How to start reading more

Posted on December 21, 2020 by meygerjos

Recently I was able to start reading more. Here is how I was able to do it.

Dopamine Detox: https://www.youtube.com/watch?v=9QiE-M1LrZk tl;dw: If you frequently do activities that produce a lot of dopamine, like going on social media, listening to music, etc., you build a tolerance to dopamine and then low-dopamine activities like reading will not be able to hold your interest. If you limit high-dopamine activities, then your tolerance will go down and reading will be more interesting. Some doubt has been cast on the scientific accuracy of this idea but I think the main point stands that if you spend a lot of time doing highly stimulating things it is hard to sustain interest in less stimulating things.
Read page-turners: Probably you have a lot of difficult books you want to read. Prioritize books that are easy, fun, and page-turners, because they will get you in the habit of reading and acclimate you to the physiology and psychology of reading (eye motion, posture, sustained focus). Once you have built the habit, it will transfer to difficult books very easily.
Set aside a separate space: Don’t try to read in front of the computer, because the computer will be calling for your attention. Don’t read in the kitchen because snacks will be calling to you. Don’t read in bed because you will get sleepy. Set aside a space that is just for reading, and remove distractions from your field of view.

How to prevent mass death

Posted on March 11, 2020 by meygerjos

Disclaimer: As people have pointed out on reddit, a logistic curve is not a very good model for this. I will post an analysis with a better model as I learn more.

tl;dr: if we model Covid-19 growth as a logistic curve, then we need a doubling time of more than 124 days to avoid overwhelming hospitals in the US.

It is becoming clear that the danger from the SARS-CoV-II coronavirus is systemic healthcare failure. This is because the virus is growing exponentially. In the US, the doubling time seems to be about 2 days, which means that if your city has 10 cases, 2 days later it will have 20 cases, 4 days later it will have 40 cases, and 20 days later it will have 10,240 cases.

This growth, of course, will not continue unabated. When enough people have gotten the virus, the growth will start to slow, as many people will be immune. So a better approximation than an exponential is a logistic curve.

Source: 3Blue1Brown. Exponential curve in red, logistic curve in yellow.

Notice that the growth accelerates until the point of inflection, when exactly half of the population is infected, and then starts to slow down.

Let $P$ be the population. I think it will be easier if we define $n=N/P$ . Then dividing both sides of the pictured equation by $P$ gives $\frac{d}{dt}\frac{N}{P}=c(1-\frac{N}{P})\frac{N}{P}$ , or $\frac{dn}{dt}=c(1-n)n$ , which is simpler and coincides with Wikipedia’s formula.

The pictured differential equation is actually an instance of the Law of Mass Action. People get infected when an infected person comes into contact with an susceptible person, so the rate of the reaction, i.e. the rate at which new people get infected, is proportional to (the number of infected people) times (the number of susceptible people). In other words, $\frac{dn}{dt}$ is proportional to $ns$ , where $s$ is the proportion of the population that is susceptible to infection. We recover the above differential equation by assuming that all people who have not yet become infected are susceptible, i.e. $s=1-n$ , and letting $c$ be the constant of proportionality.

When $n$ is very small, $1-n\approx 1$ so our equation is approximately $\frac{dn}{dt}=cn$ . The solution to this is an exponential: $n(t)=Ae^{ct}$ . Thus we identify the constant $c$ as the growth rate.

Assuming this model, we cannot prevent everyone from getting infected at some point. What we can change, through NPIs (non-pharmacological interventions like quarantines and shutdowns), is $c$ , the growth rate. If $c$ is too high, it will overwhelm hospitals.

The curve on the left is what will happen with a high $c$ , the curve on the right is what will happen with a low $c$ .

So how do we determine whether hospitals will be overwhelmed, given a certain value of $c$ ? We first need an estimate for how long patients will be in the hospital. A quick google search turns up a study saying that out of a certain sample in Jiangsu Province, the mean length of stay is 8 days.

This means that $h(t)$ , which we define as the proportion of the population who are in the hospital on day $t$ , is, on average, equal to the proportion of people who were newly symptomatic in the last 10 days times the fraction of cases which have to be hospitalized, which is about $20\%$ . So we can approximate $h(t) = 20\%(n(t)-n(t-10))$ , which in turn we can approximate as $20\%\cdot 10n'(t)=2n'(t)$ .

The number of staffed beds in US hospitals is $924,107$ . The population of the US is $327.2$ million. Thus there are $924,107/327,200,000 = 0.0028$ staffed beds per capita, or $2.8$ staffed beds per thousand people.

If we don’t want hospitals to be overwhelmed, then we can’t let $h(t)$ get higher than $0.0028$ . What will $h(t)$ be at it’s maximum? Well $h(t)=2n'(t)=2cn(t)(1-n(t))$ . This quantity is maximized when $n(t)=\frac{1}{2}$ (to see this, graph it, complete the square, or find a critical point). So at it’s maximum $h(t)=2c\frac{1}{2}\frac{1}{2}=\frac{c}{2}$ . So to prevent hospitals being overwhelmed, we need to ensure that $\frac{c}{2}\leq 0.0028$ , i.e. $c\leq 0.0056$ .

To put this in more familiar terms, let’s express it in terms of the doubling time, the amount of time that it takes $n$ to double in the early stages when it is growing exponentially. Call this doubling time $T$ . Then $e^{cT}=2$ , so $T=\frac{\ln{2}}{c}$ . The condition $c\leq 0.0056$ is equivalent to the condition $T\geq \frac{\ln{2}}{0.0056}=124$ .

So we need a doubling time of more than 124 days! Wow! That is difficult to imagine given that now the US has a doubling time of 2.15 days, and that China has the highest doubling time of any country at 29 days.

Perhaps if we slow it down now we could buy ourselves time to increase hospital capacity, so that the doubling time doesn’t have to be so high.

Or perhaps, I’m using an overly pessimistic model. Indeed, this model does assume that people who are infected are infectious forever after, and that all people who have not yet been infected are susceptible.

How to Study Math: A Brief Guide

Posted on November 4, 2019 by meygerjos

Working through math textbooks is a satisfying intellectual endeavor — however, most students don’t know how to do it. Here is a brief guide:

How to find a good textbook

The tag <reference-request> on Math.SE
Quora
Blogs (for example, Qiaochu Yuan’s blog has a reading list)
Look at the bad reviews on Amazon. If any of these reviewers have a problem you think you would also have, get a different book.
Go to a library and try out all the different books on the subject
Get a recommendation from someone

When you are selecting textbooks, you should think about pedagogy and constructing a curriculum for yourself. For example, as is widely known, Baby Rudin is a great textbook but not for the first time you learn real analysis. And if a textbook has an obvious prerequisite, you should learn that prerequisite first before attempting it. (By the way, all theoretical math textbooks have as a prerequisite proof-writing. If you haven’t already, you should learn this skill. See Appendix A for how to learn proof-writing.) And if you want to learn complex analysis, it is a good idea to read Needham’s Visual Complex Analysis before tackling a theorem-proof style work such as Ahlfors.

In general, don’t worry about finishing textbooks before moving on to other ones.

The “4 bookmark” system

Use four bookmarks, not just one. We can call them bookmarks R, C, L, and E (for reading, copying, learning, and exercises).

