If you could go back in time and tell yourself to read a specific book at the beginning of your career as a mathematician, which book would it be?
A Mathematician's Apology by G H Hardy. I did in fact read this in high school, and it raised my view of mathematics from a thing of utility to a thing of beauty and wonder. It inspired me to go on to study mathematics at Cambridge myself.
It's a pity that the "introduction" by C P Snow is longer than the original and contains a rather depressing view of Hardy's later life. I would recommend readers to skip the introduction altogether and concentrate on Hardy's own words.
When I was in my fourth year of high school I got a copy of What is Mathematics? by Courant and Robbins. That book showed to me that Mathematics is far more than a "boring tool" to do Physics and opened up new worlds. I would recommend it to any bright high school kid with an interest in math and sciences.
I am not a mathematician but Flatland: A Romance of Many Dimensions blew my mind. I read it when I was a college student in a class on Special Relativity and wish I had read it way earlier.
Not a book, but an essay: "Politics and the English Language" by George Orwell.
(I note that the original question doesn't say that the book has to help with mathematics. It also seems to conflate 'influential' with 'should be read'; as others have pointed out, there is no pressing reason for someone who wants to be a mathematician to read the influential books rather than the useful or the interesting ones.)
Anybody who wants to be a serious mathematician better read W. Rudin's "Principles of mathematical Analysis". It gives a rigorous foundation to the basic notions analysis and introduces the reader to the world of rigor, after the sloppy days of calculus courses. One must learn the notion of rigor properly if one wants to be a mathematician. More than anything else, it is an exercise in the rectitude of thought. No other book is so universally used that would teach this notion, than Rudin.
This is an extremely broad question, especially given the wide variety of mathy people here, but I'll bite.
HSM Coxeter's Introduction to Geometry is a book that was very important to the development of my interest in mathematics and inclination towards its geometric aspects.
Geometric Algebra by Emil Artin. Though not for the beginner, it can do wonders for an intermediate undergraduate in terms of expanding their horizons and helping them appreciate the beauty and interconnectedness of mathematics. It did for me and I think convinced me that I'm a geometer at heart.
I'll recommend two, which are similar in that they take fairly elementary mathematical problems and give very thorough and careful "talking out loud" illustrations of how a proper mathematician would go about thinking them through - what's really going on, what's a good example, what's a definitive counterexample, how to generalise, how to realise you've reached a dead end, and so on. "Proofs and refutations" by Imre Lakatos (just one, geometrical, problem, in glorious detail). "Mathematics and plausible reasoning Vol 1" by G. Polya (a little more advanced, and much more satisfying, than "How to solve it").
I've been rereading Littlewood's Miscellany recently. It's a very readable collection of the writings of J. E. Littlewood, carefully edited by Béla Bollobás. Any budding mathematician will draw much inspiration from it. I like A Mathematician's Apology, but if I was forced into choosing only one book, it would be Littlewood's Miscellany.
Ideally in the original languages of Ancient Greek and Latin respectively! No, just kidding. But they are true classics that any accomplished mathematician should read at some point during their career. Not because they'll teach you something you don't already know, but they provide a unique insight into the mind of these giants.
Title: The Mathematical Experience
Authors: Davis and Hersh
Short Description: A really accessible and funny introduction to the philosophy of mathematics. I think the description of the "ideal mathematician" is particularly hilarious.
Recommending one single book at the beginning of a young mathematician's career is a little like asking someone what particular vitamin they should make sure is in a child's diet. It's absurdly restrictive.
That being said-there are certainly 3 books I would recommend without reservation to any young student just getting interested in serious mathematics: Micheal Spivak's Calculus, Klaus Janich's remarkable Topology and Paul Halmos' I Want To Be A Mathematician.
The last one in particular inspired me to leave pre-med to begin the path to be a mathematician. The other 2 are remarkable works that will begin to open the edifice of modern mathematics to the novice.
I can recommend a hundred others,but those are the absolute must-reads for the beginner to me.
There are so many, and I've already seen three that I would mention. Two more of interest to lay readers:
The Man Who Knew Infinity by Robert Kanigel. Excellently written, ultimately a tragedy, but a real source of inspiration.
Goedel's Proof by Nagel & Newman. Really, a beautiful and short exposition of the nature of proof, non-euclidean geometry, and the thinking that led Goedel to his magnificent proof.
Every undergrad should read in areas outside mathematics especially in areas that can be influenced by mathematics. Theoretical physics and computer science are prominent examples. Biology and chemistry are not far. The DNA and polymers can be understood using knot theory and feynman's path integrals . Feynman's path integrals facilitated the quantization of nonabelian gauge field theories ( Quantum chromodynamics ) and is used to study complex systems and stochastic processes.
So here are two books that I found very interesting :
The road to reality by Roger Penrose Kleinert's path integrals in physics , financial markets & Stochastic processes
There are lots of books that talk about the applications of math in physical science just search Amazon.
This question does not have a unique answer. I will concur with Jonathan in that Jayne's "Probability Theory: the logic of science" is a great book.
This book changed my life as a scientist, converting me into a fervent Bayesian. For me it was a truly irreversible experience when I, for the first time, understood and comprehended that probability (as applied to understanding the real physical world) essentially stems from our lack of knowledge, our incomplete information, of reality. Fantastic book, although I admit that Jayne's style might not suit everyone's taste.
I do recommend: Hugo Steinhaus - one hundred problems in elementary mathematics Hamilton - Perelman's proof of the Poincare conjecture and the geometrization conjecture
Also there are many great books written by polish authors (especially Kuratowski, Banach, Mostowski, Steinhaus, Leja) but I am not sure is it available in english language.