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This Week's Finds in Mathematical Physics (Week 259)

  1. Dec 11, 2007 #1
    Also available as http://math.ucr.edu/home/baez/week259

    December 9, 2007
    This Week's Finds in Mathematical Physics (Week 259)
    John Baez

    This week I'll talk about the "field with one element" - even though
    it doesn't exist. It's a mathematical phantom.

    But first: the Egg Nebula.

    In "week257" and "week258" I talked about interstellar dust.
    As I mentioned, lots of it comes from "asymptotic giant branch"
    stars - stars like our Sun, but later in their life, when they're
    big, red, pulsing, and puffing out elements like hydrogen, helium,
    carbon, nitrogen, and oxygen.

    The pulsations grow wilder and wilder until the star blows off its
    entire outer atmosphere, forming a big cloud of gas and dust
    misleadingly called a "planetary nebula". It leaves behind
    its dense inner core as a hot white dwarf. Intense radiation from
    this core eventually heats the gas and dust until they glow.

    Back in "week223" I showed my favorite example of a planetary
    nebula: the Cat's Eye. And I quoted the astronomer Bruce Balick
    on what will happen here when our Sun becomes a planetary nebula
    6.9 billion years from now:

    Here on Earth, we'll feel the wind of the ejected gases
    sweeping past, slowly at first (a mere 5 miles per second!),
    and then picking up speed as the spasms continue - eventually
    to reach 1000 miles per second!! The remnant Sun will rise as
    a dot of intense light, no larger than Venus, more brilliant
    than 100 present Suns, and an intensely hot blue-white color
    hotter than any welder's torch. Light from the fiendish blue
    "pinprick" will braise the Earth and tear apart its surface
    molecules and atoms. A new but very thin "atmosphere" of free
    electrons will form as the Earth's surface turns to dust.


    Here's a "protoplanetary nebula" - that is, a planetary nebula that's
    just getting started:

    1) Rainbow image of a dusty star, NASA,

    It's called the "Egg Nebula". You can see layers of dust coming out
    puff after puff, shooting outwards at about 20 kilometers per second,
    stretching out for about a third of a light year. The colors - red,
    green and blue - aren't anything you'd actually see. They're just
    an easy way to depict three different polarizations of light. I
    don't know why the light is polarized that way.

    You can see a dark disk of thicker dust running around the star. It
    could be an "accretion disk" spiralling into the star. The "beams"
    shining out left and right are still poorly understood. Maybe they're
    jets of matter ejected from the north and south poles of the disk?
    This idea seem more plausible when you look at this photo taken by
    NICMOS, which is Hubble's "Near Infrared Camera and Multi-Object

    2) Raghvendra Sahai, Egg Nebula in polarized light, Hubble Heritage
    Project, http://heritage.stsci.edu/2003/09/supplemental.html

    This near-infrared image also shows a bright spot called "Peak A" about
    500 AU from the central star. An "AU", or astronomical unit, is the
    distance from the Sun to the Earth.

    Nobody knows what this bright spot is. Some argue that it's just a
    clump of dust reflecting light from the main star. Others advocate a
    more exciting theory: it's a white dwarf orbiting the main star, which
    exploded in a "thermonuclear burst" after accreting a bunch of dust.

    3) Joel H. Kastner and Noam Soker, The Egg Nebula (AFGL 2688): deepening
    enigma, to appear in Asymmetrical Planetary Nebulae III, eds. M. Meixner,
    J. Kastner, N. Soker, and B. Balick, ASP Conference Series. Available
    as arXiv:astro-ph/0309677.

    I hope to say more about planetary nebulae in future Weeks, mainly
    because they're so beautiful.

    But now: the field with one element!

    A field is a mathematical structure where you can add, multiply,
    subtract and divide in ways that satisfy these familiar rules:

    x + y = y + x (x + y) + z = x + (y + z) x + 0 = x

    xy = yx (xy)z = x(yz) 1x = x

    x(y + z) = xy + xz

    every element x has an element -x with x + (-x) = 0

    every element x that's not 0 has an element 1/x with x (1/x) = 1

    You'll note that the last clause is the odd man out. Addition,
    subtraction and multiplication can all be described as everywhere
    defined operations. Division cannot, since we can't divide by 0.
    This is the funny thing about fields, which is what causes the
    problem we'll run into.

