Working up to the syntax-semantics Duality

Today we reach the sacred heights of the syntax-semantics duality by repeatedly defining the monoids.

Level 1

Definition (first course in algebra): a monoid is a set $M$ equipped with a special element $e$ and a multiplication $m$, such that $e$ is the multiplicative identity, and the multiplication is associative.

The natural numbers, with $+$ and $0$, makes a monoid.

Another example is the string monoid: $M$ is all possible English strings made from the 26 letters, $e$ is the empty string, and $m$ is just putting two strings together.

Suicide notes of Yutaka Taniyama and his fiancée

Yutaka Taniyama (1927--1958) is a Japanese mathematician who proposed the Taniyama–Shimura conjecture, which was an important step in the solution of Fermat's Last Theorem. He committed suicide at the age of 31. His suicide note is as follows, taken from the Japanese Wikipedia:

昨日まで、自殺しようという明確な意思があったわけではない。 ただ、最近僕がかなり疲れて居、 また神経もかなり参っていることに気付いていた人は少なくないと思う。 自殺の原因について、明確なことは自分でもよくわからないが、 何かある特定の事件乃至事柄の結果ではない。 ただ気分的に云えることは、将来に対する自信を失ったということ。 僕の自殺が、或る程度の迷惑あるいは打撃となるような人も居るかもしれない。 このことが、その将来に暗いかげを落とすことにならないようにと、心から願うほかない。 いずれにせよ、これが一種の背信行為であることは否定できないが、 今までわがままを通してきたついでに、最後のわがままとして許してほしい。

Until yesterday I had no definite intention of killing myself. But more than a few must have noticed that lately I have been tired both physically and mentally. As to the cause of my suicide, I don't quite understand it myself, but it is not the result of a particular incident, nor of a specific matter. Merely may I say, I am in the frame of mind that I lost confidence in my future. There may be someone to whom my suicide will be troubling or a blow to a certain degree. I sincerely hope that this incident will cast no dark shadow over the future of that person. At any rate, I cannot deny that this is a kind of betrayal, but please excuse it as my last act in my own way, as I have been doing my own way all my life.

The Free Will Theorem

Today we prove Conway-Kochen free will theorem:

SPIN + TWIN + MIN + (very limited) free choice of the experimenters 

= free choice of the particles

SPIN is just a basic statement about the behavior of a spin-1 particle. It states that angular momentum operators exist and they behave in such a way to make the 101 property be true (see below).

TWIN states that it's possible to entangle two particles, such that their spins, when measured in the same direction, are opposite.

MIN is stated rather obscurely. I believe it means that it is impossible for an event to depend on another event outside of its past light cone. This is basically what special relativity states.

Free choice is defined as "non-functional", that is, not described by a function. In this interpretation, to say I have no free choice in going left or right, is to say that there is a function $f$, such that the direction I am going is $f($everything in my past lightcone$)$.

This proof goes in two steps. The first step is the Kochen-Specker Theorme uses only SPIN. The second step uses TWIN and MIN to construct an entanglement separated by a very long distance (like all those Bell-inequality experiments), and then assume (a very limited amount of) free choice of the experimenters, but not the particles, to get a contradiction.

Step 1: Kochen-Specker Theorem (1966)

This is well-known and I will direct you to plus magazine's proof. First read this, then read this. For those who know a bit more quantum mechanics, here's what 101 property means: Consider a spin-1 particle. Let $S_x$ be the operator of the angular momentum along vector $x$ for the particle, then $S_x$ has three possible eigenvalues: $\hbar, 0, -\hbar$. Normalize by setting $\hbar = 1$, we find that $S_x^2$ has two possible eigenvalues: $0, 1$. Then, it can be shown that for any triple of orthogonal vectors $x, y, z$, we have $S_x^2+S_y^2+S_z^2 = 2$, and so the measurement results must be one of $(1, 1, 0), (1, 0, 1), (0, 1, 1)$.

Notice that since $S_x^2 = S_{-x}^2$, we can safely consider a direction as defined by a line through the origin, rather than a vector.

Another note: sometimes, the configuration of 33 lines is called the Peres configuration. It's easy to verify that, if we represent each line as a vertex, and connect two vertices iff they represent orthogonal lines, then we obtain a graph with 72 edges, making up 16 triangles (corresponding to triple-orthogonal-lines) and 24 edges that do not make up any triangle.

