Month: December 2025

The Maddening Ambiguity of Mathematical Notation | December 02 2025, 15:30

If someone tells you that mathematics is an exact science, don’t believe them. Since I’m currently into data science as a hobby, I’m studying all sorts of things from different books and my brain is exploding at how this can happen in a science where every little detail should fit into a system, otherwise it goes by the wayside. Until it gets to notations. It’s a complete mess there. A set of dialects.

Take, for example, common logarithms. The “standard” for how to denote a logarithm depends on which room of the university you are in. In calculus and number theory, log(x) almost always means the natural logarithm ln(x) with base e. The derivative of e^x equals e^x. It’s “natural”. They’re too lazy to write ln. Yet, where decimal logarithms might appear (like in computer science), log(x) suddenly becomes decimal, and ln(x) is based on e.

The expected value E has an argument in square brackets. Meanwhile, the same square brackets in computer science are used for the step function 0/1.

Or if you see a vector – is it a column or a row? In classical mathematics, a vector is always a column. To multiply it by weights, we write T after the vector and then w for the weights. But in many papers, vectors are thought of as rows. And if you see y = xW+b, then x is not a column, because otherwise the dimensions wouldn’t match up. x here is a row. But in the next paper they write Wx+b. And there x is a column 🙂

Angle brackets . For the dot product, the symbol “â‹…” is used, but it is hard to see, especially on a whiteboard, and I very often see that mathematicians use angle brackets for dot product. In general, angle brackets are used for the generalized concept of inner product, where the scalar product is a special case. signifies a certain abstract way to multiply a and b and get a number. Meanwhile, in quantum mechanics this would be written as . And for the scalar product, some use a circle with a dot or x in a circle.

And just for the sake of it, in Russia tangent is tg, while in the USA it’s tan. There’s also tan^-1 and arctan, which are the same, though x^-1 generally means 1/x

In-Flight French: Building a Language App on the Fly | December 01 2025, 15:45

By the way, yesterday morning, while waiting at the gate for my flight to Miami, I quickly wrote a French language learning app using Gemini based on an idea I sketched out to a friend while driving to the airport, and then used this app during the flight.

The idea is that in an unfamiliar foreign language text, the user first marks unknown words and then sees their translations — but without the original text, and then returns to the text itself — but no longer seeing the translations. It’s as if the “dictionary was in the next room.” The hypothesis is that this method helps better memorize than when the translation is shown immediately upon clicking on a word, and when no effort is needed.

I am pleased that creating the app from scratch to the finished version took only about 35-40 minutes, and then I used it for some time during the flight, without the internet. Since all translations of all words/phrases were already made in advance.

I just deployed it on Render. It’s also nice that demonstrating the code in action was free and took another 10 minutes.

https://readandlearn.onrender.com/