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John Baez

**[SOLVED] This Week's Finds in Mathematical Physics (Week 236)**

Also available at http://math.ucr.edu/home/baez/week236.html

July 26, 2006

This Week's Finds in Mathematical Physics (Week 236)

John Baez

This week I'd like to catch you up on some papers about

categorification and quantum mechanics.

But first, since it's summer vacation, I'd like to take you on

a little road trip - to infinity. And then, for fun, a little

detective story about the history of the icosahedron.

Cantor invented two kinds of infinities: cardinals and ordinals.

Cardinals are more familiar. They say how big sets are. Two sets

can be put into 1-1 correspondence iff they have the same number of

elements - where this kind of "number" is a cardinal.

But today I want to talk about ordinals. Ordinals say how big

"well-ordered" sets are. A set is well-ordered if it's linearly

ordered and every nonempty subset has a smallest element.

For example, the empty set

{}

is well-ordered in a trivial sort of way, and the corresponding

ordinal is called

0.

Similarly, any set with just one element, like this:

{0}

is well-ordered in a trivial sort of way, and the corresponding

ordinal is called

1.

Similarly, any set with two elements, like this:

{0,1}

becomes well-ordered as soon as we decree which element is bigger;

the obvious choice is to say 0 < 1. The corresponding ordinal is

called

2.

Similarly, any set with three elements, like this:

{0,1,2}

becomes well-ordered as soon as we linearly order it; the obvious

choice here is to say 0 < 1 < 2. The corresponding ordinal is called

3.

Perhaps you're getting the pattern - you've probably seen these

particular ordinals before, maybe sometime in grade school.

They're called finite ordinals, or "natural numbers".

But there's a cute trick they probably didn't teach you then:

we can *define* each ordinal to *be* the set of all ordinals

less than it:

0 = {} (since no ordinal is less than 0)

1 = {0} (since only 0 is less than 1)

2 = {0,1} (since 0 and 1 are less than 2)

3 = {0,1,2} (since 0, 1 and 2 are less than 3)

and so on. It's nice because now each ordinal *is* a

well-ordered set of the size that ordinal stands for.

And, we can define one ordinal to be "less than or equal" to

another precisely when its a subset of the other.

Now, what comes after all the finite ordinals? Well,

the set of all finite ordinals is itself well-ordered:

{0,1,2,3,...}

So, there's an ordinal corresponding to this - and it's the first

*infinite* ordinal. It's usually called omega. Using the cute

trick I mentioned, we can actually define

omega = {0,1,2,3,...}

Now, what comes after this? Well, it turns out there's a

well-ordered set

{0,1,2,3,...,omega}

containing the finite ordinals together with omega, with the

obvious notion of "less than": omega is bigger than the rest.

Corresponding to this set there's an ordinal called

omega+1

As usual, we can simply define

omega+1 = {0,1,2,3,...,omega}

(At this point you could be confused if you know about cardinals,

so let me throw in a word of reassurance. The sets omega and

omega+1 have the same "cardinality", but they're different as

ordinals, since you can't find a 1-1 and onto function between

them that *preserves the ordering*. This is easy to see, since

omega+1 has a biggest element while omega does not.)

Now, what comes next? Well, not surprisingly, it's

omega+2 = {0,1,2,3,...,omega,omega+1}

Then comes

omega+3, omega+4, omega+5,...

and so on. You get the idea.

What next?

Well, the ordinal after all these is called omega+omega.

People often call it "omega times 2" or "omega 2" for short. So,

omega 2 = {0,1,2,3,...,omega,omega+1,omega+2,omega+3,...}

What next? Well, then comes

omega 2 + 1, omega 2 + 2,...

and so on. But you probably have the hang of this already, so

we can skip right ahead to omega 3.

In fact, you're probably ready to skip right ahead to omega 4,

and omega 5, and so on.

In fact, I bet now you're ready to skip all the way to

"omega times omega", or "omega squared" for short:

omega^2 =

{0,1,2...omega,omega+1,omega+2,...,omega2,omega2+1,omega2+2,...}

It would be fun to have a book with omega pages, each page half

as thick as the previous page. You can tell a nice long story

with an omega-sized book. But it would be even more fun to have

an encyclopedia with omega volumes, each being an omega-sized book,

each half as thick as the previous volume. Then you have omega^2

pages - and it can still fit in one bookshelf!