Bookmark R is for your reading of the textbook. You should read quickly without pencil and paper, and not worry about understanding everything. Your goal here is to see what is in the textbook, not to learn it.

Bookmark C is for copying the textbook. In general, it should lag behind bookmark R. For this bookmark, copy the textbook into your notebook in your words (and symbols). Make sure you include everything in the textbook, but only copy down things that you understand. If the textbook does something in a way you don’t like, feel free to copy it down a different way. If the textbook leaves out details, make sure to supply them yourself. If any questions occur to you, write them down and attempt to answer them soon. If you can’t figure them out yourself, look them up or ask someone.

Bookmark L is for learning the textbook. This should definitely lag behind bookmark R. Practice copying out the material which is to the left of this bookmark, referring to the textbook as little as possible. Also practice explaining the material to other people, looking at the textbook as little as possible. This is not exactly memorization, as there is a great deal of figuring out how to prove things involved in this if done properly, not simply remembering how the textbook author proved things. In many cases, the way you figure out how to prove things will be equivalent to the author’s, but in many cases it won’t. As you improve, you will have to refer to the textbook less and less. When you can write out everything to the left
of the bookmark without looking at the textbook at all, it is time to advance the bookmark. Learning the textbook will also produce questions in your mind. Just as above, you should write down these questions and attempt to answer them soon.

Bookmark E is for exercises. This bookmark should lag behind bookmark L. In other words, don’t attempt the exercises at the end of a chapter before you have learned the chapter. It is very common for students to get frustrated with exercises that are not actually difficult, simply because they violate this precept. Exercises generally assume knowledge of the chapter preceding them, so attempting them without knowledge of this chapter is a big handicap.

Appendix A: How to learn proof-writing

Approach 1: “Logic and Set Theory”

First learn logic and set theory.

Possible Curriculum:

Bergmann & Moor, The Logic Book
Halmos, Naive Set Theory

See here for a large bibiography and more curriculum advice.

Once you’ve done this, start studying math textbooks, and in step 4, translate them into symbolic logic, to learn how the mathematical language works. You will know you did it right when all the steps are valid. You can add to the small arsenal of axioms in symbolic logic by proving rules in terms of the axioms and the previously proved rules.

Approach 2: “Proofs as a Skill”

Read some book that directly teaches you the skill of writing proofs, e.g. Velleman’s How to Write Proofs or Chartrand, Polimeni, and Zhang’s Mathematical Proofs: A Transition to Advanced Mathematics.

Approach 3: “Just dive in”

Pick up a book on some mathematical subject and start trying to do the exercises even though you don’t know how to prove stuff yet. Learn from grading, office hours, and tutors. Needless to say, this approach relies heavily on other people spending a lot of time teaching you.

Assessment of Approaches

I prefer approach 1 because it makes proving a sincere expression of how you think rather than just something you learn how to do to get a desired result from a teacher. In approach 1 you are systematically convinced of the truth of the mathematical way of thinking, in approaches 2 and 3 you are merely acculturated to it. In fact, mathematical writing is really a way of describing how one would make something into a formal proof without actually doing so, and it is best learned in this context.

However, in schools the debate seems to wage between approaches 2 and 3. I think approach 3 is awful because it violates the clear social contract that students should not be graded on material that they were never taught. At least we should do approach 2, but approach 1 is best of all.

Appendix B: Impostor Syndrome

The above guide may give you impostor syndrome if you don’t yourself study textbooks in such a disciplined way. Maybe you feel like this is the right way to study, the way everyone expects you to study, and you will be embarrassed if it is discovered that you don’t study like this.

Well, yes, this is the right way to study, but no, most people don’t study like this, and no, nobody expects you to either. I think this is a sad state of affairs. How can we expect people to teach well and do good research if we don’t also expect them to learn well? So it’s not you who’s an impostor, it’s a whole discipline of impostors who claim to be scientists yet don’t value mastery of the material they hold to be central to their field.

But the solution to this problem can start with you. Start working through textbooks by following the method above (or equivalent that works for you) and use your newfound understanding to enlighten others in your department. Don’t worry about mastering everything in your path, but make sure you are mastering something.

What is a “smooth map”?

Posted on December 8, 2017 by meygerjos

I was trying to start reading John Milnor’s Topology from the Differentiable Viewpoint, but something on the first page really threw me. I think I figured it out now though, so let me explain it to you too.

Milnor writes at the very beginning of the book:

FIRST let us explain some of our terms. $R^k$ denotes the $k$ -dimensional euclidean space; thus a point $x \in R^k$ is an $k$ -tuple $latex x = (x_1, \ldots , x_k)$ of real numbers.

Let $U\subset R^k$ and $V \subset R^l$ be open sets. A mapping $f$ from $U$ to $V$ (written $f : U \rightarrow V$ ) is called smooth if all of the partial derivatives $\partial^n y/\partial x_{i_1},\cdots\partial x_{i_n}$ exist and are continuous.

More generally let $X \subset R^k$ and $Y \subset R^l$ be arbitrary subsets of euclidean spaces. A map $f : X \rightarrow Y$ is called smooth if for each $x \in X$ there exist an open set $U \subset R^k$ containing $x$ and a smooth mapping $F : U \rightarrow R^l$ that coincides with $f$ throughout $U \cap X$ .

The last part is what threw me. It really isn’t obvious what the consequences are of the general definition of a smooth map on an arbitrary subset of $R^k$ . There are two other ways I can think of defining this notion:

A map $f : X \rightarrow Y$ is smooth iff for there exists an open set $V \subset R^k$ including $X$ and a smooth mapping $F : V \rightarrow R^l$ whose restriction to $X$ is $f$ .
I’ll just give this second definition in the case of $k=1$ . It’s kind of complicated to extend it to larger $k$ , and it’s unnecessary for what comes next. Here it is: For a map $f : X \rightarrow Y$ , define $g(x)=\lim_{y\in X-\{x\}, y\rightarrow x}\frac{f(y)-f(x)}{y-x}$ , if this limit exists. The function $g$ is basically the derivative of $f$ , except that the limit is taken only over points in $X$ . Later we will just call it the derivative. Now we define $f$ to be smooth iff the limit defining $g(x)$ exists for all $x\in X$ and $g:X\rightarrow R$ is continuous.

Are these two definitions equivalent to Milnor’s definition?

Well, the first definition is. It clearly implies Milnor’s if we just choose $V=U$ for every $x\in X$ . To prove the converse is more tricky. I found on a very questionable source that the way to prove this is using partitions of unity. To understand this, I read about partitions of unity in Spivak’s Calculus on Manifolds. Basically, partitions of unity can help you stitch together the many functions $F$ to make one big function. They might disagree on some points, but you can kind of take the average of the disagreeing values to get the value of the big function. See this Math.SE question for more information.

The second definition isn’t. Milnor’s definition implies it, but it doesn’t imply Milnor’s. I found a counterexample which you can scroll down and read right now if you want, or you can continue reading about my thought process that led me to it. First of all, Milnor’s definition implies it because Milnor’s definition implies my definition 1, which clearly implies it. Now the counterexample.

