On (Not) Learning Things

I have to learn new things constantly in order to keep myself entertained. I’m rarely interested in doing things that don’t allow me to learn, and when I look back at things I have done that don’t teach me anything, I tend to feel that I have missed a good opportunity to do something sensible with my time. But it is easy for me to get confused and think that I am learning something when I am actually not.

Last weekend, I went to New York for the purpose of teaching a class on wishful thinking in mathematics, but it ended up being canceled due to too much snow. So I went to the Morgan Library for the hope of getting a good educational experience. Going to an art museum seems like a good thing to do in order to try to learn something; after all, that’s one of the things that society tells us that cultured people are “supposed” to do, and shouldn’t being “cultured” be correlated with learning things? So I thought, and so I have repeatedly thought, but now I’m reconsidering (and looking for other opinions as well).

I don’t really know what the goal of learning things ought to be, but I do have a way of determining whether I have done so: is it possible for me to write something nontrivial based on my new information that I could not have written before? If I spend two hours walking around a museum looking at paintings, then I ought to be able to say something new about art and, perhaps, write a page of two that I couldn’t have written before. But I usually can’t, and when I can, it’s generally because I’ve looked really closely at a few types of paintings and determined which sorts of brush strokes tend to result in more or less realistic-looking paintings. And I doubt that’s what I’m supposed to get out of going to an art museum. (Or is it? What am I supposed to learn from going to art museums?) I also doubt that what I’m supposed to learn from art museums is that looking at paintings teaches me nothing about art but something about how I ought to spend my time and organize my life, but that’s what happens to me some of the time.

It would probably be worthwhile to go through all the activities that I do in which I expect myself to learn something and verify that I am learning. I could try to do that by writing about everything that I do, but I would expect that to be a huge time and energy sink that would take away from other activities. I probably ought to do that more than I do (I stopped writing here for a while in my last year of graduate school, since I was already spending a lot of my time writing my thesis and didn’t want to write recreationally as well), and I will try to make it a habit to do so. However, it would also be good to have some heuristics to suggest that I am learning, even when I don’t go through with the exercise of verifying it. I could, perhaps, simply have a conversation with someone else about my experiences. That wouldn’t force me to organize my thoughts in the same way as or as carefully as writing would, but it might take less energy, and it would also have the benefit of giving me immediate feedback: if I believe I have learned something but I am wrong, I have a good chance of figuring that out as fast as possible. (And that’s something I always want: I hate being wrong, and in fact I hate being wrong so much that if I ever am, I want to know as soon as possible so that I can change and stop being wrong. I might be temporarily offended, but that’s a small price to pay for getting better at being right.)

However, asking other people to review everything I do is a lot to ask of others, even if I spread it out over quite a lot of people, and I don’t like being a burden to others any more than is necessary.

Also, much of the time, it appears to be the case that there is something interesting for me to learn, but I simply lack the creativity to work out what it is. How do I fix this?

I invite suggestions; please tell me how I can learn more and spend less time only pretending that I am learning.

My most stressful game

This past weekend, I went to Burlington to play in the Vermont Open. I had a fairly good result, scoring 2 wins, 2 draws, and 1 loss. My most interesting, and also my most stressful, game was the one from round 4, against Vermont’s top player, David Carter.

Simon Rubinstein-Salzedo (2013) vs. David Carter (2195), Vermont Open, Round 4.

1.e4 e6 2.d4 d5 3.Nc3 Bb4

The Winawer is one of the sharpest opening variations in all of chess. It can be a lot of fun to play, and I enjoy it from both sides.

4.e5 c5 5.a3 Bxc3+ 6.bxc3 Qc7

This move is annoying. The main move is 6…Ne7, when the sharpest and most common line continues with 7. Qg4 Qc7. Black sacrifices some pawns on the queenside for good piece play and (perhaps) an attack.

Black’s move could be an attempt to transpose into this line after 7. Qg4 Ne7, but there are other possibilities as well. In particular, after 7. Qg4, black could play 7…f5 or 7…f6. Both of those are quite common variations, but I didn’t know the theory for them. Thus, I thought it would be wise to stay away from them against a (presumably) well-prepared opponent. So, I just continued to develop normally.

7.Nf3 b6

This is quite a common idea in many lines of the French. Black’s biggest problem in the French is eir light-squared bishop. Thus, black does well to trade it off quickly. Of course, white should try to avoid this possibility, and I did the only really sensible maneuver that does avoid it.

8.Bb5+ Bd7 9.Bd3 Ne7 10.0-0 c4 11.Be2 Nbc6 12.a4

This move looked good at the time, and in fact it probably actually is good. The d6 square looked like such a great square to plant my bishop after I get in Ba3 and Bd6. The problem is that black gets to make moves too.

12…f6

And now I started panicking. I really have to take on f6, but then the g-file gets opened, and my king might be strongly attacked. I thought at this stage that I was probably completely lost, having fallen into some well-known (to my opponent) opening trap and would have to struggle to survive to move 30. Fortunately, this was a needlessly pessimistic view, and white is just fine.

13.exf6 gxf6 14.g3

Apparently this isn’t the sort of move that people usually make, for here my opponent started thinking for the first time. I wanted to play Bf4 next and try to control one useful diagonal, although clearly black will not let me. However, white has possibilities other than Bf4 in this position, and even ones which are consistent with 14. g3.

14…Ng6 15.Re1 0-0-0 16.Bf1

My idea was to follow up with Bh3, hitting the e-pawn, but later I opted for a different setup instead.

16…h5 17.h4 Rdg8

And here they come. I expected that black would just start piling his pieces up around my king, and that seemed very scary. Maybe I should be okay, as my king has many defenders in the area, but one is generally worried when there are so many angry pieces hovering around.

