The technique really is interesting. We have techniques that can identify patterns that are meaningful (all of cryptology, most of number theory, graph theory) but this application is neat because it is an effort to prove--rigorously--that a given set of data is just total noise.
WTF? Its that sort of freaky jedi-mind-trick that I think RUINS family gatherings. I mean really. Not only do you have to actually SEE your family, but they insist on using the occassion for charmlessly didactic "life lessons." Truly a grinchish tactic, my friend.
I'm sort of in the position you describe, or at least I was until my mid-twenties when I finally entered University to study mathematics and English literature. I learned nothing about mathematics in high school, and I had to start over--now I'm going into my senior year and may write an honors thesis. I have a couple of guidelines and then a list of suggestions.
Guidelines:
1. If you really want to understand mathematics, stay away from suggestions made my engineers; in particular, eschew books that dumb down mathematical theory in favor of the 'this is how you compute the solution' approach. Silvanus Thompson I find to be especially egrigious in this regard (those who try to learn calculus from Thompson will never understand the rigorous notion of a Limit, which is hardly pedantic since the derivative is itself a limit and the Riemann integral is the limit of a Riemann sum).
2. Be patiant with yourself. Geometry, Analysis (which includes what is called calculus) and Algebra have required centuries of constant effort to develop. If you go for the 'fast and cheap' approach to learning it, you will aquire nothing more than skills, when what you really want is knowledge.
Books:
Preliminary topics: Before you can think, you must memorize certain things and learn other things by rote. This will be hard and painful, but these fundamental topics are to mathematics as the alphabet and grammar is to Shakespeare, Milton, and Joyce. They are: the notion of a function, the laws of exponents, elementary trigonometry (sine, cosine, tangent, and their inverses), the binomial theorem, the definition of a polynomial, factoring polynomials, setting up applied problems in algebra, linear equations and their graphs, simple nonlinear equations and their graphs, slope and area, the Pythagorean Theorem. Most of these basic noitions are covered in Forgotten Algebra (which is published by Barrons for people just like us, and College Algebra, by Michael Sullivan.
Fundamental Notions:
By fundamental notions I mean ideas that form the basis for other ideas. Mathematics is all about definitions, and definitions are all about ideas; you cannot learn complicated ideas without understanding basic ideas (if you don't believe me, try explaining why every vector space has a basis to someone who doesn't understand what linear independence is). Unlike preliminary topics, fundamental notions are actually fun to learn--you get to think instead of just memorize and drill! I know of one wonderful book for this sort of thing, for someone in your position:
1. A Tour of the Calculus, by David Berlinski. This will make you think about what 'continuity' is. Good preparation for calculus, which is all about continuous functions, and good because it presents mathematics as a branch of philosophy (which it is).
Single Variable Calculus
Single variable calculus is where you will find most of the major concepts in the subject; the next time you will think this much is in linear algebra, when you study why the derivative for a n-dimensional vector space is actually representable in terms of matrix multiplication (the derivative is a linear map.) Here are some good books on calculus:
1. Calculus, Thomas and Finney. This text features a superb fusion of theory and application. The exercises are challenging, but doable for an independent student, and solution guides are available (these are indispensable as you search, at 2AM, for the mistake in your integration by partial fractions problem that required nine pages and is off by a constant).
2. Calculus, by Michael Spivak. My favorite calculus book. A brilliant synthesis of upper division real analysis and run-of-the-mill calculus. Reading it is like feeling awestruck by the beauty of someone you have known for years and years. This also has a solution manual (which you will need, because here there are proofs).
This is a real canard. On iTunes, classical music--where the tracks are often 17 minutes long--can frequently be purchased ONLY as a complete album. Metallica could have, I suspect, made the same deal. So I don't buy the "its about preserving the album" rationale. Metallica has obstructed every attempt to offer online music services since the technology's inception--this is merely the latest maifestation of their total commitment to derail ANY download based distribution model.
The dismissal says "Now Therefore, without admitting or denying liability and soley in order to save the time and expense of litigation, the Parties have determined to settle the Action in its entirety andf dismiss the complaint with prejudice..."
