The Mathematical Hacker (2012)

So what do I mean by "camps"? Well, the most obvious dividing line is between people who see "math" as something that everybody does, and that includes everything from elementary school addition up to, well whatever, and has a primary goal of computing something. Then you get the camp who see "math" as the exclusive domain of mathematicians, look down on any maths class at a level lower than abstract algebra or real analysis, and consider the primary goal of math to be more math (eg, proving things that we didn't already consider proven). And the latter often tend to look down their noses at anybody who dares talk about "learning math" but without a goal of becoming a mathematician.

sigh

My position is that "math" includes both of those notions, and that in most conversational contexts you have to explicitly define which one you are talking about up front in order to have even a chance at a productive conversation.

On the other hand, the implication that Lisp textbooks -- or other language texts that use fibonacci or like examples for recursion examples -- shouldn't be using simple recursive definitions as an illustration if there exists a closed form may well raise some eyebrows. What do you want, Collatz Sequences? Additionally, there's an air of attacking the sophistication of devs who choose Lisp specifically, which is the kind of thing that people frequently do find disagreeable (even if there's a reasonable case for it).

Fibonacci and other linear recurrence relations are used because they're simple and often familiar intros that can help conceptualize the way recursion can reduce problems to smaller versions of themselves (which is very much mathematical thinking). They're not there because they represent limits of how Lisp devs generally do or should think.

Activity-wise, mathematics is about intuiting and proving theorems to progress the frontier of knowledge about abstract mathematical objects (groups, probability distributions, triangles, matrices, knots, ...) and their interconnections. This has little to do with programming, but proofs by mathematical induction (German "vollständige Induktion" = "complete(d) induction"), which is actually - like all mathematics - a deductive method, can be better understood after understanding recursion and loops. It is also very hard.

Ontologically, for instance, "x = y" in mathematics means "the unknown value x equals the value y", and that is true always, at any time (until the context changes from one theorem/proof to another, when x and y may be 'recycled' without saying it explicitly). In contrast, "x = y" in (e.g. Python) programming means, "look up the value stored in the memory location labeled y and copy that value to the memory location labeled x." This is rarely said in textbooks and tutorials, so I tend to make a point of stressing this when teaching programming, in particular to non-computer scientists.

What the OP refers to is perhaps more applied mathematics than mathematics, though: using mathematics as a modeling tool.

I like the OP, but what is missing from it is the recognition that a lot of what he talks about is already happening in machine learning: any machine learning is mathematically a numeric optimization process, so all these tools like Tensorflow or PyTorch under the hood use what the OP calls "FORTRAN" way, often implemented in C++ and sometimes actual battle-tested FORTRAN libraries (e.g. via NumPy https://numpy.org/doc/stable/user/building.html#accelerated-...).

The only areas in CS that come close to that in terms of difficulty are discrete math/algorithms related ones and the CS chairs that are involved in such research also tend more towards the applied side of those topics. Really theoretical discrete math stuff resides in math departments most of the time.

Still, having also taken some discrete math classes, in terms of difficulty those somewhat paled in comparison to differential geometry/topology/abstract algebra kind of stuff. I can't even imagine how difficult it must be to be doing research in those areas.