### Monday, February 19th, 2018 @ 3:35 pm

Went back to NJ on Friday because my good friend Pete was having a party at his new home in Philly. Stayed with my brother Julian, who also lives in Philly, on Saturday, and made it back to Moorestown on Sunday. Now we’re here. Hopefully I can post the four stories that have been buzzing in the back of my mind today, and make up for lost time. The first story takes us from abstract algebra to n-spheres to the Bott periodicity theorem of homotopy theory and the notion of Clifford aka Geometric Algebras—thanks again John Baez! The second story takes us from factory farms to the notion of kingship in anthropology, the transformative powers of racoons, and the morality of quantum mechanics. The third story takes us from AI to the Russians to my writings circa 2014. And the fourth and final story is about what it’s like in 2018 to continue Plato’s tradition of symposia aka what it’s like to be the philosopher today at the margins of the party.

1.

So, as usual, they misled you in school, mainly by omission. Your teachers probably presented “numbers” and “algebra” like they’re fixed concepts, set in stone. *Here’s what numbers are. Here’s what the rules of algebra are. Now follow the rules to calculate the solution.*This is fine as far as it goes, but it implicitly makes you think “breaking the rules” is bad. And if you think “breaking the rules” is bad, it might prevent you from asking the simple question that lies at the heart of abstract algebra, and which represents the true path of mathematical enlightenment, aka:*What happens to the numbers when you break the rules of algebra?*

Abstract algebra thus asks you to take a new, higher vantage point. Instead of asking two separate questions once and for all time, What is number? and What is algebra?, it asks: *If you change the rules of algebra, how do the numbers change? If you change the numbers, how do the rules of algebra change?* Number and algebra are seen as two concepts that actually co-determine each other.

Now it turns out mathematicians have been exploring this subject intensely now for 200 years, and *a lot* is known. I’ll try to synthesize some of the results here, without necessarily demonstrating each step, as there are numerous excellent resources available to those who are interested in the details. I apologize for the errors I've no doubt introduced due to ignorance or overeagerness.

The basic operations of arithmetic: addition, subtraction, multiplication, and division. If you start with the counting numbers 1, 2, 3, …, you quickly find that you need to upgrade to the integers …, -3, -2, -1, 0, 1, 2, 3, … if you want subtraction to always make sense, and for every number to have an additive inverse (like -3 and 3). You also discover that you need to upgrade to the rational numbers aka ratios of integers, if you want division to always make sense, and for every number to have a multiplicative inverse (like 1/3 and 3). So even at this early stage, it’s clear: the demand that our number system be “closed” under a certain algebra naturally leads to a corresponding sophistication in the concept of number relative to that algebra.

Now suppose we demand that between any two numbers, there’s a third number (that also has an inverse, etc), then you get the “real numbers.” The real numbers can be characterized by four properties: ordering, commutativity, associativity, normitude. In other words, real numbers can always be put in some unique order aka … a < b < c < d …; the order of multiplication of real numbers doesn’t matter aka a.b = b.a; the insertion of parentheses doesn’t matter aka a.(b.c) = (a.b).c; and finally, distance between vectors of real numbers is defined by something like the Pythagorean theorem aka d.d = a.a + b.b + c.c… It’s because of this that we can define the intrinsic “length” or “norm” of a real vector, and if we have two vectors u and v, |u||v| = |u.v|.

So what happens if we relax the property of ordering? Then we get the complex numbers! (Remember, some x+y.i given in rectangular coordinates aka r.e**i.theta given in polar coordinates.) Whereas the real numbers are defined on a line, the complex numbers are defined on a plane—so there’s no single notion of orderedness to them. But they’re still commutative and associative and normie, although we need to be a little more nuanced about the Pythagorean theorem, which now appears as d**2 = a.a* + b.b* + c.c* + … where a* is a’s complex conjugate (under complex conjugation, x+y.i -> x-y.i aka r.e**i.theta -> r.e**-i.theta.) So a.a* is just x.x + y.y: the imaginary parts cancel out and we get the “real” length squared of the hypoteneuse. (You can see the old Pythagorean formula as just a special case of the new one: we just didn’t have to worry about conjugation before because the real numbers are symmetric under it.)