    Everyone who has studied math knows three examples of fields:
    the rational numbers Q, the real numbers R and the complex numbers C.
    There are a lot more, too - for example, function fields, number
    fields, and finite fields.

    Let me say a tiny bit about these three kinds of fields.

    The simplest sort of "function field" consists of rational functions
    of one variable - that is, ratios of polynomials, like this:

    (2z^3 + z + 1)/(z^2 - 7)

    Here the coefficients of your polynomials should lie in some field
    F you already know about. The resulting field is called F(z).
    If F is the complex numbers, we can think of F(z) = C(z) as consisting
    of functions on the Riemann sphere. In "week201", I explained how
    the symmetries of this field form a group important in special
    relativity: the Lorentz group!

    It's also very interesting to study the field of functions on a
    surface fancier than the sphere, but still defined by algebraic
    equations, like the surface of a doughnut or n-holed doughnut.
    Number theorists and algebraic geometers spend a lot of time
    thinking about these fields, which are called "function fields
    of complex curves".

    For example, different ways of describing the surface of a doughnut
    by algebraic equations give different "elliptic curves". This is a
    great example of terminology that's bound to mislead beginners!
    They're called "curves" even though it's 2-dimensional, because it takes one
    *complex* number to say where you are on a little patch of a surface,
    just as it takes one *real* number to say where you are on an
    ordinary curve like a circle. That's the origin of the term
    "complex curve". And, they're called "elliptic" because they first
    showed up when people were studying elliptic integrals, which are
    generalizations of trig functions from circles to ellipses.

    I explained more about elliptic curves in "week13" and "week125".
    Lurking behind this, there's a lot of fascinating stuff about
    function fields of elliptic curves.

    The simplest sort of "number field" comes from taking the rational
    numbers and throwing in the solutions of a polynomial equations.
    For example, in "week20" I talked about the "golden field", which
    consists of all numbers of the form

    a + b sqrt(5)

    where a and b are rational.

    One of the most beautiful ideas in math is the analogy between
    number fields and function fields - the idea that numbers are like
    functions on some sort of "space". I began explaining this in
    "week205", "week216" and "week218", but there's much more to say
    about what's known... and also many things that remain mysterious.

    In particular, it's pretty well understood how number fields
    resemble function fields of complex curves, and how this relates
    number theory to *2-dimensional* topology. But, there are also many
    analogies between number theory and *3-dimensional* topology, which
    I began listing in "week257". It seems these analogies are doomed to
    remain mysterious until we get a handle on the field with one element.
    But more on that later.

    The simplest sort of "finite field" comes from choosing a prime number
    p and taking the integers modulo p. The result is sometimes called
    Z/p, especially when you're just concerned with addition. But when you
    think of it as a field, it's better to call it F_p.

    The reason is that there's a finite field of size q whenever q is a
    *power* of a prime, and this field is unique - so it's called F_q. You
    build F_q sort of like how you build the complex numbers starting from
    the real numbers, or number fields starting from the rational numbers.
    Namely, to construct F_{p^n}, you take F_p and throw in the roots of a
    well-chosen polynomial of degree n: one that doesn't have any roots in
    F_p, but "wants" to have n different roots.

    Okay: that was a tiny bit about function fields, number fields and finite
    fields. But now I need to point out some slight lies I told!

    I said there was a finite field with q elements whenever q was a prime
    power. You might think this should include q = 1, since 1 is the
    *zeroth* power of *any* prime.

    So, is there a field with one element?

    If so, it must have 1 = 0. That doesn't violate the definition of
    a field that I gave you... does it? The definition said any element
    that's not 0 has a reciprocal. In this particular example, 0 also has
    a reciprocal, since we can set 1/0 = 1 and not get into any contradictions.
    But that's not a problem: in usual math practice, saying "we can divide
    by anything that's not zero" doesn't deny the possibility that we can
    divide by 0.

    Unfortunately, allowing a field with 1 = 0 causes nothing but grief.
    For example, we can define vector spaces using any field (people say
    "over" any field), and there's a nice theorem saying two vector spaces
    are isomorphic if and only if they have the same dimension. And normally,
    there's one vector space of each dimension. But the last part isn't true
    for a field with 1 = 0. In a vector space over such a field, every
    vector v has

    v = 1 v = 0 v = 0

    So, every vector space is 0-dimensional!