What comes next? Well, we have

omega^2+1, omega^2+2, ...

and so on, and after all these come

omega^2+omega, omega^2+omega+1, omega^2+omega+2, ...

and so on - and eventually

omega^2 + omega^2 = omega^2 2

and then a bunch more, and then

omega^2 3

and then a bunch more, and then

omega^2 4

and then a bunch more, and more, and eventually

omega^2 omega = omega^3.

You can probably imagine a bookcase containing omega encyclopedias,

each with omega volumes, each with omega pages, for a total of

omega^3 pages.

I'm skipping more and more steps to keep you from getting bored.

I know you have plenty to do and can't spend an *infinite* amount

of time reading This Week's Finds, even if the subject is infinity.

So, if you don't mind me just mentioning some of the high points,

there are guys like omega^4 and omega^5 and so on, and after all

these comes

omega^omega.

And then what?

Well, then comes omega^omega + 1, and so on, but I'm sure

that's boring by now. And then come ordinals like

omega^omega 2,..., omega^omega 3, ..., omega^omega 4, ...

leading up to

omega^omega omega = omega^{omega + 1}

Then eventually come ordinals like

omega^omega omega^2 , ..., omega^omega omega^3, ...

and so on, leading up to:

omega^omega omega^omega = omega^{omega + omega} = omega^{omega 2}

This actually reminds me of something that happened driving across

South Dakota one summer with a friend of mine. We were in college,

so we had the summer off, so we drive across the country. We drove

across South Dakota all the way from the eastern border to the west

on Interstate 90.

This state is huge - about 600 kilometers across, and most of it is

really flat, so the drive was really boring. We kept seeing signs

for a bunch of tourist attractions on the western edge of the state,

like the Badlands and Mt. Rushmore - a mountain that they carved

to look like faces of presidents, just to give people some reason to keep

driving.

Anyway, I'll tell you the rest of the story later - I see some more

ordinals coming up:

omega^{omega 3},... omega^{omega 4},... omega^{omega 5},...

We're really whizzing along now just to keep from getting bored - just

like my friend and I did in South Dakota. You might fondly imagine

that we had fun trading stories and jokes, like they do in road movies.

But we were driving all the way from Princeton to my friend Chip's

cabin in California. By the time we got to South Dakota, we were all

out of stories and jokes.

Hey, look! It's

omega^{omega omega} = omega^{omega^2}

That was cool. Then comes

omega^{omega^3}, ... omega^{omega^4}, ... omega^{omega^5}, ...

and so on.

Anyway, back to my story. For the first half of our half of our

trip across the state, we kept seeing signs for something called

the South Dakota Tractor Museum.

Oh, wait, here's an interesting ordinal - let's slow down and

take a look:

omega^{omega^omega}

I like that! Okay, let's keep driving:

omega^{omega^omega} + 1, omega^{omega^omega} + 2, ...

and then

omega^{omega^omega} + omega, ..., omega^{omega^omega} + omega 2, ...

and then

omega^{omega^omega} + omega^2, ..., omega^{omega^omega} + omega^3, ...

and eventually

omega^{omega^omega} + omega^omega

and eventually

omega^{omega^omega} + omega^{omega^omega} = omega^{omega^omega} 2

and then

omega^{omega^omega} 3, ..., omega^{omega ^ omega} 4, ...

and eventually

omega^{omega^omega} omega = omega^{omega^omega + 1}

and then

omega^{omega^omega + 2}, ..., omega^{omega^omega + 3}, ...

This is pretty boring; we're already going infinitely fast,

but we're still just picking up speed, and it'll take a while

before we reach something interesting.

Anyway, we started getting really curious about this South Dakota

Tractor Museum - it sounded sort of funny. It took 250 kilometers

of driving before we passed it. We wouldn't normally care about

a tractor museum, but there was really nothing else to think about

while we were driving. The only thing to see were fields of grain,

and these signs, which kept building up the suspense, saying things

like "ONLY 100 MILES TO THE SOUTH DAKOTA TRACTOR MUSEUM!"

We're zipping along really fast now:

omega^{omega^{omega^omega}}, ... omega^{omega^{omega^{omega^omega}}},...