Great! Now what happens if we relax the property of commutativity, so that a.b != b.a? Then we get the quaternions! What are the quaternions? They describe dilations and rotations in 3D space, just as real numbers represent dilations and rotations in 1D space, and complex numbers represent dilations and rotations in 2D space. And if you spent a moment with any object at hand, you will soon realize that doing rotation A followed by rotation B is not always the same as doing rotation B followed by rotation A. You can write a quaternion analogously to a complex number as some a + b.i + c.j + d.k, where a, b, c, and d are real numbers, and i, j, and k represent three orthagonal ways we can rotate around the real axis. Quaternions aren’t ordered, nor commutative, but they are still associative and normie.

If we relax the property of associativity, so that a.(b.c) != (a.b).c, we get the octonians, which can be written as a + b.i + c.j + d.k + e.l + f.m + g.n + h.o. There are 7 imaginary axes orthagonal to the real axis, which makes 8. But, due to their non-associativity, octonions have no matrix representation in themselves! Now you may have noticed that the dimensionality of our numbers has been growing like powers of 2.

Real numbers -> 2**0 dimensions Complex numbers -> 2**1 dimensions Quaternions -> 2**2 dimensions Octonians -> 2**3 dimensions...

If we continue the construction to the sedenions et al, we lose the property of normitude. There’s no way to define the intrinsic “length” or “norm” of a vector of sedenions such that |u||v| = |u.v|. Another way of saying it: we lose the classical notions of “inverse” and “division,” and along with it, the tight relationship between scalar and vector that really makes the notion of a vector space even possible or at least useful. So it’s like the universe is saying: thou shalt always define thy vector spaces over the four primal number fields: the reals, the complex numbers, the quaternions, and the octonions—because according to Hurwitz’s theorem, these are the only four normed division algebras.

But the story doesn’t end there, because rather than a limitation, this turns out to be a huge advantage! But first we need to take a new perspective on multidimensional space.

Let’s begin by defining the notion of a Clifford aka a Geometric Algebra. For this, pictures are worth a thousand words. GA(0) is the Geometric Algebra defined by 1 vector. It consists of scalars that describe the dilations and contractions of this vector, and the vector itself. GA(1) is the Geometric Algebra defined by 2 orthagonal vectors. It consists of its scalars, the two vectors, and the bivector swept out from the first vector to the second. So GA(1) is just the complex numbers in disguise: the two vectors are the real and imaginary axes, and “multiplying” by the bivector is like doing a 90 degree rotation in the complex plane. GA(2) is the Geometric Algebra defined by 3 orthagonal vectors. It consists of its scalars, the three vectors, the three possible bivectors between them, and 1 trivector. These objects provide a basis for lengths, directions, areas, volumes, etc, within the space. What I mean is that, somewhat incredibly, GA(n), the Geometric Algebra defined by n+1 orthagonal vectors, itself defines a vector space in 2**n dimensions that keeps track of all the higher geometry!

Now it turns out GA(2) is just the quaternions in disguise. But consider GA(3). It has 4 orthagonal vectors. It consists of scalars, the four vectors, the 16 possible bivectors, the 4 possible trivectors, and 1 quatrovector. But GA(3) is *not* the octonians! So instead of trying to define higher dimensional space using our old construction of reals, complex numbers, quaternions, octonions, sedenions, shedding algebraic properties as we rise in powers of 2, let’s define higher dimensional space in terms of a Geometric Algebra, because even as our dimensionality goes up by powers of 2, we retain our associativity and normitude. In retrospect, the octonions represent just one fork along the road to infinity. (PS. GA(4) actually represents the 4d relativistic space that an electron lives in (once provided with the proper metric, where time is negative--written as GA(1, 3)!)

Having perceived all this, we can now appreciate the following staggering fact known as “Bott Periodicity.”