IANAL, but it looks to my untrained eye like he IS, in fact, in violation of the dismissal. He's been running around television, his web site, etc. vigorously DENYING liability. The text SPECIFICALLY says this is not the case. I have no idea if this is illegal, etc. or if he could be cited for contempt, but I see the point those RIAA lawyers are trying to make. His actions and statements do seem to stretch, if not break, the spirit and letter of the agreement.
I've thought about this, and I think their strategy is actually sound. Thiink about this: as of April 5 2002, USA Today reported that "About 17% of adults wired to the Internet at home, work or school say they've downloaded music, according to a new CNN/USA TODAY/Gallup Poll." That number, given wwhat I know of estimates about US internet usage, is in the tens of millions. So I agree: the RIAA can't sue them all.
But they don't have to. They can just sue ultrapeers. The point of the lawsuits is to cripple the gnutella networks in the USA. Most KAZAA users I know upload almost nothing--the lawsuits won't go near them. The RIAA is interested in the heavy hitters--folks who enable the gnutella system because of their high bandwidth, and double-digit gigabyte shared folders.
Salon.com loves Ellen Ullman almost as much as I do. Read excerpts from The Bug here:
http://archive.salon.com/books/int/2003/05/16/ullm an/index_np.html
You can read articles by Ullman here:
http://archive.salon.com/directory/topics/ellen_ul lman/
Salon is free as long as you watch a little commercial (C'mon--its 10 secondds, and then you get to read Ulllman--for free!!!)
I love this service but ach, it is going to be a nasty business to be in. Apple has entered center stage with a model based on.99 cent downloads. Way cool! But the most obvious way for someone to compete with them is to offer a download service at a lower price--so expect someone to do that shortly. Apple's margins haven't been published, but I'd guess they're razor thin to begin with. Now.99 cents is already so low that there isn't much further to fall--if a price war ensues, it won't be long before corporations are running online music services as a loss-leader. In Apple's case, it promotes their hardware; other companies will have other ideas. Eventually, these services might very well lead back to where we started--corporate sponsored music-on-demand, with free content that is used as a tool to peddle something else. Look at the way Apple chases these labels--this is brand-name association for them. Think Different, Think Nirvana, etc. It entrenches Apple deep into the popular culture. So those who want free music--it may be coming sooner than you think.
Re:A serious question - i'm not trolling, honest!
on
Twin Prime Proof Erroneous
·
· Score: 3, Informative
Number theorists have proven that there exists no polynomial function f(x) such that f(x)={primes}. There is, however, a vast literature concerned with the distribution of primes. For instance:
Prime Number Theorem: "the prime number theorem gives an asymptotic form for the prime counting function pi(N), which counts the number of primes less than some integer n."
Bertrand's Postulate: If n > 3, there is always at least one prime p such that n is less than p which is less than 2n-2.
Wilson's Theorem: "if and only if p is a prime, then (p-1)! +1 is a multiple of p, that is (p-1)! congruent to -1 (mod p)." (quotations from mathworld.com)
Theorems such as these provide insight into the distribution of primes throughout the natural numbers. The Twin Prime Conjecture, if resolved, would provide additional insight into this distribution, which would be of fundamental theoretical and practical importance. For instance, it is currently regarded as hopelessly time consuming to factor large composites--public key cryptography is based on this fact. But I am not aware of any proof that factoring such numbers *must* take a long time--that they do is an interesting state of affairs, but it might not reflect the nature of the universe so much as our lack of knowledge about prime numbers. Solving the TPC would be a step in remediating that deficiency.
I liked Pattern Recognition, and found out a week ago that William Gibson has a web site; there's a (good) blog and a discussion list.
http://www.williamgibsonbooks.com/index.asp
People object to the Matrix being perceived as a philosophical film in very much the same way that Lord of the Rings is disputed as a work of literature. In 2001, the London Review of Books said that LOTR that "there's a whole little world in there, simplified and protected, like in the role-playing games to come. It is its own university, its own library, its own structure of branching knowledge. To enter it is to become a simulacrum student in a crucible of simulacrum knowledge. A simulacrum student at the simulacrum university of simulacrum life." (its at http://www.lrb.co.uk/v23/n22/turn03_.html). The Matrix fits that description even better than LOTR. Does that mean it's "junk food," as claimed by the prior post? I have no doubt that far more people willl see the Matrix movies and read LOTR than will ever struggle with the Meditations or Beowulf. But who cares? Why does the valuable lesson about philosophy/literature have to be written by somebody you'll never meet, in a language you'll never learn? To me, the vitality of those subjects is in their ability to make you grasp frantically for understanding of *your existence*--not to comprehend what some guy meant in Europe 400 years ago. Rigor is important, but so is inspiration.