The Stanford Encyclopedia contains more variations and ways to escape the conclusion of the Kochen-Specker theorem.

Tech stack/hierarchy in mathematics

What are layers?

Consider the OSI model. In technology, in building a complex system, one often divides the system into layers with simple exterior appearance, but complex interiors. The layers are basically modules that have a direction: whereas "modules" are egalitarian, and can be assembled in many directions, layers have a fixed pecking order, and can only be assembled in one fixed direction.

Layers = Modules affected by gravity

Kolmogorov complexity is Turing equivalent to the halting problem

This post describes several Turing equivalent problems to the halting problem: the busy beaver problem, the busy beaver runtime problem, the minimal program problem, and the Kolmogorov complexity problem.

But really, only one part of the proof is hard, and that is proving that the halting problem can be Turing-reduced to the problem of calculating Kolmogorov complexity.

The core of the argument is this: there exists short strings with short minimal programs, whose runtime is insanely long. That's really it.

Aitken's trick for calculating decimal expansions of rationals

Aitken's Method

Professor Aitken was both a mental calculator and a master mathematician, and a detailed account of his mental math abilities is found in An exceptional talent for calculative thinking, (IML Hunter, 1962)

I'd just like to sketch out one amazing trick, which is illustrated thus:


The proof is  $x = 5/23 = 15/69$, then, $x = (15 + x) /70$, so the trick is to calculate $15/70$ by short division, to generate the digits of $x$ one at a time, then immediately feed to the top to continue. A kind of "just-in-time" algorithm!

Geometry quickie: volume of a tetrahedron

I saw this formula on Wikipedia for the volume of tetrahedron and was intrigued.

Any two opposite edges of a tetrahedron lie on two skew lines, and the distance between the edges is defined as the distance between the two skew lines. Let $d$ be the distance between the skew lines formed by opposite edges... 

then it gave a volume formula. I want to derive it. This post gives my thought process, where I keep using the trick of reducing a general problem to a good example. This is a general strategy for solving problems, often written as "wlog".
(I write it as "wolog" and pronounce as "volog", as if it's a German adverb. So that I can write "We assume wolog that...", "... and thus wolog we have...")

How would a true geometer approach this problem? She would first do a dimension analysis. The volume has dimension $meter^3$, and the lengths have dimension $meter$, so it ought to look like
$$Volume = k \cdot AB \cdot CD \cdot h$$
where $h$ is the distance between the two skew lines of $AB$ and $CD$, and $k$ is some dimensionless number yet to be determined.

The binomial numbers, the Rule 60 celluar automata, and the Sierpinski triangle

Consider the pattern of the parity of the binomial numbers. Color the even numbers white and odd numbers black. You get a Sierpinski Triangle.

We will prove this using spacetime-geometric reasoning. It would help a lot if you have done some geometry of special relativity.

Define the parity function $f(t, n) = C(t, n) \mod 2$, defined on $\mathbb{N}\times\mathbb{Z}$.

We can consider $f(t, n)$ as encoding a 1-d celluar automata, with $t \ge 0, n \in \mathbb{Z}$. Then we have the evolution rule:

$$ f(t+1, n) = f(t, n) + f(t, n-1) \mod 2$$

This is just the Rule 60 celluar automata. 

Then it's easy to prove the following:

Representing a complex line as a directed ellipse

Consider nonzero $v = v_r + iv_i \in \mathbb{C}^n$, It can be thought of as an ordered 2-tuple of vectors $(v_r, v_i)\in \mathbb{R}^n\times\mathbb{R}^n$.

The complex line generated by $v$ is
$$\{r[(\cos(\theta) v_r -\sin(\theta) v_i) + i(\sin(\theta) v_r +\cos(\theta) v_i)]:\\ r\ge 0, \theta\in[0, 2\pi]\}$$
So, essentially, it is a set of concentric ellipses. We consider one of them:
$$\{(\cos(\theta) v_r -\sin(\theta) v_i) + i(\sin(\theta) v_r +\cos(\theta) v_i): \theta\in[0, 2\pi]\}$$
As $\theta$ increases, both $(\cos(\theta) v_r -\sin(\theta) v_i)$ and $(\sin(\theta) v_r +\cos(\theta) v_i)$ rotate in the ellipse, always being conjugate to each other.