n+1 vectors | GA(n) | Vector Space | Dimensionality 1 GA(0) R 2**0 = 1 2 GA(1) C 2**1 = 2 3 GA(2) H 2**2 = 4 4 GA(3) H, H 2**3 = 8 5 GA(4) H(2) 2**4 = 16 6 GA(5) C(4) 2**5 = 32 7 GA(6) R(8) 2**6 = 64 8 GA(7) R(8), R(8) 2**7 = 128 9 GA(8) GA(n-8)(16) 2**8 = 256 ~ 1 10 GA(9) 2**9 = 512 ~ 2 11 GA(10) 2**10 = 1028 ~ 3 12 GA(11) 2**11 = 2056 ~ 4 13 GA(12) 2**12 = 4096 ~ 5 14 GA(13) 2**13 = 8192 ~ 6 15 GA(14) 2**14 = 32768 ~ 7 16 GA(15) 2**15 = 65536 ~ 8 17 GA(16) 2**16 = 131072 ~ 9 ~ 1 18 GA(17) 2**17 = 262144 ~ 10 ~ 2 19 GA(18) 2**18 = 524288 ~ 11 ~ 3 20 GA(19) 2**19 = 1048576 ~ 12 ~ 4 21 GA(20) 2**20 = 2097152 ~ 13 ~ 5 22 GA(21) 2**21 = 4194304 ~ 14 ~ 6 23 GA(22) 2**22 = 8388608 ~ 15 ~ 7 24 GA(23) 2**23 = 16777216 ~ 16 ~ 8 ... . . . . . .

It’s ironic: the octonions were originally called the “octaves,” but it’s by graduating to GA(3) instead of to O, that we come to perceive what may rightly be called God’s true “octave.” Notationally, “H, H” means two quaternions “concatenated” together; and H(2) means 2x2 matrices whose elements are quaternions. So Bott Periodicity is the statement that after the first octave, GA(n) = GA(n-8)(16) aka 16x16 matrices whose elements come from GA(n-8). Concretely, GA(8) = GA(1)(16) = R(16) aka 16x16 matrices of real numbers. GA(16) = GA(8)(16) = GA(1)(16)(16) = R(16)(16) aka 16x16 matrices of 16x16 matrices of real numbers. So higher dimensional space actually wraps back around itself, as the original abstract “octave” is recursively embedded within it in ever more sophisticated ways.

It’s all beautifully analogous to number theory/musical harmony.

**1** **2** 3 **4** 5 6 7 **8**

Between adjacent powers of two, we always find some new primes. We also find every 2nd number divisible by 2, every 3rd number divisibly by 3, etc. This means that new primes always appear situated between old primes, and so can be seen not merely as new notes, but as “higher dimensional” corrections to old notes, approximating the universal division of the octave. This is where Arnold Schoenberg, bless his heart, went astray.

Now I’m going to blow your mind and tell you that a quaternion is actually just a qubit: it is, physically speaking, a spin 1/2 particle. The easiest way of realizing this is to realize that that some quaternion t + x.I + y.J + z.K can be written as a 2x2 matrix of the form:

[[t+i.x y+i.z] [ -y+i.z t-i.x]]

The quaternion basis elements are:

T [[1,0] [0,1] I [[i,0] [0,-i]] J [[0, 1] [-1, 0]] K [[0,i] [i,0]]

Compare the Pauli matrices:

X [[0,1] [1,0]] Y [[0,-i] [i, 0]] Z [[1,0] [0,-1]]

So we find that, I = iZ, J = iY, K = iX. And in any case, all these are just different ways of writing bivectors. If we switch to X, Y, Z, we get a Hermitian matrix, as opposed to a skew-symmetric matrix. If we find the two eigenvalues and two eigenvectors of that Hermitian matrix, and constellate those two eigenvectors using the Majorana representation, we get two antipodal stars on the 2-sphere. The ratio of the eigenvalues determines a division of the antipodal line aka a point in the interior or surface. And, quantum mechanically, we know that pure qubit states are points on the surface of the 2-sphere and mixed qubit states are points in the interior of the 2-sphere, up to complex phase. So there you go. Now you can look at the bivectors of GA(2) as defining the X, Y, and Z measurements in QM, generalizing to higher dimensional analogues accordingly. We recall the crazy plot twist of QM whereby just as light can be broken into its component colors, sound into its component frequencies, composite numbers into their primes, so spin states can be broken down into orthagonal spin states, as in the Stern Gerlach apparatus, where the electrons emerge at random in either one of two locations, with their spins either completely parallel or completely anti-parallel with the magnetic field they just traveled through, with probabilities given by the projection of their former spin state against the axis defined by the magnetic field. The “probablistic” or “free” behavior of a spin in the Stern-Gerlach apparatus reveals that the “generators of motion,” the Hermitian matrices, each of which defines a “fixed pole” around which you can rotate another axis in a circle unitarily by some U(t) given by exponentiating the H, also defines a possible measurement/bifurcation/free choice of an eigenstate of H. And you rediscover what Dirac rediscovered, which is that an electron lives in GA(1,3) aka H(2) aka: an “electron” breaks into two entangled spins in the singlet state, a positive energy spin 1/2 and a negative energy spin 1/2, a positron part and an electron part--from which one can derive the Pauli exclusion principle on first grounds.

