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

  1. Nov 4, 2006 #1
    [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 lets you prove other stuff you
    can't prove with just Peano arithmetic. For example, it lets 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 gonna 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 lets 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 lets 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!

    -----------------------------------------------------------------------
    Previous issues of "This Week's Finds" and other expository articles on
    mathematics and physics, as well as some of my research papers, can be
    obtained at

    http://math.ucr.edu/home/baez/

    For a table of contents of all the issues of This Week's Finds, try

    http://math.ucr.edu/home/baez/twfcontents.html

    A simple jumping-off point to the old issues is available at

    http://math.ucr.edu/home/baez/twfshort.html

    If you just want the latest issue, go to

    http://math.ucr.edu/home/baez/this.week.html
     
  2. jcsd
  3. Nov 4, 2006 #2
    In article <ea7eq3$jk4$1@glue.ucr.edu>,
    baez@math.removethis.ucr.andthis.edu (John Baez) writes:

    > 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.


    Anyone who enjoyed John's post on this should read INFINITY AND THE MIND
    by Rudy Rucker. This is one of the few books I have read cover to cover
    more than once (along with ALICE IN WONDERLAND, THROUGH THE LOOKING
    GLASS (both in the editions annotated by Martin Gardner), ZEN AND THE
    ART OF MOTORCYCLE MAINTENANCE and LILA).

    Author's web page:

    http://www.mathcs.sjsu.edu/faculty/rucker/

    Book's web page:

    http://pup.princeton.edu/titles/5656.html

    He also has a tongue-in-cheek science-fiction novel called WHITE LIGHT
    which deals with infinity (in a mathematical sense). Apart from these
    two books, I have also read and recommend his novel SOFTWARE. I would
    be interested (via email, not via the newsgroup) on comments on his
    other books.
     
  4. Nov 4, 2006 #3
    Omegad!

    Rgds

    Ian Macmillan
     
  5. Nov 4, 2006 #4
    In article <ea8l01$27t$1@online.de>,
    Phillip Helbig wrote:

    >Anyone who enjoyed John's post on this should read INFINITY AND THE MIND
    >by Rudy Rucker.


    I enjoyed my post, so maybe I should read this book!

    These large ordinals are a bit off topic from physics, but I need
    to make some corrections:

    In article <ea83ig$qmq$1@news.ks.uiuc.edu>,
    John Baez <baez@math.removethis.ucr.andthis.edu> wrote:

    >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!


    Kevin Buzzard pointed out a typo here. The sequence is:

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

    Also, while I got the huge number above from this website:

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

    he pointed out they actually say the sequence "can increase for
    approximately 2.6 * 10^{60605351} steps", not that it reaches
    zero at this point.

    I'm not sure what it means to say the "can increase" for this long -
    it either does or doesn't, right?

    Anyway, Kevin worked out the details himself, and I checked his
    calculations. We now seem to agree that the sequence increases
    until the ith term, where

    i = (1/4) 24 2^{24} 2^{24 2^{24}} - 2 ~ 1.72 x 10^{121210694}

    Then it levels off, and eventually it decreases, reaching zero at
    the ith term for

    i = 24 * 2^24 * 2^{24 * 2^{24}} - 2 ~ 6.9 * 10^{121210694}

    I wish some people would check and see if we've done the calculation
    correctly. It's basically just algebra, somewhat intimidating at
    first - but it gave me quite a sense of power when I got into it.

    Here is Kevin's email, prettied up by me, but perhaps with some
    mistakes added:

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


    You write this as if it were some kind of mystery. I remember working
    out this number explicitly when I was a graduate student! There is
    some nice form for it, as I recall. Let's see if I can reconstruct
    what I did.

    If I've understood the sequence correctly, it should be (where "n)"
    at the beginning of a line denotes we're working in base n on this
    line, so strictly speaking it's probably the n-1st term in the sequence)

    2) 2^2 = 4
    3) 3^3-1 = 2.3^2+2.3+2 = 26 [note: base 3, ends in 2, and 3+2=5]
    4) 2.4^2+2.4+1 = 41 [note: base 4, ends in 1, and 4+1=5]
    5) 2.5^2+2.5 = 60 [we're at a limit ordinal here, note 3+2=4+1=5]
    6) 2.6^2+2.6-1 = 2.6^2+6+5 = 83 [note: base 6, ends in 5]
    7) 2.7^2+7+4 [note: base 7, ends in 4]
    8) 2.8^2+8+3 [note: base 8, ends in 3, so we next get a limit ordinal at...]
     
  6. Nov 4, 2006 #5
    In article <eahpuk$bli$1@glue.ucr.edu>,
    baez@math.removethis.ucr.andthis.edu (John Baez) writes:

    > These large ordinals are a bit off topic from physics


    To bring things back on topic, what branches of physics, if any, use
    large ordinals, various kinds of infinity etc? Of course, not ALL
    mathematics is used in physics. On the other hand, some branches were
    once thought to be completely irrelevant to physics---group theory, for
    example. (I don't recall if it was Rutherford or Lord Kelvin who
    claimed this.)
     