What comes after all these?

At this point we need to stop for gas. Our notation for ordinals

runs out at this point!

The ordinals don't stop; it's just our notation that gives out.

The set of all ordinals listed up to now - including all the ones

we zipped past - is a well-ordered set called

epsilon_0

or "epsilon-nought". This has the amazing property that

epsilon_0 = omega^{epsilon_0}

And, it's the smallest ordinal with this property.

In fact, all the ordinals smaller than epsilon_0 can be drawn as

trees. You write them in "Cantor normal form" like this:

omega^{omega^omega + omega} + omega^omega + omega + omega + 1 + 1 + 1

using just + and exponentials and 1 and omega, and then you turn

this notation into a picture of a tree. I'll leave it as a puzzle

to figure out how.

So, the set of (finite, rooted) trees becomes a well-ordered set

whose ordinal is epsilon_0. Trees are important in combinatorics

and computer science, so epsilon_0 is not really so weird after all.

Another cool thing is that Gentzen proved the consistency of the

usual axioms for arithmetic - "Peano arithmetic" - with the help

of epsilon_0. He did this by drawing proofs as trees, and using

this to give an inductive argument that there's no proof in Peano

arithmetic that 0 = 1. But, this inductive argument goes beyond

the simple kind you use to prove facts about all natural numbers.

It uses induction up to epsilon_0.

You can't formalize Gentzen's argument in Peano arithmetic: thanks

to Goedel, this system can't proof itself consistent unless it's *not*.

I used to think this made Gentzen's proof pointless, especially since

"induction up to epsilon_0" sounded like some sort of insane logician's

extrapolation of ordinary mathematical induction.

But now I see that induction up to epsilon_0 can be thought of as

induction on trees, and it seems like an obviously correct principle.

Of course Peano's axioms also seem obviously correct, so I don't know

that Gentzen's proof makes me *more sure* Peano arithmetic is

consistent. But, it's interesting.

Induction up to epsilon_0 also let's you prove other stuff you

can't prove with just Peano arithmetic. For example, it let's you

prove that every Goodstein sequence eventually reaches zero!

Huh?

To write down a Goodstein sequence, you start with any natural

number and write it in "recursive base 2", like this:

2^{2^2 + 1} + 2^1

Then you replace all the 2's by 3's:

3^{3^3 + 1} + 3^1

Then you subtract 1 and write the answer in "recursive base 3":

3^{3^3 + 1} + 1 + 1

Then you replace all the 3's by 4's, subtract 1 and write the

answer in recursive base 4. Then you replace all the 4's by

5's, subtract 1 and write the answer in recursive base 5. And so on.

At first these numbers seem to keep getting bigger! So, it seems

shocking at first that they eventually reach zero. For example,

if you start with the number 4, you get this Goodstein sequence:

4, 26, 41, 60, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, ...

and apparently it takes about 3 x 10^{60605351} steps to reach zero!

You can try examples yourself on this applet:

1) National Curve Bank, Goodstein's theorem,

http://curvebank.calstatela.edu/goodstein/goodstein.htm

But if you think about it the right way, it's obvious that every Goodstein

sequence *does* reach zero.

The point is that these numbers in "recursive base n" look a lot

like ordinals in Cantor normal form. If we translate them into

ordinals by replacing n by omega, the ordinals keep getting smaller

at each step, even when the numbers get bigger!

For example, when we do the translation

2^{2^2 + 1} + 2 |-> omega^{omega^omega + 1} + omega^1

3^{3^3 + 1} + 1 + 1 |-> omega^{omega^omega + 1} + 1 + 1

we see the ordinal got smaller even though the number got bigger.

Since epsilon_0 is well-ordered, the ordinals must bottom out at zero

after a finite number of steps - that's what "induction up to epsilon_0"

tells us. So, the numbers must too!

In fact, Kirby and Paris showed that you *need* induction up to

epsilon_0 to prove Goodstein sequences always converge to zero.

Since you can't do induction up to epsilon_0 in Peano arithmetic,

thanks to Goedel and Gentzen, it follows that Peano arithmetic is

unable to prove the Goodstein sequences go to zero (unless Peano

arithmetic is inconsistent).

So, this is a nice example of a fact about arithmetic that's obvious

if you think about it for a while, but not provable in Peano arithmetic.