I am enormously skeptical that MS will actually generate something worthwhile here. Why? I bought a new wintel laptop about 1.5 years ago for the express purpose of digitizing as much of my life as possible--this project included scanning in all personel records, setting up a digital camera, and burning every CD I owned and listening to them *exclusively* via the computer itself.
Now there are various music clients available for winXP but, in that way the these things tend to happen, I gradually found that it was easier to just tolerate the MS music client rather than working around its interuptions with another piece of software. To make a long story short, this was a disaster. The MS software was ok for ripping and playback, but file organization was a big fat mess--I could edit files using the player but then the actual files on the disk wouldn't be changes, or I could change the files directly but the player wouldn't update the new data. I see a number of/. users saying they don't use the player to manage files--they just work with the files directly--and I totally get it; file management for music in XP sucks. Finally, I gave up and sold the damn thing (at a substantial loss).
Now a month ago, on a whim, I setup an apple loan (I'm a starving math student), bought a powerbook, and tried to do the same thing as in the windows case. There is just no comparison to be made between the slipshod wintel music software and iTunes. iTunes is simply amazing--not just for 'look and feel' or the online store, but for FILE MANAGEMENT. The system they devised seems obvious, but having used other products I am acutely aware of how much thought actually went into it. I have no dificulty believing that, long term, my music will continue to be well organized. In less then a month, Apple has earned not only my money and my respect, but also my trust.
To conclude, it seems like MS is grasping here the way they were back in the mid-nineties with the internet; they are late to take this market seriously, and it looks like the strategy is to rely on a huge bank account, the installed user base, and general complacency among users. Apple, on the other hand, has clearly been deliberately moving in this direction for years, devised solutions to both the legal and technical problems, and is really on a roll. From the sound of it, we all might be in for a replay of the browser wars, only this time it will be the music wars.
In his book "A Tour of the Calculus," David Berlinski quotes Alonzo Church: "Any idiot can learn anything in mathematics. It requires only patiance. Now to create something, that is another matter." I think this is the distinction we're grasping for here. Its easy for me to decompose an arbitrary natural number into its unique prime factorization; its even fairly straightforward to prove that I will *always* be able to find such a unique decomposition (fairly easy inductive proof). Being the first person to notice/prove something like that is on a completely different level of difficulty. The difference, I think, is all about the contrast between looking at he construction of a system of ideas vs. examing their actual chronological development. Look at the way math texts are written: a few definitions, some axioms, followed by theorems and their proofs. They lay it all out for you to follow like the yellow brick road. Very few writers make any effort to motivate ideas they way they occurred at the moment of their origination--when they do, it seems insanely pedantic. Example: look at Thomas Apostol's calculus. He introduces the integral PRIOR to the derivative, which is the historically correct development, but tends to confuse students (like me), who find it easier to understand the integral (initially) as an antiderivative. To conclude: I think both of these/. posts are right: contemporary mathematicians have the advantage of perspective--a construction of the original ideas that makes them follow from simpler ideas we may take for granted. Euler, Newton, Goedel, etc were extraordinary because they were able to work without that construction--they could just navigate Oz without the yellow brick road. Pretty amazing to consider, if you've ever tried it yourself.
I once read an article (maybe by Keith Devlin) about the millenium prize problems (solve one, you get a million dollars--get to work!) Anyway, he argues that only one of them--P=NP?--is likely to be solved by a prodigy because all the others require years of study to even formulate, whereas P=NP might yield to somebody with one really clever idea and a solid grasp of the first principles involved.
