"At some future period, not very distant as measured by centuries, the civilized races of man will almost certainly exterminate, and replace, the savage races throughout the world. At the same time the anthropomorphous apes, as Professor Schaaffhausen has remarked, will no doubt be exterminated. The break between man and his nearest allies will then be wider, for it will intervene between man in a more civilized state, as we may hope, even than the Caucasian, and some ape as low as a baboon, instead of as now between the Negro or Australian and the gorilla."
Thursday, 7 April 2011
Monday, 7 March 2011
1. After the Packers' Super Bowl victory, an exuberant Aaron Rogers Shook Hands with Everyone in the Stadium.What is wrong with them? Well, I don't expect anyone reading this not to have noticed that shaking 40,000 hands would take at least 10 hours, that $2 billion comes to about $10 per person or that Foreign Aid is such a tiny portion of the US deficit that even eliminating it entirely wouldn't make a big dent (this doesn't, of course, mean that it shouldn't be eliminated, only that if you're obsessed with the deficit, you have bigger fish to fry).
2. Experts Fear Total US Housing Costs (Rents plus Mortgage Payments) Will Top $2 Billion in 2011.
He then moves onto the following headline:
4. Number of Americans with Alzheimer's Believed to Be 5,451,213.The supposed problem?
4. The problem here is that the number is ridiculously precise. Definitions of Alzheimer's vary and it's difficult to determine whether a single individual is suffering from it, much less whether five million plus are. Such impossible precision is common.Well, yes, the number is ridiculously precise. No, no-one does think that we can measure the number of Americans with Alzheimer's to that degree of accuracy, but so what? If you do a survey of Americans, do some calculations, and your best estimate of the number of Americans with Alzheimer's comes out as 5,451,213, what number, exactly, does Paulos want you to report?
Assuming that you've done your sums correctly, 5,451,213 is an unbiased estimator of the number of Americans with Alzheimer's. Rounding your guess to 5.5 million does systematically worse than just reporting the estimator you got out of your calculations, so what exactly is the rationale behind it?
Yes, numbers like this should probably be reported along with some estimate of variance, and maybe it's a convention that we assume the number of signficant figures of a number to be a proxy for the size of its error bars, but it doesn't have to be that way: I look forward to a day when numbers like "5.5 million" get scoffed at by popular mathematics writers for being "overly round" or "not accurate enough".
Thursday, 3 March 2011
Now, as you can see all of the money in the system comes from the universities. Universities pay the wages of the researchers and the reviewers directly, and they pay the wages of the editors indirectly (through journal subscriptions). So, here's an idea; why don't the universities club together to buy the journals, employ the editors directly, and publish all the content for free?
Note that buying the journals doesn't cost the universities (as a group) anything in the long-run, as the entire current value of the journal companies comes from the amount of money they expect to be paid in journal subscriptions by universities in the future. And there's no need for the journals to charge "submission fees", as those were all being paid by the universities in the first place: they can just come out of the communal pot.
So far as I can see, there is literally no downside to this - assuming coordination can be achieved, you have the same universities paying the same amount of money to the same people to produce the same articles, but the articles are now all available open-access. I admit that "assuming coordination can be achieved" is a fairly hefty assumption, but given the massive upsides, why isn't anyone at least suggesting this sort of approach?
There seems to be a general trend towards open-access publishing anyway, which is a Good Thing, but I don't undestand why this model isn't a strict Pareto improvement on the current system.
Tuesday, 15 February 2011
From Wolfram's talk (trancript available here):
I want to see a completely renewed, changed math curriculum built from the ground up, based on computers being being there, computers that are now ubiquitous almost. calculating machines are everywhere and will be completely everywhere in a small number of years. Now I'm not even sure if we should brand the subject as math, but what I am sure is it's the mainstream subject of the future.From Lockhart's Lament:
The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs— you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.So, Lockhart thinks we should be teaching mathematics as an art form, and Wolfram thinks we should be introducing more computers into mathematics lessons so that people can concentrate on doing the bits that are actually useful. Which of them is right? Well... both, but mostly Wolfram.
The question we have to ask ourselves before we can even begin to compare the two essays is why do we teach mathematics at all? So far as I can see, the only sensible justification for having mathematics as a subject that everyone in the world should be taught up to a relatively high level is because it's useful. You can't do physics, or engineering, or any sort of science, or do derivatives trading, or even decide which mortgage to get, without knowing quite a lot of mathematics. For this reason, everyone should be taught the basic mathematics that they need to know in order to do these things (or the basics they need to know in order to learn the specific maths they wnat to use).
