MrManatee

As you already realized, my example was supposed to be in 2D, but I forgot to mention it. Sorry about that.

But your point about steradians is still spot on. If I had been talking about the 3D case, there would have been a 1/(2pi) factor in the front of the pdf. Using tau would not have helped anyone realize that this particular 2pi is most naturally thought of as 1/2 of the full 4pi steradians. Using tau is not a substitute for careful thinking.

And I am not even actually advocating switching to tau; I am not sure if it is worth the effort. But I still think that tau is a good way of conveying the idea that the current definition of pi as basically a historical accident. This can be enlightening even if one doesn't start using tau. As a more extreme example, I think it is useful to recognize that the decimal base (as opposed to binary, hex, etc.) is a historical accident, but I wouldn't advocate changing it.

I have found the idea of tau useful even though I have never used it in writing.

One argument in favor of tau is that in many formulas pi often has the multiplier 2 in front of it. If these formulas are written in terms of tau, they may become slightly easier to memorize and manipulate. Perhaps so, but I don't really care about this. It’s not a big difference. Besides, there are also lots of formulas that are easier to memorize and manipulate using pi instead of tau. The probability density function of the standard Cauchy distribution f(x) = 1/pi * 1/(1 + x^2) is one example.

However, to get a deep understanding of mathematics, I want to understand the connections between different parts of mathematics. If I see a mathematical formula with the constant pi in it, I ask myself: "How is this connected to circles?" The idea of tau taught me that that 2pi is the natural state of affairs. If I see pi by itself, I need to ask myself: "How is this connected to half-circles? Or has the multiplier 2 been cancelled away?”

So, why does the pdf of the standard Cauchy distribution above contain pi instead of 2pi? What is the standard Cauchy distribution anyway? Take a gun that shoots particles in random directions, and place it one unit distance away from an infinitely long wall. Standard Cauchy distribution is where the particles will hit the wall. The particle will only hit the wall if it is shot in a direction towards it. This corresponds to 180 degrees - and there you have it: the connection to half-circles. Of course, you also have to work out the technical details. But on an intuitive level, when I see pi in the pdf of the standard Cauchy distribution I don’t think about how the missing multiplier makes it easier to remember a bunch of symbols; I think of particles hitting a wall.

One aspect of teaching everyone to code is what Seymour Papert talked about in Mindstorms: it can be used as a medium for carrying ideas. For example, I think Mindstorms had this anecdote about a child who previously had trouble grasping the idea of classifying parts of speech (nouns, verbs, etc.). Later, she was writing a program for creating random nonsensical sentences, when suddenly it clicked, and it all made sense to her.

The point of this anecdote is not that every child should write such a program to understand parts of speech. It's about something much more general, but I have no idea how to express it briefly. I'm not even sure if I fully understand it. So, unfortunately, this idea might just be too subtle to be properly executed on a national scale.

Yeah. The main thing people probably want is "TeX quality" typesetting of mathematical formulas inside web pages. Bundling the entirety of TeX inside a browser for this purpose would be kind of a hack, and suboptimal in many ways. I think that instead of just using TeX as a black box, it would be more interesting to actually learn what makes its formulas look good, and apply these lessons in modern software.

There's a related article called A Functional Description of TeX's Formula Layout [1] that describes a more modern reimplementation of a part of TeX.

[1] http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.39....

I also use a password manager, and enable two-factor authentication when possible. I mark my personal devices as trusted, so I only have to do two-factor authentication whenever I get a new device.

I can think of a few examples of threat vectors where an attacker could get one of my passwords without being able to completely take over any of my devices.

1. There might be a browser vulnerability that allows a malicious web page to get one of my passwords - let's say it gets to read memory that was deallocated but not zeroed, or it gets to see the contents of my clipboard, or something like that.

2. The provider of the service I'm using might accidentally leak some of their passwords. In the worst case, they might not even notice it.

3. In some special circumstances I might have to check my mail on a computer that I don't own, and there might be a keylogger.

I don't necessarily consider these threats to be very likely, but I get new devices so rarely, that even a minor improvement in security makes it is worth the inconvenience for me.

I have sometimes wondered whether it would be useful to have two different types for indices, depending on if we are indexing the elements themselves or the "gaps" between them. Let's say I'm thinking about some language like Haskell, where types are already a big deal.

That way the compiler would be able to tell if I accidentally mixed the two. Every conversion would have to explicit: for example, there might be two functions, "before" and "after", that take a gap index, and return an element index.

I think I might actually enjoy programming this way, but perhaps others would find it needlessly bureaucratic.

I would agree that the examples can be somewhat misleading, but not because of computability. Universal approximation theorem for neural networks doesn't care about computability, but it does require that the domain of the function is finite (or to be more precise, compact).

For example, suppose that the function to be approximated is simply f(x) = x. For any real numbers a < b we can produce an approximation that works well for a ≤ x ≤ b, but it cannot be a good approximation for all real numbers x. (The sigmoid in the hidden layer means that every approximation has to be a bounded function, and a bounded function cannot be a good global approximation of f(x) = x.)

Therefore, the translation example works if we assume that the number of different Chinese texts is finite, but otherwise nothing is guaranteed.

To put my words in context, I was thinking about mathematical questions, such as "does there exist a curve that goes through every point in a square?". Everyone has some kind of intuitive understanding of what's a "curve", but mathematicians need a precise definition in order to make this question clear enough for mathematical study. (There is such a curve: the Hilbert curve.)

It's not at all easy to recognize when a question is stated "clearly enough", and your questions provide an excellent example of this. The question "is this question more than eight words long?" refers to question itself (as a string), and in general, this kind of self-reference is perfectly harmless. The question "Is the answer to this question 'no'?" refers to its own answer, and this kind of self-reference is potentially problematic.

With regards to 2), there's a difference between trying to find an answer to a single yes/no question, and trying to find a general method for solving an infinite class of such questions.

If a single yes/no question is clearly stated, then it does have an answer. This just a tautology - that's what it means to be clearly stated. For example, does there exist integers x, y, and z such that x^3 + y^3 + z^3 = 33? No one knows, but at least the question is clearly stated. The answer is definitely yes or no.

More generally, x^3 + y^3 + z^3 = 33 is an example of a Diophantine equation (an equation between multivariate polynomials with integer coefficients and integer variables). Suppose we are not so interested in this particular equation, but instead want to find an algorithm that can always tell if a Diophantine equation has a solution. This kind of question might not have a solution, since such an algorithm might not exists. And in fact, it doesn't (Hilbert's tenth problem).

The term "isomorphic JavaScript" also sounds strange to my ears. I think it’s because in traditional use (in mathematics, linguistics, biology, etc.) you don't say "X is isomorphic", you say that "X and Y are isomorphic".

Grammatically, it’s kind of like the word "similar". I would feel silly saying "JavaScript is a similar language", and I feel equally silly saying "JavaScript is an isomorphic language". Similar/isomorphic to what?