I don't know any results in mathematical physics that use induction

up to epsilon_0, but these could be one - after all, trees show up

in the theory of Feynman diagrams. That would be pretty interesting.

There's a lot more to say about this, but I hear what you're asking:

what comes after epsilon_0?

Well, duh! It's

epsilon_0 + 1

Then comes

epsilon_0 + 2

and then eventually we get to

epsilon_0 + omega

and then

epsilon_0 + omega^2,..., epsilon_0 + omega^3,... , epsilon_0 + omega^4,...

and after a long time

epsilon_0 + epsilon_0 = epsilon_0 2

and then eventually

epsilon_0^2

and then eventually...

Oh, I see! You want to know the first *really interesting* ordinal

after epsilon_0.

Well, this is a matter of taste, but you might be interested in

epsilon_1. This is the first ordinal after epsilon_0 that satisfies

this equation:

x = omega^x

How do we actually reach this ordinal? Well, just as epsilon_0

was the limit of this sequence:

omega, omega^omega, omega^{omega^omega}, omega^{omega^{omega^omega}},...

epsilon_1 is the limit of this:

epsilon_0 + 1, omega^{epsilon_0 + 1}, omega^{omega^{epsilon_0 + 1}},...

In other words, it's the *union* of all these well-ordered sets.

In what sense is epsilon_1 the "first really interesting ordinal" after

epsilon_0? I'm not sure! Maybe it's the first one that can't be

built out of 1, omega and epsilon_0 using finitely many additions,

multiplications and exponentiations. Does anyone out there know?

Anyway, the next really interesting ordinal I know after epsilon_1 is

epsilon_2. It's the next solution of

x = omega^x

and it's defined to be the limit of this sequence:

epsilon_1 + 1, omega^{epsilon_1 + 1}, omega^{omega^{epsilon_1 + 1}},...

Maybe now you get the pattern. In general, epsilon_alpha is the

alpha-th solution of

x = omega^x

and we can define this, if we're smart, for any ordinal alpha.

So, we can keep driving on through fields of ever larger ordinals:

epsilon_2,..., epsilon_3,..., epsilon_4, ...

and eventually

epsilon_omega,..., epsilon_{omega+1},..., epsilon_{omega+2},...

and eventually

epsilon_{omega^2},..., epsilon_{omega^3},..., epsilon_{omega^4},...

and eventually

epsilon_{omega^omega},..., epsilon_{omega^{omega^omega}},...

As you can see, this gets boring after a while - it's suspiciously

similar to the beginning of our trip through the ordinals, with

them now showing up as subscripts under this "epsilon" notation.

But this is misleading: we're moving much faster now. I'm skipping

over much bigger gaps, not bothering to mention all sorts of ordinals

like

epsilon_{omega^omega} + epsilon_{omega 248} + omega^{omega^{omega + 17}}

Anyway... so finally we *got* to this South Dakota Tractor Museum,

driving pretty darn fast at this point, about 85 miles an hour...

and guess what?

Oh - wait a minute - it's sort of interesting here:

epsilon_{epsilon_0},..., epsilon_{epsilon_1},..., epsilon_{epsilon_2}, ...

and now we reach

epsilon_{epsilon_omega}

and then

epsilon_{epsilon_{omega^omega}},...,

epsilon_{epsilon_{omega^{omega^omega}}},...

and then as we keep speeding up, we see:

epsilon_{epsilon_{epsilon_0},...

epsilon_{epsilon_{epsilon_{epsilon_0}}},...

epsilon_{epsilon_{epsilon_{epsilon_{epsilon_0}}}},...

So, by the time we got that tractor museum, we were driving really fast.

And, all we saw as we whizzed by was a bunch of rusty tractors out in

a field! It was over in a split second! It was a real anticlimax -

just like this little anecdote, in fact.

But that's the way it is when you're driving through these ordinals.

Every ordinal, no matter how large, looks pretty pathetic and small

compared to the ones ahead - so you keep speeding up, looking for a

really big one... and when you find one, you see it's part of a new

pattern, and that gets boring too...

