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# Football+Maths

If you have read the past 5 post­s, you saw it was com­ing, right? ;-)

Via Slate here's a pa­per ap­ply­ing game the­o­ry to penalties:

http://www.e­con.brown.e­du/­fac/i­pala­cios/pdf/pro­fes­sion­al­s.pdf

The con­clus­sion (and they made me re­type this, be­cause of stupid DR­M):

The im­pli­ca­tions of the Min­i­max the­o­rem are test­ed us­ing nat­u­ral da­ta. The tests use a unique da­ta set from penal­ty kicks in pro­fes­sion­al soc­cer games. In this nat­u­ral set­ting ex­perts play a one-shot two-per­son ze­ro-­sum game. The re­sults of the tests are re­mark­ably con­sis­tent with equi­lib­ri­um play in ev­ery re­spec­t: (i) win­ning prob­a­bil­i­ties are sta­tis­ti­cal­ly iden­ti­cal across strate­gies for play­er­s; (i­i) play­er­s' choic­es are se­ri­al­ly in­de­pen­den­t. The tests have sub­stan­tial pow­er to dis­tin­guish equi­lib­ri­um play from dis­e­qui­lib­ri­um al­ter­na­tives. These re­sults rep­re­sent the first time that both im­pli­ca­tions of von Neu­man­n's Min­i­max the­o­rem are sup­port­ed un­der nat­u­ral con­di­tion­s.

In hu­man:

• Play­ers are pret­­ty good at mak­ing de­­ci­­sions ac­­cord­ing to game the­o­ry

• Game the­o­rists call that good

How­ev­er (this I am mak­ing up as I go)...

• A well kicked pe­nal­­ty is a goal, be­­cause there are places the keep­­er sim­­ply can't reach in time.

• Since there is a win­n­ing strat­e­­gy, it makes no sense to ap­­ply min­i­­max: there is a glob­al max­i­­mum (yes, I know, it makes sense, you just need to con­sid­er mis­s­ing the goal as the chance of fail­ure, then it is not a win­n­ing strat­e­­gy... and I am not go­ing to read the 21-­­page pa­per to see if he thought about it).

Of course the win­ning strat­e­gy (strong kicks to the top an­gles of the goal) is im­prac­ti­cal for mere hu­man­s. All the more rea­son to con­sid­er RoboCup the most im­por­tant tour­na­ment for the fu­ture.

# More on .9999... and 1

Read the com­ments. I am ashamed of math­e­mat­i­cal ed­u­ca­tion, right now. If these peo­ple has passed any math­e­mat­ics tests (and some even claim to have gone to col­lege), maths are hope­less­ly dif­fi­cult.

Some choice quotes:

1/3 is a sym­bol for a set of 4 word­s, it is not a NUM­BER.

On­ly a sin­gle num­ber CAN POS­SI­BLY = 1. Oth­er num­bers may ADD UP to 1, but they don't EQUAL 1. Since 1 clear­ly = 1, .99999 re­peat­ing sim­ply can­not equal 1.

.33(repet­ing) is ir­ra­tional.

.99999 does not equal 1. It might in the CUR­RENT UN­DER­STAND­ING of math­e­mat­ic­s, but that don't make it true.

Math­e­mat­ics can­not even prove that .99999 ... is not equal to 1.

Right now, math re­al­ly can't deal with in­fi­nite num­bers

.9 re­peat­ing, an ir­ra­tional num­ber, is AB­SO­LUTE­LY EQUAL to the ra­tio­nal num­ber 1. Can this be used as proof to show there is no such thing as ir­ra­tional num­ber­s?

I'm half tempt­ed to say there is­n't re­al­ly a right or wrong an­swer

I think I've come to the con­clu­sion that .999... = 1 in the same sense that .333... = 1/3. Which is to say, it does­n't, quite, but we treat it like it does be­cause our dec­i­mal sys­tem has prob­lem­s.

0.9 re­cur­ring does not equal 1. Why? Be­cause it's 0.9 re­cur­ring.

1 = .9 re­peat­ing IF WE WANT IT TO.

This is an ex­ploita­tion of our nu­mer­ic sys­tem, to ar­rive at an out­come that is in­deed very close to be­ing true, but the clos­er it gets to be­ing true the fur­ther away it ac­tu­al­ly is.

2.9 re­peat­ing plus 2.9 re­peat­ing equals 5.9 re­peat­ing 8

And this last one is amaz­ing. The poster pro­pos­es a num­ber that is a 5.9 (an in­fi­nite num­ber of 9s... and an 8). Right. An 8. Af­ter in­fi­nite 9s. At the end of them. Right there. Go to in­fin­i­ty po­si­tion, then one more. There's the 8.

My mind bog­gles. And it's a mind that ac­tu­al­ly ac­cepts .99(re­peat­ing) is 1.