  7. Nov 4, 2006 #6
    In article <eamp71$t2f$1@online.de>, Phillip Helbig wrote:

    >In article <eahpuk$bli$1@glue.ucr.edu>,
    >baez@math.removethis.ucr.andthis.edu (John Baez) writes:


    >> These large ordinals are a bit off topic from physics


    >To bring things back on topic, what branches of physics, if any, use
    >large ordinals, various kinds of infinity etc?


    As for the large countable ordinals I was mentioning, I don't know
    any work in physics that uses them beyond omega or maybe omega^n
    for finite n. Certainly no physicist with any brains would object
    to the proof that Goodstein sequences converge to zero, and you can
    only prove this by induction up to epsilon_0. But, I've never
    seen Goodstein sequences used in physics.

    However, it's worth emphasizing that these ordinals are not exotic
    entities. The ordinals I mentioned are all countable ordered sets,
    and you can describe them all as subsets of the rational numbers.

    More precisely: any countable ordered set is isomorphic, by an
    order-preserving map, to a subset of the rational numbers.

    Moreover, I believe that for any countable ordinal up to (but not
    including) the Church-Turing ordinal, you can write a computer
    program that will decide whether or not any given fraction is in
    this subset. As a consequence, you can also write a computer program
    that lists the fractions in this set.

    It's pretty obvious how to do this for omega^2:

    http://math.ucr.edu/home/baez/omega_squared.png

    But, I believe that you can do it for all the ordinals I mentioned,
    except for the Church-Turing ordinal. David Madore has drawn a picture
    of epsilon_0, for example - see sci.math.research for more on that.

    In short: countable ordinals below the Church-Turing ordinal are
    nothing scary. But, I haven't seen them used in physics, and I
    don't really expect it.

    What about cardinals?

    Well, physicists routinely use real numbers, which have a cardinality
    much larger than anything mentioned so far. Indeed, unless you add
    extra axioms to ordinary ZFC set theory, there's no telling how large
    the cardinality of the real numbers is!

    But, I'm also pretty sure that any calculation that predicts a concrete
    physical result can be done using only finite sets. And, the set of
    of *computable* real numbers is countable.

    So, for the purposes of physics, cardinals above aleph_0 are more a
    mathematical convenience than a necessity.

    However, every sufficiently convenient convenience eventually becomes
    a necessity.

    For example, flush toilets. In the West it might almost seem to be a
    *necessity* that public places have flush toilets - as opposed to, say,
    holes in the floor that you squat over. But here in Shanghai, many don't.

    In short, infinite sets resemble flush toilets. This is not to say that
    they're full of... oh, never mind, I think I've taken this analogy far
    enough.

    >Of course, not ALL
    >mathematics is used in physics. On the other hand, some branches were
    >once thought to be completely irrelevant to physics---group theory, for
    >example. (I don't recall if it was Rutherford or Lord Kelvin who
    >claimed this.)


    Lord Kelvin is mainly noted for having dismissed *vectors* as
    unnecessary to physics. He wrote:

    Quaternions came from Hamilton after his really good work had been
    done; and though beautifully ingenious, have been an unmixed evil
    to those who have touched them in any way, including Maxwell.
    Vector is a useless survival, or offshoot from quaternions, and has
    never been of the slightest use to any creature.

    To understand this, remember that J. Willard Gibbs, the first person
    to get a math PhD in the USA, introduced the modern approach to vectors
    around 1881, long after Hamilton's quaternions first became popular. He
    took the quaternion and chopped it into its "scalar" and "vector" parts.

    Vectors are another great example of a convenience that's so convenient
    that they're now seen as a necessity.

    It's mainly the American physicist John Slater, inventor of the "Slater
    determinant", who is famous for having dismissed groups as unnecessary
    to physics. He wrote:

    It was at this point that Wigner, Hund, Heitler, and Weyl entered the
    picture with their "Gruppenpest": the pest of the group theory [actually,
    the correct translation is "the group plague"] ... The authors of the
    "Gruppenpest" wrote papers which were incomprehensible to those like
    me who had not studied group theory... The practical consequences
    appeared to be negligible, but everyone felt that to be in the mainstream
    one had to learn about it. I had what I can only describe as a feeling
    of outrage at the turn which the subject had taken ... it was obvious
    that a great many other physicists we are disgusted as I had been with the
    group-theoretical approach to the problem. As I heard later, there were
    remarks made such as "Slater has slain the 'Gruppenpest'". I believe
    that no other piece of work I have done was so universally popular.

    And now, of course, it's categories that some physicists dismiss, just
    as they're catching on.

    So, judging by the history, you can be almost sure that if a bunch of
    physicists angrily dismiss a branch of mathematics as useless to physics,
    it's useful for physics. The branches of math that don't yet have
    applications to physics don't arouse such controversy!
     