18.Kh1 Rf8

This move is completely incomprehensible to me. In fact, I guessed very few of my opponent’s moves in the next stage of the game. Generally, I get most of them right.

19.Bg2

My idea had been to play 19. Bh3, but after 19…e5, black probably benefits from having the position opened up more. As a French player myself, I know that nothing makes a French player happier than seeing the central pawns move down the board. So, I wanted to discourage that.

19…Kb7 20.Ba3

I didn’t really know what to do here. It’s not easy to find an ideal setup for my pieces, and attacking the black king also doesn’t look easy. I figured I’d at least try to rearrange my pieces a bit somehow.

The difficulty in positions like this one is that white’s strengths are not immediately relevant: I have the two bishops, perhaps good scope for my pieces once a few of them get traded, and perhaps more space. On the other hand, black is very active and has what seems to be an obvious plan: pile up on the g-file and attack my king.

20…Re8 21.Qd2

One idea I was considering was to play for something like Bb4 and a5, breaking open the queenside. But if I move my bishop to b4, black will just play a5, shutting down my play on that side of the board for a while. And then it will take me a while to get my bishop back to a sensible location of the board.

21…Nge7 22.Reb1 Ka8 23.Bc1 Nf5

Now, the move I wanted to play was 24. Bf4, and I tried a while to make 24. Bf4 e5 25. dxe5 fxe5 26. Nxe5 Rxe5 27. Bxd5 work. Unfortunately, it doesn’t work at all, so I gave up and looked for something else. The next move I might want to play is 24. Qe2, but that loses to 24…Qxg3!!, so I had to avoid that as well.

24.Qd1 Rhg8 25.Bf4 Nd6

That was a big surprise. I thought he would play Qd8 and transfer the queen to the kingside. My first thought was to try to exploit the pin on the knight with Qc1-a3, and I thought for a while that this was an unstoppable winning idea. Fortunately I then woke up out of fantasyland and realized that black would just play e5 after my queen got to a3, and then my queen would look really stupid on a3.

26.Qe2 Rg4

Okay, so black wants to sacrifice the exchange on f4. I didn’t know which one of us would benefit from that sacrifice, but I thought that at least I could encourage it under slightly better circumstances for me. Also worth noting was that the time control at move 40 was looking perilously far away, even though I still had half an hour or so remaining at this point.

27.Nh2 Rxf4 28.gxf4 Nf5 29.Qxh5 Qxf4 30.Bh3

 

Black was threatening 30…Qxf2 followed by 31…Ng3#. So, my first inclination was to defend with 30. Rf1. But then I thought I could just trade off the knight, and then black would have fewer attackers in the vicinity.

The computer, however, does not like my move and instead recommends 30.  Qh7 Qxf2 31. Ng4 Ng3+ 32. Kh2 Qf4 33. Qxd7 Ne2+ with a draw. But black could also try 30…Rd8, and maybe black is a little bit better.

30…Qxh4

This move was quite shocking for me. I thought black would play 30…Nxh4 or 30…Qxf2, since it doesn’t make so much sense to trade queens while attacking. The best line seems to be 30…Qxf2 31. Bxf5 exf5 32. Rg1 Re2 33. Nf3 Rxc2, with a small edge for black.

30…Nxh4 turns out to be less successful, as 31. Rg1 Qxf2 32. Raf1 Qxc2 33. Qxh4 Qxc3 34. Rg7 looks good for white: black’s pawns aren’t sufficiently far advanced as to be dangerous, while white’s pieces are actively placed.

My opponent, however, saw the logjam of pieces on the h-file and thought that it would cause me tactical problems once the queens were traded. I had thought about this as well and was worried about it, but I didn’t see anything concrete, so I suspected that I wasn’t in any danger.

31.Qxh4 Nxh4 32.Rg1 Rh8 33.Rg7 Bc8 34.Rag1 Nf3 35.Nxf3 Rxh3+ 36.Kg2 Rh8

Now white is winning. The question is how to convert this position most effectively. I had two ideas, based on two different principles. My first idea was to double my rooks on the 7th rank, since they can be quite dangerous there, and I might be able to arrange to tie up black’s pieces enough that I can get my knight in and win a few pawns.

My other idea was to trade a set of rooks off. The principle behind this idea is that it is beneficial to get rid of redundant pieces when there is a material imbalance. The point is that my two rooks serve the same purpose as each other, so the second rook might be slightly less valuable to me than to my opponent.

With these two principles at odds, I had to choose which one to follow. I decided to follow the first one for now and try to double on the 7th, but I was prepared to change that at a moment’s notice.

37.Kf1 e5

And I meant a moment’s notice. With this change in the pawn structure, black can no longer defend the f6 pawn. So, it’s time to trade rooks and attack some pawns.

38.Rg8 Rxg8 39.Rxg8 Kb8 40.Rf8 e4

And we made time control, although it wasn’t a serious issue in the end; the last several moves were fairly quick.

I looked at 41. Rxf6 exf3 42. Rxc6 Bd7 43. Rf6 Bxa4 44. Rxf3 for a while. I think it should be a fairly straightforward win, as I can get my king over to the queenside in time to stop the a-pawn from causing trouble, while I have my own passed f-pawn. But why bother getting fancy and risking it?

41.Nh4 f5 42.Nxf5 Kc7 43.Rf7+ Bd7 44.Ne3 Kd6

 

And now I had a long think, trying to work out whether the knight and pawn endgame was winning. If it is, that’s definitely the way to play. But if not, it would be very sad to sacrifice the exchange back and only get a draw. I concluded that it was indeed winning, although I missed some ideas.

In fact, I now think that black can draw the knight endgame. But we’ll get to that shortly.

45.Rxd7+? Kxd7 46.Nxd5 Ke6

Forced, or else 47. Nf6 picks up the e-pawn.

47.Ne3 Na5?

