Wiles meets his match | Gowers's Weblog

]]>Because the extreme silliness of this passage spoilt it for me *as fiction*: it made me too aware of the hand of the author.

Can’t remember exact wording: She giggles and then there is something to the effect that it is more a question for philosophers or something like that.

“She GIGGLES” – it is a joke – it is funny! – get it?

… as in 1+1=”a window” all kids learn that – get it yet?

Well, fine, ok: what do horrible, unsolvable maths questions do? they put you to sleep, right? and z3 is in fact “zzz” which any child can tell you means “sleep” so x3 + y3 = “zzz”.

x2 + y2 = z2 is Pythagoras theorem for a right angled triangle and completely solvable: apparently x3 + y3 = z3 is not.

I don’t know the maths of trying to solve it but late at night reading the book, she is a maths genius who could not solve the problem, until she “gets the joke and giggles” – I actually just came on line to see if someone else had “discovered” the answer and reached the same conclusion I did – I laughed out loud reading it because my sleep deprived brain actually made the connection between being tired and “zzz”.

x3 + y3 = z3 is a parody and the answer is funny.

there is a Simpsons episode that had a maths joke with the solution being rdr2 (which is rdrr, or – say it – r d r r – “ha de ha ha”)

ps just in case: ( 1+1=window: if you write 1 with + touching the 1 then write another 1 touching the other side of the 1 then write the = as one line above the + and one below, you have drawn a window )

of course I could have it completely wrong but I don’t think so and I like it! She is a maths genius and recognises it can’t be solved and that the mathematician was having a laugh, all these serious people trying with great seriousness to solve the problem and they don’t realise it is a joke – she has just got the joke!

“The answer was so disarmingly simple. A game

with numbers that lined up and then fell into place in a simple formula that was most

similar to a rebus.

Fermat had no computer, of course, and Wiles’ solution was based on mathematics

that had not been invented when Fermat formulated his theorem. Fermat would never

have been able to produce the proof that Wiles had presented. Fermat’s solution was

quite different.

She was so stunned that she had to sit down on a tree stump. She gazed straight

ahead as she checked the equation.

So that’s what he meant. No wonder mathematicians were tearing out their

hair.

Then she giggled.

A philosopher would have had a better chance of solving this riddle.

She wished she could have known Fermat.

He was a cocky devil.”

The author refers to it as a riddle or a puzzle so you need to think laterally. ie the answer is z3, z3 “lined up” is zzz which we are told is a rebus, the rebus of “zzz” is sleep, an unsolvable maths problem puts you to sleep. The exponent of 3 is used as any child will tell you what “zzz” is. to me the joke is also that the larger the exponent the more z’s you get. The harder the equation the more it puts you to sleep.

I think that the author expected more of the general population to get the joke.

Sadly mathematicians try to solve it or pick holes in it, and non-mathematicians skip over it thinking they wouldn’t understand.

]]>(at the very end)

]]>That’s a good one there Paul. Was actually scouring the net over solutions for Fermat’s Last Theorem to figure out what Lisbeth had in mind. Interesting perspective to look at.

]]>This is a clever attempt at resolving the problem, but I don’t find it very plausible. It’s been a while since I watched the episode, but as I recall, none of the verbal discussion of Strategema (as opposed to the video footage of people playing the game) suggested that computational speed was the issue. If it had been, surely Data would have realized that he simply did not have enough time to calculate the correct strategy, and would not have scurried off to debug himself. Furthermore, Data’s problem was solved not by a technical suggestion that he adopt a computationally more tractable strategy, but by wise-sounding words that changed his psychological mindset.

It seems clear to me that the script writers were just trying to send the message that for a *human being* to be a true champion, technical virtuosity is not enough, even in pure strategy games like chess and go. A chess match is also a psychological match, and one must be properly prepared psychologically in order to play one’s best. This is a fine message as far as it goes; it’s just that it doesn’t apply to *perfect* players. Hence the contradictions.

