mstoehr

To say that the sequence is computable means that you must present a Turing machine, call it T, that will produce BB(n) in finitely many steps after taking input n. Our specific Turing machine T will have a fixed number of rules call it N rules, then by the definition of the Busy Beaver T(N+1) <= BB(N) = T(N) but BB(N+1) > BB(N) so T(N+1) does not compute BB(N+1). The flaw in your inductive proof is that you have shown there is a Turing machine that can compute BB(N) for each N, but you haven't shown that the same Turing machine can compute all numbers in the sequence.

Employers are probably unconvinced that younger applicants will be able to make them more profits. This is caused in part by the uselessness (for businesses) of the work students do in high-school and college--this is a group of people whose life has primarily consisted in consuming profits. Employers are also probably more risk-averse because the recession means that profit margins are tighter (if existent at all).

Most commenters are focusing on relatively high level features of decoding speech. It is important to also be aware that there is still great debate about what are the acoustic correlates of linguistic events in speech. It seems that our words are composed of subunits (usually taken to be phones out of the IPA--but there's work on alternatives) but what exact acoustics correspond to the phones are is still unsettled: lots of debate and mediocre recognition performance

Undoubtedly there is much room for improvement on these higher-level features but computers are still well behind humans in large vocabulary isolated keyword spotting: this is a task where one word from a very large corpus of words is spoken and the human or computer has to guess what that word was. Computers do poorly relative to humans (particularly in noise), which suggests that many of the mistakes that computers make is in not being able to interpret the acoustics correctly.

I agree that its at the heart of it (and I'm presently writing a paper where I'm using articulatory-phonetic features rather than phonemes). Unfortunately, there is no large-vocabulary speech recognizer that uses articulatory phonetics (yet!). Every large scale speech recognizer and most small scale use phonemes and are trained using speech that has been transcribed into phonemes. There is almost no data that is annotated with articulatory phonetics (a problem I'm working on right now).

It's not actually clear what we hear at this point, there is evidence that we respond to something like frequencies, pitch, volume, and other things. But the jury is still out on the low-level signal processing that is occurring in the cochlea and the primary auditory cortex. What's happening further downstream in the brain is even less clear.

It is true that humans do use situational context. In the cases where semantics is important and complex for understanding an utterance a computer will fail even more because it won't get the semantics or the speech signal.

On the topic of dialog, this is arguably the area that speech recognition has gained in over the last nine years. Prior to 2001 there were not many usable dialog systems and (depending on your definition of "usable") there are many usable dialog systems deployed in call centers around the world.

Most call center dialog systems have a rudimentary system asking for people to repeat things when it doesn't understand. Although, if it asks more than once the callers tend to get very angry.

Actually most research effort in speech is more on the language side rather than the signal processing of the speech signal. So I think many people have a similar intuition as yourself.

Bear in mind though, that humans significantly outperform machines in tasks where isolated or streams of non-sense syllables are said: i.e. "badagaka" is said and humans can pick out the syllables whereas computers can have a lot of difficulty (in noise in particular).

Computers start approaching human performance most when there is a lot of linguistic context to an utterance. So it appears that humans are doing something other than using semantics.

He didn't make any advance that has made its way to a full word recognizer, he's merely recognizing phonemes (which are linguistic subunits of words) several researchers in the field have criticized his methods. Additionally, none of the top five phoneme recognizers have ever been deployed as a word recognizer, and there is little chance that they even will be in the next few years.

I'm not any sort of expert on the neural network literature but these were some papers in the last three years that caught my eye, Yann LeCun also does work on neural nets, but I haven't been all that impressed by his results. One of the main advances has been ways of developing 'deep' architectures with multiple layers rather than the traditional shallow neural networks (arguably the SVM, for instance, is actually a very cleverly trained single layer neural network)

Here's a Geoffrey Hinton paper on training deep belief networks: http://www.cs.toronto.edu/~hinton/absps/ncfast.pdf

Here's some stuff from Andrew Ng's group: This paper shows how his deep belief network was able to 'learn' in an unsupervised manner certain plausible image primitives http://robotics.stanford.edu/~ang/papers/nips07-sparsedeepbe... This won a best paper award (application paper) and its about fast ways of building a deep belief network: http://robotics.stanford.edu/~ang/papers/icml09-Convolutiona...