Note that one corollary of this mode of thinking is that most of the mathematics you learn probably shouldn't be taught in a maths class. It's much easier to learn how to get from acceleration to speed than it is to learn how to differentiate a function. Yes, it is useful to then point out the possible generalisations (getting from acceleration to speed is the same as getting from jerk to acceleration) but I don't see any reason why these topics can't be introduced concretely. I personally have serious trouble doing any calculus that I can't do using physical intuition, and I think it's fair to say that I am an above-average student when it comes to learning maths. Calculus should be taught when you're doing engineering, statistics should be taught when you're analysing the results of experiments, graph theory should be taught when you're trying to solve scheduling problems.
Lockhart, on the other hand, seems to think that we should be learning mathematics because, essentially, mathematics is awesome. I happen to agree with him that mathematics is an exceptionally beautiful art form. I'm happy to sit back and bask in the glory of Cantor's diagonalisation argument, or the ingenuity of Karp's reductions between NP problems, but I'm not sure that I'm willing to contend that everyone should be forced to. Yes, if you want to be a mathematician you have to learn that mathematics is actually an art, but most people who study mathematics don't want to be mathematicians, and most people who study mathematics *shouldn't* want to be mathematicians. For these people, learning about the art of mathematics is little more than an intellectual curiosity, on a par with learning about Titian or Shakespeare.
In other words, Lockhart is right, inasmuch as we want people to study mathematics for it's own sake. Wolfram is right, inasmuch as we want people to study mathematics because it's useful.
Now, I happen to tend strongly towards the idea that the only things we should be teaching in schools are things which are potentially useful, but that obviously isn't the prevailing wisdom - everyone in this country is still forced to do an English Literature GCSE. Lockhart-style mathematics is a perfectly good substitute for art class, or critical theory. Wolfram's mathematics is a necessary prerequisite for doing just about anything else.
Monday, 14 February 2011
A recent experience of mine in my own research seems particularly relevant here, and I'm going to try and explain what happened (this will, at the very least, satisfy my promise to Michael Brough at about the same time of getting some more serious mathematics into my blog). This is pretty much the first time I've ever tried to explain something technical in a non-technical setting, so let's see how it goes...
Friday, 11 February 2011
I have read quite a large percentage of the books that are currently widely available on the topic of writing mathematics, and several that aren't. I've also been involved in teaching the Mathematical Writing course run at the QMUL. There is one serious issue that I've encountered: the main problem seems to be that most people start with the assumption that you already know everything that you're going to write, and the basic structure of your proof, but don't know the best order to put the words on the paper. In my experience, this is far from true.
I'm currently writing up my thesis, and have several new ideas which I've never written down in complete formal detail before. This process is difficult. The ideas in are generally simple enough that I think I could explain them to reasonably bright 15 year old in about 10 minutes given a piece of paper, but sufficiently abstract that to write them down formally has involved pages of writing that even I barely understand. These are proofs that I can (and have) run through in the pub with non-experts on the back of a napkin, and they translate into 10 pages of symbols that even I can barely follow. (I'm working on a post containing an example, but it will require drawing some pictures, so I'll probably get round to it some time over the weekend).
It is all very well saying that one should write an informal verbal summary of what you are going to do before you do it, but when the formalism is so far removed from the key idea, this becomes difficult. Also, when you are writing a maths paper, you have to make damn sure that the "summary" is still technically accurate. There can be no hand-wavey 'look, it just works' and (importantly) no interaction with the audience - you dont' know if they 'get it', so you have to put down everything it might take for them to get it. In my experience, the heavy formalism has often meant that the proofs I write down end up so complicated that I'm not even sure I would 'get' them if I had to pick up the ideas directly from reading the papers. The formalism masks the intuition, and the intuition isn't quite formal enough to be appropriate for the paper.
This is an interesting topic. It's one that several people have no doubt spent a lot of time thinking about, but it's not one that seems to be discussed. Even in a mathematical writing course, the tendency is to focus on technicalities. To be fair, this is usually adequate for undergraduates, and you do need to get the technical stuff right even when writing down simple ideas, but this is something that needs addressing: exposition of mathematics is a difficult skill, and probably one that there's not enough focus on in training academics.
Maybe not everyone has this problem - maybe not everyone has elementary ideas which don't translate nicely into formalisms - maybe their ideas are inherently more technical, or maybe they're just better at finding formalisms than I am - but I do find it interesting that it's happened to me almost every time I've tried to write out a formal version of a proof that I've created myself.
PS - the vast majority of this post was written over a year ago - the posting of it was sparked by having a discussion with Andy in which he mentioned that he is currently having exactly the same problem, and by me finally figuring out how to access the drafts of my old posts...
My vision of a university was succintly described by David L. King, writing in The Times Higher on 9 April 2004. It is:
I've no idea what a "confederacy of self-seekers" is, so I'm not going to address that point. I don't think anyone believes in a professor as an entrepreneur, one thing that I think people can sensibly expect professors to be, at least in the current system, is teachers. Incidentally, I happen to think that Rosemary is a very good teacher, and one who thinks a lot about doing the best for her students, so I hope she won't be offended by any of this.
