Let us suppose for a moment that mathematics is invented and not discovered. Just how far do you want to push this.
Was the concept of 0 and 1 discovered or invented? You can argue that because of quantum mechanics and the probability that there is a nonzero (though immeasurably small) chance that any particle (or group of particles) could exist at any point in space and time, that the idea of 0 and 1 cannot truly be represented in nature (if you tried to show me 0 blocks and 1 block, quantum mechanics would say that there is a small probability that my 0 block might actually be a small fraction of a block -- of course the odds against this are ridiculously small, but that is not the point). So even for something this simple, you can claim that mathematics is only a model that is not truly represented in nature.
The existence of 0 and 1 is an "axiom" in mathematics (for set theorists, they usually describe this as the existence of Set with nothing -- the nothing is 0 and 1 is the Set that contains nothing). It is not provable, but it does not mean that it cannot assume it to be true and work from there.
I want to find anybody rational that believes that there can be "intelligence" of any reasonable complexity and sophistication that does not intuitively understand the difference between the absence of something and something. I want to go further and say that this idea was discovered and not invented. The symbols and notations that we use to represent this idea were invented but the underlying idea exists and is true even if all of existence were to vanish and nothing ever existed anywhere.
Once you believe in 0 and 1, there is a nicely built up sequence of logic that will lead you to circles and PI. Some of it requires advance graduate mathematics to fully understand, but there is an unavoidable discoverable chain of logic. For example, the existence of numbers following 1. This is one of the Peano axioms. Again you have to assume it is true, but nothing breaks down in logic if you do and you work from there. Again, in Set theory, which is one way to build up the axiomatic foundations of mathematics, if 1 is the Set that contains nothing, then 2 is the Set that contains the Set that contains nothing. Once you believe in positive integers as being discovered and not invented, then the rest of the big construct called mathematics followed and was "discovered" just as much as 0 and 1 were "discovered".
I have more to say. I forgot to mention another reason why I think this result is so exciting.
Assume you have a game being played by N people (N > 2, but think of N > 8 as a typical example). By what I said in the prior post, everybody is playing a BAD strategy, a strategy that is far from the Nash Equilibrium because it is IMPOSSIBLE to effectively calculate a better strategy in a reasonable amount of time.
Although it is impossible to create a perfect strategy, it is possible to create an exploiting strategy. If you can accurately predict the strategies of your opponents (where the accuracy does not have to be that great -- the strategies of your opponents are that intrinsically BAD), it is easy to create an exploiting strategy in any game of sufficient complexity involving enough players. I will give a simple example. Suppose you play poker and after a sequence of bets (with the same type of sequence occurring reasonably often), you notice that an opponent tends to fold to large bets almost all the time, you would then exploit that player by always bluffing after that sequence of events and "slow calling" (that is not raising) when you have a monster hand. In a two player game, that opponent would correct his play successfully by calling more often, but in multi-player games, the information can be more muddled because the opponent does not notice the correlation between a sequence of prior bets involving multiple players and his tendency to fold. In other words, the opponent may think that he is correctly calling big bets half the time but not notice that if a second player raises, he tends to fold more often when another player enters the pot with a big bet (there is more I can say here, but I don't want to get into the strategy of poker in great detail -- but I will say that the folding player is not being stupid).
But here is where things get interesting. In order to effectively exploit the strategies of others, you have to adopt a highly exploitable strategy yourself (for example, "always bluffing after a certain sequence of events" is a fairly exploitable strategy). In stock market terms, it would be the equivalent of going into a highly vulnerable "short position" on a stock. This brings up the issue of "counter-exploitation" and the need to disguise your actions when playing in the game.
So what happens in such games.
The games are highly unstable. People are always doing crazy exploiting moves that create great risk for themselves (and potentially for the infrastructure of the game itself).
The games are highly exploitative by colluding players (the "old boys network" as some would call it).
Getting "inside information" on the actual strategies pursued by other players is highly valuable.
Likewise, hiding what you are doing is very important.
A big danger is acting in a predictable manner that is too similar to what others are doing (the "herd" mentality problem).
There is no safe way to play the game.
Researching potential ways to exploit the current state of the game is a must, particularly to see if anybody else is pursuing such strategies against you (example: there might be an opportunity to artificially squeeze somebody else's short position on a stock that you own).
So what do I conclude from this. The more we make our "markets" (stock & other assets) more "perfect" the crazier they are going to become. Derivatives made the "market" more perfect because it allowed people to act on "risk factors" in a more precise way. It is not a surprise that they have also caused problems. Derivatives removed "frictions" in the market and made the market better. But it is the friction in the "markets" that prevents greater craziness. This is not because people are irrational. Far from it. It is the nature of game theory itself that says this will be the out come. To put it another way, cleverness and rationality are actually the enemy. If we were stupider and less rational there would be less of a problem.