I thought this move was required in order to hold onto the c-pawn, but 47…Ne7! is much better. I guess we both missed this possibility. In fact, I cannot see how to win after 47…Ne7. One possible line is 48. Ke1 Nd5 49. Kd2 Nxe3! 50. Kxd3 Kd5 51. Ke2 a5!!. This last move prevents the black king from getting through on the queenside, and the kingside is too far away: if the white king tries to get through on the kingside, black will play b5! and promote the a-pawn and win.

So, instead white might try to play 51. a5, with the idea of breaking up black’s queenside pawns and then collecting them with the king. The problem is that the black king can chase the white f-pawn and promote the e-pawn. For example, play might continue 51…bxa5 52. Kd2 Ke6 53. Kc1 Kf5 54. Kb2 Kf4 55. Ka3 Kf3 56. d5 Kxf2 57. d6 e3 58. d7 e2 59. d8Q e1Q, and the queen endgame is surely drawn.

Alternatively, I can take the c-pawn immediately with 48. Nxc4, but black just plays 48…Nd5 and 49…Nxc3, and it’s hard to see why white should expect to win that.

48.Kg2 Kf6 49.Kg3

Gaining a tempo.

49…Kg5 50.f3 exf3 51.Kxf3 Kf6 52.Ke4 Ke6 53.d5+ Kd6 54.Kd4 a6 55.Nxc4+?!

This move wins, but 55. Nf5+! wins much more easily. Play can continue 55…Kd7 56. Ng3 Kd6 57. Ne4+ Ke7 58. Ke5 Nb7 59. d6+ Kd7 60. Kd5 a5 61. 62. Nf6+ Kd8 62. Kc6 Nc5 63. Kb5 Nb7 64. Ne4, eventually winning all the pawns.

55…Nxc4 56.Kxc4 b5+?

The most testing line, by far, is 56…Ke5!. Then I have to find 57. d6! Kxd6 58. Kd4 Kc6 59. c4 Kd6 60. c5! bxc5 61. Ke4! Kc6 62. c4! Kd6 63. Kf5 Kc6 64. a5 Kd6 65. Kf6, and I’ll eventually win both of the black pawns. But after 56…b5+, it’s trivial.

57.axb5 axb5+ 58.Kxb5 Kxd5 59.c4+ Kd6 60.Kb6 1-0

It wasn’t a perfect game, and trading down from a winning late middlegame/early endgame to what now appears to be a completely drawn knight and pawn endgame was inexcusable, especially after spending 15 on that move. But I’m mostly pleased with my play. However, I’m probably going to dump the 7. Nf3 line against 6…Qc7, because it just seems too easy to find ideas for black and too hard to find them for white. Also, I’ll probably get a heartattack if I keep trying to play games like this. (Well, either that or I’ll get used to them and they’ll stop causing me such anxiety.) I was a wreck until at least half an hour after the game had finished. That couldn’t have helped me much in the next round, where I lost against a master without much of a fight.

I shall have to be contented with a tulip or lily

I came to the Labor Day tournament this year full of high expectation. I’d just done a large overhaul of my openings and was feeling quite well-prepared.

In my first game, I faced Philipp Perepelitsky, whom I had known since high school. (Philipp beat me once in a high-school league game back in 2002. Back then, Philipp was rated around 1800, and I around 1400.) I played his (very) identical twin brother Edward in February, and we got a draw in a poorly-played game.

Simon Rubinstein-Salzedo (2047) vs. Philipp Perepelitsky (2151), Round 1

1. e4 d6 2. d4 Nf6 3. Nc3 g6 4. Be2

Well, I didn’t prepare for the Pirc, so already I was making it up. I wanted to play 4. Be3 but then thought that 4…Ng4 might be a bit annoying, so I might as well put a stop to that move first.

4…Bg7 5. Be3 Nc6

I was surprised to see this move; the knight gets kicked around while white gains space. It’s probably okay, but I was under the impression that black intends to play …c6 instead of …Nc6 in the Pirc.

6. d5 Ne5

I’d love to play 7. Nf3 here, but after Neg4, I thought black would be quite happy. In retrospect, this seems like a stupid thing to have been concerned about: 8. Bd4 c5 9. dxc6 bxc6 10. h3 Nh6 looks excellent for white.

7. h3 c6 8. dxc6 bxc6 9. Nf3 Qa5 10. Bd2 Qc7?

After 10…Nxf3+, black has equalized. Now, white is close to winning.

11. Nxe5 dxe5 12. 0-0 0-0

White has a position with no weaknesses, and all the white pieces have at least decent scope. On the other hand, black has weak pawns on a7 and especially c6, and the bishop on g7 has nowhere to go. My plan is to get my pieces to good squares on the queenside, and especially to put either a knight or bishop on c5, and start pressuring the c6 pawn.

At this point during the game, I thought it might make more sense to neutralize some of black’s pieces on the kingside, so I played

13. Bg5.

This move is okay; my idea was to prevent the knight from moving to a better square. (Black would probably like to play something like Nd7-b6.) But the knight isn’t the most dangerous black piece at the moment, so I should discourage play with a black rook on the d-file instead. 14. Be3 with the idea of Bc5 is better.

13…Be6 14. Qc1 Rfd8

Black wants to place the rook on d4 to gain counterplay against the e-pawn. I can stop this by playing f3 at some point, but then black can play Nh5-f4, so I’d prefer not to have to do that.

15. Be3

At this point, I was expecting my opponent to play 15…Rd4 anyway; if 16. Bxd4 exd4 17. Na4 Nxe4, I think black has fantastic compensation for the exchange: he has managed to get his bishop on g7 active, and it’s now a strong piece. Furthermore, black’s center pawns are now strong; it’s hard to stop ideas of c5-c4. So, on 15…Rd4, I was planning to play 16. Bd3 (now threatening to take on d4). After 16…Rd6 17. Na4 Rad8 18. Nc5, white is doing well.