How can we choose $X$ to make a counterexample? I want to make a stronger counterexample than necessary: I want $f$ to be continuously differentiable on $X$ , but have no extension to an open set containing $X$ which is even continuous, not to mention continuously differentiable. I think a stronger counterexample like this will be easier to find. Well, $X$ clearly can’t be open. Also, $X$ can’t be closed, because then it could be continuously extended to $R^k$ by the Tietze extension theorem. Furthermore, $X$ can’t be closed in any open set $V$ , because then it could be continuously extended to $V$ . Thus, it can’t be the intersection of a closed set and an open set.

But we need even more than this. We need there to be an $x\in X$ such that for all open neighborhoods $U$ of $x$ , $U\cap X$ is not closed in $U$ . Otherwise, we could pick a neighborhood $U_x$ of each $x\in X$ such that $U_x\cap X$ is the closed in $U_x$ , and then we could extend $f|_{U_x\cap X}$ to $U_x$ using the Tietze extension theorem. Then we would have satisfied Milnor’s definition.

I thought about how to do this. We need it to be so no matter how small we make the neighborhood $U$ of $x$ , we still find limit points missing from $X\cap U$ . How about $X=[0,1)-\{\frac{1}{2},\frac{1}{3},\frac{1}{4},\ldots\}$ ? This works: in every neighborhood of $0$ , there are limit points $\frac{1}{n}$ missing from $X$ . Now we can define a function on $X$ . We define it piecewise:

For $\frac{1}{2}<x<1$ , let $f(x)=\frac{1}{4}(x-\frac{1}{2})$ .

For $\frac{1}{3}<x<\frac{1}{2}$ , let $f(x)=\frac{1}{9}(x-\frac{1}{3})$ .

For $\frac{1}{4}<x<\frac{1}{3}$ , let $f(x)=\frac{1}{16}(x-\frac{1}{4})$ .

In general, for $\frac{1}{n}<x<\frac{1}{n-1}$ , let $f(x)=\frac{1}{n^2}(x-\frac{1}{n})$ .

And, let $f(0)=0$ .

This function is continuous at $x\in X, >0$ because it is linear on a neighborhood of $x$ . It is continuous at $0$ because its value gets arbitrarily close to $0$ as $x$ approaches $0$ . It is differentiable at $x\in (\frac{1}{n},\frac{1}{n-1})$ with derivative equal to $\frac{1}{n^2}$ . It is differentiable at $0$ because $\frac{f(h)}{h}$ approaches $0$ as $h$ approaches $0$ (while staying in $X$ of course). Its derivative is clearly continuous for $x\in X, >0$ because here it is constant on a neighborhood of $x$ . And $0\leq f'(x)\leq x^2$ , so $f'$ is continuous at $0$ as well.

However, $f$ does not continuously extend to an function $g$ on an open set $V$ . If it did, then $V$ would have to include a neighborhood of $0$ , and this neighborhood would include infinitely many points $\frac{1}{n}$ . The function $g$ would have to assign a value to these points, and there is no way of doing this continuously, since at such points the left- and right-hand limits are unequal. Infinitely many jump discontinuities would be inevitable.

I understand quaternions!

Posted on December 2, 2017 by meygerjos

So I just read the first chapter of John Stillwell’s Naive Lie Theory, and now I understand how quaternions can be used to represent spatial rotation! Previously, I could do the calculation but I didn’t understand why it was true. Here’s the way I understand it, which is inspired by but not identical to Stillwell. First of all, what exactly is the claim that I now understand?

A purely imaginary quaternion $x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$ can be thought of as a vector $\mathbf{u}=(x,y,z)$ in $\mathbb{R}^3$ . Thus we can think of a general quaternion as the sum of a scalar and a vector: $w+x\mathbf{i}+y\mathbf{j}+z\mathbf{k}=w+\mathbf{u}$ . Thus any unit quaternion can be written $q=\cos\theta+\mathbf{u}\sin\theta$ for some angle $\theta$ and unit vector $\mathbf{u}\in\mathbb{R}^3$ .

The claim concerns rotations in $\mathbb{R}^3$ . Apparently, the rotation by the angle $2\theta$ about the axis $u\in\mathbb{R}^3$ is represented by the unit quaternion $q=\cos\theta+\mathbf{u}\sin\theta$ . To rotate a vector $v\in\mathbb{R}^3$ by this rotation, we write $qv\overline{q}$ . People really like this because it makes 3D spatial rotation into quaternion multiplication, which is much easier and less messy than using $3\times 3$ matrices. Computer graphics use quaternions for this reason. Also, this gives a nice interpretation for quaternions, which otheriwse have unclear meaning: rotations of $\mathbb{R}^3$ . Later in this post we will see an even nicer interpretation.

Now why does this work?

First let’s see what happens when two vectors (purely imaginary quaternions) are multiplied.

$(a\mathbf{i}+b\mathbf{j}+c\mathbf{k})(x\mathbf{i}+y\mathbf{j}+z\mathbf{k})=-(ax+by+cz)+(bz-cy)\mathbf{i}+(cx-az)\mathbf{j}+(ay-bx)\mathbf{k}$

We can recognize the right-hand-side as minus the dot product plus the cross product of the two vectors:

$\mathbf{u}\mathbf{v}=-\mathbf{u}\cdot\mathbf{v}+\mathbf{u}\times\mathbf{v}$

Two interesting special cases of this:

1) If $\mathbf{u}$ and $\mathbf{v}$ are parallel, then their cross product is $0$ , so $\mathbf{u}\mathbf{v}=-\mathbf{u}\cdot\mathbf{v}$ . In particular, $\mathbf{u}^2 = -\mathbf{u}\cdot\mathbf{u}=-|\mathbf{u}|^2$ . If $\mathbf{u}$ is a unit vector, then its square is $-1$ , just like $\mathbf{i}$ , $\mathbf{j}$ , and $\mathbf{k}$ . This means that the set of quaternions $a+b\mathbf{u}$ are a replica of the complex plane: they add and multiply just like complex numbers $a+b\mathbf{i}$ .

2) If $\mathbf{u}$ and $\mathbf{v}$ are orthogonal, then their dot product is $0$ , so $\mathbf{u}\mathbf{v}=\mathbf{u}\times\mathbf{v}$ . In this case, let’s define $\mathbf{w}=\mathbf{u}\times\mathbf{v}$ . Now the vectors $\mathbf{u}$ , $\mathbf{v}$ , and $\mathbf{w}$ are mutually orthogonal. Let’s assume that $\mathbf{u}$ and $\mathbf{v}$ are both unit vectors, and see how $\mathbf{u}$ , $\mathbf{v}$ , and $\mathbf{w}$ multiply:

$\mathbf{u}^2=-1$ (by the first special case)
$\mathbf{v}^2=-1$
$\mathbf{w}^2=-1$ (since $\mathbf{w}=\mathbf{u}\times\mathbf{v}$ is a unit vector)
$\mathbf{u}\mathbf{v}=\mathbf{w}$
$\mathbf{v}\mathbf{u}=-\mathbf{w}$ (because cross product is antisymmetric)
$\mathbf{v}\mathbf{w}=-\mathbf{v}\mathbf{v}\mathbf{u}=\mathbf{u}$ (by previous line, associativity of quaternion multiplication, $\mathbf{v}^2=-1$ )
$\mathbf{w}\mathbf{v}=-\mathbf{u}$
$\mathbf{w}\mathbf{u}=-\mathbf{v}\mathbf{u}\mathbf{u}=\mathbf{v}$
$\mathbf{w}\mathbf{u}=-\mathbf{v}$

Lo and behold, these are the exact same relations that hold among $\mathbf{i}$ , $\mathbf{j}$ , and $\mathbf{k}$ and that are used to define the quaternions! In other words, it turns out that we have made a copy of the quaternions using $\mathbf{u}$ , $\mathbf{v}$ , and $\mathbf{w}$ in place of $\mathbf{i}$ , $\mathbf{j}$ , and $\mathbf{k}$ . This is an automorphism of the quaternions.