    To prevent such problems, people add one extra clause to the definition
    of a field:

    1 is not equal to 0

    This clause looks even more tacked-on and silly than the clause
    saying everything *nonzero* has a reciprocal... but it works fairly

    However, the field with one element still wants to exist! Not the
    silly field with 1 = 0, but something else, something more mysterious...
    something that Gavin Wraith calls a "mathematical phantom":

    4) Gavin Wraith, Mathematical phantoms,

    What's a mathematical phantom? According to Wraith, it's an object
    that doesn't exist within a given mathematical framework, but
    nonetheless "obtrudes its effects so convincingly that one is forced
    to concede a broader notion of existence".

    Like a genie that talks its way out of a bottle, a sufficiently powerful
    mathematical phantom can talk us into letting it exist by promising to
    work wonders for us. Great examples include the number zero, irrational
    numbers, negative numbers, imaginary numbers, and quaternions. At one
    point all these were considered highly dubious entities. Now they're
    widely accepted. They "exist". Someday the field with one element
    will exist too!


    I gave a lot of reasons in "week183", "week184", "week185", "week186"
    and "week187", but let me rapidly summarize.

    It's all about "q-deformation". In physics, people talk about q-deformation
    when they're taking groups and turning them into "quantum groups". But it
    has a closely related aspect that's in some ways more fundamental. When
    we count things involving n-dimensional vector spaces over the finite field
    F_q, we often get answers that are polynomials in q. If we then set q = 1,
    the resulting formulas count analogous things involving n-element sets!

    So, finite sets want to be finite-dimensional vector spaces over the
    (nonexistent) field with one element... or something like that. We can
    be more precise after looking at some examples.

    Here's the simplest example. Say we count lines through the origin
    in an n-dimensional vector space over F_q. We get the "q-integer"

    q^n - 1
    ------- = 1 + q + q^2 + ... + q^{n-1}
    q - 1

    which I'll write as [n] for short.

    Setting q = 1, we n. This is the number of points in an n-element set.
    Sure, that sounds silly. But, I'm trying to make a point here! At
    q = 1, stuff about n-dimensional vector spaces over F_q reduces to stuff
    about n-element sets, and the q-integer [n] reduces to the ordinary
    integer n.

    This may not be the best way to understand the pattern, though.
    Lines through the origin in an n-dimensional vector space are the
    same as points in an (n-1)-dimensional projective space. So, the
    real analogy may be between "points in a projective space" and
    "points in a set".

    Here's a more impressive example. Pick any uncombed Young diagram D
    with n boxes. Here's one with 8 boxes:

    X 1 box in the first row
    X X 3 boxes in the first two rows
    X X X 6 boxes in the first three rows
    X X 8 boxes in the first four rows

    Then, count the "D-flags on an n-dimensional vector space over F_q".
    In our example, such a D-flag is:

    a 1-dimensional subspace
    of a 3-dimensional subspace
    of a 6-dimensional subspace
    of a 8-dimensional vector space over F_q

    If you actually count these D-flags you'll get some formula, which is
    a polynomial in q. And when you set q = 1, you'll get the number of
    "D-flags on an n-element set". In our example, such a D-flag is:

    a 1-element subset
    of a 3-element subset
    of a 6-element subset
    of a 8-element set

    For details, and a proof that this really works, try:

    5) John Baez, Lecture 4 in the Geometric Representation Theory
    Seminar, October 9, 2007. Available at

    These examples can be generalized. In "week187" I showed how to get
    one example for each subset of the dots in any Dynkin diagram! This
    idea goes back to Jacques Tits, who was the first to suggest that there
    should be a field with one element. Dynkin diagrams give algebraic
    groups over F_q... but he noticed that these groups reduce to "Coxeter
    groups" as q -> 1. And, if you mark some dots on a Dynkin diagram you
    get a "flag variety" on which your algebraic group acts... but as q -> 1,
    this reduces to a finite set on which your Coxeter group acts.

    If you don't understand the previous paragraph, don't worry - it's
    over now. It's great stuff, but my main point is that there seems to
    be an analogy like this:

    q = 1 q = a power of a prime number

    n-element set (n-1)-dimensional projective space over F_q
    integer n q-integer [n]
    permutation groups S_n projective special linear group PSL(n,F_q)
    factorial n! q-factorial [n]!