My understanding was that it was a pure strategy game, but real time. It turned out that Data — while trying to win — was unable to respond fast enough to his opponent in the first game, and so he lost. In the second game (after the talk with various people), he adopted the simpler strategy of avoiding a loss. The resulting pruning of the possibilities enabled him to compute plans in time to keep up with his opponent (which would eventually result in a win for Data when his opponent succumbed to the effects of fatigue and slowed down enough for Data to explore winning possibilities safely).

]]>Abalone is a good example of what the scriptwriters may have been thinking of (at least if you use the traditional starting position rather than the Belgian daisy, which was introduced about ten years ago precisely to mitigate the problem Joel alludes to). It all sounds plausible to us imperfect human players. If Data were human then it is easy to imagine Data thinking he could win, but being mistaken, and getting crushed, until he realized he was being too aggressive. But it doesn’t make any sense for perfect players. A perfect Abalone player would not lose.

The plot is fine as long as we don’t take Data too literally, but just realize that he’s a character (like Spock) that allows the writers to explore what happens when a human being acts “overly logically.” Perhaps the Star Trek writers should write a self-referential episode where Data watches the Strategema Star Trek episode and gets confused the way I’ve gotten confused!

]]>A real-life game comes to mind here. From what my friends tell me of the game Abalone, it is relatively easy to play not to lose and to keep the game going indefinitely; but if you try to win, you open yourself up to losing.

Joel

I was also not impressed at the math writing in this book in general when I read it, but I think your reading of the passage above misses the point entirely. Though the last statement may sound harsh, it is perhaps better taken as praise of your good taste rather than criticism: the point is groanworthy and not particularly well made.

Cairo above was trying to explain the correct reading, which you again misunderstood: Salander hasn’t realized a proof, or think that Fermat is encoding some scret message; she thinks Fermat is making a silly joke Saying the margin does not have enough space is not saying that his proof is too long – the “square is converted to a cube” line is not suggesting the case n=3 is hard, but suggesting that n is acting as a dimension. The “joke”, then, is that Fermat is saying that the margin of the paper is two dimensional, and thus cannot contain a cube.

Reread the passage above with this in mind and I think a lot of supporting evidence jumps out.

]]>To Patrick: I always assumed that Wiles wrote TeX, based on a remark Knuth made in an interview:

“One of the most important somehow to me was last month when I went to the library and saw Andrew Wiles’s solution to Fermat’s Last Theorem. I think a lot of you know that it was front-page news. . . Just as people can remember where they were when they heard about Kennedy being assassinated, I know mathematicians can all remember where they were when they ﬁrst heard that Fermat’s Theorem was solved. The paper came out in the Annals of Mathematics last month; it arrived in our library and I saw it sitting there, and I looked at it and **it was just wonderful for me because it was in TEX and it looked gorgeous! [laughter] This to me was the . . . you know, it was so . . . I mean, I almost felt like I had helped to solve the Theorem myself!**”

But, of course, that doesn’t say who actually typed the paper.

]]>If both players in Strategema can force a draw, then if Data had indeed played perfectly the first time around, then he would not have lost. He would have at worst drawn. Therefore Data did not play perfectly the first time around. If Data had sufficient computational power to solve Strategema (and we are given no indication that he didn’t) then he was correct to infer that he had a bug. Yet the plot indicates that he didn’t have a bug; his problem was some kind of psychological one that was best solved by Counselor-Troi-like wise words.

]]>with a big computer program. I don’t think he even writes TeX. ]]>

The average viewer probably saw nothing at all amiss, but like Data, my brain dropped everything in an attempt to “debug” the plot until I finally snapped out of it by telling myself, “Look, it’s just a story! Logical coherence is beside the point!”

Other examples that come to mind off the top of my head include the botched explanation of Nash equilibria in the movie “A Beautiful Mind,” and Chinghiz Aitmatov’s conflation of the solar system with the Milky Way galaxy in “A Day Lasts More than a Hundred Years.” (The latter is not math, I guess, but it has the same feel.)

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