How much math you need for programming (2014)

It depends on the scale and latency requirements: for a startup creativity and pace of execution/person is the most important (just like Facebook in the early times).

But when you're working on a huge project with many people and many users, lots of data (while slowing down the number of new features), math gets very important for modelling the behaviour of that huge system and getting every extra $$$ out of the network effect.

I myself love math, and a very bad fit for small companies, while people who just like to get dirty in throwing together features thrive there.

As an example Facebook was dropping messages for a long time, and when Google started Google+, Facebook got serious in fixing bugs with consistency of its features, and started to focus on latency (using Google hires of course).

There are two different problems actually : what you write, and the quantitative aspects of your code.

What you write may or may not have a strong maths component ( game engine vs crud app say, or PDE solver vs typical 'plumbing' app).

However, the moment you start having performance issues (and this is everywhere), maths comes in very handy : big o, back of the envelope calculations, orders of magnitude, etc. You may call it engineering (and to a large extent it is), but I tend to lump quantitative reasoning on the math side and not on the compsci/engineering side. It's not necessarily extremely sophisticated, but having a 'quantitative view' of your code can help a lot.

I remember many many years ago developing a sudden interest in algorithmic complexity (and what quadratic meant..) after having used a bubble sort to sort polygons in my 3d engine. Needless to say, log became a fairly interesting function after reading a bit more on sorting!

Having a good math basis keeps you from jumping down rabbit holes.

The question seems a bit backwards to me (who come to programming from maths). I am more familiar with the opposite question "how much, if any, programming do you need for math?" Every year I meet lots of math students who love math but hate the very idea of programming, as if it was a dirty and unnoble thing to do. I have to convince them that sometimes it may be useful for them. The best course of action is to start by offering the game: "you don't need any programming for math, except in very particular cases". This gives them a clear starting point where they feel safe. Then we work from there, showing them inductive experiments that would be impossible without computing. For example, it is well known that Gauss would check, by hand computation, many theorems about prime numbers before attempting to prove them. This entailed many days of tedious hand computation that he was very proficient at. Today, most of these experiments can be programmed in a handful lines of code and the computer verifies them in a second. Then some students are eager to actually see these lines of code, but more importantly, to see their result.

So basically in programming, mathematics is abstracted...

This can be generalised even further!

In programming X is often abstracted, so you don't need to know much of X to know how to program...