    This opens up lots of questions. For example, if projective spaces over
    F_1 are just finite sets, what should *vector spaces* over F_1 be?

    People have thought about this, and the answer seems to be "pointed
    sets" - sets with a distinguished point, which you can think of as
    the "origin". A pointed set with n+1 elements seems to act like
    an n-dimensional vector space over F_q.

    For more clues, and an attempt to do algebraic geometry using the
    field with one element, try this:

    6) Christophe Soule, On the field with one element, Talk given at the
    Arbeitstagung, Bonn, June 1999, IHES preprint available at

    Soule tries to define "algebraic varieties" over F_1, namely curves
    and their higher-dimensional generalization. And, he talks a lot about
    zeta functions for such varieties. He goes into more detail here:

    7) Christophe Soule, Les varietes sur le corps a un element, Moscow
    Math. Jour. 4 (2004), 217-244, 312.

    The theme of zeta functions - see "week216" - is deeply involved in
    this business. For more, try these papers:

    8) N. Kurokawa, Zeta functions over F_1, Proc. Japan Acad. Ser. A
    Math. Sci. 81 (2006), 180-184.

    9) Anton Deitmar, Remarks on zeta functions and K-theory over F1,
    available as arXiv:math/0605429.

    But instead of talking about zeta functions, I'd like to talk about
    two approaches to giving a formal definition of the field with one
    element. Both of them involve taking the concept of field and
    modifying it so it doesn't necessary involve the operation of addition.
    The first one, due to Deitmar, simply throws out addition! The second,
    due to Anton Durov, allows for a wide choice of operations - and thus
    a wide supply of "exotic fields".

    For Deitmar's approach, try these:

    10) Anton Deitmar, Schemes over F1, available as arXiv:math/0404185.

    F1-schemes and toric varieties, available as arXiv:math/0608179.

    The usual approach to fields treats fields as specially nice commutative
    rings. A "commutative ring" is a gadget where you can add and multiply,
    and these rules hold:

    x + y = y + x (x + y) + z = x + (y + z) x + 0 = x

    xy = yx (xy)z = x(yz) 1x = x

    x(y + z) = xy + xz

    every element x has an element -x with x + (-x) = 0

    Deitmar throws out addition and treats fields as specially nice
    commutative monoids. A "commutative monoid" is a gadget where you can
    multiply, and these rules hold:

    xy = yx (xy)z = x(yz) 1x = x

    For Deitmar, the field with one element, F_1, is just the commutative
    monoid with one element, namely 1. A "vector space over F_1" is just
    a set on which this monoid acts via multiplication... but that amounts
    to just a plain old set. The "dimension" of such a "vector space"
    is just its cardinality.

    All this so far is quite trivial, but Deitmar makes a nice attempt at
    redoing algebraic geometry to include this field with one element.
    One reason to do this is to understand the mysterious 3-dimensional
    aspect of number theory.

    To explain this, I need to say a bit about "schemes". In ordinary
    algebraic geometry, we turn commutative rings into spaces to think
    about them geometrically. I explained this back in "week199" and
    "week205", but let me review quickly, and go further:

    We can think of elements of a commutative ring R as functions on
    certain space called the "spectrum" of R, Spec(R). This space has
    a topology, so we can also talk about functions that are defined,
    not on all of Spec(R), but just *part* of Spec(R) - namely some open set.
    Indeed, for each open set U in Spec(R), there's a commutative ring O(U)
    consisting of those functions defined on U. These commutative
    rings are related in nice ways:

    A) If the open set V is smaller than U, we can restrict functions from
    U to V, getting a ring homomorphism O(U) -> U(V)

    B) If U is covered by a bunch of open sets U_i, and we have a function
    f_i in each O(U_i), such that f_i and f_j agree when restricted to
    the intersection of U_i and U_j, then there's a unique function f in O(U)
    that restricts to each of these functions f_i.

    Something satisfying condition A) is called a "presheaf" of commutative
    rings; something also satisfying condition B) is called a "sheaf" of
    commutative rings.

    So, Spec(R) is not just a topological space, it's equipped with a
    sheaf of commutative rings. People call this a "ringed space".

    Whenever we have a ringed space, we can ask if it comes from a
    commutative ring R in the way I just sketched. If so, we call
    it an "affine scheme". Affine schemes are just a fancy geometrical way
    of talking about commutative rings!

