While I appreciate the effort the author has clearly put in here, Im not sure the visualization provides much in the way of practical intuition. These seem to essentially be mechanistic explanations of the rote bits of matrix math.
There are already rich geometric interpretations which provide useful intuition for that generalizes, rather than just demonstrating mechanical details.
Used it for like a year before switching to opensuse tumbleweed. For rolling release Ive found it to be a lot more stable, though some of more obscure packages I use are harder to obtain.
tbh if you find this "condescending", you have to reevaluate your stance towards your general attitude. If you really wish to do so, you can find offensive and condescending wordings literally everywhere, it is always in the eye of the beholder to be offended or not.
Classic problem of trying to draw definitive lines between these things. I would guess this is more motivated by the social aspects of academics today than any real issue.
In any event reading the work of philosophers has certainly made me a better thinker and mathematician. Without a broad base of thought one can easily get sucked into many intellectual traps and tar pits. In my case, I found Penelope Maddy and Wittgenstein to be helpful when I was figuring out what math "is".
That theorem is not valid constructively, so no way to make a computer program out of it. I.e., there is no program that given types A, B and injections (f : A -> B) and (g : B -> A) produces a bijection A -> B.
But from reading the first bit of the paper I think they are talking about something stronger, not any bijection but one that only requires inspecting and manipulating a bounded number of layers of the trees.
Ah, I think I see. Open sets are just how we encode information topologically. Theyre sort of the "atoms" of topology; the types of opens sets you have say how the topology behaves.
In particular open sets are neighborhoods of all of the points they contain. This means they contain all the topological information about all of their points.
Closed sets are not neighborhoods of their points in general. Eg [0,1] contains no neighborhood of 0 or 1. Then we would require knowledge of the space around those points to know how a function behaves just on [0,1].
In standard calculus this amounts to "taking the left and right limits".
> Open sets are just how we encode information topologically. Theyre sort of the "atoms" of topology; the types of opens sets you have say how the topology behaves.
This is an interesting perspective, thank you. It reminds me of NAND gates