What makes this automorphism work? Well, it is essential that $\mathbf{u}$ , $\mathbf{v}$ , and $\mathbf{w}$ be mutually orthogonal and that each of them have length $1$ . But this is not all, they must also have the correct orientation. For example, $\mathbf{u}=i$ , $\mathbf{v}=j$ , $\mathbf{w}=-k$ would not work. The automorphisms of the quaternions are therefore exactly the proper rotations of $\mathbb{R}^3$ .

Now what happens to all the quaternions when we multiply them on the left by a unit vector $\mathbf{u}$ ? To answer this, let’s first construct $\mathbf{v}$ and $\mathbf{w}$ as we did above. We let $\mathbf{v}$ be any unit vector orthogonal to $\mathbf{u}$ , and we let $\mathbf{w}=\mathbf{u}\mathbf{v}$ . Now any quaternion can be expressed as a linear combination of $1$ , $\mathbf{u}$ , $\mathbf{v}$ , and $\mathbf{w}$ , and this is more convenient for our purposes than the conventional expression of a quaternion as a linear combinations of $1$ , $\mathbf{i}$ , $\mathbf{j}$ , and $\mathbf{k}$ . So:

$\mathbf{u}(a+b\mathbf{u}+c\mathbf{v}+d\mathbf{w})=-b+a\mathbf{u}-d\mathbf{v}+c\mathbf{w}$

What happened? Well, we simultaneously rotated by $90$ degrees in the $1,\mathbf{u}$ -plane and in the $\mathbf{v},\mathbf{w}$ -plane. This is called a double rotation, and it is not possible in 3 dimensional space because it requires two orthogonal planes. Here is a projection onto 3-space of a 4-dimensional hypercube undergoing a double rotation.

Tesseract

It is simultaneously rotating and turning inside out. The turning inside out, however, is really a projection of a rotation in an orthogonal plane. The small cube in the center is not really smaller than the cube around it, it is just further away from the viewer. As the hypercube turns inside out, each constituent cube comes closer and goes farther away.

Now let’s take the general case of multiplying quaternions on the left by a unit quaternion $q=\cos\theta+\mathbf{u}\sin\theta$ . For a quaternion $x$ , we get

$qx = x\cos\theta+ \mathbf{u}x\sin\theta$

This is a linear combination of $x$ and $\mathbf{u}x$ , $x$ doubly rotated by 90 degrees. When we combine these two things this way, we end up rotating $x$ by $\theta$ . Indeed, if $\theta=0$ then $qx=x$ and if $\theta=90^\circ$ then $qx=\mathbf{u}x$ . If you think about this you’ll see that as $\theta$ increases $qx$ doubly rotates by the angle $\theta$ in the $1,\mathbf{u}$ -plane and in the $\mathbf{v},\mathbf{w}$ -plane.

It is important to note that $qx$ rotates at the same rate in both planes. A double rotation where both rotations are of the same angle is called an isoclinic rotation.

Now what about multiplying quaternions on the right by a unit quaternion $q=\cos\theta+\mathbf{u}\sin\theta$ ? Quaternions aren’t commutative right, so maybe it will do something different? Well it does! First let’s just multiply on the right by $\mathbf{u}$ :

$(a+b\mathbf{u}+c\mathbf{v}+d\mathbf{w})\mathbf{u} =-b+a\mathbf{u}+d\mathbf{v}-c\mathbf{w}$

This is just like the multiplying on the left except the signs of the $\mathbf{v}$ and $\mathbf{w}$ terms have flipped. It is a double rotation again, but now we are rotating the planes opposite ways. The general case works as expected:

$xq = x\cos\theta + x\mathbf{u}\sin\theta$

So we doubly rotate $x$ by the angle $\theta$ in the $1,\mathbf{u}$ -plane and by the angle $-\theta$ in the $\mathbf{v},\mathbf{w}$ -plane.

We distinguish isoclinic rotations by whether they rotate the planes in the same or opposite directions. If they rotate the planes in the same direction, they are called left isoclinic rotations, and if opposite, they are called right isoclinic rotations. Left isoclinic rotations have come about through left quaternion multiplication, and right one have come about through right quaternion multiplication, but is it possible to do the reverse? No. Every left multiplication yields a left isoclinic rotation, as we have seen, and a left isoclinic rotation can’t also be a right one. This is related to the previous point that $\mathbf{u}$ , $\mathbf{v}$ , and $\mathbf{w}$ must have the same orientation as $\mathbf{i}$ , $\mathbf{j}$ , and $\mathbf{k}$ .

Now, finally, let’s justify the claim above. What happens to a quaternion $x$ when we left-multiply it by $q$ and right-multiply it by $\overline{q}$ ? If $q=\cos\theta+\mathbf{u}\sin\theta$ , then $\overline{q}=\cos\theta-\mathbf{u}\sin\theta=\cos(-\theta)+\mathbf{u}\sin(-\theta)$ . Thus the two isoclinic rotations performed on $x$ are in the same planes and by the same angle but they are in different directions. Also, one of them is left and one is right. Let’s write down what they do:

Left-multiplication by $q$ : Rotates by $\theta$ in the $1,\mathbf{u}$ -plane and by $\theta$ in the $\mathbf{v},\mathbf{w}$ -plane.

Right-multiplication by $\overline{q}$ : Rotates by $-\theta$ in the $1,\mathbf{u}$ -plane and by $\theta$ in the $\mathbf{v},\mathbf{w}$ -plane.

So the combined effect is to rotate by $2\theta$ in the $\mathbf{v}, \mathbf{w}$ -plane and do nothing else! Since we don’t touch the real component of $x$ , we are then rotating $\mathbb{R}^3$ exclusively. Since we don’t touch the $u$ -component of $x$ , the vector $u$ must be our axis of rotation. Thus we did it! This is why conjugation by $q$ is rotation by angle $2\theta$ about $u$ !

Before I go, here’s some more cool stuff about this. Notice the factor of $2$ . This is interesting because it means that if $\theta=180^\circ$ we won’t rotate at all. In this case, $q=\cos 180^\circ + \textbf{u}\sin 180^\circ=-1$ . So the unit quaternions $1$ and $-1$ both correspond to not rotating at all. These are actually the only such unit quaternions, because every other unit quaternion has an angle $\theta$ such that $2\theta\neq 0$ . More generally, there are exactly two unit quaternions that correspond to any rotation: $q=\cos\theta+\mathbf{v}\sin\theta$ and $-q$ . This means that the 3-sphere of unit quaternions is a double cover of the space of proper rotations of 3-space. This fact was exploited in the very cool game Hypernom.

