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Timothy Gowers
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Timothy Gowers

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Yesterday a new world record for ski jumping was set. It's not a sport that hugely interests me, but there's something about this video that I find very compelling. About two thirds of the way through his jump, the Slovenian ski jumper Peter Prevc is clearly about to land, but then the laws of physics appear to be suspended so that instead of travelling in a parabolic arc, he continues in a straight line more or less parallel to the track and only very slightly above it. Of course, this must have a lot to do with aerodynamics but it is still extraordinary to look at.
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For those interested in the official specification of hill parameters: http://www.fis-ski.com/mm/Document/documentlibrary/Skijumping/03/19/96/ICRSkiJumping2014_English.pdf (pages 46-51).
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This is Christopher Gunning's wonderful obituary of my father, who died just under a month ago. I posted in November 2013 about what it felt like to turn 50. But that pales into insignificance next to losing my father, as those who have experienced both will know well. At some point I may try to say something about his interest in science, mathematics and computing, and how his scientific way of thinking deeply influenced his composition (not to mention deeply influencing my whole life). 
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sorry to hear the bad news. :(
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I thought I'd seen it all with apparent spirals that turn out to be circles. But this piece of weaving that isn't weaving is pretty disturbing. Sorry not to be able to make it full size but if you click on the link you'll see what I'm talking about.
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I've since forgotten the source (appropriately enough) but there was some study [citation needed] indicating that when possible we don't actually make new memories, we just relabel and patch together old ones. It saves space to just recontextualize. I walk basically the exact same path to work every day, and at the end of it I have a "memory" of walking to work but no idea of when that memory was created. I noticed a similar thing happening during highschool, so it kicks in at least by then.
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Another very nice simulation via +François Dorais. This one demonstrates that very small and apparently harmless amounts of individual bias can have big and harmful effects when put together. Roughly speaking, if nobody wants to be in a small minority in their immediate surroundings, the global effect is a large amount of segregation. With the simulations, you get to play with various parameters and see what happens.
 
"Parable of the Polygons" is a playable post on how harmless choices can make a harmful world.
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This model is known as "Schelling segregation" and it's been studied mathematically in some detail. See e.g. my blogpost here: http://richardelwes.co.uk/2013/06/18/schelling-segregation-part-2/
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Michelle Brook has just published figures, obtained by FOI requests, for what British Russell Group universities have spent on journals published by Wiley, OUP and Springer for the last five years. As one might expect, it's a lot, but less than they pay to Elsevier. In the blog post linked to below, Michelle also discusses some of the details, and also points out reasons for being cautious when interpreting them. The figures themselves are linked to from the post. 
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Timothy Gowers

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A couple of days ago, much amusement was caused after a meeting between the Greek finance minister, Yanis Varoufakis, and his German counterpart, Wolfgang Schäuble, when Schäuble summed it up by saying that they agreed to disagree, and Varoufakis summed it up by saying that they did not even agree to disagree.

This reminded me strongly of the liar paradox, or more precisely its cousin that consists of the following two sentences.

1. The next sentence is true.

2. The previous sentence is false.

That reformulation is considered important by philosophers because it shows that certain explanations of the liar paradox are insufficient. The explanations in question are ones that say that it boils down to problems with the notion of a sentence that refers to itself. But these sentences do not refer to themselves, and are often unproblematic. For instance, consider the following two sentences.

1. Mitt Romney is the president of the United States of America.

2. The previous sentence is false.

I tried thinking about whether the statements by the finance ministers could be turned into a genuine liar-type paradox, and have run into problems, though I haven't managed to convince myself that it cannot be done. Here are a couple of failed attempts.

1. This sentence agrees with the next sentence.

2. This sentence disagrees with the previous sentence.

This isn't a paradox, because the first sentence is false and the second one is true, or at least we can consistently take that to be the case. 

I've tried a few other things, and am coming to the conclusion that we probably can't make a paradox if we just talk about agreement of sentences and not their actual truth values. For instance, consider the first sentence above in a slightly different form.

1'. The truth value of this sentence is the same as the truth value of the next sentence.

If that sentence is true, then the next sentence is true, and if that sentence is false, then the next sentence is again true. So we can decide that the next sentence is true and then whatever it implies about the truth of 1' will be compatible with that decision. A similar problem occurs with this sentence.

1'. The truth value of this sentence is different from the truth value of the next sentence.

This time we obtain that the next sentence is false either way, so again we can choose the truth value of 1' to make everything work.