Let me give an example of this "stupid irrationality". Suppose we decided to deliberately sabotage the economy by telling Big Banks that they had to hold large assets and they were barred from doing anything interesting with them. This would sabotage the economy because those assets would sit moribund and do nothing to grow the economy in an efficient fashion (as any Big Bank lobbyist would be sure to tell you). We would do this because of an "irrational" fear of economic collapse. This was how we used to treat banks and our own assets. This latest result from game theory maybe implies that this type of "paranoid thinking" may be "irrational" for any one person, but it might be "rational" thinking for the world as a whole. Though it may be heresy to some to say this, but maybe we do not want the economy and its markets to work too efficiently.
I am pleased to see this result. It agrees with some of my own suspicions. Let me describe a simple three person game (players A, B, and C). Here is the rule for each player when it is their turn.
A player must take $1.00 away from one opponent and give at least half of it to the other opponent. Whatever is not given to the other opponent can be kept by the acting player.
Each player in turn gets to choose which player to steal a $1.00 from and how much of it they will keep (they can keep at most half) and how much to give to the other opponent. Assume you do 50 rounds of this game (each player is visited 50 times).
Here is the problem: construct an optimal strategy for each player.
What makes this problem so difficult is the issue of collusion. If two players decide to gang up on another player, they can both profit at that other player's expense. But if you assume that none of the players have friendships or other factors that might induce collusion, then the only way to get collusion is to offer "bribes". A bribe can be both an offer to not take away a $1.00 and it can be an offer to give more than the standard half of the $1.00 to the other opponent. An optimal strategy is based on deciding who is likely to be giving you better bribes in the future and how do you induce such behavior with your own bribes.
Now, here is the part I consider exciting. The claim is that calculating the Nash Equilibrium is hard (in a computer calculating sense) for three person games, this one being an example. The claim is actually stronger than that. Getting even somewhat close to a Nash Equilibrium is hard (if it were not, then you would evolve slowly towards the Nash equilibrium by slowly refining "good" solutions into "better" ones -- that would be most likely doable in polynomial time). In other words, not only is it hard to calculate the perfect strategy, it is hard to even calculate a good one.
To see why that might be the case, let us assume that the three players have chosen a strategy which causes player A to think "it really does not matter if I choose to steal $1.00 from one opponent or another -- my expected outcome is the same." Then player B could offer just 0.01 more of a bribe to induce player A to favor B over C consistently. In other words, a very small adjustment in strategy by one player can have a huge potential impact on their future expected out come. It is this instability of outcome which is the mark of an NP computational problem. A small wiggle in inputs creates huge "chaotic" changes in outputs.
Side note: This "large change" in outcome based on small change in input is at the heart of what makes factoring large numbers hard. If you multiply two large numbers together, and change just one digit in one number, it will have a large and somewhat unpredictable (until you actually calculate it -- but then that means you are still having to try out all different products to factor a number) outcome.
As usual when I hear a debate like this (debate about activities in the middle east being another good example), I believe both sides to be wrong. Or I agree (at least partially) with the accusations made by each side against the other. The problem becomes trying to sort out the mess and figure out who the greater offender might be.
If I understand correctly, RMS claims that Miguel is willing to overlook the fact that Microsoft may have devious and evil intent behind C# (and Silverlight) and in their general relationship with the open source community. In particular, Microsoft has been made it clear that they have in no way released any patent claims on the technologies underlying C# and all the libraries built on top of it. Proof to the contrary is welcome. In fact, they have made it clear that there may be patentable material by giving Novell a special "free-usage-of-patents" dispensation to allow the shipping of Microsoft technologies without being particularly clear about what that exactly means. Both Novell and Microsoft are being disingenuously ambiguous on this point (in a way that reminds me of how some politicians talk about what they would do to balance the budget).
Miguel's claim is that RMS has a "no compromise" position towards open source. RMS would prefer a clear firewall between the non commercial "free" software and the "commercial" for sale software. RMS has attacked software developers (or more precisely the things that they do) who take a more pragmatic approach to the usage of open source code (which means he is attacking a large part of the open source community). RMS is right that breaching this firewall can compromise the future of some open source projects, but he is unwilling to balance that against the potential virtues that such "breaches" might give you.
I agree with both of the above claims, but only up to a certain point. In a better world, I would have Miguel admit that some of the practices of Microsoft and Novell are highly suspect and threaten to quit if some of the ambiguity about patent usage are not made clearer. Miguel has said that there are no known "patent" infringements and that all software can have unknown patent infringements. This is a misleading statement, because he is cooperating with a very large Monopolistic company with a vested interest in the software he is developing that is known to sue for patent infringement. The risks of using some of the libraries built on C# is much greater than software that was not written to the specification outlined in a commercial software package. Both Microsoft and Novell need to make clearer (that passes muster with lawyers) statements about patent risks of using C# libraries (note the focus on libraries and not C# language syntax -- this is deliberate -- usually trouble comes from the things built on top of a technology not the underlying root technology itself).