How do you think that the character traits of the writer are relevant to his case against Google?

And even the most fancy statistical methods are generally about proving that some mild generalization of linear regression is sufficient to solve a particular problem. Non-linear models often result in a great increases in model complexity which leads to issues of overfitting, computational intractability, and instability.

There really isn't very much available on the practical side. So if you are looking to implement algorithms I suggest that you make use of the machine learning at ocw.mit.edu

Alternatively, if you want a good dose of a theoretical explanation of algorithms currently in use I highly recommend "Pattern Recognition and Machine Learning" by Christopher Bishop. It is definitely the best machine learning (and statistics) textbook that I have ever come across.

The orthodox school of thought in financial economics was the efficient market hypothesis, which more or less states that the best valuation on current assets is the market price. I suppose there is something to be said for pointing out that the "best valuation" may not be very good. Taleb certainly wasn't the first to bring this up, but we probably aren't worse off for being reminded of that fact.

If you're looking to enjoy the finer things in life, Chile is probably not the best place. Argentina, Peru, or Brazil would all probably be better picks.

"This was really Godel's point: mathematics is not identical with formalism. They stand and fall separately. This is not to say that mathematics could never collapse for any reason, only that it would take a lot more than finding a paradox at the center of ZFC to make it happen."

That's significant because another great mathematician David Hilbert challenged mathematicians to come up with a complete and consistent set of axioms for all of mathematics. This was a great hope at one time that was shattered by Godel. Most people don't really talk about this old program anymore (except as a historical curiosity) because there is utterly no hope in it at all. The other implications that people try to draw from Godel's work is probably a consequence of the fact that his work sounds like it says so much more than it actually does when it's translated into normal English (and out of math-speak).

Although it does lead to a mathematically uninteresting paradox: if you let A be the axioms of set theory and you add an axiom P which states that A proves x and not x, (i.e. set theory is inconsistent) then A' = A and P is still consistent.

I was originally an economics major, but I've managed to switch out into a major in math and a minor in CS (the study of AI was just too seductive). One area of CS theory that is pretty straightforward to pick up for an economist is algorithmic game theory: (http://www.amazon.com/Algorithmic-Game-Theory-Noam-Nisan/dp/...). There are also some strong connections to reinforcement learning and multi-agent systems (indeed much new research in computational economics is focused on multi-agent systems).

The clearest description of what CS researchers talk about when they talk about "intelligence" (see http://www.vetta.org/definitions-of-intelligence/) is probably a production function from economics.

I've definitely seen evidence that the school one goes to correlates with success, but are you sure that the causality flows in the direction: academic credential --> success? It seems plausible at least that if the parent is wealthy, seeing to it that their kid is well-educated is a good way of signaling that their kid is decent material, it's a costly signal so only the wealthiest of parents will probably be able to afford it. And since the kids of wealthier parents tend to do better, the education signal, by some accident ends up being used a great deal to gauge future success.

I actually think that it used to just be a straightforward signal of wealth, but these days we don't want to admit that that's the case, we instead want to justify it as producing some useful economic product.

The solution uses something called common knowledge, which is that everyone knows that everyone knows that every one knows, ad infinitum. Before the queen makes the announcement they may all know that the other husbands are cheating but they don't know if everybody else knows. To see the difference consider the case where there are only two husbands (H1, H2) and two wives (W1,W2). So, every man has cheated on his wife, and when the queen announces that a man has been unfaithful, then consider the situation from W1's perspective. She knows that H2 has cheated on W2, she doesn't know that H1 has cheated on her, and she can't talk to W2 about it. With the announcement, from her perspective, there are two possibilities (both H1 and H2 are cheaters), or (just H2 is a cheater). Now, to continue we need to think about what W1 will think about these two possibilities given her incomplete information: If it were possible world where (just H2 is a cheater) and H1 is honest, then since W2 would know that H1 hasn't cheated then W2 would conclude that (just H2 is a cheater) since at least one husband had cheated. Thus, by the laws of the island W2 would kill H2 the day of the announcement.