Going back to the Greek and German finance ministers, let's assume that one of the items on their agenda was the question of whether they agreed to disagree. The German minister said yes, and the Greek minister said no. So they disagreed about it. But did they agree to disagree about it? A reasonable interpretation of the evidence is that they did not agree to disagree about it (or else the Greek finance minister would have said that they had agreed to disagree), that the German finance minister is not telling the truth, and that the Greek finance minister is telling the truth. But another interpretation is that they did agree to disagree, and it is the German finance minister who is telling the truth.
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The paradox of the liar vanishes if we treat truth and falsehood as a local and temporal condition or value and not some Platonic universal.
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I can't say I'm too impressed by the final-year economics students at Sheffield referred to in this article. The question they complained about is one that you don't need to have learnt any serious economics to be able to do: common sense and A-level calculus are sufficient. Basically it's what Americans call a word problem: once you've stripped out the words, it asks you to maximize sigma N^{1/2} - gamma N^2, which I would hope an about-to-graduate economist would know how to do. With the words back in again, N represents the number of people in a city, sigma N^{1/2} is the output per person, and gamma N^2 is the "coordination cost" associated with the city. The question also asks for a bit of qualitative discussion, such as why the exponent of N in the coordination cost might be expected to be greater than 1. (Answer: because the number of other people each person have to coordinate with will tend to grow as the size of the city grows. The exponent of 2 would suggest that it grows linearly, which seems highly implausible, but the question doesn't ask you to defend that.)
Final year economics students at Sheffield University are furious after an exam this week contained "impossible" questions.
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Leaving aside the question of the merits of their cause, I am curious whether these students calculated the expected return of complaining so publicly before they did it. It might have been better for everyone involved to keep it quiet.
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The Universities Superannuation Scheme, the fund that pays pensions for university staff in the UK, announced a few months ago that it had a huge deficit and that changes were going to be necessary, which, to put it bluntly, mean that we'll get smaller pensions than we thought we were going to. I'm amazed that it's legal -- I thought I was entering into some kind of contract when I signed up for the scheme -- but I suppose there must be some small print somewhere that enables them to do it. 

Recently they put out a document that has a FAQ. One of the answers they give could, it seems to me, be abbreviated to "yes". The question is "Will I lose my final salary benefits?" The answer they give is this.

The final salary section of the scheme will close, but existing final salary benefits will be protected at the point of change, calculated on pensionable salary and service at that date and increased each year in line with the CPI.

Suppose, then, that you are in mid-career. Then you will have put in several years of pension contributions on the understanding that your reward will be 1/80 of your final salary for each year's worth of pension. Now, if I understand correctly, the reward will for those contributions that you made on that understanding in fact be 1/80 of your current salary, adjusted for inflation, which is likely to be considerably less.

I'm posting this not because I have a deeper point to make, but simply to urge UK academics to look closely at what is on offer and not be fooled by the gentle sounding language in which it is explained: these changes are likely to make you thousands or even ten thousands of pounds worse off per year when you retire.

They promise to provide an "online benefits calculator" from next Monday. I'm guessing it will tell you what your benefits are likely to be, but not the amount by which they are less than the benefits you would have received under the current system. But we shall see about that.
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"I suppose there must be some small print somewhere that enables them to do it."  I wouldn't be too sure.  In the US it's become quite common for governments to renege on payments owed to workers, on the grounds that they just don't have the money.  Sometimes the courts allow this, sometimes they don't.
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I've just found this in my mailbox. One of the things that makes me slightly uncomfortable about boycotting Elsevier is the fact that I am thereby not performing a service to my fellow mathematicians that they may need. I try to make up for it in other ways, but nevertheless it is reassuring to know that despite my refusal to do any work of any kind for Elsevier, they nevertheless regard me as a valued referee. Unfortunately, I'm not going to be at the Joint Mathematical Meeting, so am unable to accept the invitation to share ideas with other valued Elsevier referees. Another time maybe.
As a valued Referee for our journals in Mathematics and Statistics, we would like to take this opportunity to thank you for your efforts. Your dedication and commitment are very much appreciated by both the Editors of the journals and by us. We are organizing a Referee Reception at the ...
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I got the same.  There is an "unsubscribe" link at the end, which hopefully worked...
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Something good was announced today in George Osborne's autumn statement. A pet peeve of mine is taxes or benefits that do not depend continuously on circumstances. On the benefits side, for instance, there are situations where to obtain a particular benefit you have to demonstrate that you are disabled, but it's an all-or-nothing benefit, so person A can be very slightly less disabled than person B and get no benefit when person B gets the full benefit.

