adult, you do it because you love it and for no other reason. You are free to use better texts than you could as a student, so if you want to brush up on calculus, you can use Spivak or Courant rather than being forced to use low-quality texts that are optimized for the convenience of the professor rather than the insight of the student. There are also so many learning resources available now that weren't available when I was a student -- things like wikipedia, planet math, and physicsforums.com, not to mention software like Octave and Sage.
I'd like to add that one of the nice things about mathematics as a choice is that you can avoid credentialing issues. I'm sure we've all read Hanson on how most of the value of a college degree is in that the college is certifying your abilities, and stamping you.
If you chose world-class ability in economics/financial trading, say, and you are a poor student, then what are you going to do with it? You can't make a killing on the market to prove your abilities; you can't go work as an intern for a firm to prove your knowledge, etc.
Similarly with genetics. If I suddenly gained world-class genetic knowledge, I cannot walk up to Cold Spring Harbor and ask them to let me use some multi-million dollar equipment for a year because I have this awesome bit of research I'd like to do. I simply don't have any proof that I'm not a random bozo who has memorized a bunch of textbooks and papers. I'd have to get lucky and convince a professor or somebody to take me on as an assistant and slowly build up my credentials until I can do the bit of research that will irrefutably establish me as a leading luminary.
Or how about physics? If I specialize in experimental or practical physics, I have the same chicken-and-egg problem; if I specialize in theoretical, then I run the risk of simply being ignored, or written off as a crank (and the better my contribution, the more likely I am to be seen as a crank!).
But with mathematics, I can just crank out a bunch of theorems and send in a paper. If people are still unconvinced, being a mathematical genius, I can just formalize it and send in a Coq/Isabelle/Twelf file consisting solely of the proof.
While mathematics certainly appears to me to be more of a meritocracy than the sciences, it's still the case that the notion of proof has changed over time -- and continues to change (witness Coq and friends) --, as have standards of rigor and what counts as mathematics. There are social and other non-mathematical reasons that influence how and why some ideas are accepted while others are rejected only to be accepted later, and vice versa.
It's an interesting question whether this will always be the case or if it will converge on something approaching unanimously accepted truth and aesthetic criteria. Personally, I think mathematics is intrinsically an artistic endeavor and that the aesthetic aspect of it will never disappear. And where there is aesthetics, there is also politics and other sausage-making activities...
The gold standard of what is a proof and what is not was achieved with the first-order predicate calculus a century ago and has not changed since. Leibniz' dream has been realised in this area. However, no-one troubles to explicitly use the perfect language of mathematical proof and nothing else, except when the act of doing so is the point. It is enough to be able to speak it, and thereafter to use its idioms to the extent necessary to clearly communicate one's ideas.
On the other hand, what proofs or theorems mathematicians find important or interesting will always be changing.
I don't really think the question is whether mathematics is more meritocratic - it's an economic question of credentialing. You need credentialing when you cannot cheaply verify performance. If I had a personal LHC and wrote a paper based on its results, I don't think anyone would care too much about whether I have 2 PhDs or just a GED - the particle physicists would accept it. But of course, nobody has a personal LHC.
With mathematics, with formal machine-checkable proofs, the cost of verification is about as low as possible. How long does it take to load a Coq proof and check it? A second or two? Then all someone needs to do is take a look at my few premises; either the premises are dodgy (which should be obvious), or they're common & acceptable (in which case they know I'm a math genius), or I'm exploiting a Coq flaw (in which case I'm also a math genius). Once they rule out #1, I'm golden and can begin turning the genie's gift to good account.
By meritocracy, I meant what you explain by credentialing: the idea that the work alone is absolutely sufficient to establish itself as genius or crackpottery or obvious or uninteresting or whatever, that who you are, who you know, where you went to school and who your advisors were, which conferences you've presented at, the time and culture in which you find yourself, whether you're working in a trendy sub-discipline, etc., that all that is irrelevant.
How much of mathematics is machine-checkable now? My (possibly mistaken) understanding was that even the optimists didn't expect most of existing mathematics for decades at least. And how will we formalize the new branches of mathematics that have yet to be invented? They won't spring forth fully formed as Coq proofs. Instead, they'll be established person-to-person at the whiteboard, explained in coffee shops and over chinese food in between workshop sessions. And much, much later, somebody will formalize the radically revised descendant of the original proof, when the cutting edge has moved on.
I'll know you're right and I'm wrong if I ever begin to hear regular announcements of important new theorems being given in machine-checkable format by unaffiliated non-professionals and their being lauded quickly by the professionals. And that is the easier task, since it is the creation of new branches and the abstraction and merging of seemingly unrelated or only distantly related branches that is the heart of mathematics, and that seems even less likely to be able to be submitted to a theorem prover in the foreseeable future.
I'm not sure how one would measure that. The Metamath project claims over 8k proofs, starting with ZFC set theory. I would guess that has formalized quite a bit.
I think that only follows if genius outsiders really do need to break into mathematics. Most math is at the point where outsiders can't do Fields-level work without becoming in the process insiders. Consider Perelman with Poincare's conjecture - he sounds like an outsider, but if you look at his biography he was an insider (even just through his mother!).
This is what I thought everyone was going to say. I don't see why you'd be concerned about the paycheck though, a strong mathematics background could land you a job as a banker or trader or something. But looking at your upvotes it seems like plenty of people agree with you.
My next question would be what you'd like to have a basic introduction to. Plenty of LW posts tend to assume a grounding in subjects like maths, economics or philosophy - which is fine, this is a community for informed people - but it probably shrinks LW's audience somewhat, and certainly shrinks the pool of people who are able to understand all the posts. We probably miss this because nobody's going to jump into the middle of a thread and say, "I lack the education to understand this." espiecally not a casual reader.
My upvotes are probably due to the fact that I said mathematics, rather than any agreement concerning my potential lack of paycheck. I know that a mathematics background could supply me a paycheck at this point in my life, but I was urged against it by some other people when I was choosing my major.
Basic introduction to? Is this in addition to the expertise I got from the genie, or if the genie was only offering me a basic introduction? Do I only get to choose one? Gee that's hard. I'm pretty much working on having a basic introduction to everything already. So, given my existing basic introductions... I think I'd like to get a basic introduction to quantum physics... but that's kind of cheating because I'd have to know a lot of physics and mathematics in the basic intro. I choose that one because I want to know it for purely vain reasons, and it would be nice to save the time of learning it for more "useful" studies.
This blog definitely is going to appeal to a minority of people. I personally do not have the proper education to follow the bayesian/frequentist debate, though I want to hear about it. I think that the healthy practice of linking to information is fantastic, as well as the lesswrong wiki. That way if you know what the person is talking about, you don't have to follow the link, but if you need to learn, it's right there at your fingertips.
Edit: Oh right, and I can't contribute to quantum physics discussions very well either.