# How satanic messages work (with video)

Ev­ery­one knows about the hid­den sa­tan­ic mes­sages in songs.

You take a song, you play it back­ward­s, and in cer­tain places, you will have the singer say­ing some­thing evil, like "I like eat­ing pup­pies with cin­na­mon".

I have al­ways as­sumed that this hap­pens be­cause our brains try to rec­og­nize pat­terns in the sounds they get, and they are a bit too good in that job, but now I have proof.

Here's a video Rosario (my wife, hi dear!) sent me:

In it you can hear pieces of pop songs, in eng­lish (and lat­in), and sub­ti­tles of what they seem to say in span­ish.

Now, un­less you be­lieve Avril Lav­i­gne ac­tu­al­ly says "Lei­va quiso ven­derme el Ford" (Lei­va tried to sell me a Ford), and Mar­ley sings about "Where is Ju­li­a", the "pick­ing too much sig­nal" the­o­ry seems true.

Spe­cial­ly, if you are told what you should hear, it works much bet­ter!

I had heard these songs a mil­lion times, and I had nev­er thought they said that, but with the sub­ti­tles... some of them are pret­ty close :-)

The is­sue of why sub­lim­i­nal mes­sages en­cod­ed back­wards in songs make no sense in the first place is an­oth­er top­ic.

# Infinite Amateurism on Maths

I stum­bled up­on Thomas Thur­man's post on his blog (here) where he com­ments about a dis­cus­sion on The Guardian about how .99(re­cur­ring) is or is not the same as 1.

Of course to any­one who knows what a ra­tio­nal num­ber is, .99(re­cur­ring) is sim­ply a very long way to write 1. Hel­l, to any­one who both­ered learn­ing his frac­tion­s, that should be ob­vi­ous!

But any­way, one of the com­ments men­tions Hilbert's Hotel, which is a pet toy of mine.

If you are un­com­fort­able or an­noyed by the con­cept of in­fin­i­ty, you may want to avoid the rest of this post.

Hilbert's Ho­tel is this para­dox (Thanks Wikipedia!):

A ho­tel with an in­fi­nite num­ber of rooms (1, 2, 3 and so on, so it's a nu­mer­able in­fin­i­ty) is ful­l. A guest ar­rives. Yet he still gets a room. How?

The an­swer is that you ask the guest in room 1 to move to room 2, from 2 to 3 and so on. Then the new guest goes to room 1, which is now free.

Be­cause of the na­ture of in­fin­i­ty, this work­s, while on a fi­nite ho­tel it would­n't.

• Un­in­­tu­i­tive things about in­­fin­i­­ty: If you add any num­ber to it, the re­­sult is the same in­­fin­i­­ty.

Now as­sume an in­fi­nite (nu­mer­able) num­ber of guests ar­rives. You have to ask the guest in room 1 to go to room 2, the guest in room 2 to go to room 4, and guest n to go to n*2.

Now you have an in­fi­nite (nu­mer­able) num­ber of free room­s: all the odd room­s.

• Un­in­­tu­i­tive things about in­­fin­i­­ty: If you mul­ti­­ply it by any num­ber, the re­­sult is the same in­­fin­i­­ty.

• Un­in­­tu­i­tive things about in­­fin­i­­ty: If some­thing is in­­finite, it con­­tains a part the same size as the whole (fol­lows from the pre­vi­ous two things, and is, in fact an "if and on­­ly if").

How­ev­er, here's where it gets trick­y. You could get a cer­tain num­ber of new guests and there could be no way to fit them in the rooms even if the ho­tel was emp­ty.

Be­cause there's in­fin­i­ty, and then there is in­fin­i­ty. You saw that when­ev­er I men­tioned the num­ber of rooms I men­tioned they were in­fi­nite (nu­mer­able)? That's be­cause you can put an in­te­ger num­ber to each, and num­ber them al­l.

There are in­fi­nite sets of things that are big­ger, they are lit­er­al­ly un­count­able. You can't put a num­ber to each, even with an in­fi­nite amount of time (and yes, I know that in­fi­nite amount of time there is a big prob­lem).

The sim­plest set imag­in­able that large is that of the re­al num­ber­s. The re­al num­bers are all the num­bers you can imag­ine, al­low­ing for in­fi­nite dec­i­mal­s, and al­low­ing that those dec­i­mals may not ev­er be re­cur­ring (so you have things like 2, 1/3, and pi).

Show­ing there are more of those that there are in­te­ger num­bers is not sim­ple enough for this but go along with me for a while.

• Un­in­­tu­i­tive things about in­­fin­i­­ty: There are dif­fer­­ent sizes of in­­­fi­nite. Go blame Georg Can­­tor.