This has been bugging me since Wed. nite. First idea: use symmetric groups. Remember that 'neo' is a permutation of 'one'? Maybe they're referring to S_3, the group of permutations for {o, n, e}. Then for the identity permutation, obviously, we get f_0(ONE)=ONE (the identity element--kinda neat) and then for the other five we get:
f_1(ONE)=EON,
f_2(ONE)=OEN,
f_3(ONE)=NOE,
f_4(ONE)=ENO,
f_5(ONE)=NEO,
Any reason to design the Matrix using symmetric groups? Also, 6 is the first perfect number. But that's all I've got. Anybody else have any ideas?
Slashdot Mirror
The technique really is interesting. We have techniques that can identify patterns that are meaningful (all of cryptology, most of number theory, graph theory) but this application is neat because it is an effort to prove--rigorously--that a given set of data is just total noise.
WTF? Its that sort of freaky jedi-mind-trick that I think RUINS family gatherings. I mean really. Not only do you have to actually SEE your family, but they insist on using the occassion for charmlessly didactic "life lessons." Truly a grinchish tactic, my friend.
Guidelines:
1. If you really want to understand mathematics, stay away from suggestions made my engineers; in particular, eschew books that dumb down mathematical theory in favor of the 'this is how you compute the solution' approach. Silvanus Thompson I find to be especially egrigious in this regard (those who try to learn calculus from Thompson will never understand the rigorous notion of a Limit, which is hardly pedantic since the derivative is itself a limit and the Riemann integral is the limit of a Riemann sum).
2. Be patiant with yourself. Geometry, Analysis (which includes what is called calculus) and Algebra have required centuries of constant effort to develop. If you go for the 'fast and cheap' approach to learning it, you will aquire nothing more than skills, when what you really want is knowledge.
Books:
Preliminary topics: Before you can think, you must memorize certain things and learn other things by rote. This will be hard and painful, but these fundamental topics are to mathematics as the alphabet and grammar is to Shakespeare, Milton, and Joyce. They are: the notion of a function, the laws of exponents, elementary trigonometry (sine, cosine, tangent, and their inverses), the binomial theorem, the definition of a polynomial, factoring polynomials, setting up applied problems in algebra, linear equations and their graphs, simple nonlinear equations and their graphs, slope and area, the Pythagorean Theorem. Most of these basic noitions are covered in Forgotten Algebra (which is published by Barrons for people just like us, and College Algebra, by Michael Sullivan.
Fundamental Notions:
By fundamental notions I mean ideas that form the basis for other ideas. Mathematics is all about definitions, and definitions are all about ideas; you cannot learn complicated ideas without understanding basic ideas (if you don't believe me, try explaining why every vector space has a basis to someone who doesn't understand what linear independence is). Unlike preliminary topics, fundamental notions are actually fun to learn--you get to think instead of just memorize and drill! I know of one wonderful book for this sort of thing, for someone in your position:
1. A Tour of the Calculus, by David Berlinski. This will make you think about what 'continuity' is. Good preparation for calculus, which is all about continuous functions, and good because it presents mathematics as a branch of philosophy (which it is).
Single Variable Calculus
Single variable calculus is where you will find most of the major concepts in the subject; the next time you will think this much is in linear algebra, when you study why the derivative for a n-dimensional vector space is actually representable in terms of matrix multiplication (the derivative is a linear map.) Here are some good books on calculus:
1. Calculus, Thomas and Finney. This text features a superb fusion of theory and application. The exercises are challenging, but doable for an independent student, and solution guides are available (these are indispensable as you search, at 2AM, for the mistake in your integration by partial fractions problem that required nine pages and is off by a constant).
2. Calculus, by Michael Spivak. My favorite calculus book. A brilliant synthesis of upper division real analysis and run-of-the-mill calculus. Reading it is like feeling awestruck by the beauty of someone you have known for years and years. This also has a solution manual (which you will need, because here there are proofs).
Advanced Mathematics
Don't stop learning math just because you
http://www.phuber.com/huber/js/js.htm
You may also find interesting materials on his web site:
http://www.phuber.com/
IANAL, but it looks to my untrained eye like he IS, in fact, in violation of the dismissal. He's been running around television, his web site, etc. vigorously DENYING liability. The text SPECIFICALLY says this is not the case. I have no idea if this is illegal, etc. or if he could be cited for contempt, but I see the point those RIAA lawyers are trying to make. His actions and statements do seem to stretch, if not break, the spirit and letter of the agreement.