    More interestingly, whenever we have a ringed space, we can ask if
    it's *locally* isomorphic to one coming from a commutative ring.
    In other words: does every point have a neighborhood that, as a
    ringed space, looks like Spec(R) for some commutative ring R?
    Or in other words: is our ringed space *locally* isomorphic to an
    affine scheme? If so, we call it a "scheme".

    A classic example of a scheme that's not an affine scheme is the
    Riemann sphere. There aren't any rational functions defined on the
    whole Riemann sphere, except for constants - the rest all blow up
    somewhere. So, it's hopeless trying to think of the Riemann sphere
    as an affine scheme.

    But, for any open set U there's a commutative ring O(U) consisting of
    rational functions that are defined on U. So, the Riemann sphere
    becomes a ringed space. And, it's *locally* isomorphic to the complex
    plane, which is the affine scheme corresponding to the commutative ring
    of complex polynomials in one variable. So, the Riemann sphere is a

    For more on schemes, try this nice introduction, which actually
    has lots of pictures:

    11) David Eisenbud, The Geometry of Schemes, Springer, Berlin, 2000.

    Now, we can talk about schemes "over a field F", meaning that each
    commutative ring O(U) is also a vector space over F, in a well-behaved
    way, giving us a "sheaf of commutative rings over F". For example,
    the Riemann sphere is a scheme over C.

    There's a secret 3-dimensional aspect to the affine scheme Spec(Z),
    where Z is the commutative ring of integers. As explained in the
    Addenda to "week257", we might understand this if we could see Spec(Z)
    as a scheme over the field with one element! For more, see this:

    12) 7) M. Kapranov and A. Smirnov, Cohomology determinants and
    reciprocity laws: number field case, available at

    So, we really need a theory of schemes over the field with one element.
    The problem is, F_1 isn't really a field. In Deitmar's approach, it's
    just a commutative monoid.

    So, let me sketch how Deitmar gets around this. In a nutshell, he takes
    advantage of the fact that a lot of basic algebraic geometry only requires
    multiplication, not addition!

    He starts by defining a "commutative ring over F_1" to be simply a
    commutative monoid. The simplest example is F_1 itself.

    Now, watch how he gets away with never using addition:

    He defines an "ideal" in a commutative monoid R to be a subset I for
    which the product of something in I with anything in R again lies in I.
    He says an ideal P is "prime" if whenever a product of two elements in
    R is in P, at least one of them is in P.

    He defines the "spectrum" Spec(R) of a commutative monoid R to be the
    set of its prime ideals. He gives this the "Zariski topology". That's
    the topology where the closed sets are the whole space, or any set of
    prime ideals that contain a given ideal.

    He then shows how to get a sheaf of commutative monoids on Spec(R).
    He defines a "scheme" to be a space equipped with a sheaf of
    commutative monoids that's *locally* isomorphic to one of this sort.

    If you know algebraic geometry, these definitions should seem very
    familiar. And if you don't, you can just replace the word "monoid"
    by "ring" everywhere in the previous three paragraphs, and you'll get
    the standard definitions in algebraic geometry!

    Deitmar shows how to build a scheme over F_1 called the "projective
    line". The projective line over C is just the Riemann sphere. The
    projective line over F_1 has just two points (or more precisely, two
    closed points). This is good, because the projective line over the
    field with q elements has

    [2] = 1 + q

    points, and we're doing the q = 1 case.

    Deitmar's construction seems like a lot of work to get ahold of the
    2-element set, if that's all it secretly is. But, I need to think
    about this more. After all, he doesn't just get a space; he gets a
    sheaf of commutative monoids on this space! And what's that like?
    I should work it out.

    Deitmar also shows how to relate schemes over F_1 to the usual sort
    of schemes.

    From a commutative ring, we can always get a commutative monoid just
    by forgetting the addition. This process has a kind of reverse, too.
    Namely, from a commutative monoid, we can get a commutative ring
    simply by taking formal integral linear combinations of elements.
    Using this, Deitmar shows how we can turn ordinary schemes into
    schemes over F_1... and conversely. He says an ordinary scheme is
    "defined over F_1" if it arises in this way from a scheme over F1.

