TITLE: Is computer science a branch of mathematics?
QUESTION [211 upvotes]: I have been wondering, is computer science a branch of mathematics? No one has ever adequately described it to me. It all seems very math-like to me. My second question is, are there any books about computer science/programming that are very rigorous and take an axiomatic approach? Basically, putting computer science and programming on a rigorous foundation.

REPLY [3 votes]: the two fields of CS and mathematics are becoming increasingly intertwined especially on the theoretical side and once "more sharp" boundaries are getting blurred by various active/ongoing research programs and developments, and one would expect this trend to continue and heighten gradually over the long term, this century. possibly a whole essay, paper or book could be written on this subject. some examples/highlights where there are strong connections or overlap:

automated theorem proving and its major successes/conquests eg the 4 color theorem. see eg a recent paper A fully automatic problem solver with human-style output of Fields medalist Gowers who collaborated with a computer scientist. he has a remarkable/unusual paper Rough structure and classification musing/predicting a sort of mathematical AI arising in the next century.
Erdos is a figure who didnt really work in (T)CS directly per se but his "intensely combinatorical" research is given new life in TCS eg the probabilistic method, sunflower theorem, etc.
P vs NP is a problem that was originally considered to be a CS problem but seems to have deep ties to combinatorics and extremal set theory. it is one of the Clay mathematics problems. in other words it was chosen by mainly an advisory board of elite mathematicians.
the properties of undecidability found all over mathematics. the Matiyasevitch-Davis-Robbins-Putnam proof is striking and has computational proof mechanics. the Turing proof of undecidability and Godels proof share a common core mathematical structure

The halting theorem, Cantor's theorem (the non-isomorphism of a set and its powerset), and Goedel's incompleteness theorem are all instances of the Lawvere fixed point theorem

experimental mathematics is a significant/emerging research program combining computer experiments and mathematical analysis.
extremal combinatorics/set theory and extremal graph theory have applications in proving TCS algorithmic complexity lower/upper bounds.
theorems that connect deep properties of one field to another are sometimes called "bridge theorems" and there are many between TCS/mathematics. some highlights mined from tcs.se:
Examples of “Unrelated” Mathematics Playing a Fundamental Role in TCS? ("bridge theorems"), Mathematical analysis and computational complexity?, Applications of TCS to classical mathematics?, geometric complexity theory, Fourier analysis of boolean functions, algebraic structures in computer science, representation theory of the symmetric group, math background for complexity theory
discrete mathematics! combinatorics! boolean algebra! mathematical logic!
Mandelbrot is an example of another notable/pioneering figure who spans the divide between math/TCS eg with the study of fractals. worked at IBM research for 3½ decades. Chaitin, Shannon are other examples.
RJ Lipton is an expert on the connections between math and TCS and has also given Godel credit as founding the basic concept of computational complexity theory in a 1956 letter to von Neuman ("Godels lost letter" when he mused on the limiting steps required by a machine to find proofs).
more evidence of this trend can be found in science funding organizations/foundations such as the Simons Institute [founded by Phd mathematician Simons] and Clay Math foundations.