- How type theory can prove mathematical theorems. I can limp through the first exercises of Software Foundations, but eventually realize I'm just "making it work" instead of knowing how it works. Only made it halfway through My Little Typer and even implemented STLC and used it to prove simple propositions with implications. But I still couldn't tell you how dependent types magically make it possible to prove real theorems.
- How opamps work and how they're used to make a differentiator.
- How quantum computing works. I want understand it to understand why factoring and discrete log fall to QC but not the post-quantum algorithms. I never make it past the intro chapters.
- What exactly makes cryptographic hash functions have desirable properties like collision resistance? Bits go in, bits go out, why can't full observation of the process not yield a method to construct inputs for a target output? Similar questions for block ciphers. It is much easier to "see" the impossibility in public key schemes because you can play and experiment with the math. In symmetric algorithms it looks like just a bunch of arbitrary bit operations.
EDIT:
- Relativity. I accept things like time dialation and nothing can go faster than the speed of light. But I have no idea why you would experience less time passing than me after a high speed rocket trip when, from your perspective, you saw me on a high speed planet trip. Or what prevents a rocket in space with 3m/s^2 acceleration from reaching the speed of light after 100,000,000 seconds.
- How opamps work and how they're used to make a differentiator.
- How quantum computing works. I want understand it to understand why factoring and discrete log fall to QC but not the post-quantum algorithms. I never make it past the intro chapters.
- What exactly makes cryptographic hash functions have desirable properties like collision resistance? Bits go in, bits go out, why can't full observation of the process not yield a method to construct inputs for a target output? Similar questions for block ciphers. It is much easier to "see" the impossibility in public key schemes because you can play and experiment with the math. In symmetric algorithms it looks like just a bunch of arbitrary bit operations.
EDIT:
- Relativity. I accept things like time dialation and nothing can go faster than the speed of light. But I have no idea why you would experience less time passing than me after a high speed rocket trip when, from your perspective, you saw me on a high speed planet trip. Or what prevents a rocket in space with 3m/s^2 acceleration from reaching the speed of light after 100,000,000 seconds.