If you're doing the research and building the tools. Using the tools however does not require much more than very basic knowledge. I know the mathematics used for graphics. But I've never needed to write a library for it, so I've never needed to use it. I know Fourier transforms. But I only care about the output, so I've never had the need to do the calculations myself.
You could never be competent graphics developer if you didn't understand things like lighting models or 3D geometry. Similarly, even understanding what time and frequency domains are, orthogonal bases, implications of using sin/cos waves as bases, etc., requires fairly advanced maths. Many people are surprised to find out that it exists for generic inner product spaces and not just function spaces.
Applied mathematics is, well, applied, so it is not pure and certainly not abstract.
Applied maths is very much built on pure maths - I do agree, not the pure maths where you move into things like analysis/topology, etc., but certainly the first few years of pure maths (calculus, basic algebra, linear algebra, etc.) in order to move on to advanced calculus, DEs/PDEs, modelling, convex optimization, etc.
Software engineering is hugely complex. Bad software happens everywhere and a lot of the time it is driven by financial imperatives rather than any deficiency in the ability of the people making the code.
Agreed, but beyond a certain level of technical complexity, it is exactly driven by the people writing the code.
Avoiding as many conceptual errors as possible will require strong language skills. Mathematics is entirely irrelevant. Mathematics may use logic, but it is certainly required neither to learn nor to use logic. Mathematics only comes into it if the product being developed involves mathematics.
Yes, and no. I do agree that the idea that understanding the difference between and abelian and a non-abelian groups, isn't going to somehow allow someone to build a cleaner class hierarchy. Working through things like performance analysis (tails, fat tailed distributions, understanding how the performance distribution of a chain of sequential events, where you know the individual event distributions, would generate a new distribution, and how the overall distribution would differ from expectation if they're not independent on each other, etc.), would be an example of how someone with more advanced maths would think about understanding something like just the performance of a system. If someone is simply timing things end to end, and just reporting the mean or median, as people tend to do, you're just never going to resolve the more complex problems as efficiently.
Point being that mathematics frequently doesn't show up, not because it's not useful, but because the people doing the work don't have the background for it. In some case it slows down development, and in some cases it can outright prevent the building of tractable solution without changing the problem.
... holy carp, the pathing challenge in 3D space...