Also, I’d like to mention rotations in the 4-space that quaternions live in. We know how to do left and right isoclinic rotations in this space, but how do we do general rotations? Well, it turns out we can put together a left and a right isoclinic rotation to do any rotation we want. This can be written $qxr$ , where $q$ and $r$ are unit quaternions which are performing left and right isoclinic rotations on $x$ respectively. I’ll show how to make any single rotation, and you can put these together to make double rotations (not necessarily isoclinic).

Let’s say we want to rotate by the angle $2\theta$ in the plane spanned by quaternions $y$ and $z$ . If $y$ and $z$ are both fully imaginary, we already know how to do this. So let’s consider the case wherein $y$ has non-zero real component. If $y=1$ , then $z$ must be fully imaginary. We then follow a procedure very similar to the previous. We do a left and a right isoclinic rotation, both of which rotate in the same direction by $\theta$ in the $1,z$ plane and which cancel in the orthogonal plane. This can be written $qxq$ . Notice the absence of the conjugate.

Now if $y\neq 1$ , but still has non-zero real component, we can turn it to be $1$ , perform the rotation in the way just described, and then turn it back. In more detail:

Step 1. Singly rotate in the $1,y$ -plane so that $y$ maps to $1$ . Let $z'$ be the rotated version of $z$ .

Step 2. Rotate in the $1,z'$ -plane by $2\theta$ .

Step 3. Perform the same rotation as step 1 but in reverse.

Now, I claimed that every rotation could be written as $qxr$ , but now we have multiple steps. What’s going on? Well, we can write each step this way. Step 1 can be written $q_1xr_1$ , step 2 $q_2xr_2$ , and step 3 is just the inverse of step 1: $q_1^{-1}xr_1^{-1}$ . Doing all these steps sequentially yields $q_1^{-1}(q_2(q_1xr_1)r_2)r_1^{-1}=(q_1^{-1}q_2q_1)x(r_1r_2r_1^{-1})$ . This is of the required form because $q_1^{-1}q_2q_1$ and $r_1r_2r_1^{-1}$ are both unit quaternions performing left and right isoclinic rotations on $x$ respectively.

So every rotation in 4-space can be accomplished with two unit quaternions, $q$ and $r$ . Rotating a quaternion $x$ then yields $qxr$ . This means that rotations in 4-space can be represented by pairs of quaternions. However, just like in the case of rotations in 3-space, there are actually exactly two pairs of quaternions corresponding to every 4-spatial rotation: $(q,r)$ and $(-q,-r)$ . We can see this because $(-q)x(-r)=qxr$ . But why are these the only two? Well, suppose that we had another pair $(q\alpha, \beta r)$ . Then $q\alpha x\beta r=qxr$ , which simplies to $\alpha x\beta=x$ . This must hold for all $x$ , so we can choose $x=1$ in particular, to attain $\alpha\beta=1$ . This means that $\beta=\alpha^{-1}=\overline{\alpha}$ , so $\alpha x\beta=\alpha x \overline{\alpha}$ is a rotation of $\mathbb{R}^3$ . Since it doesn’t change anything, $\alpha=\pm 1$ .

Should I go to Graduate School?

Posted on September 5, 2017 by meygerjos

If I went to graduate school, I would go to get a PhD in pure mathematics. I have been contemplating lately whether to go. Here are the pros and cons, with counterarguments. I could also have given counterarguments to the counterarguments, but I decided to stop at just counterarguments.

First the cons:

If I went to graduate school, I would be removing myself from the real world for a few years of my life. I have a sense of urgency about the political situation and I feel irresponsible cloistering myself away and just doing math all day for years while people are suffering and the environment is quickly being destroyed.
1. Counterargument: I’m not really politically active right now anyway. Why do I think I would be politically active if I wasn’t in graduate school?
2. Counterargument: I wouldn’t necessarily be doing math all day if I went to graduate school; I could do other things too with my time. I would just be doing math with most of my time.
3. Counterargument: Perhaps my sense of urgency is exaggerated. I have my whole life to be political, and there are some ways of being political that academic credentials really help with (e.g. Noam Chomsky, David Graeber).
I have a problem communicating with people, and one of the reasons for this is my immersion in mathematics. For example, I often put form over content — I will often critique the form of what someone says and leave the content alone, and people will often reply to the content of what I’m saying when really the form is the important part. Years more of immersion in mathematics would exacerbate my communication problem.
1. Counterargument: Maybe my communication problem is not due to mathematics.
I generally have a problem with school. I don’t like being forced to do things, and even if I like something I am forced to do in school, I don’t have time to do it in the face of all the other things I am simultaneously being forced to do. Each semester of college, I had a really bad time but I blamed myself rather than Stony Brook and I thought that the next semester would be better — I was consistently wrong. Why would I not be making this same mistake again by going to graduate school? I also anticipate that I will also be uncomfortably limited in my choice of thesis topic.
1. Counterargument: This is really only a problem for the first year of graduate school. After that, I will pretty much be able to focus on a single topic that I choose and go deeply into it.
2. Counterargument: By doing enough research beforehand, I can choose a graduate school and advisor that will let me write about what I want to write about.
3. Counterargument: It is arguable that Stony Brook wasn’t a good fit and I would have been happier if I had transferred to some other college. So maybe the problem was just Stony Brook and not college in general.
4. Counterargument: Graduate school would be an entirely different thing than college; maybe my problem is just with college and not with school in general.

Now the pros:

I will be able to find people who I can relate to mathematically. Mathematics is a big part of my identity and it is hard to find people who can relate to this part of me. I wouldn’t get along with everybody in graduate school, but I would have much better luck among them than among the general population.
1. Counterargument: I could find people to relate to among the general population. Though it would take more work to find such people, my perspective would be broadened through them because they wouldn’t think in the narrow mathematical way.
2. Counterargument: I could find people to relate to in anything I do, not just graduate school. It doesn’t matter if they are math people, the important thing is that I am around them a lot and get to know them in the context of a years-long shared experience.
I have found that I am happiest when I am immersed in something involving other people. Graduate school would be a deep immersion in mathematics, which I love deeply. I technically could become immersed in mathematics without being in graduate school, but it would not be socially acceptable, i.e. I wouldn’t get a stipend and my parents and potential employers wouldn’t approve as much — so it would be way harder. The difficulty of this would be distracting and diminish the immersion.
1. Counterargument: I could be deeply immersed in something involving other people other than mathematics in graduate school. (Not sure what.)
2. Counterargument: Happiness is not the most important thing.
I need to be part of a higher purpose — a project with other people to do something. Unfortunately the people with good work-ethic are mostly situated in the well-established ones, and all the well-established ones are evil. Because of this I have lately been trying to contribute to projects that are not well-established. For these projects there is no existing reservoir of commitment and work-ethic, so I need to come up with it myself, and this is really difficult. I have been feeling lately like I don’t have the energy to come with with all this work-ethic myself. Perhaps doing something less good just to be doing something at all is a valid option. Moreover, I would not have to be evil myself just because I was part of an evil institution. I could use the resources of of the institution for my own ends. I probably couldn’t be fully good, but I would certainly be less evil than the institution.
1. Counterargument: Don’t be evil!
2. Counterargument: There is a third option that Momo brought up. I could work on a project just by myself, or with just 2 or 3 people. It would be easier to summon the requisite work-ethic that way, because there would be less people to coordinate. In a sense I am already doing this with this very blog, as well as with my website socialmath.github.io. This is nice, but doing a project with a whole bunch of other people would be even nicer.
3. Proargument: I seem to want to be pure and not let any evil inside myself. This is stupid. Most people don’t even have the privilege to be that pure. Since the oppression of others is facilitating my purity, it really isn’t that pure at all. My desire for purity really comes from white guilt and not from genuine goodness. I should sacrifice my purity to be more good. As Miles quoted from some anime, “is it better to remain pure from evil or to take evil into yourself in order to purge it form the world?”
Having a PhD would make things way easier in the future for me, and I would have way more time to be political.
1. Counterargument: This would be at the expense of indoctrination. I would basically be making a bargain with the authorities. They’ll give me credentials and make my life easier in exchange for me buying into a lot of academia bullshit and distancing myself from real life.
I have a very strong desire to learn mathematics because I love it. I want to have the society’s knowledge of mathematics under my belt. I think of myself as a mathematician, and when I learn mathematics I feel like I am learning about myself. This is an aesthetic end rather than an moral one, but maybe that is ok (cf. this ContraPoints video around 6:15 but watch the whole thing for context).
1. Counterargument: Life is the only end in itself and I am unethical and religious if I set up another end in itself, mathematics. If I want to be aesthetic in my life (as per the ContraPoints video) I should pick something more widely accessible than mathematics. If only a few people understand a beautiful thing I made, is it really beautiful? (On the other hand, I could be a mathematics educator like Martin Gardner or betterexplained.com.)
It would be really nice to have the option to make money with scholarly work. To do this I really need credentials, i.e. a PhD.
1. Counterargument: ??
I have found that I like to have a large living space, and I feel trapped and depressed when my living space is small, like an apartment. I love being at Stony Brook because I feel like the entire campus is my house. (I’m not saying I like being enrolled in the college, I just like being at the campus. I’m there right now and I’m not enrolled in any classes.) I can walk around, bump into people I know, sit down and study, etc. In Manhattan there is really nowhere that I can sit down and study because there’s music playing in every single establishment (except the Hungarian Pastry Shop, thank you for existing!) and people talking in the libraries. I am very easily distracted: I can’t concentrate in the face of meaningful sound, such as music or people talking. I can concentrate in the Hungarian Pastry Shop because so many people are talking that I can’t make out what anyone is saying, plus there’s the morale boost of having other people around my also studying things. At Stony Brook, however, I can go to pretty much any building at any hour of the day or night and find a place to study, it’s great. I’m not sure what places would meet my criteria other than a university campus. A city doesn’t work, and suburbs wouldn’t really work either. If I went to graduate school, then I would get to be on a campus. Similarly if I became a professor at some time in the future.
1. Counterargument: Maybe there’s some other place that meets the criteria, such as the secular monastery Jimmy and his friends are thinking about if/when it comes to fruition? Or if my parents would let me go to Nesin Mathematics Village or if I get independent enough that I don’t need their permission anymore. Of course these are not mutually exclusive with going to graduate school, and perhaps going to graduate school now would facilitate their occurrence in the future….
2. Counterargument: If I had my own apartment and I really made it nice, maybe I could enjoy being there. I would only need a large campus to run away to if my own apartment was deficient. My problem with a small living space is that I am densely surrounded by various reminders of myself — it is like being in a room with mirrors on all the walls; and this makes me sink into neurosis. I might be able to alleviate this effect if I threw out a lot of stuff and organized and decorated really well (perhaps according to the advice of Mari Kondo) —- if I engineered a place that is healthy for me to live.

Please give me advice. More arguments and counterarguments are welcome.

The Idea of God Offends Me on a Spiritual Level (Red Notebook)

Posted on August 30, 2017 by meygerjos

A selection that I particularly like from the red notebook. I like the first paragraph more than the rest:

The idea of God offends me on a spiritual level. I have a deep loyalty and spiritual connection to this world, the material world that I live in, and the other people in it. I live my life for this world and the people in it, not for the post-death approbation of God. God is oft invoked to give credence to us: a beautiful natural thing is said to be the work of God — is nature not beautiful enough as it is? A good outcome is ascribed to the goodness of God — are we not good enough to achieve a good outcome of our own accord by working together? The genesis of life over millions of years as an amazing and sublime fact; God takes credit for this as well: of course this world could never achieve such greatness without his help. Everything good about this world is attributed to God, who is not of this world, but who belongs to a different superior world. Without God, this world would be terrible. Thus love of God and hatred of this world go hand in hand. God is like a boss who would have his workers believe that they are nothing without him. In reality the boss is nothing without his warriors, just as God is nothing without the people who believe in him and worship him.

People claim to love others out of love of God. The argument goes something like this: I love God, and $x$ also loves God / is loved by God / is a child of God / is a creation of God. Therefore I love $x$ . Is the love of one’s neighbor so fragile that it needs the authority of a king to legitimize it? And what if God stopped loving $x$ , or $x$ stopped loving God, or God disowned $x$ , etc. Would I then stop loving $x$ ? I wouldn’t. The spiritual connection between me and $x$ is strong, and it will not be impacted by something as feeble and evanescent as $x$ ‘s relationship with a king in another world. The idea that it is weak and must rest on this for its very existence is an insult to the dignity of this world.

Things that [people claim] God explains:

Love

Beauty

Morality

Fortuitousness

Life

Afterlife

Consciousness

Creation

Some context: in the beginning of high school (2010) I was a militant atheist and believed that religion was responsible for all of the world’s problems. I would argue with people about the existence of God all the time as a way to make friends. I had a year-and-a-half long argument with a Muslim student that took place during lunch, after school, and on Facebook. Eventually I realized that religion wasn’t actually the cause of the world’s problems, rather it was a pretext used for lots of bad things. Really, religion was just a tool which could be used to justify good things or bad things. Around the same time that I realized this (sophomore year of high school), I figured out that people didn’t actually believe in religion because of all the complicated philosophical arguments that they gave to justify their beliefs. Rather, they just had faith — they believed for non-rational reasons. Now a lot of atheists think that the argument stops there — when a religious person admits that their beliefs are founded on faith, there’s nothing more that can be done: they have renounced their rationality. Well I didn’t think so! I figured that the argument could be continued to a discussion of the merits of faith. There are various benefits of faith in religion: community, perceived love of God, the security of feeling that you know the answers to the “big questions”, etc. I had a precise list of these benefits in mind when I thought of this, but I don’t remember them now. Anyway my argument was that all of these benefits could be attained without religion, so religion was unnecessary. The above is a refinement of this argument — it argues the stronger point that faith in God is actually wrong, not just unnecessary.

However, faith is not wrong in general. I think that it makes a lot of sense to have faith in people. I have faith in my friends (which includes myself, since I am one of my friends), which means that I have unjustified beliefs that they are trustworthy and that the things they try to do will succeed. I believe these things not because of rational justification, but because of ethics and pragmatics — ethics because believing in your friends is the right thing to do, and pragmatics because my friends are more likely to succeed at what they do if their friends (e.g. me) believe that they will. I do not have a belief that my friends will succeed easily, without trying — such a belief is dangerous and statements like “don’t worry about it ur rly smart you can get anything you want in life without trying” are unhelpful. I don’t ignore the obstacles and the things that make it hard to succeed, but I retain a belief that my friends will succeed in spite of these things. (Statements relying on faith in God like “don’t worry about it God will provide” are unhelpful in a similar way.) And obviously I’m not talking about “success” in the narrow capitalistic sense.