A very unexpected 333 that I found in math

While studying about dynamical systems theory for my ergodic theory course, I came across the extremely unexpected number 333. From Resonances and small divisors (Etienne Ghys, 2007), chapter 10 of Kolmogorov’s Heritage in Mathematics:

In 1962, Moser succeeded in accomplishing the feat of proving the theorem in the space of infinitely differentiable functions [Moser, J. On invariant curves of area-preserving mappings of an annulus (1962)]. 

In fact, Moser used functions which are 333 times differentiable and the topology of uniform convergence on these 333 derivatives... The mere fact that it is necessary to use as many derivatives shows the difficulty of the proof. Nowadays, it is known that the theorem is true with 4 derivatives and false with 3 [Sur les courbes invariantes par les difféomorphismes de l'anneau Herman, Michael R. Astérisque, no. 144 (1986)].

333 is half the number of the devil. Moser probably cut half of a deal with the devil.

The heart and soul of elliptic, flat (Euclidean), and hyperbolic spaces

... is constant, polynomial, and exponential.

Think about building cellphone towers in a Euclidean plane. First, we choose the central station arbitrarily at point $O$, and call it the "origin". To cover all space that's within a distance $R$ to the origin, we need $O(R^2)$ stations.

This is polynomial growth.

Then think about building cellphone towers on a spherical surface, on a little asteroid. Then it's clear that there's only a constant number of towers needed no matter how big $R$ is chosen, since the whole surface has finite area. So we need $O(1)$ stations.

This is constant "growth".

Then, if you know how to play in hyperbolic space (if not, go play Hyperrogue!), you'd know that the number of stations grows as $O(\exp{(CR)})$, where $C=(-K)^{-1/2}$, and $K$ is the Gaussian curvature of the plane. This can be derived by using the formula of the area of a hyperbolic circle of radius $R$:
$$\frac{4\pi}{-K} \sinh^2 \frac{r}{2\sqrt{-K}} $$

We normalize $K = -1$ to get the growth rate $O(\exp{R})$.

The best proof of Clairaut's theorem on equality of mixed partials

This is the best proof of Clairaut's theorem on equality of mixed partials that I have ever thought up. And I bet it's the best proof of the theorem you'll ever see!

I'll prove the 2-variable version, but the generalization to $n$-variables is obvious.

Clairaut's Theorem: Suppose $f$ is a real-valued function of two variables $x,y$ and $f(x,y)$ is defined on an open subset $U$ of $\mathbb{R}^2$. Suppose further that both the second-order mixed partial derivatives $\partial_x\partial_y f(x,y)$ and $\partial_y\partial_x f(x,y)$ exist and are continuous on $U$. Then, we have: $$\partial_x\partial_y f = \partial_y\partial_x f$$ on all of $U$.

Cox-Zucker

The Cox-Zucker Machine is an algorithm created by David A. Cox and Steven Zucker. This algorithm determines if a given set of sections provides a basis (up to torsion) for the Mordell–Weil group of an elliptic surface $E \to S$ where $S$ is isomorphic to the projective line.

The algorithm was first published in Intersection numbers of sections of elliptic surfaces (1979), by Cox and Zucker, and it was later named the "Cox–Zucker machine" by Charles F. Schwartz in A Mordell-Weil group of rank 8, and a subgroup of finite index (1984).

And I did check the paper by Charles Schwartz, and indeed:

We will find, for a specific equation of this form, $$y^2 = 4(x^3 - u^4x + 1),$$ 8 solutions that generate a subgroup of index 4 in the Mordell-Weil group of the fibration given by this equation. We do this using the Cox-Zucker Machine. We then use this result to draw certain conclusions concerning the general case, and to make certain conjectures. 

§ 1. The algorithm of Cox and Zucker (AKA, The Cox-Zucker Machine)...

$\omega$-inconsistent life

I was sad and wanted to die, but didn't want to die yet because I would like to think of a good reason to.

I said, "I know that there's a proof of life's meaninglessness, but I'd like to be justified in my belief. So I would kill myself on the next day iff I find a proof of life's meaninglessness."

Ten Lessons for Mathematicians, according to Gian-Carlo Rota

Ten Lessons I wish I had been Taught (1994), by Gian-Carlo Rota.

7. Use the Feynman method

You have to keep a dozen of your favorite problems... Every time you hear or read a new trick or a new result, test it... to see whether it helps. Every once in a while there will be a hit, and people will say: "How did he do it? He must be a genius!"