Anyway, when we reach the limit of this sequence

epsilon_0,

epsilon_{epsilon_0},

epsilon_{epsilon_{epsilon_0},

epsilon_{epsilon_{epsilon_{epsilon_0}}},

epsilon_{epsilon_{epsilon_{epsilon_{epsilon_0}}}},...

our notation breaks down, since this is the first solution of

x = epsilon_x

We could make up a new name for this ordinal, like eta_0.

Then we could play the whole game again, defining eta_{alpha} to be

the alpha-th solution of

x = epsilon_x

sort of like how we defined the epsilons. This kind of equation, where

something equals some function of itself, is called a "fixed point"

equation.

But since we'll have to play this game infinitely often, we might

as well be more systematic about it!

As you can see, we keep running into new, qualitatively different types

of ordinals. First we ran into the powers of omega, then we ran into

the epsilons, and now these etas. It's going to keep happening! For

each type of ordinal, our notation runs out when we reach the first

"fixed point" - when the xth ordinal of this type is actually equal to

x.

So, instead of making up infinitely many Greek letters, let's use

phi_gamma for the gamma-th type of ordinal, and phi_gamma(alpha) for

the alpha-th ordinal of type gamma.

We can use the fixed point equation to define phi_{gamma+1} in terms

of phi_gamma. In other words, we start off by defining

phi_0(alpha) = omega^alpha

and then define

phi_{gamma+1}(alpha)

to be the alpha-th solution of

x = phi_{gamma}(x)

We can even define this stuff when gamma itself is infinite.

For a more precise definition see the Wikipedia article cited below...

but I hope you get the rough idea.

This defines a lot of really big ordinals, called the "Veblen hierarchy".

There's a souped-up version of Cantor normal form that can handle

every ordinal that's a finite sum of guys in the Veblen hierarchy:

you can write them *uniquely* as finite sums of the form

phi_{gamma_1}(alpha_1) + ... + phi_{gamma_k}(alpha_k)

where each term is less than or equal to the previous one, and each

alpha_i is not a fixed point of phi_{gamma_i}.

But as you might have suspected, not *all* ordinals can be written

in this way. For one thing, every ordinal we've reached so far is

*countable*: as a set you can put it in one-to-one correspondence

with the integers. There are much bigger *uncountable* ordinals -

at least if you believe you can well-order uncountable sets.

But even in the realm of the countable, we're nowhere near done!

As I hope you see, the power of the human mind to see a pattern

and formalize it gives the quest for large countable ordinals a

strange quality. As soon as we see a systematic way to generate

a sequence of larger and larger ordinals, we know this sequence

has a limit that's larger then all of those! And this opens the

door to even larger ones...

So, this whole journey feels a bit like trying to run away from

your own shadow: the faster you run, the faster it chases after you.

But, it's interesting to hear what happens next. At this point we

reach something a bit like the Badlands on the western edge of South

Dakota - something a bit spooky!

It's called the Feferman-Schuette ordinal, Gamma_0. This is just

the limit, or union if you prefer, of all the ordinals mentioned

so far: all the ones you can get from the Veblen hierarchy. You

can also define Gamma_0 by a fixed point property: it's the smallest

ordinal x with

phi_x(0) = x

Now, we've already seen that induction up to different ordinals

gives us different amounts of mathematical power: induction up

to omega is just ordinary mathematical induction as formalized by

Peano arithmetic, but induction up to epsilon_0 buys us more -

it let's us prove the consistency of Peano arithmetic!

Logicians including Feferman and Schuette have carried out a detailed

analysis of this subject. They know a lot about how much induction

up to different ordinals buys you. And apparently, induction up to

Gamma_0 let's us prove the consistency of a system called "predicative

analysis". I don't understand this, nor do I understand the claim

I've seen that Gamma_0 is the first ordinal that cannot be defined

predicatively - i.e., can't be defined without reference to itself.

Sure, saying Gamma_0 is the first solution of

phi_x(0) = x

is non-predicative. But what about saying that Gamma_0 is the union

of all ordinals in the Veblen hierarchy? What's non-predicative

about that?

If anyone could explain this in simple terms, I'd be much obliged.

As you can see, I'm getting out my depth here. That's pretty typical

in This Week's Finds, but this time - just to shock the world -

I'll take it as a cue to shut up. So, I won't try to explain the

outrageously large Bachmann-Howard ordinal, or the even more

outrageously large Church-Turing ordinal - the first one that can't

be written down using *any* computable system of notation. You'll

just have to read the references.