Since W1 considers it a possibility that her husband, H1, was faithful, she won't do anything the first day; instead she'll just wait to see whether W2 kills H2.

By symmetry W2 will go through the same line of reasoning about W1. Thus, both will do nothing the first day. So then on the second day, W1 will realize that (just H2 is a cheater) must be false, since W2 didn't kill H2. So, she'll go ahead and kill W1. (Apply the same reason symmetrically to W2). Therefore, they'll both kill their husbands on the second day.

The rest of the details are pretty easy to establish. The point is that 99 days later all the men are killed.

The author might have done well to read pg's essay on disagreement since his attacks were mostly vague (I too agree that the characterization of London and Paris could have been better supported), and the attacks were executed merely by bringing up an example point and then not attacking it. On the point of cities, I think that there is good economic evidence (not cited by pg) that Cambridge and Boston are very much an intellectual center because some of the greatest universities of the world and they are leaders in terms of patents and scientific citations (from "Who's Your City" by Richard Florida).

This article claims that venture capital is lacking in innovation, but the article (a) sites no evidence to back the claim (b) does not provide support that any perceived lack in innovation might be the result of the recent financial crises (which would be a better-supported conclusion).

The idea that people from these selective schools are too conformist to innovate disregards the fact that most innovations are incremental. Furthermore, if one is taking a risk by innovating in one area of a business, it's better risk management to be more risk averse in other areas of your business. That will allow a greater chance that the business will survive in the longer terms and that you'll be able to test out new innovations and really contribute something.

So, I would guess any skill in conforming would certainly be an asset in innovation, and that this would at least counter-balance any negative effects.

It wouldn't be an internet discussion without at least one ad hominem argument.

It may be "dangerous", but basing causation just on correlation (or statistical data in general) is actually a pretty prominent idea: http://plato.stanford.edu/entries/causation-probabilistic/. Chris Anderson obviously glosses over many tricky details, but Judea Pearl among others have written extensively on it. Ultimately, all these ideas can probably be traced back to David Hume (at least here in the West).

Hank's argument that Chris is suggesting that we don't try to find spurious correlations is rather simplistic since a probabilistic theory of causation can account for "spurious" and "real" correlations quite well. Indeed, the only reason why we know that "spurious" correlations exist is because they eventually show up in the data (i.e. the economist finds out that the stock market does not predict sun spots). Thus, a statistical definition of causation (more or less one that only uses a notion of correlation), can actually be quite robust.

I agree. I mean human minds usually forget things unless they're reminded more than once.

I'm surprised at how strongly held most of the opinions are argued in the forum. The reason why is that I know of very little scientific evidence that would lend people such certainty. Firstly, as I understand it, to date, there has been no longitudinal study of mathematics talent in people who develop in interest after adolescence. Secondly, the neurological basis of learning is a rapidly growing field, but the field right now may not have much to say about the development of mathematics talent. The points that have been made: i.e. that you can't do it may be true, it's just that you probably should take what they say with a grain of salt since they are talking about a subject that is not very well understood.

I would wager, though, that going from having a small knowledge of math to going to a level of knowledge of comparable size to a mathematics researcher is going to be very hard. People who focus on math starting freshman year of high school go through 4 years of high school + 4 years of college + 5-7 years of grad school = 13-15 years before they become research mathematicians. So it might take you 10+ years before you know enough to start contributing.

Additionally, if you read psychological research on expert performance (which you probably should, and Eric K. Andersson is one of the experts in that area so perhaps start with an article by him), it normally takes 10 years to develop expert skills in a field, and that expertise comes about from something he terms "deliberate practice". Check out: (http://projects.ict.usc.edu/itw/gel/EricssonDeliberatePracti...) The Role of Deliberate Practice in the Acquisition of Expert Performance.

Most evidence would indicate that the abilities of a Putnam fellow are probably not out of reach to somebody who uses a strict deliberate practice regimen to develop their working memory (google that with 'mathematics') of and the ability to execute the thousands of tricks that mathematicians employ to solve problems. I am not very confident in this particular assertion, and I believe there is only weak evidence for it, however, there is only weaker evidence against it (and most people who argue against it make use of hand-waving has their primary argumentative technique).