Similarly, I believe that we will overthrow capitalism and create a better world. I don’t believe this because I’ve weighed everything and figured out that it is likely, and in fact it is really unlikely. I believe it because it definitely won’t happen if we don’t believe that it will.

An Announcement (Red Notebook)

Posted on August 29, 2017 by meygerjos

I’m currently going through my notebooks (2014 to present), and I’m gonna be posting and discussing stuff from them here. These notebooks have a lot of ideas that I intended to think about more but never did because I was in school and I didn’t have time/energy. Well now I do have time/energy, because I am not in school!

This is from the red notebook, which I wrote in roughly in the Spring semester of 2015. It is a joke announcement that I thought it would be funny if someone said at the Bronx Science commencement, which I attended with Alex to watch Sid graduate.

Excuse us, may we have your attention please. We would like to make an important announcement. Welcome to the commencement of 2015 of the Bronx High School of Science. We apologize to all students, faculty, family, and friends for any inconvenience that may be caused by this interruption to the commencement proceedings. We assure you that this announcement will not take a significant amount of time away from the commencement proceedings, and we implore you to momentarily divert your attention from the proceedings in order to understand this message. We thank you for your time and attention, and we apologize again for any inconvenience that may have been caused on account of this announcement. The commencement will now continue.

Anybody is welcome to (and encouraged to) adapt this announcement to any ceremony they might attend and hijack the PA system to say it. If it has the desired effect, everybody, including the people in charge, will think that someone in charge is making the announcement until the end when they realize it’s just a dumb joke.

Planning Ahead and Self-control

Posted on March 23, 2017 by meygerjos

For a long time, I have been adverse to planning ahead, because I felt that it limited the freedom of my future self. Suppose I planned to do something, but when it came time to do it, I wanted to do something else? I equated freedom with the ability to be spontaneous and do whatever I thought of at any moment.

I have known for a while that I am biased against planning ahead, because most of the planning ahead I encountered in my life was done to me by authorities, like parents or school, not by myself. During the summer of 2011 (I think) I tried to get over my bias and schedule myself. I decided to spend 2 hours on studying each of 5 subjects for a total of 10 hours per day. It failed miserably. I got behind, and needed to make up lost time. I felt like I was moving very slowly, because I advanced at most 2 hours on each subject per day. I also was trying to get on a polyphasic sleep cycle at the time, of which my parents were very unsupportive (I would miss my naptimes arguing with them about the sleep cycle), and hence my morale was further weakened. After this experiment I decided that planning ahead and making schedules wasn’t for me, and that the best way to get things done is to just be driven by passion and choose at the moment what thing to do.

Now I am reevaluating this conclusion. I am realizing that if I make a reasonable plan to do something at a future time, I will want to do it when it comes to be that time. I won’t feel like doing something else instead. If I feel like doing something else instead, that is an indication that my plan wasn’t reasonable.

To demonstrate this, let us speak about the enjoyment cycle, in analogy with the sexual response cycle. The stages of the enjoyment cycle are 1) anticipating the event, 2) enjoying the event itself, 3) enjoying the memory or result of the event. So our enjoyment of an event is not localized in time to when the event itself takes place, but also occurs before and after the event. Here is a simple thought experiment to prove this:

“I can give you ultimate enjoyment for a duration of 10 minutes. It will be the best feeling you have ever experienced in your life. And I can do this for just $20. What a great deal! Here’s the catch: These 10 minutes could happen at any time, and you’ll have no way of predicting it. Furthermore, after these 10 minutes have happened, you won’t remember them at all, and they will have no lasting effect.”

Is this service worth $20? Most people would say no, demonstrating that an event is not really enjoyed if it is only enjoyed when it takes place. Not just stage 2 of the enjoyment cycle, but also stages 1 and 3 are necessary.

So being spontaneous all the time is not the best way to enjoy life, because it neglects stage 1 of the enjoyment cycle, so spontaneity always gives incomplete enjoyment.

Another experience explained by the enjoyment cycle is the disappointment of anticipating something that doesn’t happen. If I make plans with someone and they don’t show up, I am disappointed because the enjoyment cycle is cut short after stage 1. I have trouble doing something else instead of seeing them, because my whole mind was set up to see them and now I have to suddenly change tracks and prepare to do something else, which doesn’t have a stage 1 setup, so to speak.

So a reasonable plan will set in motion stage 1 of the enjoyment cycle: anticipation. You will anticipate the thing you plan to do, so that when it comes time to do it, you won’t feel like doing something else, but rather you will feel disappointed if you do something else.

An unreasonable plan puts unreasonable demands on the enjoyment cycle. You might anticipate something impossible, setting yourself up for disappointment. Or you might misconceive the length of the enjoyment cycle, by planning too far in advance, trying to do an activity that takes too long before it gives a reward, or commencing anticipation for an event too soon before it will actually happen.

Having an unreasonable plan feels horrible. It feels like you have no control over yourself, because you consistently do something other than what you plan to do. When your actions don’t correspond to your plan, you feel out-of-control. There are two ways of fixing this and making your actions correspond to your plan. You can either bend your actions to the plan, or you can bend the plan to your actions. Most people do the former way, and don’t ever consider the latter. For example, suppose someone in school plans to get all their homework done on time, but never does. Most people in this situation will just resolve to push themselves harder to follow their plan, but nothing will really change significantly. What about the other option? Why not change your plan, so that you plan to not get all your homework done on time? This option might seem crazy, but it is really much more doable than the previous option. But of course, a plan mustn’t just be negative, it mustn’t just be a plan to not do something, but it must also have a positive component. So a plan really would be something like “I’ll get these specific homeworks done and not these other ones, and I’ll spend the bulk of my time for the next five days learning Arabic”.

A good plan is a response to what you are already doing, not a fantasy of what you could be doing. With a good plan, your actions determine the plan. A bad plan reverses things and tries to determine your actions.

Caution: I am worried that people will take this the wrong way. What if you are doing really badly? Am I saying that you should plan to do just as bad? No. You should plan to do better, but in a way that addresses the reasons that you are doing bad. It should be a plan designed for you, who has certain problems, not a plan for the person you would be if the problems had already been fixed by magic. Don’t make the error of math teachers who teach based on what their students are supposed to know (i.e. everything in prior math classes) rather than what they actually know.

Grothendieck wrote in his beautiful Recoltes et Semailes (pp. 64-65) about how mathematical objects, just as a good plan, must reflect reality rather than vice versa:

One cannot invent the structure of an object. The most we can do is to patiently bring it to the light of day, with humility – in making it known it is “discovered”. If there is some sort of inventiveness in this work, and if it happens that we find ourselves the maker or indefatigable builder, we aren’t in any sense “making” or “building” these structures. They hardly waited for us to find them in order to exist, exactly as they are! But it is in order to express, as faithfully as possible, the things that we’ve been detecting or discovering, to deliver up that reticent structure, which we can only grasp at, perhaps with a language no better than babbling. Thereby are we constantly driven to invent the language most appropriate to express, with increasing refinement, the intimate structure of the mathematical object, and to “construct” with the help of this language, bit by bit, those “theories” which claim to give a fair account of what has been apprehended and seen. There is a continual coming and going, uninterrupted, between the apprehension of things, and the means of expressing them, by a language in a constant state [of] improvement, and constantly in a process of recreation, under the pressure of immediate necessity.