Also in a better world, I would have RMS be more accepting of Linux and the general compromise that it has taken between commercial interests and non commercial interests. Programmers want to make money, but some would like to do it in a way that allows them to contribute something (but not all) back to the community at large. RMS has limited utility for such people in his vision of the future of software.
As to who is the greater offender. I would say Miguel. RMS is a known entity with a public position and nothing in his writing suggests a particular antipathy towards Miguel in particular (counter examples of unproductive inflammatory rhetoric are welcome). I may disagree with RMS, but I have never felt that he was trying to fool or mislead me. I am not so sure about Miguel.
This whole issue of home schooling is complex. Since there have been multiple child A vs. child B, let me give my own.
Child A -- given standard curriculum surrounded by some kids brighter and some kids not so bright. Once in a while the curriculum is challenging, but most of the time it is easy and a little bit boring.
Child B -- Has a teacher that constantly challenges the child to push themselves so that at no point are they allowed to coast doing repetitive easy things.
I don't think I would get much disagreement if we were to say that Child B got the better education.
Who is one of the best people to give a "Child B" style education? It is clearly the parent and this is where home schooling can be a big winner. But this assumes that the parent has the intelligence, patience, and time (I have a full time job -- so this is not a trivial thing to ask for) to do this for their child. Most home schooled children I have met have clearly been quite well ahead of their peers and generally did quite well with their lives. But, like most Slashdotters, I tend to hang around people who know how to give a "Child B" style education and they are hardly representative of the general slice of humanity.
There is one point that is being missed here. I believe that the "child B" education is so dramatically effective, that probably it only needs to be done a few hours of the day leaving a lot of time for the child to pursue whatever interests they like. It can be so effective that a clever parent can sneak in such an education over the course of the day without even the child knowing it. It is because of this that some times "Home schooling" and "Unschooling" can seem to be the same.
I feel uncomfortable with the idea that any child can opt out of public education without any type of mechanism to confirm that dropping out was actually beneficial for that child. I can see certain religious cult groups taking advantage of this in ways that would make worry about what is being inflicted on the child. I am also uncomfortable with the idea that it would be a generally good practice for a random normal child to let them "pursue" their own agenda in education and pretty much let them do whatever they want. I have never met a child that succeeded this way that was not discernibly gifted at an early age. But my "sample set" is small, so maybe it is possible to let a "sports" oriented child to idle away their time choosing how they would like to be educated. But I don't think so.
I always had a question about MySQL's limitations on use.
Suppose I created a GPL module that wrappered MySQL and created a more standardized SQL interface (one that used tricks to support nulls for dates, and so on) and when I installed MySQL I also installed this GPL library. Then if I created an application that would work with SQL Server and using exactly the same SQL calls would work with my GPL MySQL wrapper module, I don't think this would be a breach of the MySQL license.
I didn't see anybody give their experience with the "new math" experiment that was done a while back. If you think about this from a little distance you can see that there are two camps in mathematics education, the "Creatives" and the "Pedantics", the "new math" was an attempt by the "Creatives" (who Lockhart is clearly a member of) to inject "thinking" and "creative thought" into the mathematics curriculum. It was a total bust, primarily because the teachers teaching it really didn't understand the intentions behind this new curriculum and they reduced it to rote. Those in favor of "back to basics" would be in the "Pedantics" camp and have been making a comeback recently.
So here in a nutshell is the two opposing camps arguments.
Creatives argument against the Pedantics -- The Pedantic curriculum is a soul destroying exercise in rote and memorization leaving no room for a child to feel any inspiration or creativity.
Pedantics argument against the Creatives -- The Creatives assume the world to be filled with inspired teachers that won't reduce any curriculum to a pedantic exercise. If the quality of teachers is such that they can only teach pedantic material, you might as well have the children learn something useful and constructive even if it is boring and soul destroying.
I am an ex-mathematician and I am firmly in the "Pedantics" camp. I hate to see children that cannot add two digit number to two digit numbers without a calculator. That is the world that well meaning "Creatives" create.
Also, is there really that strong a correlation between the percentage of students that pass standardized tests on calculus and the overall success of the community? Russia has a very strong educational system, see what that got them. The general population of the U.S. would be considered be woefully uneducated by the standards of many other countries. But if you were to take any country with as large an immigrant population, I suspect you would see similar numbers. Over time the immigrants are absorbed into the main stream and their children do better. But could it be possible that these immigrants are also the source of the vitality of the U.S. economy and their education (or lack of it) is not the primary reason for why they make this nation so successful?