This is the best advice. I must immediately think up at least one problem to get my chance of being a GENIUS!

1. TODO

Pointless Notes: Murder and Complex Analysis

Theodore John Kaczynski, BA (Mathematics, Harvard University, 1962), PhD (University of Michigan, 1967), is famous for being a series bomber, but to mathematicians, he's also known as a complex analyst. Not a (mathematically) famous one, but a competent one at least, and some also said he was brilliant. In 1969, he resigned as an assistant professor at Berkeley.

There was a news article about his math, and it contained the usual kind of flailing about how research level math is too hard for layponies. No need to be so dramatic about it... 

I was amused by this:

Writing in the first person plural is common in mathematics, said Kenneth Ribet of UC Berkeley. So in hindsight, the Unabomber's manifesto may provide an obscure hint at its alleged author's former profession. 

The mathematician's habit of writing in the Royal We has been exposed!

Let's Read: In Praise of Lectures by T. W. Körner

Alright, let's read! I'm on a roll of venting against lectures.

In Praise of Lectures (2004), Thomas William Körner

The essay is off to a strong start that lifted my mood.

For these students attendance at lectures has a magical rather than a real significance. They attend lectures regularly (religiously, as one might say) taking care to sit as far from the lecturer as possible (it is not good to attract the attention of little understood but powerful forces) and take complete notes. Some lecturers give out the notes at such speed (often aided by the technological equivalent of the Tibetan prayer wheel — an overhead projector) that the congregation is fully occupied but most fail in this task... 

The notes of the lecture are then kept untouched until the holidays or, more usually, the week before the exams when they are carefully highlighted with day-glow yellow pens (a process known as revision). When more than 50% of the notes have been highlighted, revision is said to be complete, the magical power of the notes is exhausted and they are carefully placed in a file never to be consulted again.

The invention of the Xerox machine [or Internet] means that only one student need attend each lecture, the remainder being freed ... Nor would this student need to take very extensive notes since everything done in the lecture is better done in the textbooks.

Stop math lecturing

There are many bad things and few good things about teaching math by lectures.

Personally I have not learned anything from math lectures since the end of high school. As I said before, lectures are bad for higher math. It's either too fast, then it's not understandable, or too slow, then it's boring. If it's written in a book, it's much better presented, than if it's written on a board live with a human hand. Using slides is mildly better, but slides are ugly, and why even use slides when you have a book?

And what's the point of taking notes? If the lecturer has prepared a lot about the lecture, to the point that the lecture notes are already done and so they could just refer to it during the lecture, why not print it out and give to the students so that they don't have to waste their time writing down their own lecture notes? To allow some idiosyncrasy for the notes, large margins can be put into the printed out notes so that the students can put in their own stuff there.

Let's read: $C^*$-algebra, quantum mechanics, Pascual Jordan

A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details. ---- Hermann Weyl

After 3 courses in analysis I have finally reached Banach algebras, I feel exhausted by the climb and yet there's more theory ahead until the true pinnacle: the theory of $C^*$-algebra, for quantum mechanics!

Let's read, what is being promised?

$C^*$-algebras were first considered in quantum mechanics to model algebras of physical observables... began with Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establish a general framework for these algebras which culminated in a series of papers on rings of operators. These papers considered a special class of $C^*$-algebras which are now known as von Neumann algebras.

So experiments in quantum mechanics → weird behavior of observables in quantum mechanics → Heisenberg matrix mechanics → Jordan purified it into Jordan algebra → von Neumann generalized to rings of operators → von Neumann algebra generalized to $C^*$-algebras.

Let's Read: Nielsen and Ninomiya (2007)'s backward causation, LHC, and the end of mathematics

I considered writing a blog because it's better than writing things on the wood paper, which is hard to search, even more prone to be forgotten than electronic writing, and almost impossible to be read by anyone but me.

I mean, there's a slight chance that my papers will get buried by volcanic ash and become dug up by future archeologists (like the scrolls in the Pompei library), while there's pretty much no hope of any server surviving such a treatment, but that's too unlikely to be considered a valid argument for using wood paper.

Today we write about LHC, SSC, and Nielsen and Ninomiya's crazy proposal in 2008 about why LHC shut itself down. Then it digresses to a depressing thought about the ends of scientific and mathematical fields.