Likewise, there should be a continual coming and going, uninterrupted, between the doing of things, and the means of expressing what we are doing in the form of a plan. Our plans are must be in a constant state of improvement, and constantly in a process of recreation, under the pressure of immediate necessity. The end goal, which we may never reach, is for our plan to conform to our actions perfectly.

Grothendieck continues:

As the reader must have realized by now, these “theories”, “constructed out of whole cloth”, are nothing less than the “stately mansions” treated in previous sections: those which we inherit from our predecessors, and those which we are led to build with our own hands, in response to the way things develop. When I refer to “inventiveness” (or imagination) of the maker and the builder, I am obliged to adjoin to that what really
constitutes it soul or secret nerve. It does not refer in any way to the arrogance of someone who says “This is the way I want things to be!” and ask that they attend him at his leisure, the kind of lousy architect who has all of his plans ready made in his head without having scouted the terrain, investigated the possibilities and all that is required.

In planning ahead, we should make a plan that suits us, not a plan that suits some idealization of ourself. We shouldn’t say “This is the way I want things to be!” and expect that they will simply happen as such; we shouldn’t have “all [our] plans ready made in [our] head”, but we should first “investigate the possibilities and all that is required”.

Grothendieck continues:

The sole thing that constitutes the true “inventiveness” and imagination of the researcher is the quality of his attention as he listens to the voices of things. For nothing in the Universe speaks on its own or reveals itself just because someone is listening to it. And the most beautiful mansion, the one that best reflects the love of the true workman, is not the one that is bigger or higher than all the others. The most beautiful mansion is that which is a faithful reflection of the structure and beauty
concealed within things.

(This passage is so beautiful. I love how Grothendieck talks about “the voices of things”. This is the humility and serenity of doing mathematics.)

The sole thing that determines our effectiveness at planning ahead is the quality of our attention to ourselves.

Another Caution: Paying attention to yourself and basing a plan on your nature doesn’t mean defining yourself, saying “I am this type of person and only like x, y, and z, and my favorite color is orange and I can’t do math”. This is not paying attention to yourself, this is imposing a definition on yourself. The yourself I am actually talking about has no precise definition, but is constantly changing with infinite flexibility. It has positive attributes, it is not merely defined by what it isn’t (“I only like x” is actually a negative attribute because it is really saying that you don’t like anything that is not x. “I like x” would be a positive attribute). So, when I say to base a plan on your nature, I’m not talking about a false image of your nature, I’m talking about who you actually are.

There is an analogy that I really want to make with Marx’s discussion of democracy in his Critique of Hegel’s Philosophy of Right, but this post is long enough already so I’ll make it another time and add a link to it in this sentence.

Learning through Immersion

Posted on March 21, 2017 by meygerjos

I was talking with Jimmy on Friday, March 10 and I asked him for advice on time management. I asked something like, “do you ever plan ahead of time to do one thing for a certain amount of time, then do another thing for a certain amount of time, etc?” I was thinking of making a schedule for each day, like “from 1-4 I”ll read this book, from 4-5 I’ll eat, from 5-9 I’ll write about x, etc.” He said something like, “yeah sometimes I’ll plan to spend a few weeks on one thing, then a few weeks on another, etc.”

This made me reconsider. I realized I’m trying to do too much at once. For the next week I only did one major thing per day. I allowed myself to do other, minor things, like answering emails, but the day was centered around a single thing. On Saturday and Sunday I just did cleanup, organizing and administrative things. I packed all of my clothing here at Stony Brook to bring to Manhattan except for predetermined quantities — 7 shirts, 7 pyjama shirts, 10 underwear, etc. — because I have way too much here. Monday and Tuesday were just devoted to reading Karl Marx’s Critique of Hegel’s Philosophy of Right with Russell Dale’s notes (I am taking Russell’s class). Wednesday and Thursday were for computer science, which I am studying with AnarchoTechNYC.

I didn’t spend weeks on a single thing like Jimmy had mentioned, though I do hope to get to the point where I can do that. This was still much preferable to my prior method, however. I was able to focus on things I needed to do without constantly feeling guilty for not doing some other thing I also needed to do. I also felt in control of myself, which is a very nice feeling. My plans aligned with my actions much more than I am used to. I required much less sleep than normal because I was excited to get up in the morning.

The two days of Marx (Monday and Tuesday) were really helpful. Marx is very difficult and slow to read. On Monday I read pages 5-19 (and Russell’s much easier notes concurrently), and it took me about 10 hours total. On Tuesday I read to page 36 (without Russell’s notes, which didn’t make it to page 36 yet, now they do so I will catch up in them shortly), and it took me about as long. So I got faster from Monday to Tuesday. I also got faster on the short time scale. I noticed that when I first sat down to read, it would take about a half-an-hour of reading really slowly before I got into the groove, and then I could read at a workable pace. This meant that every time I stopped reading and my flow was broken, I needed a half-an-hour to get back into it.

The reason I am mentioning all this is to illustrate how indispensable it was that I read Marx intensively, rather than, like, a little every day. Reading a little every day, it would have taken much longer, I wouldn’t have understood it as well, and it would have been more frustrating because it would take longer to see any progress.

I think this is a key principle of learning. To learn a new field, you must at first immerse yourself in it. Once you have done this, you can then later do it more intermittently. Most people already know this about language learning. The best way to learn a language is immersion. Learn French by going to Paris for a month. Then when you come back, you will still be able to read a French news article when you need to. Immersion first and a little a day later. In a sense, all learning is language learning. To learn Marx is to learn how to understand his language. To learn math is to learn the language of math. To learn computer science is to learn computer languages. When considered this way, the principle that languages are learnt best by immersion is not just analogous to this key principle of learning, but identical with it.

School does not allow this. In school, you must divide yourself between many classes, and you cannot ever immerse yourself in any of them. Moreover, each class switches violently between topics, so even within a class you are split between many topics. Thus you cannot ever really learn. School does not help you learn, it prevents you from learning. In school your mind is split between many different things. You are schizophrenic in the literal sense of the word: split (schizo, from Ancient Greek σχίζω, split) – brained (phrenic, from Ancient Greek φρήν, mind, but seemingly used in English for words about “brain”). You cannot learn when you are split-brained. To learn you must be focused.

If you are in school and you want to learn, I recommend either 1) drop out of school and find teachers who care about your education, 2) take very few classes (even just 1 class if you can pull it off!), or 3) bullshit your way through school with as little effort as possible and devote your time and energy to actually learning by immersing yourself in topics with teachers who care about your education. More information on these approaches.

There are more things I want to talk about relating to this, but they belong in separate posts. I will update the following bulleted list with links when I write these posts.

The learning as digging analogy
How and why I unfairly maligned making schedules
I want to write something about this article I just found, which gives an extremely convincing argument illustrating why it is more efficient to do things one at a time than to multitask. I think it is a decisive result. I can’t see any way of getting around the logic of it.

Revolutionary Mathematics

Math for Liberation