I always wonder what the parents are thinking when they push their kid through the entire K-12-BA school system at such a fast rate. Do they really believe that the kid is better set up for life at that point then if they were to take their time and send the kid to a better college where the kid would get an early exposure to professors who are leaders in research in their respective fields? Better yet, have the kid skip out on the normal curriculum and find some more challenging instructional texts that the kid can do with remote supervision by a professor at a local college.
There is a quote that I remember from one of my professors in college. When talking about a particularly bright kid, he said "that kid is sufficiently bright that our classes and curriculum are more of a hindrance than an aid". The implication is that accomplishing a BA degree is many times more about grinding through material than something that shows a real difference between those who are truly educated than those who are not. I can still remember Jay Leno's "Jay Walks" where he would ask college graduates questions that many (admittedly unusually smart) 12 year old kids (still not in high school) without a BA could answer easily.
I have read enough biographies of those who end up with great achievements in their fields (Einstein and Newton being good examples). Almost none of them had abnormally accelerated the normal learning track. They almost always just side-stepped it. For example, in his teen age years Einstein was thinking thoughts that only specialized masters of those fields would be able to fully comprehend. Did his track record in school reflect this? No. (And if you follow Newton as a model, then you should force your child to be a farmer for a while so that they really will do anything they can to avoid that fate).
This common fallacy comes up again. Poker is not just an estimation of probabilities. Poker is a raw and pure example of a very complicated multi-way "game theory" problem. Such problems are far from NP-complete (not computable in polynomial time relative to number of inputs on a computer) and much more interesting than they might first appear. Nobel prizes in economics have been given for insight into such problems. There is a book called "Mathematics of Poker" (those interested can do a Google search) which uses Poker to introduce game theory. Please read that before trivializing the theory that underlies poker.
A good poker player models his or her opponents and then creates optimized exploiting strategies -- strategies which are far more sophisticated than you might first imagine (once you have started giving up chances to raise up pots with strong hands in order to catch bluffs you have started down the first step of this long road). This optimization takes into account that your opponents are trying to exploit you and how well you believe you have misled them. The skills to master this are not too different from the skills required by a chess grandmaster (the analogy is not exact -- poker is more like a large look up into a vast reservoir of experience than a deep computational think, but an expert can beat up an non expert very quickly -- even a non expert who knows all the odds).
The one thing I found strange about String theory is that it made Physicists study Algebraic Geometry (with Sheaves and such). Algebraic Geometry got started as a field when mathematicians tried to link up the algebraic properties of polynomial equations (what "algebraic solutions" does it have is one of the questions you might ask) with the differential/topological (how much curvature does it have, how many "multi-dimensional" pseudo-holes does it have -- think about the questions of curvature and "holes" you might ask about multi-holed doughnut in many dimensions -- this is a gross simplification, but I am grasping for intuitive analogies).
What I remember about Algebraic Geometry is that it was one of the harder fields of study in all of mathematics and only a few mathematicians in the world could wield the theory with any real authority and skill (Faltings is a famous such Mathematician). At the time it made me worry that maybe humans would never be clever enough to truly figure out the rules of the universe. Because if we are already have to understand some Algebraic Geometry to get a handle on the current most respected "theory of everything", what would happen if the "theory of everything" required one level of abstraction complication beyond that? There has been a constant progression of theories in Physics from the less abstracted to the crazier highly abstracted (quantum mechanics and general relativity already can only truly be understood by at most a few hundred people in the world). Maybe this time we are going beyond the ability of us poor human mortals to understand.
In defense of String theory, though it may give no predictions, it does give those who study it a feeling of "enlightenment", as if they are getting a potential intuitive understanding of how the universe is put together. Studying mathematics in general can create such a feeling (I think in general that is why mathematicians love their field of study), but it is way cooler to think that the theory and the real world might have some linkage. Also, from what I understand, competing theories all have the feeling of artificial glitchy repairs to existing theories without granting much enlightenment. If you give me a bunch of data, I can create an equation which will spit out the data. But if the equation does not offer insight into the nature of the data (for example, you cannot see that it is actually a "repeating wave pattern of visual distortions"), then though it may be useful, it really does not offer much in the way of "enlightenment".
Some may view the usage of the word "enlightenment" as an allusion to some type of religious feeling. That may be, but it is NOT connected to any type of statement that could be read as "this vision that I see is true and those who disagree with me are morally inferior beings and will be viewed as a lesser person by the higher powers that rule the universe". In fact I suspect that those who disagree with me about the worthiness of understanding mathematics may have spent more time worrying about their morality (as opposed to their "faith") and may actually be superior human beings (and may be viewed as such by the higher powers in the universe).
The cost of feathers has risen, even down is up!