I believe that university students should be able to be confident that they are being taught by people who are immersed in the subject in other ways than teaching. I collaborate with a range of scientists on the design of their experiments and the analysis of their data, so I teach Statistics. I still prove theorems in Combinatorics and Algebra, so I also teach those subjects.
- “the belief in a community of scholars and not a confederacy of self-seekers;
- the idea of openness and not ownership;
- the professor as a pursuer of truth and not an entrepreneur;
- the student as an acolyte whose preferences are to be formed, not a consumer whose preferences are to be satisfied.”
The problem I have mostly is with the last point in the bulleted list: "the student as an acolyte whose preferences are to be formed, not a consumer whose preferences are to be satisfied.". Now, that might be a nice model for a university if the point of going to university was to learn about the subject you were studying. The problem is that for the vast majority of people who go to university currently this just isn't the case. They go because in order to get the better paid jobs you have to have a little piece of paper which said that you went to university. They go to university to get this piece of paper and, and I think this is the important part. This system is not the students' fault.
I personally would love to live in a world where universities could all be like Cambridge and Oxford, and where the vast signalling game that is university education as it exists now didn't exist. I admit that this world would probably be a world in which less "blue skies" research got done, and I find it hard to believe that would be a particularly bad thing.
Thursday, 10 February 2011
Now, whatever the merits of having someone check the ID of everyone entering a university building, whatever the point of this exercise is supposed to be, surely it is rendered *entirely* useless if you don't do it every day. If the purpose is to stop people stealing things (there are plenty of old computers which must be worth tens of pounds on the black market) then surely they'll just come in and steal them on the days when you don't do the checks. If the point is... well, I actually have no idea what else the point could be, it's a university campus, not a Government Intelligence building.
There can be literally no point whatsoever in having these security checks unless they're done regularly (in much the same way as there can be no point whatsoever having full-body scanners in airports unless you also perform cavity searches). Security which is this easy to circumvent is not security, it's theatre.
Tuesday, 8 February 2011
One catch, you can only get one of these cards by signing up for a twelve-month contract, and I have almost no idea what I'll be doing in three months time, never mind twelve. I therefore made the completely rational and considered decision to get the card anyway, and let future me worry about paying for it. I then figured that: hey, I only need to go about 20 times and it's paid for the entire year. I then figured: hey, why not go 20 times in the first 20 days, then I'll have 345 days worth of free cinema!
Yes, I realise that this doesn't actually make any sense, I do understand the concept of sunk costs, but it sounded like it might be fun, and I'm all for fun. So, in the past 10 days I have been to the cinema 10 times, to see 10 different films. After each one I took brief notes: my reviews come after the fold, along with my reflections on the somewhat ridiculous project.
By the way, for those of you who know me, and know that I'm currently supposed to be writing my thesis, and are wondering what impact this has had on my progress, well I can say for certain that I've not made any less progress in the past 10 days that I did in the previous 10. For those of you who know me a little better, no, that isn't actually a trivial statement.
Having a conversation, and then realising that you're providing amusement for everyone else on the carriage, as otherwise everyone just sits there in complete silence.
Finding exactly the right place to stand on the platform so that you don't have to walk at all when you get to your destination (and pondering where the optimal place to stand is when going to an unfamiliar destination station).
Waiting two minutes for the next train, and getting a seat instead of piling into a crowded carriage.
The sheer inanity of the announcements: "we're being held at a red signal", really? what colour do you use for "go"?
Saturday, 5 February 2011
Marginal Revolution linked recently to this paper which claims that overuse of Instrumental Variables (IVs) in econometric studies has led to them being less useful. So far as I can tell, the argument given in the paper is exactly backwards... what am I missing?
For the uninitiated, an IV is a variable which is correlated with the thing we want to study but is unequivocally not caused by it. Let's say we want to study the effect of x on y, then an instrumental variable z is useful if its only effect on y is through its effect on x. To use the language of Pearl's causal graphs, this is equivalent to saying that every path in the causal graph from z to y passes through x.
For example, suppose we want to measure the effect of a change in the price of apple on the demand for apples. It may seem difficult to do this directly, as an increase in demand is likely to lead to an increase in supply, and so price is affected by demand. A potential instrumental variable is the weather. If the the weather is favourable for growing apples then more apples will end up being grown, and this is presumably independent of demand - the increase in apples will have a predictable increase in the supply (and therefore the price) and we can measure the effect on demand.
Now, the authors of this paper claim that:
A Tragedy of the Commons has led to overuse of instrumental variables anda depletion of the actual stock of valid instruments for all econometricians. Each time an instrumental variable is shown to work in one study, that result automatically generates a latent variable problem in every other study that has used or will use the same instrumental variables, or another correlated with it, in a similar context. We see no solution to this. Useful instrumental variables are, we fear, going the way of Atlantic Cod.