On the actual task of becoming a mathematician I suggest you read some essays by Gian-Carlo Rota (http://web.archive.org/web/20070630211817/www.rota.org/hotai...), in particular look at the "10 Lessons I wish I had been Taught" and the reflections on math and mathematicians. He points out that mathematicians seem to have only a couple of tricks up their sleeve which they apply over and over and over again.

  The points made by yelsgib are good since mathematics is about a community of researchers, and most problems have been looked at by hundreds of researchers.  At the very least you will need to attend conferences at Universities.

And, if you plan on making 'significant' contributions you'll have to do more or less the same preparation that a math grad students goes through and you'll have to put in the same, if not more time as grad students do in preparation to do research.

I admit that I am curious to see what happens if you go through with such a project. It would take a great deal of perseverance on your part as you come across all the barriers I am facing (as a math major) as well as those that come about because of your age and station in life.

My main suggestion is to try to get into correspondence with mathematicians to try to get more guidance. This can be difficult because you are probably just starting out, but you will most likely find one or two gracious ones if you start trying to correspond with mathematics departments.

I lack the experience to be a good person to talk to but I am nearly always willing to discuss this sort of thing so feel free to send me an e-mail to [my sn without the 'm'] [the a with a circle around it] uchicago [period ] edu.

Most econometricians would admit it, it's just that the linear models, and Gaussian models are easier to handle since they require much less data to calibrate, and the results have more or less confirmed economic theory. There is no doubt that economic theory does not have near the predictive power that physical sciences have attained, but that is due in part to how recent new developments have been, and how sparse the data available to economists is. Indeed, estimating all the parameters necessary for non-linear models is significantly trickier. Anybody that developed robust nonlinear economic models could safely bet on receiving a good academic appointment at a top-20 school, and very probably an economic prize.

I can't think of a single econometrician who claims to have all the answers through Gaussian statistics.

These two are my favorite (that are reasonably priced!). Samuelson is definitely a formidable thinker and lays the groundwork for much later economic thinking.

http://www.amazon.com/Worldly-Philosophers-Lives-Economic-Th... http://www.amazon.com/Price-Theory-Milton-Friedman/dp/020206...

However if you go with the Samuelson, Fridman, and Heilbroner you'll be missing out on some key concepts relating to information and games, which are essential for understanding modern developments in economics.

So, I would also look at Myerson's Game Theory http://www.amazon.com/Game-Theory-Analysis-Roger-Myerson/dp/...

Finally, the empirical side of economics is quite important, and it is essential for investigating whether the theoretical conceptions (say in Samuelson) actually hold up. There are many bad texts, and the seminal ones (Greene) are quite difficult in the beginning. So, I recommend a pedagogical text that uses the likelihood approach. Although, this is, by no means, a comprehensive text, you can certainly gain an understanding of a basic econometric study http://www.amazon.co.uk/Econometric-Modeling-David-Hendry-Ni...

That should give you a broad introduction. Also, checkout http://papers.ssrn.com/sol3/DisplayJournalBrowse.cfm, which is an excellent resource with many free papers.

Good Luck!

I somewhat agree with sah in that Roth's essay did not fully address the central point of Graham's essay: the effect that large hierarchical structures have on people. The discussion of ad hominem arguments aside, I would point out that you could phrase Roth's argumentative method as, "you've argued for x given method y, and I can use method y to also prove z, you don't believe z, therefore you can't prove x". A rough sketch might look like this: y --> x y --> z therefore z <--> x (this step is hard to sketch out because it's unclear) ~z therefore ~x.

The difficulty with this argument is a fallacy about implication. Roth states that the "evolutionary argument" supports pg's view, but that he can then use that same "evolutionary argument" to prove an absurd point of view. Then he holds that he has disagreed with pg's central viewpoint.

This method of argumentation is fundamentally flawed because pg's view might be supported by a multitude of arguments, the truth or usefulness of the "evolutionary argument" is not a necessary condition for the truth of pg's view of organizations, bosses, and human nature. Hence, Roth has merely attacked pg's method of proof while leaving the central claim untouched. I have other criticisms of Roth's argument, but even if I am wrong in such criticism the argument would fail to disagree with pg.