Now it gets re­al­ly weird. Sup­pose we call the size of the in­fi­nite in Hilbert's Ho­tel A0 (I have no idea how to do an Ale­ph, sor­ry), and the size of the re­al num­bers C. Can­tor showed how to build, once you have an in­fi­nite set, a larg­er in­fi­nite set called his pow­er set.

That means we now have a whole in­fi­nite (nu­mer­able) "sizes" of in­ifi­nite things. Those are called the trans­fi­nite num­ber­s.

• Un­in­­tu­i­tive things about in­­fin­i­­ty: There are in­­­fi­nite dif­fer­­ent sizes of in­­­fi­nite. Go blame Georg Can­­tor some more.

Which brings a lot of ques­tion­s:

• Are there on­­ly those? Is­n't there some­thing be­tween A0 and C which is some in­­-­be­tween size?

• Is there an in­­­fi­nite set that's smal­l­­er than the in­­te­ger­s?

• Ok, in­­­fi­nite in­­finites... in­­­fi­nite (nu­mer­able) in­­finites, or in­­­fi­nite (some­thing else) in­­finites?

Well... I have no idea. And last I checked, which was long ago, and my mem­o­ry is no good, noone else knew.

This is the kind of things that will tell you whether you could be a math­e­ma­ti­cian. Do you find all this talk about trans­fi­nite num­bers in­trigu­ing and mys­te­ri­ous, or just dull and bor­ing and im­prac­ti­cal?

If you find it dull and bor­ing, it may be my writ­ing, or you may be un­suit­ed for math­s.

If you find it in­trigu­ing and/or mys­te­ri­ous, it cer­tain­ly is not my writ­ing, and you would prob­a­bly en­joy maths in your life. Where else are you go­ing to run in­to "noone knows that" this quick­ly?

The prob­lem is, of course, that most of the fun math has al­ready been done at least a cen­tu­ry ago, but there is al­ways a chance of some­thing fun and in­trigu­ing and new com­ing along.

The last I know of was Gödel's the­o­rem, which is re­al­ly sim­ple enough for any­one with knowl­edge of arith­metic to fol­low, but weird enough for 99.99% of the peo­ple to go crazy about (and for those who don't re­al­ly un­der­stand it to write whole books about it ap­ply­ing it to to­tal­ly im­prop­er sub­ject­s).

But you know, noone re­al­ly had thought of such thing as "larg­er than in­fin­i­ty" quite as Can­tor did, and noone thought about Gödel's sub­ject quite as he did be­fore him.

Maybe we are miss­ing some­thing ab­so­lute­ly sim­ple, in­cred­i­bly el­e­gan­t, awe­some­ly shock­ing some­where in ba­sic math­s. Not like­ly. But pos­si­ble. Would­n't it be fun to find it?

BTW: Gödel starved him­self to death and Can­tor "suf­fered pover­ty, hunger and died in a sana­to­ri­um".

# Happy post

I have not post­ed in a few days, be­cause I have been very busy.

I have not even been able to see all match­es (I missed US­A/Czech, Italy/Ghana).

But man, did watch­ing Ar­genti­na/S&M pay of­f! :-)

A very good match by Ar­genti­na, which of course brings up the ob­vi­ous ques­tion­s...

Al­leged­ly it's one of the best de­fens­es in Eu­rope, but their best play­er was out. So it's not con­clu­sive. My bet is that they are not be­head­ed chick­en­s, and that Ar­genti­na made them look bad. Or rather: 25% be­head­ed chick­en­s, 75% Ar­genti­na's mer­it.

On the oth­er hand, the play­er that said they should have at­tacked against the dutch was wrong. It's not called be­ing a cow­ard, it's called know­ing you can't score.

• Was the good Dutch per­­for­­mance against S&M re­al?

Who knows. We will fig­ure it out for sure af­ter they play Ivory Coast. My bet? Tough match for the dutch, pos­si­bly a tie.

Now for the oth­er team­s:

Eng­land has noth­ing. They suck much more than you think now. Wait un­til they play a team that ac­tu­al­ly has a for­ward who can score (and is not play­ing 5 like Yorke). Set pieces for Beck­ham at mid­field and Crouch try­ing to head it in is not some­thing that's go­ing to work a lot. For their sakes, Owen and Rooney bet­ter start show­ing some­thing com­plete­ly dif­fer­ent than against T&T. Which still has a chance. I would pre­fer them and not Swe­den to stay in the cup!

Ger­many has some­thing. The de­fense was im­proved (although Poland has a lame at­tack­), and their for­wards are not bad.

Spain has a lot. Spe­cial­ly go­ing for­ward, and they made Ukrayne look like am­a­teurs. We'll see if that was Ukrayne's fault of Spain's mer­it lat­er on.

Brazil has some­thing big. His name is Ronal­do. Re­move him and they are go­ing to kick as­s.