A particularly notable discontinuity occurred with stamp duty, a tax on buying houses (and a somewhat unfair tax as it penalizes, for no obviously good reason, people who have to move house a lot -- one could get round that by taxing the difference in price between someone's old house and their new house). It used to be the case, and in fact will continue to be the case for the next two and a half hours or so, that the rate of stamp duty was a percentage of the entire price of the house, and that percentage jumped when the price of the house passed certain values. For example, a house costing £200,001 would be liable for a lot more stamp duty than a house costing £199,999. From midnight tonight, that will finally be replaced by a sensible system, where the function (house price) --> (stamp duty) is continuous. It will be zero up to £125,000 then have derivative 0.02 up to £200,000, then derivative 0.05 up to £925,,000, then derivative 0.1 up to £1,500,000, and derivative 0.12 beyond that.

One effect is that the total amount raised in stamp duty will go down by quite a bit. I'm not saying that that is a good idea -- in general I'm in favour of tax rises rather than spending cuts to deal with the deficit, though maybe I'd prefer to see different tax rises such as income tax and anything that is closely linked to carbon emissions (another of Osborne's announcements is a cut on flight tax for young people) -- but the general principle of continuous dependence of tax and benefits on circumstances is a good one and it's good to see it applied here.

Another principle is that the tax and benefits system should be organized so that you always gain enough by working to make work worth doing. Successive governments have tried and failed to achieve this. I had hopes of Ian Duncan Smith, but he seems to have messed things up pretty badly, so we may have to wait several more years for it.
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A term I've heard for the cutoff point of a benefit program, or multiple programs one is signed up for or might possibly sign up for, is a "welfare cliff."
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This is gorgeous, if you haven't seen it. Well worth a look even if you skip most of the text and just look at the animations.
 
Thanks to +Peter Krautzberger for pointing this wonderful visualization by Mike Bostock.

http://bost.ocks.org/mike/algorithms/
Algorithms are a fascinating use case for visualization. To visualize an algorithm, we don't merely fit data to a chart; there is no primary dataset. Instead there are logical rules that describe behavior. This may be why algorithm visualizations are so unusual, as designers experiment with ...
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Thanks for sharing. This is amazing
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Latest update on the intellectual development of my two young children. As I've said before, with the older one (who has just turned seven) I try to encourage his enjoyment of mathematics, but not to the point where I interfere with his discovering things for himself. From time to time this policy bears fruit. The most recent example was when he announced that to multiply a number by five you could multiply it by ten and take half the result. I suggested that he try it on a couple of examples such as 7 and 9, which he did successfully. I then asked him why his method worked. His answer was not wholly clear but appeared to be not much more than the observation that 15 is half of 30. What pleases me about it is that I think that to be good at mathematics it is a huge help to be friends with numbers. For example, if somebody applying to Cambridge didn't know that 512 was a power of 2, I would be worried about that person. This observation of my son's was good evidence that he is developing that kind of friendship.

It's an interesting question whether his justification of his rule constitutes a proof. At first one would want to say that it obviously doesn't: it's just checking one instance. But my son has no real concept of a general mathematical argument, so it may be, and I think probably is, the case that he somehow intuited that this observation was more than just a coincidence, but lacked the means of articulating that intuition. An interestingly similar phenomenon happened in ancient mathematics, where we see evidence that people could solve quadratic equations and find Pythagorean triples, but no evidence that they had a language in which to express their methods in abstract form. If I understand correctly, historians of mathematics argue that it is wrong to claim that they did not have a good general understanding of what is going on, whether or not they had anything that resembles a modern proof.

Another thing I might mention is that a transition has occurred in my son that seems to occur in everybody but for no clear reason: he now knows, and seems to find it obvious, that multiplication is commutative. I haven't asked him why, and am not sure I can without interfering in the way that I don't want to, though I am curious to know whether he has anything like an explanation in mind or whether this is another case of observing several instances and inducing the rule.

Meanwhile, also within the last few weeks, something seemed to click in my daughter, who will be four soon, and now she completely gets simple three-letter words. There are several things that have changed.

1. She is much more reliable (but still not 100%) if you ask her what something like P-O-T spells.

2. She can now do the reverse process: that is, she can spell words like "pot".

3. She likes inventing her own questions of the "What does P-O-T spell?" variety.

4. She has made short words using magnetic letters on our fridge.

The plan now is to let all that consolidate for a little while, and also try to iron out a few kinks in her letter recognition, before moving to the next stage and teaching her about letter combinations such as CH, EE, OO, AI, etc. After that it will be on to common short words. From past experience, I know that these stages may take months each.
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