But they don't have to. They can just sue ultrapeers. The point of the lawsuits is to cripple the gnutella networks in the USA. Most KAZAA users I know upload almost nothing--the lawsuits won't go near them. The RIAA is interested in the heavy hitters--folks who enable the gnutella system because of their high bandwidth, and double-digit gigabyte shared folders.
The RIAA has lousy public relations. Sending the former CEO to do commentary on a major cable TV news network sounds a lot like a PR move...
Salon.com loves Ellen Ullman almost as much as I do. Read excerpts from The Bug here: http://archive.salon.com/books/int/2003/05/16/ullm an/index_np.html
You can read articles by Ullman here:
http://archive.salon.com/directory/topics/ellen_ul lman/
Salon is free as long as you watch a little commercial (C'mon--its 10 secondds, and then you get to read Ulllman--for free!!!)
I love this service but ach, it is going to be a nasty business to be in. Apple has entered center stage with a model based on .99 cent downloads. Way cool! But the most obvious way for someone to compete with them is to offer a download service at a lower price--so expect someone to do that shortly. Apple's margins haven't been published, but I'd guess they're razor thin to begin with. Now .99 cents is already so low that there isn't much further to fall--if a price war ensues, it won't be long before corporations are running online music services as a loss-leader. In Apple's case, it promotes their hardware; other companies will have other ideas. Eventually, these services might very well lead back to where we started--corporate sponsored music-on-demand, with free content that is used as a tool to peddle something else. Look at the way Apple chases these labels--this is brand-name association for them. Think Different, Think Nirvana, etc. It entrenches Apple deep into the popular culture. So those who want free music--it may be coming sooner than you think.
Number theorists have proven that there exists no polynomial function f(x) such that f(x)={primes}. There is, however, a vast literature concerned with the distribution of primes. For instance: Prime Number Theorem: "the prime number theorem gives an asymptotic form for the prime counting function pi(N), which counts the number of primes less than some integer n." Bertrand's Postulate: If n > 3, there is always at least one prime p such that n is less than p which is less than 2n-2. Wilson's Theorem: "if and only if p is a prime, then (p-1)! +1 is a multiple of p, that is (p-1)! congruent to -1 (mod p)." (quotations from mathworld.com) Theorems such as these provide insight into the distribution of primes throughout the natural numbers. The Twin Prime Conjecture, if resolved, would provide additional insight into this distribution, which would be of fundamental theoretical and practical importance. For instance, it is currently regarded as hopelessly time consuming to factor large composites--public key cryptography is based on this fact. But I am not aware of any proof that factoring such numbers *must* take a long time--that they do is an interesting state of affairs, but it might not reflect the nature of the universe so much as our lack of knowledge about prime numbers. Solving the TPC would be a step in remediating that deficiency.
I liked Pattern Recognition, and found out a week ago that William Gibson has a web site; there's a (good) blog and a discussion list. http://www.williamgibsonbooks.com/index.asp
People object to the Matrix being perceived as a philosophical film in very much the same way that Lord of the Rings is disputed as a work of literature. In 2001, the London Review of Books said that LOTR that "there's a whole little world in there, simplified and protected, like in the role-playing games to come. It is its own university, its own library, its own structure of branching knowledge. To enter it is to become a simulacrum student in a crucible of simulacrum knowledge. A simulacrum student at the simulacrum university of simulacrum life." (its at http://www.lrb.co.uk/v23/n22/turn03_.html). The Matrix fits that description even better than LOTR. Does that mean it's "junk food," as claimed by the prior post? I have no doubt that far more people willl see the Matrix movies and read LOTR than will ever struggle with the Meditations or Beowulf. But who cares? Why does the valuable lesson about philosophy/literature have to be written by somebody you'll never meet, in a language you'll never learn? To me, the vitality of those subjects is in their ability to make you grasp frantically for understanding of *your existence*--not to comprehend what some guy meant in Europe 400 years ago. Rigor is important, but so is inspiration.