But backing up a bit, what makes the “Bloch representation” of a qubit—a point on a 2-sphere up to complex phase—legal anyway? The answer is the Hopf Fibration, and this brings us both full circle and to the denouement.

Consider the 2-sphere. It’s defined by some x**2 + y**2 + z**2 = 0. Consider the 3-sphere. It’s defined by some t**2 + x**2 + y**2 + z**2 = 0. Just as the 2-sphere is a 2d surface embedded in a 3d space, a 3-sphere is a 3d surface embedded in a 4d space. Hence the quaternions. Now suppose we’re in 4d real space. We can map 4d real space into 2d complex space with:

(t, x, y, z) -> (t+i.x, y+i.z) -> (r, s)

Then we can define the 3-sphere as r.r* + s.s* = 1. Now suppose we’re in a 3d real space, we can also map it into 2d complex space with

(x, y, z) -> (x+i.y, z) -> (u, v)

so that the 2-sphere is defined by u.u* + v.v = 1. Then the Hopf fibration p

p(r, s) -> (2.r.s*, r.r* - s.s*) -> (u, v)

“stereographically” projects the 3-sphere onto the 2-sphere. If two points on the 3-sphere project to the same point on the 2-sphere, aka

if p(r, s) = p(r’, s’), then (r’, s’) = (ar, as) where a is a complex phase aka a.a* = 1.

And vice versa, if two points on the 3-sphere differ only in complex phase, then they map to the same point on the 2-sphere.

So this is exactly what’s going on in quantum mechanics, because, as they say, the rotation group SO(3) of rotations in 3 dimensions has a “double cover,” the spin group Spin(3), which is diffeomorphic to the 3-sphere. In essense, thinking of elements of SO(3) as points in a space, you can move continuously from one SO(3) rotation to the next, and there’s another group Spin(3) that also lets you move continously in the same way, but for every one element in SO(3), there’s two elements in Spin(3). And finally, it turns out you can map this “space” formed by Spin(3) to the 3-sphere. Furthermore, there are two ways of interpreting Spin(3): as either Sp(1), the group of quaternions with norm/length/magnitude = 1; or SU(2), the group of unitary matrices with determinant 1. But we already knew that: a quaternion is a qubit is a rotation in 3d. In quaternion language, we look at quaternions with g.g* = 1 as defining the 3-sphere. Now, a point (x, y, z) in 3d can be expressed as an imaginary quaternion: h = x.i + y.j + z.k. You can confirm that, for some unit quaternion g, and an imaginary quaternion h,

h -> g.h.g*

expresses a rotation in 3 dimensions. If you square it, you get h.h*, so it’s length/distance/norm preserving. So the unit quaternions are just the group of rotations in 3d, except that it turns out that g and -g determine the same rotation: hence the double cover business. Now if we try to represent a 3d rotation *in* 3d space, we find we can use the Hopf fibration to project our unit quaternion to a point on the 2-sphere, with a circle’s worth of freedom left over: as if to say, we can always rotate around the very axis defined by the point on the 2-sphere and its antipode, and we get the same projection. And this is the meaning of the Hopf fibration:

fiber spacebase space = total spaceS(1)S(2) = S(3)

The “total space,” which is a 3-sphere, can be broken down into an 2-sphere “base space,” and above every point on the 2-sphere, there’s a 1-sphere “fiber,” a circle of freedom that lives atop each point of the fabric of the 2-sphere.

Now, remarkably, according to the theorems of Hopf and Adams, we find that in all of multidimensional space there are exactly four such fibrations and what’s more, they correspond to the 4 normed division algebras: R, C, H, and O.