I urge you to start by reading the Wikipedia article on ordinal

numbers, then the article on ordinal arithmetic, and then the one

on large countable ordinals - they're really well-written:

2) Wikipedia, Ordinal numbers,

http://en.wikipedia.org/wiki/Ordinal_number

Ordinal arithmetic,

http://en.wikipedia.org/wiki/Ordinal_arithmetic

Large countable ordinals,

http://en.wikipedia.org/wiki/Large_countable_ordinals

The last one has a tempting bibliography, but warns us that most

books on this subject are hard to read and out of print. Apparently

nobody can agree on notation for ordinals beyond the Veblen hierarchy,

either.

Gentzen proved the consistency of Peano arithmetic in 1936:

3) Gerhard Gentzen, Die Widerspruchfreiheit der reinen Zahlentheorie,

Mathematische Annalen 112 (1936), 493-565. Translated as "The

consistency of arithmetic" in M. E. Szabo ed., The Collected Works

of Gerhard Gentzen, North-Holland, Amsterdam, 1969.

Goodstein's theorem came shortly afterwards:

4) R. Goodstein, On the restricted ordinal theorem, Journal of

Symbolic Logic, 9 (1944), 33-41.

but Kirby and Paris proved it independent of Peano arithmetic

only in 1982:

5) L. Kirby and J. Paris, Accessible independence results for Peano

arithmetic, Bull. London. Math. Soc. 14 (1982), 285-93.

That marvelous guy Alan Turing wrote his PhD thesis at Princeton

under the logician Alonzo Church. It was about ordinals and their

relation to logic:

6) Alan M. Turing, Systems of logic defined by ordinals, Proc.

London Math. Soc., Series 2, 45 (1939), 161-228.

This is regarded as his most difficult paper. The idea is to

take a system of logic like Peano arithmetic and throw in an

extra axiom saying that system is consistent, and then another

axiom saying *that* system is consistent, and so on ad infinitum -

getting a new system for each ordinal. These systems are recursively

axiomatizable up to (but not including) the Church-Turing ordinal.

These ideas were later developed much further...

But, reading original articles is not so easy, especially if you're

in Shanghai without access to a library. So, what about online stuff -

especially stuff for the amateur, like me?

Well, this article is great fun if you're looking for a readable

overview of the grand early days of proof theory, when Hilbert was

battling Brouwer, and then Goedel came and blew everyone away:

7) Jeremy Avigad and Erich H. Reck, "Clarifying the nature of the

infinite": the development of metamathematics and proof theory,

Carnegie-Mellon Technical Report CMU-PHIL-120, 2001. Also

available as http://www.andrew.cmu.edu/user/avigad/Papers/infinite.pdf

But, it doesn't say much about the newer stuff, like the idea that

induction up to a given ordinal can prove the consistency of a logical

system - the bigger the ordinal, the stronger the system. For work

up to 1960, this is a good overview:

8) Solomon Feferman, Highlights in proof theory, in Proof Theory,

eds. V. F. Hendricks et al, Kluwer, Dordrecht (2000), pp. 11-31.

Also available at http://math.stanford.edu/~feferman/papers.html

For newer stuff, try this:

9) Solomon Feferman, Proof theory since 1960, prepared for the

Encyclopedia of Philosophy Supplement, Macmillan Publishing Co.,

New York. Also available at

http://math.stanford.edu/~feferman/papers.html

Also try the stuff on proof theory, trees and categories mentioned

in "week227", and the book by Girard, Lafont and Taylor mentioned

in "week94".

Finally, sometime I want to get ahold of this book by someone who

always enlivened logic discussions on the internet until his death in

April this year:

10) Torkel Franzen, Inexhaustibility: A Non-Exhaustive Treatment,

Lecture Notes in Logic 16, A. K. Peters, Ltd., 2004.

The blurb sounds nice: "The inexhaustibility of mathematical

knowledge is treated based on the concept of transfinite

progressions of theories as conceived by Turing and Feferman."

Okay, now for a bit about the icosahedron - my favorite Platonic solid.

I've been thinking about the "geometric McKay correspondence" lately,

and among other things this sets up a nice relationship between the

symmetry group of the icosahedron and an amazing entity called E8.