    Okay, that's a taste of Deitmar's approach. For Durov's approach,
    try this mammoth 568-page paper:

    13) Anton Durov, New approach to Arakelov geometry, available as

    or read our discussions of it at the n-Category Cafe, starting here:

    14) David Corfield, The field with one element,

    Durov defines a "generalized ring" to be what Lawvere much earlier
    called an "algebraic theory". What is it? Nothing scary! It's just a
    gadget with a bunch of abstract n-ary operations closed under composition,
    permutation, duplication and deletion of arguments, and equipped with
    an identity operation.

    So, for example, if our gadget has a binary operation

    (x,y) |-> f(x,y)

    we can compose this with itself to get the ternary operation

    (x,y,z) |-> f(f(x,y),z)

    and the 4-ary operation

    (w,x,y,z) |-> f(w,f(x,f(y,z)))

    and so on. We can then permute arguments in our 4-ary operation
    to get one like this:

    (w,x,y,z) |-> f(z,f(x,f(w,y)))

    or duplicate some arguments to get a binary operation like this

    (x,y) |-> f(x,f(x,f(y,y)))

    From this we can then form a 3-ary operation by deleting an argument,
    for example like this:

    (x,y,z) |-> f(x,f(x,f(y,y)))

    If you know about "operads", this kind of gadget is just a specially
    nice operad where we can duplicate and delete operations.

    Now, a generalized ring is said to be "commutative" if all the
    operations commute in a certain sense. (I'll let you guess what
    it means for an n-ary operation to commute with an m-ary operation.)
    We get an example of a commutative generalized ring from a commutative
    ring R if we let the n-ary operations be "n-ary R-linear combinations",
    like this:

    (x_1, ..., x_n) |-> r_1 x_1 + ... + r_n x_n

    We also get a very similar example from any "commutative rig", which is
    a gizmo satisfying rules like those of a commutative ring, but without

    x + y = y + x (x + y) + z = x + (y + z) x + 0 = x

    xy = yx (xy)z = x(yz) 1x = x

    x(y + z) = xy + xz

    And, we get an example from any commutative monoid, where we only
    have 1-ary operations, coming from multiplication by elements of
    our monoid:

    (x_1) |-> r x_1

    So, Durov's framework generalizes Deitmar's! But, it includes a lot
    more examples: exotic hothouse flowers like the "tropical rig", the
    "real integers", and more. He develops a theory of schemes for all
    these generalized rings, and builds it "up to construction of algebraic
    K-theory, intersection theory and Chern classes" - fancy things that
    algebraic geometers like.

    What I don't yet see is how either Deitmar's or Durov's approach
    helps us understand the secret 3-dimensional nature of Spec(Z).
    I may just need to read their papers more carefully and think about
    them more.

    Finally, here's yet another approach to the field with one element:

    15) Bertrand Toen and M. Vaquie, Under Spec(Z), available as

    In short, a mathematical phantom is gradually materializing before our
    very eyes! In the process, a grand generalization of algebraic
    geometry is emerging, which enriches it to include some previously
    scorned entities: rigs, monoids and the like. And, this enrichment
    holds the promise of shedding light on some otherwise impenetrable
    mysteries: for example, the deep inner meaning of q-deformation, and
    the 3-dimensional nature of Spec(Z).


    Quote of the Week:

    "The analogy between number fields and function fields finds a basic
    limitation with the lack of a ground field. One says that Spec(Z)
    (with a point at infinity added, as is familiar in Arakelov geometry)
    is like a (complete) curve, but over which field?" - Christophe Soule

    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


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


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


    If you just want the latest issue, go to

  2. jcsd
  3. Dec 12, 2007 #2
    baez@math.removethis.ucr.andthis.edu (John Baez) writes:
    > So, for example, if our gadget has a binary operation
    > (x,y) |-> f(x,y)
    > we can compose this with itself to get the ternary operation
    > (x,y,z) |-> f(f(x,y),z)
    > and the 4-ary operation
    > (w,x,y,z) |-> f(w,f(x,f(y,z)))
    > and so on.

    That's not an 'and so on'. 'And so on' would have been
    (w,x,y,z) |-> f(f(f(w,x),y),z)

    Or similarly you could have seeded the process with
    (x,y,z) |-> f(x,f(y,z)))

    Dear aunt, let's set so double the killer delete select all.
    -- Microsoft voice recognition live demonstration
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