S(0)S(1) = S(1)S(1)S(2) = S(3)S(3)S(4) = S(7)S(7)S(8) = S(15)

There’s a fibration of the 15-sphere (which lives in 16 dimensional space, etc) which breaks it down into a 8-sphere base space, and a 7-sphere fiber space. But then! a 7-sphere can be fibered into a 4-sphere base space and a 3-sphere fiber space. But also! a 3-sphere can be fibered into a 2-sphere base space and 1-sphere fiber space. And lo! a 1-sphere can be fibered into a 1-sphere base space, and a 0-sphere fiber space, aka just two points. There can’t be any more such fibrations, because, after all, n-spheres rely on the whole a.a* + b.b* + c.c* + d.d* + … = 1 thing to work, with |u|v|’s ==ing |uv|’s, and the octonions are the last normed division algebra!

But consider that the 15-sphere lives in 16 real dimensional space. If we have 16 orthagonal vectors, we can consider its Geometric Algebra GA(15). Continuing the pattern,

GA(0)GA(1) ~ GA(1)GA(1)GA(2) ~ GA(3)GA(3)GA(4) ~ GA(7)GA(7)GA(8) ~ GA(15) akaRC = CCH = H, HH, HH(2) = R(8), R(8)R(8), R(8)R(16) = [R(8), R(8)](16)

So the vectors in one of the higher GA(n)’s can be fibered into vectors in the lower dimensional GA(n)’s. In the Hopf Fibrations, we have GA(0; 1; 2; 3; 4; 7) of the first octave, and GA(8; 16) aka the first and last notes of the second octave. Of the first octave, we’re only missing GA(5) = C(4) and GA(6) = R(8), and C(4) can seen as R(8) complexified, so it’s really like we’re missing just one GA(5/6). Furthermore, it’s interesting how GA(15) is conveniently GA(7)(16). It’s as if doing GA(n-8) is like dropping down to the fiber space.

And indeed, the thing is: if we have some n-dimensional vector space, we can construct it’s Geometric Algebra, and we have this “subtract 8 while making 16x16 dimensional matrices rule.” But precisely because we stuck to using normed division algebras, we can always collapse each of the elements in the 16x16 grid to get back a 16x16 matrix that acts on states that live on the 15-sphere and so can be fibered down earth! Meanwhile, at the deepest level of the hierarchy, we find only GA(0; 1; 2; 3; 4; 5; 6; 7), and so, except for GA(5/6), we can also begin the fibration there. Therefore, putting the exception to the side, we can begin the fibration anywhere! In this way, anything can be projected “down to earth” (and/or sent back up again) through the hierarchy of “celestial spheres” from 2 points, to circle, to surface, to sphere, etc, … and back.

What do we do with GA(5/6) then? Well, the algebra of C(4) is isomorphic to the complexification of GA(1,3), the space in which the electron lives. But now it can describe a photon too? I'm not sure. I'll just say now that C(4) reminds me of the dimensionality of Penrose’s “twistor space,” which describes photons traveling *between* spheres--although I guess twistor space is really more like GA(3) aka H, H. In any case, in twistor theory, a point in spacetime can be identified with a Hermitian matrix H:

1/sqrt(2) . [[t-z x+i.y x-i.y t+z]]

A twistor is a complex 4 dimensional vector that breaks into two 2d parts: r and s. If the twistor satisfies the relationship:

r = i.H.s

then that twistor is incident with that point in Minkowski space. From the r and s, you can calculate the position, momentum, and helicity of the photon, etc. By considering all the twistors incident with a given point in spacetime, you can build up a picture of the night sky of the observer.

And now: Links!

- Bott Periodicity
- Clifford Algebra: A visual introduction, and also many other fascinating things
- Cayley-Dickson Construction
- Hopf Fibration
- A visualization of the Hopf fibration, by Niles Johnson who also has a very good lecture about it.
- Versor
- Versor.js
- All Hail Geometric Algebra!
- Geometric Algebra for Physicists
- Ganja.js
- PS. Someone needs to write/probably has already written some linguistics code that creates a concordance matrix of symbols over a corpus, vectorizes the symbols using SVD, and then constructs the Geometric Algebra over the symbol vectors. Symbol juxtaposition in a sentence is interpreted simply as the geometric product, collapsing and expanding dimensions accordingly. The sentence (or its complement), considered as a transformation, is then applied to the semantic space as a whole, altering the background against which the next sentence will be interpreted. Then test hypotheses about quantum gravity in the world of linguistics, where there is a constant feedback whereby the semantic vectors guiding thought are defined relative to the current symbol frequencies, but the use of those vectors, in general, changes the very symbol frequencies used to define them in the first place, aka shifting ground on which you stand.