As I said, I think this is exactly backwards. It is not the fact that new papers are produced which use these instrumental variables in a new context which introduces the latency problem: it reveals a latency problem which already existed. The previous studies were already invalid. The new studies just reveal the fact.
E.g. Imagine that people buy more apples when they have high levels of vitamin D in their blood (because apples are a substitute for fish oil). Then you have to correct for the effect of the Sun on vitamin D levels when you're using the sun as a proxy for apple demand. The problem here though, is that the weather conditions already weren't a good IV for demand in apples. The fact that a new study appears to demonstrate this is not a "Tragedy of the Commons" in any meaningful sense - the study has not been made any worse, we've just found out that it was bad.
On the other hand, Alex Tabarrok, who is more of an economist than I am, appears to be taking the paper vaguely seriously, and not to have noticed this. As I said before: am I missing something?
Wednesday, 2 February 2011
I can think of several reasons why they would exaggerate times: it's a pleasant surprise when your book arrives before it is supposed to (but then why not claim that all deliveries will take "up to 2 weeks"; it also covers them in case delivery takes longer than they might expect (but, again, why not give an even more exaggerated time). I can also think of reasons why they might want to introduce a "second-class" delivery service, in order to justify charging for their "first class" service (ie, the normal delivery rates that one would expect them to charge anyway). Similar to the way that second class customers on trains are deliberately inconvenienced in order to make first class more attractive. However I don't think this is what's going on here.
I think Amazon is indulging in some good old-fashioned price discrimination. As I said earlier, I'm a pretty regular customer; I order something on the order of £50 worth of books a month. As such, I know how long it takes for my books to be delivered, and I'm not even vaguely tempted to press the "standard delivery" option when it comes to the payment page. I'm exactly the sort of person who is most price-sensitive when it comes to deliveries, and I'm in a position to know that I can get the same service more cheaply.
Contrast Dear Aunt Doris, who orders a book off Amazon once a year for her nephew's Christmas present. She needs to get her delivery when she wants it, and she has no idea that "standard delivery" is the same as "super saver delivery". I mean, why would it be? That would just be crazy, surely? So Dear Aunt Doris pays the extra £2 for her one book a year, because she's not the sort of person who gets the chance to notice that she's paying for exactly the same service she could have gotten for free.
This is almost textbook price discrimination. You charge a lower price for the same service to those customers who are most likely to go look elsewhere, and most likely to be in a position to know exactly how much the service is worth. It's fun noticing phenomena you've read about in books in real life. I vaguely wonder if this theory of Amazon delivery pricing is testable (well, it obviously is if you work for Amazon...) but either way, I'm fairly confident it explains what's going on.
Monday, 31 January 2011
"If I'm selling to you, then I'll speak English. If you're selling to me dann muessen Sie Deutsch sprechen"
"If he knew, he's too evil to be Prime Minister, and if he didn't know, he's too stupid to be Prime Minister."
"The Noah Principle: predicting rain doesn't count, building arks does."
"It's hard to make a man understand something when his livelihood depends on him not understanding it"
Why am I not taking a StickK contract out? Partly because I haven't quite figured out their business model - how do they make money if the users don't pay them any? Partly because I'm pretty sure one has no control over who the money goes to if you do lose. And partly because I want to see if it's necessary. If you have a blog, and if enough of your friends read it, does that replace the need for StickK?
Monday, 17 January 2011
It took me about 10 hours to learn to touch type, very possibly less, and it must save me several hours a week in time spent not looking at the keyboard and, more importantly, not having to correct mistakes that I didn't notice because I was looking at the keyboard instead of the screen.
I propose that teaching kids to touch type would be more useful than literally everything that they learn in school after the age of eleven. Anyone got any counter examples?
Friday, 7 January 2011
With absolutely no fanfare, TFL have extended the YPR discount to off-peak single fares, as well as the previous discount on the off-peak daily cap. See here for details. So now an off-peak zones 1-2 single will only cost you £1.25.
For those of you who are wondering, the off-peak daily cap applies as long as your first journey in the day came after 09.30. That is, you don't have to make all journeys in off-peak times in order for the off-peak cap to apply. For single fares, peak is 6.30-9.30 and 16.00-19.00. (For anyone who wants to stay out late, the "day" finishes at 04.30 the following morning).
For some reason, this seems to have been completely buried by TFL, I even had to edit the wikipedia page myself just now to contain the relevant information! My post, which comes from a relatively tiny blog is still in the top 10 hits on Google for "Young persons railcard oyster", for some reason. This information could be saving quite a lot of people quite a lot of money if they knew about it. Please tell your friends!