Mathematical Notation Is Awful

Funny, I said the same about Andrew Ng's ML course.

Ultimately, if what you're teaching is going to end up in software, why use math at all? Use code or pseudo code. I don't think it's bad to just give the working algorithm without having to prove the math.

Really how many students will end up being computer scientists anyway, and research and write about new methods of doing AI, and do the actual math? So few. I guess that's a simple criticism of academics.

It's just easier to work with code than mathematical notation most of the time, in my view. You can't replace math, of course, but when things are simple enough, it could be avoided. It's a matter of making math accessible to the most people.

Code is amazing because a computer can check to see if it works. A computer doesn't understand math.

I understood math better once I used it for coding.

Linear algebra and matrices, for example, by creating a simple 3D simulation and later studying a circuit simulation.

Same with physics and other sciences, which are typically taught more like math calculation and/or memorization classes.

It would be nice to see more math documented and taught as code.

If you think probability theory is bad...

The old joke that "differential geometry is the study of properties that are invariant under change of notation" is funny primarily because it is alarmingly close to the truth. Every geometer has his or her favorite system of notation, and while the systems are all in some sense formally isomorphic, the transformations required to get from one to another are often not at all obvious to the student. —John M. Lee, "Introduction to Smooth Manifolds"

> Now that's a fine thing in a computer language -- and there are philosophers who say mathematical truth is no more than computation. But mathematical notation is for humans to talk about things that have meanings that humans can understand.

Then how come an unfamiliar piece of source code is easier to follow than a mathematics paper using unfamiliar notation?

You can look up the definition of each function and learn precisely what it does, without ambiguities. You can open up a debugger and trace every step of the algorithm. Source code tends to be formatted to emphasise its division into units that can be analysed separately. Even if the code uses a framework you don't recognise, you can probably understand some of the underlying logic just fine, thanks to a consistent syntax and (often) descriptive function names.

In a mathematical paper, often there's some notation that isn't defined and there isn't even any pointer where to look for definitions. Important objects are given one-letter names. Theorems are often only numbered; good luck remembering what lemma 2.17.3b was about. Proofs are written in prose (instead of something like Leslie Lamport proposed[0]). You have to fill in conceptual gaps and remind yourself of unstated assumptions. Notation and terminology is often ambiguous — what does "exponential time" mean? Is it DTIME(2^(c*n)) or DTIME(2^polynomial(n))? Does "increasing" mean "strictly increasing" or "non-decreasing"? Is zero a natural number? And so on.

A mathematical formalism that could be interpreted in a purely mechanical way would be a huge improvement.

[0] http://research.microsoft.com/en-us/um/people/lamport/pubs/p...

I started doing a lot better in calculus when I started using longer notation (e.g. f = x -> x³ instead of f(x) = x³) and making sure that things "type checked". For instance, this tendency to use f(x) to refer to a function, rather than just f, was very confusing to me, because f(x) is an element of the co-domain while f is a function (typically from real to real in my undergrad classes). I had to figure this out by myself because the textbook I was using and the prof all went with the notation that wouldn't type check. When I finally realized that dy/dx should instead be (d/dx)(f), things started being a lot clearer to me: derivation takes a function and returns a function and f is a function so everything checks out.