E8 is the largest of the exceptional Lie groups - it's 248-dimensional.

It's related to the octonions (the number "8" is no coincidence) and

it shows up in string theory. It's very beautiful how this complicated

sounding stuff can be seen in distilled form in the icosahedron.

I have a lot to say about this, but you're probably worn out by our

road trip through the land of big ordinals. So for now, try "week164"

and "week230" if you're curious. Let's talk about something less

stressful - the early history of the icosahedron.

I spoke about the early history of the dodecahedron in "week63".

It's conjectured that the Greeks got interested in this shape

from looking at crystals of iron pyrite. These aren't regular

dodecahedra, since normal crystals can't have 5-fold symmetry -

though "quasicrystals" can. Instead, they're "pyritohedra".

The Greeks' love of mathematical perfection led them to the

regular dodecahedron...

... and it also led them to invent the icosahedron:

11) Benno Artmann, About the cover: the mathematical conquest of

the third dimension, Bulletin of the AMS, 43 (2006), 231-235.

Also available at

http://www.ams.org/bull/2006-43-02/S0273-0979-06-01111-6/

According to Artmann, an ancient note written in the margins of a copy

of Euclid's Elements says the regular icosahedron and octahedron

were discovered by Theaetetus!

If you're a cultured sort, you may know Theaetetus through Plato's

dialog of the same name, where he's described as a mathematical

genius. He's also mentioned in Plato's "The Sophist". He probably

discovered the icosahedron between 380 and 370 BC, and died at an

early age in 369. Euclid wrote his construction of the icosahedron

that we find in Euclid's Elements:

12) Euclid, Elements, Book XIII, Proposition 16, online version

due to David Joyce at

http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII16.html

Artmann says this was the first time a geometrical entity appeared

in pure thought before it was seen! An interesting thought.

Book XIII also contains a complete classification of the Platonic

solids - perhaps the first really interesting classification

theorem in mathematics, and certainly the first "ADE classification":

13) Euclid, Elements, Book XIII, Proposition 18, online version

due to David Joyce at

http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII18.html

If you don't know about ADE classifications, see "week62".

I got curious about this "ancient note written in the margins of a

copy of Euclid" that Artmann mentions. It seemed too good to be true.

Just for fun, I tried to track down the facts about this, using only

my web browser here in Shanghai.

First of all, if you're imagining an old book in a library somewhere

with marginal notes scribbled by a pal of Theaetetus, dream on.

It ain't that simple! Our knowledge of Euclid's original Elements

relies on copies of copies of copies... and centuries of detective

work, with each detective having to root through obscure journals

and dim-lit library basements to learn what the previous detectives

did.

The oldest traces of Euclid's Elements are pathetic fragments of

papyrus. People found some in a library roasted by the eruption

of Mount Vesuvius in 79 AD, some more in a garbage dump in the

Egyptian town of Oxyrhynchus (see "week221"), and a couple more in

the Fayum region near the Nile. All these were written centuries

after Euclid died. For a look at one, try this:

14) Bill Casselman, One of the oldest extant diagrams from Euclid,

http://www.math.ubc.ca/~cass/Euclid/papyrus/

The oldest nearly complete copy of the Elements lurks in a museum

called the Bodleian at Oxford. It dates back to 888 AD, about a

millennium after Euclid.

More copies date back to the 10th century; you can find their stories

here:

15) Thomas L. Heath, editor, Euclid's Elements, chap. V: the text,

Cambridge U. Press, Cambridge, 1925. Also available at

http://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Euc.+5

16) Menso Folkerts, Euclid's Elements in Medieval Europe,

http://www.math.ubc.ca/~cass/Euclid/folkerts/folkerts.html

All these copies are somewhat different. So, getting at Euclid's

original Elements is as hard as sequencing the genome of Neanderthal

man, seeing a quark, or peering back to the Big Bang!

A lot of these copies contain "scholia": comments inserted by

various usually unnamed copyists. These were collected and

classified by a scholar named Heiberg in the late 1800s:

17) Thomas L. Heath, editor, Euclid's Elements, chap. VI: the scholia,

Cambride U. Press, Cambridge, 1925. Also available at

http://www.perseus.tufts.edu/cgi-bin/ptext?lookup=Euc.+6

One or more copies contains a scholium about Platonic solids in

book XIII. Which copies? Ah, for that I'll have to read Heiberg's

book when I get back to UC Riverside - our library has it, I'm

proud to say.

