I’m good at maths. I won several math and physics competitions on primary and high school. Although finally I’m become a programmer, I’m still interested in physics and maths, and I try to understand things mainly in physics.
The problem I see is that many things in papers and lecture notes are described in very terse, plain rigorous way. Even the maths lecture at the college was presented in this way. I often had the feeling that the lecturer is ‘showing off’, and I often thought they are playing a game of telling everything to a students in a way they have no chance to understand it.
In constrant with this, when something is described by examples, problems, analogies, intuition and induction or by telling the patterns, even understanding the most sophisticated concept becomes easy.
After the intuitive foundations are built then we can generalize to N, other fields, etc, and discover the corner cases and gotchas, and backtrack towards the axioms (or towards the desired level).
But in real life I see that things are explained the other way around: axioms, definition, definition, definition, definition, definition, theorem, proof, theorem, proof, theorem, proof, theorem, proof, definition, definition, definition, definition, definition, theorem, proof, theorem, proof, theorem, proof, theorem, proof. It’s very easy to get lost at the very beginning. Because all look like meaningless symbol folding. It’s very easy to get lost in this. All that’s missing is: what is this all about? What can I do with this?
For me this’s something like deciphering what does a program do by starting from the silicon and learning how to build an x86 CPU, then reading the machine code. While you can do the same by reading the nicely commented source code, without even knowing anything about the gory details of the hardware.
Personally I cannot think of differentiation without visualizing the slope or thinking of a rate of change; also I cannot think of integration without visualizing the area under the graph, or thinking of summation. Visualizing matrix determinant as an area of a parallelogram (22) or a parallelepiped (33) was a great eye opener, now I know why the determinant of 0 is so special.
I’ve also found saying ‘x and y have the same sign’ in plain Enlish much easier to understand than simply writing down ‘xy > 0’.
I like maths, I like puzzles, I like solving problems, I like understanding world. I can prove many theorems. I can use mathematical tools when I need to. But excess rigor is not for me.
Probably I’m thinking this way because I’m an engineer, and I use maths, not inventing it. But heck I discovered, reinvented many things way before I learned about it.