15…Rab8 16. Bc5 Ne8

Black’s plan is to play Nd6, followed by either Nc4 or Bc4, getting a bit of activity. I thought about playing 17. b3 in order to control the c4-square, but in the end I decided on

17. b4,

which I think is a stronger move. My idea was to fix the pawn on c6 as much as possible so that I can attack it and win it when the time is right. This move also has the advantage of making the rook on b8 less powerful.

17…Nd6 18. Qe3 f5 19. f3 f4 20. Qf2 a5 21. a3 Bf6 22. Rfd1 Kh8 23. Na4 Nc8 24. Bd3 g5 25. Qe2 Nd6 26. Qf2 Nc8

I didn’t know how to make progress here if my opponent “passed” for the rest of the game (perhaps, playing Kg8-h8 endlessly). But the computer suggests that I play for bxa5; for instance, I can play 27. Ba6 h5 28. Bxc8 Qxc8 29. Ba7 Rxd1+ 30. Rxd1 Rb7 31. bxa5 Rd7 32. Rb1, and white has a delightful position. It’s not completely straightforward to win, since black is going to open up the kingside and develop threats, but with careful play, white should be winning.

27. Be2 Rxd1+ 28. Rxd1 axb4 29. Bxb4 Nd6 30. Nc5 Bc8

At this point, I felt much more comfortable about my position than I had at move 26. I have a passed pawn, a weakness on c6 to pressure, and black’s dark-squared bishop is still dismally placed.

31. Na6

My opponent looked a bit surprised when I played this move. I guess that’s understandable: the knight had a great square on c5, and the bishop was serving a purely defensive role on c8. But getting the two bishops was appealing to me, and I also reasoned that the queen would be as well-placed on c5 as the knight had been before.

31…Bxa6 32. Bxa6 Rd8 33. Qc5 Nf7 34. Rxd8+ Qxd8 35. Qxc6

And the rest should be easy, with two passed pawns on the queenside. Right? (Apparently not.)

35…Qd1+ 36. Bf1?!

I saw that after 36. Kh2 g4? I had 37. Qc8+ and 38. Qxg4(+), but I rejected it for some reason I now cannot understand. I think that’s the most sensible way to play though: I should then continue with Bd3!, cutting the queen off, and then try to get in Qg1 with a queen trade if possible. Black will make it difficult to do these things, but they might be possible.

36…Kg7 37. Qc3 h5 38. Qd2 Qa1 39. c4 e6 40. c5 Bd8

This can’t be bad either. The problem is that black’s king is frustratingly safe. Another problem is that 41. c6?? loses to 41…Bb6+, so that makes it more difficult to advance the c-pawn.

41. Kf2 g4 42. hxg4 hxg4 43. fxg4

At this point, I didn’t believe that black had any counterplay to speak of. And, in fact, that’s correct, assuming that I avoid being a complete idiot. (Sadly, I proceeded to fail spectacularly at that task from this point on.)

43…Qb1 44. Qd3 Qc1 45. Kg1 Qb2 46. Qc3 Qb1 47. c6 Bc7 48. Ba5 Bb8

So far, I’ve still been doing everything right, but now black threatens 49…Ba7+. The best move, which I strongly considered, is 49. Qc5. I was scared, however, of putting my queen and king on the same diagonal, where the queen might get pinned by the bishop. But there’s no way for black to do that. I was starting to get low on time here, though, so it’s somewhat understandable that I didn’t want to enter into something where I thought I might fall for a cheap tactic.

One possible continuation is 49. Qc5 Bd6 50. Qc4 Bxa3 51. c7 Nd6 52. Qxe6 Bc5+ 53. Kh2 Qxf1 43. Qxe5+ Kg6 55. Qh5+ Kf6 56. Qxc5 Qd3 57. e5+ Ke6 58. Qxd6+ where black’s only move is to resign.

But instead, I played the hideous blunder

49. Qc4??,

and after

49…Ba7+

I can get a draw with 50. Kh2. But, not sensing any danger, I played

50. Kh1??,

completely missing black’s next move.

50…Qb8!

with the threat of Qh8#!

51. Qd3 Be3 52. g3

I can prolong things with 52. Bd8, but that loses as well.

Qh8+ 53. Kg2 Ng5 54. Qd7+ Kg6 55. Qxe6+ Nxe6 56. gxf4 Nxf4+ White resigns

A horrific end to what seemed certain to be a well-played win for me against a rather strong player. On the plus side, though, I get to use a G&S quote for a post title.

I was devastated by this loss and proceeded to play uncharacteristically bad chess the rest of the tournament, losing 25 ratings points in the process. But that happens sometimes. I’ll try to recover my form in the next one!

Book review: Good and real

We don’t have to look very hard to find aspects of the real world that appear at first glance not to make sense or that are perplexing. With careful analysis, however, we can frequently make sense of weird scenarios.

Gary Drescher’s book Good and real is devoted to demystifying some of these paradoxes in as clean a manner as possible. The book is divided between two classes of paradoxes: those arising from physics, and those arising from ethics. According to Drescher, though, these two classes actually have a sizable overlap.

Early in the book, Drescher discusses a paradox arising from looking at mirrors: it appears that mirrors do not switch up and down, but they do switch right and left. That is, if I were to look at myself in the mirror while wearing a watch on my left hand, I would find that reflected-Simon is wearing a watch on his right hand. However, he would not be standing on his head. However, this seems unlikely to be the end of the story: how can the mirror favor one axis parallel to its surface over another?