And, it turns out that another scholar named Hultsch argued

that this scholium was written by Geminus of Rhodes.

Geminus of Rhodes was an astronomer and mathematician who may have

lived between 130 and 60 BC. He seems like a cool dude. In his

Introduction to Astronomy, he broke open the "celestial sphere",

writing:

... we must not suppose that all the stars lie on one surface,

but rather that some of them are higher and some are lower.

And in his Theory of Mathematics, he proved a classification theorem

stating that the helix, the circle and the straight line are the only

curves for which any portion is the same shape as any other portion

with the same length.

Anyway, the first scholium in book XIII of Euclid's Elements, which

Hultsch attributes to Geminus, mentions

... the five so-called Platonic figures which, however, do not

belong to Plato, three of the five being due to the Pythagoreans,

namely the cube, the pyramid, and the dodecahedron, while the

octahedron and the icosahedron are due to Theaetetus.

So, that's what I know about the origin of the icosahedron!

Someday I'll read more, so let me make a note to myself:

18) Benno Artmann, Antike Darstellungen des Ikosaeders, Mitt.

DMV 13 (2005), 45-50. (Here the drawing of the icosahedron in

Euclid's elements is analysed in detail.)

19) A. E. Taylor, Plato: the Man and His Work, Dover Books, New

York, 2001, page 322. (This discusses traditions concerning

Theaetetus and Platonic solids.)

20) Euclid, Elementa: Libri XI-XIII cum appendicibus, postscript

by Johan Ludvig Heiberg, edited by Euangelos S. Stamatis,

Teubner BSB, Leipzig, 1969. (Apparently this contains information

on the scholium in book XIII of the Elements.)

Now for something a bit newer: categorification and quantum mechanics.

I've said so much about this already that I'm pretty much talked out:

21) John Baez and James Dolan, From finite sets to Feynman diagrams,

in Mathematics Unlimited - 2001 and Beyond, vol. 1, eds. Bjoern

Engquist and Wilfried Schmid, Springer, Berlin, 2001, pp. 29-50.

22) John Baez and Derek Wise, Quantization and Categorification,

Quantum Gravity Seminar lecture notes, available at:

http://math.ucr.edu/home/baez/qg-fall2003/

http://math.ucr.edu/home/baez/qg-winter2004/

http://math.ucr.edu/home/baez/qg-spring2004/

As I explained in "week185", many basic facts about harmonic

oscillators, Fock space and Feynman diagrams have combinatorial

interpretations. For example, the commutation relation between

the annihilation operator a and the creation operator a*:

aa* - a*a = 1

comes from the fact that if you have some balls in a box, there's one

more way to put a ball in and then take one out than to take one out

and then put one in! This way of thinking amounts to using finite

sets as a substitute for the usual eigenstates of the number operator,

so we're really "categorifying" the harmonic oscillator: giving it a

category of states instead of a set of states.

Working out the detailed consequences takes us through Joyal's

theory of "structure types" or "species" - see "week202" - and

on to more general "stuff types". Some nice category and

2-category theory is needed to make the ideas precise. For a

careful treatment, see this thesis by a student of Ross Street:

23) Simon Byrne, On Groupoids and Stuff, honors thesis,

Macquarie University, 2005, available at

http://www.maths.mq.edu.au/~street/ByrneHons.pdf and

http://math.ucr.edu/home/baez/qg-spring2004/ByrneHons.pdf

However, none of this work dealt with the all-important *phases*

in quantum mechanics! For that, we'd need a generalization of

finite sets whose cardinality can be be complex. And that's what

my student Jeffrey Morton introduces here:

24) Jeffrey Morton, Categorified algebra and quantum mechanics,

to appear in Theory and Application of Categories. Also available

as math.QA/0601458.

He starts from the beginning, explains how and why one would

try to categorify the harmonic oscillator, introduces the

"U(1)-sets" and "U(1)-stuff types" needed to do this, and shows

how the usual theorem expressing time evolution of a perturbed

oscillator as a sum over Feynman diagrams can be categorified.

His paper is now *the* place to read about this subject. Take

a look!

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