I am enormously skeptical that MS will actually generate something worthwhile here. Why? I bought a new wintel laptop about 1.5 years ago for the express purpose of digitizing as much of my life as possible--this project included scanning in all personel records, setting up a digital camera, and burning every CD I owned and listening to them *exclusively* via the computer itself. Now there are various music clients available for winXP but, in that way the these things tend to happen, I gradually found that it was easier to just tolerate the MS music client rather than working around its interuptions with another piece of software. To make a long story short, this was a disaster. The MS software was ok for ripping and playback, but file organization was a big fat mess--I could edit files using the player but then the actual files on the disk wouldn't be changes, or I could change the files directly but the player wouldn't update the new data. I see a number of /. users saying they don't use the player to manage files--they just work with the files directly--and I totally get it; file management for music in XP sucks. Finally, I gave up and sold the damn thing (at a substantial loss).
Now a month ago, on a whim, I setup an apple loan (I'm a starving math student), bought a powerbook, and tried to do the same thing as in the windows case. There is just no comparison to be made between the slipshod wintel music software and iTunes. iTunes is simply amazing--not just for 'look and feel' or the online store, but for FILE MANAGEMENT. The system they devised seems obvious, but having used other products I am acutely aware of how much thought actually went into it. I have no dificulty believing that, long term, my music will continue to be well organized. In less then a month, Apple has earned not only my money and my respect, but also my trust.
To conclude, it seems like MS is grasping here the way they were back in the mid-nineties with the internet; they are late to take this market seriously, and it looks like the strategy is to rely on a huge bank account, the installed user base, and general complacency among users. Apple, on the other hand, has clearly been deliberately moving in this direction for years, devised solutions to both the legal and technical problems, and is really on a roll. From the sound of it, we all might be in for a replay of the browser wars, only this time it will be the music wars.
In his book "A Tour of the Calculus," David Berlinski quotes Alonzo Church: "Any idiot can learn anything in mathematics. It requires only patiance. Now to create something, that is another matter." I think this is the distinction we're grasping for here. Its easy for me to decompose an arbitrary natural number into its unique prime factorization; its even fairly straightforward to prove that I will *always* be able to find such a unique decomposition (fairly easy inductive proof). Being the first person to notice/prove something like that is on a completely different level of difficulty. The difference, I think, is all about the contrast between looking at he construction of a system of ideas vs. examing their actual chronological development. Look at the way math texts are written: a few definitions, some axioms, followed by theorems and their proofs. They lay it all out for you to follow like the yellow brick road. Very few writers make any effort to motivate ideas they way they occurred at the moment of their origination--when they do, it seems insanely pedantic. Example: look at Thomas Apostol's calculus. He introduces the integral PRIOR to the derivative, which is the historically correct development, but tends to confuse students (like me), who find it easier to understand the integral (initially) as an antiderivative. To conclude: I think both of these /. posts are right: contemporary mathematicians have the advantage of perspective--a construction of the original ideas that makes them follow from simpler ideas we may take for granted. Euler, Newton, Goedel, etc were extraordinary because they were able to work without that construction--they could just navigate Oz without the yellow brick road. Pretty amazing to consider, if you've ever tried it yourself.
I once read an article (maybe by Keith Devlin) about the millenium prize problems (solve one, you get a million dollars--get to work!) Anyway, he argues that only one of them--P=NP?--is likely to be solved by a prodigy because all the others require years of study to even formulate, whereas P=NP might yield to somebody with one really clever idea and a solid grasp of the first principles involved.
This has been bugging me since Wed. nite. First idea: use symmetric groups. Remember that 'neo' is a permutation of 'one'? Maybe they're referring to S_3, the group of permutations for {o, n, e}. Then for the identity permutation, obviously, we get f_0(ONE)=ONE (the identity element--kinda neat) and then for the other five we get: f_1(ONE)=EON, f_2(ONE)=OEN, f_3(ONE)=NOE, f_4(ONE)=ENO, f_5(ONE)=NEO, Any reason to design the Matrix using symmetric groups? Also, 6 is the first perfect number. But that's all I've got. Anybody else have any ideas?