Also, we can see by experiment that this can’t possibly be right: if I lie down facing the mirror so that my left hand is on the floor, then reflected-Simon’s right hand is on the floor; however, his head is on the same side as my actual head. In short, something different is going on. Actually, the mirror doesn’t flip the left/right axis or the up/down axis; it flips the axis perpendicular to its surface (the front/back axis). However, our method of interpreting what’s going on suggests that it flips the left/right axis when we look at our own reflections because that’s the only axis of near-symmetry that we have. Drescher’s explanation for what’s going on isn’t revolutionary, but I found it well thought-out.

Less familiar to me was his discussion about the asymmetry of time: why does moving forward in time (which is easy to do) feel so different from moving backward in time (which we can only do in some suitable metaphorical sense)? The laws of physics don’t distinguish one direction of movement in time, so why shouldn’t we be able to use similar mechanisms to move forward or backward in time, as we do to move right or left?

Drescher demonstrates, using a toy model, that time-symmetric models can easily yield time-asymmetric effects. Imagine a small universe consisting of a bunch of balls. Most of the balls are very small and move slowly; a few of the balls are very large and move quickly. Furthermore, assume that there are a lot of these balls, so that they fill up a substantial portion of the universe.

As they move around, these balls routinely collide. Eventually, the large balls slow down, and the small balls might speed up, but let’s say we don’t get to watch for long enough to get much of a sense of that. Let’s also say we’re taking a video of these balls moving around, and we’re allowed to play this video forward or backward. Are there features of this movie that will allow us to distinguish between the forward and backward versions?

Yes, there are. In particular, if we look at the forward version and we look just behind one of the large balls, we’ll see empty space, because the small balls can’t move quickly enough to fill in the space. As we look at the backward version, the empty space is in front of the balls, which is inconsistent with the way of the physics of this system works.

What’s going on here is that we’ve started with a configuration that cannot have had much of a past in this universe: we start with a configuration with low entropy, but entropy always increases. Drescher explains that the feeling we have of moving forward in time comes from an innate understanding of the increase in entropy of the forward-direction of motion in time. This section was my favorite part of the book.

Drescher then moves on to a discussion of quantum mechanics, which I found less satisfactory. His gripe with the standard (Copenhagen) interpretation of quantum mechanics is that we need a collapse of superposition when we observe a particle, but this collapse doesn’t seem to show up in the equations. So, he advocates for Everett’s “many-worlds” model, where we’re really living in a much larger configuration space and where observers are superpositions of observers. This rids quantum mechanics of its nondeterminism.

As far as I can tell, this does nothing to clear up any paradoxes we might have had: any nondeterministic system can be converted into a deterministic system if we’re willing to pass to a much larger state space, as I learned when I studied finite automata. Such a conversion is purely formal, so I fail to see how it can help to explain any paradoxes that we can’t explain at least as easily without leaving a nondeterministic world.

Then Drescher moves on to ethical paradoxes; here I start to disagree with his views sharply. The central problems he tackles are Newcomb’s problem and the (non-iterated) prisoner’s dilemma.

Here’s the setup for Newcomb’s problem. A benefactor with great predictive powers presents me with two boxes. One is a transparent box containing $1000; the other is an opaque box which is either empty or else contains $1000000. I am allowed to choose between taking both boxes and taking just the opaque box. If the benefactor has predicted that I’ll take just the opaque box, then ey has placed $1000000 in it; if ey has predicted that I’ll take both, then ey has left it empty. What should I do? (Assume I’m just trying to maximize our payoff; the benefactor has no preference about which option I choose.)

To me, the answer is clear: regardless of what the opaque box contains, I do better to take both boxes than to take just the opaque one. The content of the box doesn’t change once I decide what to do or while I’m deciding.

Strangely, Drescher disagrees with my analysis. His line of reasoning, roughly, is that I should be the type of person who takes only the opaque box, so that the benefactor will predict this. My counter, then, is that I should be the type of person who takes only the opaque box, but then I should still take both boxes. Drescher would probably say that that’s contradictory.

Drescher provides a reasonably careful analysis, but I think his error is that he subtly assumes that this is a repeated game. If we’re going to be presented with this scenario many times (and we anticipate that), then it makes sense to take only the opaque box, because that will influence the benefactor’s predictions in the future. But that doesn’t hold if we’re only playing once.

Here’s the setup for the prisoner’s dilemma: Two criminals are caught for committing a crime and are held separately. The police ask each prisoner separately to report the other. If they both stay silent, they each get 5 years in prison; if both talk, they each get 10 years in prison. If one talks and the other doesn’t, then the one who talks goes free, and the other one gets 20 years in prison. What should they do?

For each prisoner, the better option is to talk, but it both stay silent, that’s better for both of them than if both talk. Drescher claims that they should stay silent, again roughly because he subtly interprets this as a repeated game, where this is (to a first approximation) the correct strategy.

But here, there’s an interesting point that deserves consideration. Hofstadter proposes an approach to rational decision making known as superrationality. Since the prisoner’s dilemma is a symmetric game, both prisoners should realize that only symmetric plays really make sense; therefore, they shouldn’t even consider asymmetric options in their analysis of the game and should thus stay silent.

How can we justify this way of looking at the problem? Well, suppose that both prisoners are extremely confident in the rationality of the other. If I’m one of the prisoners, I should realize that, since I’m extremely confident about both my rationality and the rationality of the other prisoner, then whatever conclusion I come to will also be the conclusion that the other prisoner comes to. So, we can’t possibly end up with differing conclusions. And it’s obvious that I prefer to stay silent, so the other prisoner much prefer that as well.

Is it convincing? Not exactly, but it’s a start. It would certainly be interesting if people could regularly come to such conclusions (the ramifications of which would be fantastic), but I don’t believe that it works as well as Drescher believes it does.

All in all, this is an interesting book, with a lot of provocative ideas. (There are lots more that I didn’t talk about here.) Some of them I think don’t make complete sense, but I’m generally more interested in people throwing out ideas than in making sure that they’re all perfectly worked out.