  8. Nov 4, 2006 #7
    John Baez wrote:
    >
    > Well, physicists routinely use real numbers, which have a cardinality
    > much larger than anything mentioned so far. Indeed, unless you add
    > extra axioms to ordinary ZFC set theory, there's no telling how large
    > the cardinality of the real numbers is!
    >
    > But, I'm also pretty sure that any calculation that predicts a concrete
    > physical result can be done using only finite sets. And, the set of
    > of *computable* real numbers is countable.
    >
    > So, for the purposes of physics, cardinals above aleph_0 are more a
    > mathematical convenience than a necessity.
    >
    > However, every sufficiently convenient convenience eventually becomes
    > a necessity.
    >
    > For example, flush toilets. In the West it might almost seem to be a
    > *necessity* that public places have flush toilets - as opposed to, say,
    > holes in the floor that you squat over. But here in Shanghai, many don't.
    >
    > In short, infinite sets resemble flush toilets. This is not to say that
    > they're full of... oh, never mind, I think I've taken this analogy far
    > enough.
    >
    >
    > Lord Kelvin is mainly noted for having dismissed *vectors* as
    > unnecessary to physics. He wrote:
    >
    > Quaternions came from Hamilton after his really good work had been
    > done; and though beautifully ingenious, have been an unmixed evil
    > to those who have touched them in any way, including Maxwell.
    > Vector is a useless survival, or offshoot from quaternions, and has
    > never been of the slightest use to any creature.
    >
    > To understand this, remember that J. Willard Gibbs, the first person
    > to get a math PhD in the USA, introduced the modern approach to vectors
    > around 1881, long after Hamilton's quaternions first became popular. He
    > took the quaternion and chopped it into its "scalar" and "vector" parts.
    >
    > Vectors are another great example of a convenience that's so convenient
    > that they're now seen as a necessity.
    >
    > It's mainly the American physicist John Slater, inventor of the "Slater
    > determinant", who is famous for having dismissed groups as unnecessary
    > to physics. He wrote:
    >
    > It was at this point that Wigner, Hund, Heitler, and Weyl entered the
    > picture with their "Gruppenpest": the pest of the group theory [actually,
    > the correct translation is "the group plague"] ... The authors of the
    > "Gruppenpest" wrote papers which were incomprehensible to those like
    > me who had not studied group theory... The practical consequences
    > appeared to be negligible, but everyone felt that to be in the mainstream
    > one had to learn about it. I had what I can only describe as a feeling
    > of outrage at the turn which the subject had taken ... it was obvious
    > that a great many other physicists we are disgusted as I had been with the
    > group-theoretical approach to the problem. As I heard later, there were
    > remarks made such as "Slater has slain the 'Gruppenpest'". I believe
    > that no other piece of work I have done was so universally popular.
    >
    > And now, of course, it's categories that some physicists dismiss, just
    > as they're catching on.
    >
    > So, judging by the history, you can be almost sure that if a bunch of
    > physicists angrily dismiss a branch of mathematics as useless to physics,
    > it's useful for physics.


    No. In the examples above, there _were_ already significant uses
    in physics, and only those unfamiliar with the techniques called
    then superfluous.


    Arnold Neumaier
     
  9. Nov 4, 2006 #8
    In article <44D5C38D.3010105@univie.ac.at>,
    Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:

    >John Baez wrote:


    >> Lord Kelvin is mainly noted for having dismissed *vectors* as
    >> unnecessary to physics.


    >> It's mainly the American physicist John Slater, inventor of the "Slater
    >> determinant", who is famous for having dismissed groups as unnecessary
    >> to physics.


    >> And now, of course, it's categories that some physicists dismiss, just
    >> as they're catching on.


    >> So, judging by the history, you can be almost sure that if a bunch of
    >> physicists angrily dismiss a branch of mathematics as useless to physics,
    >> it's useful for physics.


    >No. In the examples above, there _were_ already significant uses
    >in physics, and only those unfamiliar with the techniques called
    >then superfluous.


    Exactly my point. If there _weren't_ significant uses of those
    mathematical techniques in physics, there wouldn't be physicists
    running around using that math, so there wouldn't be famous physicists
    getting upset and claiming those techniques were superfluous.

    For example, you don't have famous physicists claiming that large
    cardinals are superfluous to physics, precisely because these mostly
    *are* superfluous to physics - so nobody uses them, so nobody in his right
    mind complains about someone else using them. If tomorrow I read in
    sci.physics.research that Hawking wrote a polemic decrying the use
    of large cardinal hypotheses in physics, I'd immediately guess someone
    had found a use for them.

    ........................................................................

    Puzzle 28: Why should you be careful if you meet someone whose passport
    is from the British West Indies?

    If you get stuck, try:

    http://math.ucr.edu/home/baez/puzzles/
     
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