I learned programming and math mostly by self-driven exploration. But for computer science there was a book that hit at the right place and time: Feynman Lectures on Computation.

It's bounced around between publishers over the years and is generally a pain to get. I have the 1996 edition[1] which is out of print. Then there's a 2002 edition[2] and a 2000 edition[3].

[1] https://www.amazon.com/Feynman-Lectures-Computation-Richard-...

[2] https://www.amazon.com/Feynman-Computation-Frontiers-Physics...

[3] https://www.amazon.com/Feynman-Lectures-Computation-Frontier...

Why not? He was into computing. His lectures on computation (transcribed from his course in California Institute of Technology) are published:

https://www.amazon.com/Feynman-Lectures-Computation-Frontier...

Homotopy Type Theory by Vladimir Voevodsky is another possibility. This is an attempt to link Topology and Computer Science

https://homotopytypetheory.org/book/

Back in the day there was Feynman's Lectures on Computation. Hint: pdf can be found by searching

https://www.amazon.com/Feynman-Lectures-Computation-Richard-...

See also nLab

https://ncatlab.org/nlab/show/higher+category+theory

one should never forget Jacob Lurie's "Higher Topos Theory" which is 1000 pages just like that

http://www.math.harvard.edu/~lurie/papers/croppedtopoi.pdf

Actually I recommend against readin it as it only covers 2 of the 4 topics you discuss (Topology and Logic). However it certainly has applications to the other two.

I haven't read it, but I saw it on a friend's bookshelf:

https://www.amazon.com/Feynman-Lectures-Computation-Richard-...

If you enjoyed this, may I also recommend Feynman's Lectures on Computation. [1] It consists of transactions of a lecture series that begins with computer architecture and theory of computation and considers interesting subjects like the physical limits to computational capability--all through the unique lens of Feynmans' mind.

Feynman worked on this subject working on the Connection Machine supecomputer. [2][3]

[1] http://www.amazon.com/Feynman-Lectures-Computation-Richard-P...

[2] https://en.wikipedia.org/wiki/Connection_Machine

[3] http://longnow.org/essays/richard-feynman-connection-machine...

Feynman discusses this problem in his "Theory of Computation" book. If I remember correctly, its actually part of a joke he plays on the reader too. In one of the earlier chapters he brings up the problem and then assigns it to the reader as homework they should complete before moving on. I spent maybe a week or two on that problem, discussed it with coworkers... and we came up with nothing.

So I gave up and continued reading. Then somewhere in the 4th or 5th Chapter he says something like: Oh I hope you had fun with the Firing Squad Problem, I still work on it from time to time and hope to come up with a solution myself one day.

Facepalm.

Edit: Here is a link to the book, its enjoyable for experts and laymen alike. http://www.amazon.com/Feynman-Lectures-On-Computation-Richar...