Cognitive bias while reading

Recently I read Steven Pinker’s book The blank slate, where he claims that genetics is responsible for a large part of a person’s personality and abilities. Since I disagree with this position, I found myself picking many holes in Pinker’s arguments and generally trying to discredit it.

I do think that the right way of reading a book advancing a controversial viewpoint is to look at it with a critical eye and to try to notice its errors (as well as its convincing arguments). However, I do that less when I read books whose conclusions agree with my prior beliefs. Naturally, I still try to find holes in the arguments, and I tend to notice when things don’t make sense. But surely I put in much less effort when I already agree with the author, for example when I read Delusions of gender, which roughly takes the opposite viewpoint, and which I found much better thought out. Presumably, Cordelia Fine also made many unjustified or false claims in her book (it’s hard to avoid doing this when writing a book!), but since I was on her side to begin with, I only found a few of them, and I have since forgotten what they were.

It is essential that we have mechanisms for changing our beliefs when presented by strong evidence that discredits our prior beliefs, or else we can never improve our opinions and will thus be wrong frequently. However, we’re frequently going to be presented with weak evidence against our priors, and we’d like to have good defenses against changing our opinions too readily when presented with an argument that doesn’t quite work or that might be flawed for subtle reasons. (But there seem not to be many people who change their minds too quickly, and at any rate, I’m definitely not one of them.)

It is sensible, then, for us to analyze arguments whose conclusions we disagree with very carefully so that we can reject them when they are flawed and accept them only if we are quite certain that they are not.

What would be good, though, would be to learn how to be more critical when judging arguments whose conclusions I already agree with.  This can be tricky because we might have a tendency of throwing away the conclusion when an argument in support of it fails, even though that’s unnecessary. But once we realize this is unnecessary, we have little to lose by carefully analyzing arguments whose conclusions we already agree with, dismissing those that are unconvincing while keeping and strengthening those that make sense.

Careful analysis of arguments whose conclusions we already agree with is also good practice for another reason: doing so helps us make solid arguments when talking to others. Those of us who have strong opinions (hi there!) frequently want to share them with others, and ideally, we’d like other people to adopt similar opinions. Getting better at analyzing possible objections to our arguments allows us to present solid arguments to others when the opportunity arises.

So, how do I teach myself to attack arguments whose conclusions I already agree with more thoroughly? What techniques do other people have for doing so? I’d be delighted to hear any suggestions!

Book review: Triumph of the city

It’s easy to see some of the benefits of city living: being the middle of a bunch of excitement, things to do, shorter distances to points of interest. But, according to Harvard economist Edward Glaeser in his book Triumph of the city, the advantages are more numerous and more important than I expected.

Provocatively subtitled “How our greatest invention makes us richer, smarter, greener, healthier, and happier,” this book makes the case that not only are the advantages of city life overwhelming, but also that American policy that encourages home ownership is a tremendous detriment to our society.

Throughout the book, Glaeser challenges various myths that people hold about how cities work. Among the most important false belief that people might hold is that cities make people poor. Of course, there tend to be many poor people in cities, more so than in suburbs, so one might be tempted to believe that suburbs cause people to be better off than do cities. As is so often the case, Randall Munroe explains the problem best:

The correct line of reasoning, the one that accurately explains why cities contain more poor people than do suburbs, is that poor people choose to move to cities, where there are the greatest number of opportunities for advancement. They would be still poorer in rural areas, and they would be unable to afford any of the suburban comforts were they to live in the suburbs.

This point wouldn’t matter if it didn’t otherwise alter people’s perceptions of cities, but it does. Somehow, the conclusion becomes that people should be encouraged to move to the suburbs, where they’ll be better off. So, instead of investing resources in making cities as habitable and pleasant as possible, we neglect them and focus our efforts on suburbs.

Let’s get to Glaeser’s claims in his subtitle.

Is the city our greatest invention? I don’t know, but I think Glaeser does a good job of presenting his case. The existence of cities certainly makes future innovation possible in the way that rural living does not: when everyone has to take care of all eir basic needs for emself, then there isn’t much time left to invent new things that improve quality of life. On the other hand, the city on its own without any future innovations is not impressive at all: it’s just a bunch of people living close to each other. In the past, I’ve said that farming, irrigation, writing, usable electricity, and cohomology (and possibly also hummus) are our greatest inventions, but I’m willing to reconsider. So, I’ll say this is a matter of definitions, and I’ll hand Glaeser the benefit of the doubt.

Richer. This one is too easy. Technology and innovation allow us to specialize and become more efficient, which in turn allows for surplus output, which then leads to more money for everyone. Technology and innovation are most likely to show up in densely populated areas where people are constantly exposed to the ideas of others and are able to develop them further.

Smarter. Similarly, here, being around other people with exciting new ideas tends to cause us to think more. But even in modern times, it’s likely that people will do more thinking if they live in cities compared to suburbs. In the suburbs, a lot of time is wasted in getting from one place to another (generally in cars), and some of that time could be spent thinking and coming up with new ideas if one were instead to live in a big city.

Greener. This point is the heart of the book, and I think it’s his most important statement. Since I just want to give a brief overview now, I’ll postpone this conversation for a bit later.

Healthier. This was his weakest point. Glaeser claims that people in cities live longer, but that this was not always the case, and in fact was not the case even within the past century. Without proper regulation, cities can be fertile breeding grounds for diseases in a way that rural areas cannot, but on the other hand, cities can be more easily regulated to provide clean water and other hygienic advantages to keep people living longer. I suspect that, if Glaeser is right that currently people in cities are healthier, this is an effect that will go away as we get better at providing clean water and other such things to rural areas. In other words, this claim is a function of the fact that Glaeser published this book in 2011 rather than in 1911 or 2111.

However, the only time Glaeser made me really angry in this book was during his discussion of why New York City once had a much lower life expectancy than the country as a whole (a difference of 2.7 years for men). Glaeser writes “This gap didn’t appear for women, in part, because the great majority of murder victims are men.”

While it’s presumably true that the great majority of murder victims are (young) men, this statement is simply a lie, and I hate it when people lie to me. The fact that a larger proportion of men were being murdered in New York City than in the rest of the country does nothing at all to explain a 2.7-year difference in life expectancy, and I expect someone with Glaeser’s economic background to find this totally trivial to understand.

Happier. Glaeser claims that a larger proportion of people living in cities claim to be very happy with their lives than do people living in rural areas. The difference isn’t huge though. However, in light of the other advantages that cities give us, it’s clear that our lives are improved tremendously by the existence of cities, whether we live in them or not.

I guess we should understand exactly what we’re comparing here. Are we comparing cities to rural areas? To suburbs? Glaeser sometimes glosses over this point more than he probably should.

It seems plain enough that people generally (but not always) prefer to live in cities than in rural areas, which accounts for much of the migration from rural areas to cities and the lack of migration in the reverse direction. But what happens when we compare cities to the suburbs? Now it’s far from clear. The general trend is for young people to want to live in cities and then eventually move back to the suburbs to raise families. The benefits for suburban living when raising a family are enough that even Glaeser himself abandoned city life for the suburbs of Boston after starting a family.

The appeal for parents to want to live in the suburbs, at least in America, is clear. Government subsidies (or relative lack of taxation) on gas make living in the suburbs and commuting relatively inexpensive. The suburbs are also safer for kids and tend (with some notable exceptions) to have better schools (which should be highly embarrassing, given the lack of quality of my supposedly-good suburban high school “education”). Also, larger houses are available, and at much lower prices than in nearby cities.

While large housing is clearly going to be difficult to come by in large cities, the others need not be insurmountable problems. They are only because, in the US at least, we have made them so through subsidies and misplaced attentions. Glaeser says that, in France, the best schools are in Paris, and there’s no reason that the US couldn’t follow suit and arrange to have good schools in cities.

Why should we want to encourage people to live in cities rather than suburbs? One of the best reasons is the environmental factor. People who live in cities frequently walk or take public transportation to get from one place to another; this is rare in the suburbs. As a result, people save a substantial amount of energy from transportation by living in cities. On top of that, houses in cities tend to be smaller than those in suburbs; a consequence of this is that per household energy use for heating and air conditioning are lower in cities than in suburbs.

Furthermore, people who live in suburbs are often under pressure from their communities to do stupid things like maintain lawns, even in dry places like California. In many parts of the country, there are water shortages. Instead of using water sensibly, however, we’re expected to use large amounts of it to water lawns, for no reason other than blind conformity. People in cities generally don’t have enough yard space for this to be a serious issue there, so that’s one more problem that doesn’t arise for many city dwellers.

People in rural areas can live in an environmentally friendly manner if they so choose. However, few people want to live like pre-industrial farmers. Much more frequently, they want access to the benefits of modernity while having a large chunk of land and being closer to nature. Too bad that living near nature doesn’t equate well at all to environmental friendliness.

Of course, not all cities or all suburbs are equal with respect to energy usage. This is especially true for energy used for heating and cooling. Certain parts of the country, such as most of California, have relatively little need of cooling, and indeed, it’s rare for houses in the Bay Area to have air conditioning. It also doesn’t get cold enough to require heating all that frequently (although some people seem to find discomfort in being in a room with a temperature of 65F or 18C; this confuses me endlessly). On the other hand, in Texas, where temperatures might exceed 90F or 34C on a quarter of the days of each year or more, living without air conditioning is relatively unbearable.

As a result, we ought to encourage people to move to cities on the West Coast, rather than cities in places with harsher climates. Unfortunately, we’re not doing that. Rules against constructing new buildings in Santa Clara County are extremely stringent, and as a result, there are far more people who would like to live here than there are people who do.

A common theme in Californian environmentalism seems to be to oppose new building projects, perhaps on the grounds that fewer buildings leads to less pollution. But we’re not really preventing them from being built; we’re just causing them to be built elsewhere, in less efficient places that have less stringent building rules. Smarter policy decisions in California would be quite helpful for reducing greenhouse gas emissions.

I understand that it’s hard to see our local communities turn from bucolic to industrial. My hometown of Sunnyvale was once (admittedly before I was born) largely cherry orchards; now it’s a key location in the heart of Silicon Valley, being the headquarters for companies like Yahoo!. There are two main remnants of Sunnyvale’s past: a giant can reminding us of canning company Libby’s former presence in the city, and a cherry stand called Olson Cherries that has miraculously managed to remain open despite tremendous pressure to sell the land to, well, everyone else. The Libby can now rests atop a water tower in a tech company parking lot. It’s Sunnyvale’s most notable landmark.

But even those people who would like to live in California (or New York) and are able to do so might actually prefer to live in Houston or Atlanta. Glaeser has a section devoted to comparing the economic realities of living in Houston versus New York, and I found it a fascinating and enlightening read. While people generally earn quite a bit more in New York than in Houston, this is more than offset by the far higher cost of living in New York. And, of course, the houses in these two places will not be even remotely comparable: the one in Houston will undoubtedly by far larger and more pleasant. Economic differences explain why the populations of Houston, Atlanta, and Phoenix are rapidly rising, despite the fact that they seem so much worse than New York and San Francisco. Parents in Houston can send their kids to private schools so that they’ll still be taught about evolution.

Another problem with suburban living is that it’s so good at promoting harmful NIMBYism. In California, there’s a project at the moment to build a high-speed train between San Francisco and Sacramento, and San Diego. Unfortunately, there are many residents in Palo Alto and Menlo Park who oppose this project on the grounds that it is expected to go through these cities; I see many copies of this sign walking around town:

It’s unclear if these people will prevent the project from being completed, but they might.

One point here is that people frequently expect to abhor any new developments in their neighborhoods, preferring to keep them the way they are. But, in practice, people are good at adapting to new circumstances, and after a short amount of time, these things stop being at all problematical. A few new buildings or trains won’t destroy a city. In fact, in a thriving city, they’re more likely to further enliven the downtown area and help businesses. It isn’t reasonable to allow communities to stagnate because some of the current residents are fearful of change, when the changes will almost certainly be of great help to the community. People should have a certain amount of control over the development in their communities, but they must not be allowed the power to veto any new projects, especially those that have the potential to be of great benefit to society.

Sometimes, I find Glaeser’s conclusions a bit bizarre. He seems to find any increase in division of labor and specialization to be desirable, to the extent that he claims that the fact that he sometimes cooks for his family is a crushing condemnation of suburbia. I find it alarming that people are so content to dispose of all self-sufficiency in the name of specialization. We can sort of get on well enough by asking other people to do everything we don’t know how to do, but I find it very hard to believe that this is the ideal way to live. Or am I the only one who finds the idea of being unable to perform simple tasks that so many more people knew how to do in the not-so-distant past disturbing?

Glaeser also talks about many other interesting facets of cities in this book, but I’ll leave the rest for the reader to discover on eir own. Glaeser discusses why some cities don’t work out and what we should do about them, and how certain cities, such as Gaborone, have surprisingly managed to be quite successful and prosperous.

I found this book to be extremely interesting, and I learned a great deal from it. I don’t agree with everything Glaeser says, although I do agree with most of it. While I was reading, there were things I kept wishing he would talk about; inevitably, he did so at a later point in the book. This book is excellent, and I strongly recommend it.

Mathematics in art

A few months ago, Manjul Bhargava came to Stanford to give a public lecture on mathematics in the arts. I was skeptical of the topic — is there anything of content to be said? — but I’d heard him give some lectures on higher composition laws and on Selmer ranks of elliptic curves before, and they had all been very impressive talks. So, I went without much of an idea of what to expect.

Bhargava’s claim in the talk was that, while many people think that mathematics is inspired by science (or, perhaps, only by other mathematics; the latter is presumably more likely to be the opinion of mathematicians), there’s actually plenty of mathematics inspired by the arts. Since Bhargava has some background in Sanskrit poetry, he told three stories of mathematics being developed in order to solve concrete problems that arise from poetry.

In Sanskrit poetry, there are two types of syllables: long syllables and short syllables. The long syllables are twice as long as the short syllables. So, suppose we wish to write a line of poetry that takes up eight beats. How many different rhythms are there that allow us to realize a line of eight beats? If we write L for a long syllable and S for a short one, here are a few of the possible rhythms:

• LLLSS
• SLSSSL
• LLLL
• SSSSSSL

And there are quite a few more.

How do we do this? Well, suppose is the number of rhythms possible for a line of poetry consisting of beats. Suppose that is at least 2. Then we can remove the last syllable; either it’s an L or an S. If it’s an L, then when we remove it, we’re left with a line consisting of beats. If it’s an S, then when we remove it, we’re left with a line consisting of beats.

Hence, the number of rhythms with beats is equal to the number of rhythms with beats plus the number of rhythms with beats. In symbols: . Since we can easily check that , we then have:

• 
• 
• 
• 
• 
• 
• 

So, there’s our answer: there are 34 different rhythms for a line of eight beats.

You probably recognize these numbers: they’re the Hemachandra numbers, named after the 12th-century Indian scholar Acharya Hemachandra.

Oh? You object to that name? Fibonacci wrote about this sequence of numbers in the context of the growth of rabbit populations roughly 50 years after Hemachandra considered the sequence in the context of poetry patterns. (Incidentally, rabbit mating has inspired a completely different branch of mathematics in the past few years. I’d like to write about this at some point, when I understand it better.)

He also discussed how poetry led to the development of what we usually refer to as Pascal’s triangle and de Bruijn sequences.

After the talk, I noted that it makes sense that combinatorial problems arise naturally arise from the arts, but I wondered if there were such problems in other branches of mathematics. One can imagine various topics in mathematics that look as though they might have something to do with art, but they were actually developed for entirely different reasons.

For example, identification spaces in topology sort of look like art, but I doubt that they were first studied for that reason. Similarly, one can imagine that there’s plenty of mathematics that one can get out of Escher’s paintings and wood carvings. But, the causality is the opposite in this case: Escher was inspired by the geometry of the Poincaré disc to study hyperbolic tessellations, such as this one.

And then, of course, there’s the dance.

But there’s one other obvious instance of mathematics being developed to answer questions of artists, although I wasn’t able to come up with it on my own. (I remembered only when it came up, in a completely different context, in Edward Glaeser’s Triumph of the city, which I’m currently reading.) And that’s projective geometry.

Early paintings tend not to look very realistic when portraying depth, since they lack perspective. As a result, we end up seeing a very flat rendition of what ought to be a 3-dimensional scene.

While various attempts were made to correct this problem, it wasn’t until 1413 that Filippo Brunelleschi solved this problem conclusively by working out the right way to draw perspective paintings, and this was done by using some projective geometry. One key observation is that lines that ought to be parallel should not necessarily be drawn parallel to each other; rather, they should intersect at a point which we understand to be very far away. Here’s a sample of his paintings:

Since this was done well before mathematicians such as Desargues studied projective geometry, I think it’s fair to say that projective geometry got off the ground because of its study by artists rather than mathematicians.

Finally, knitters who make socks need to understand something of geodesics on Riemannian manifolds.

Are there any other examples of topics in mathematics that were studied in order to solve concrete problems in the arts? If there are, I’d be delighted to hear about them!