Intereting Posts

Operator vs function
Need help with proof for Dedekind cuts on $\mathbb{Q}^+$
Induction, 0'1 and 1's sequence fun question
Classification of general fibre bundles
Can one construct a non-measurable set without Axiom of choice?
Prove that the Gaussian Integer's ring is a Euclidean domain
Initial Segments of Modular Arithmetic
Motivation behind topology
Strehl identity for the sum of cubes of binomial coefficients
What is the inverse of the Carmichael-function?
What is a maximal abelian extension of a number field and what does its Galois group look like?
How to prove that $B^\vee$ is a base for coroots?
If $f$ is continuous, $f(x) \to -\infty$ as $x \to -\infty$, and $f(x) \to \infty$ as $x \to \infty$, then $f(\mathbb{R}) = \mathbb{R}$.
An example about finitely cogenerated modules
Why does it suffice to check the geodesic equation to leading order?

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?

- Is there a mathematical theory of physical knots?
- Approximating continuous functions with polynomials
- Why is the 3D case so rich?
- Generalized Logarithmic Integral - reference request
- Alternative proof that the parity of permutation is well defined?
- Green's functions of Stokes flow
- For which categories we can solve $\text{Aut}(X) \cong G$ for every group $G$?
- Proof of Nesbitt's Inequality: $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}\ge \frac{3}{2}$?
- A topology on the set of lines?
- Looking for examples of Discrete / Continuous complementary approaches

Polya’s “How To Solve It”

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.

William Dunham’s “Journey through Genius.”

Well, rather that is the book I read that made me want to be a mathematician.

Gödel, Escher, Bach: An Eternal Golden Braid by Douglas Hofstadter:

alt text http://upload.wikimedia.org/wikipedia/en/thumb/f/f1/GEBcover.jpg/200px-GEBcover.jpg

Proofs from the Book

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.

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.

Not a book, but an essay: “Politics and the English Language” by George Orwell.

What? What?

(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.)

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.

Men of Mathematics by E T Bell

T.W.Körner, *The Pleasures of Counting*. It shows how mathematics is alive.

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’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*.

Following Noah’s lead I will mention;

“The Man Who Loved Only Numbers”

and

“How to Read and Do Proofs”

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”).

Euclid’s Elements

Newton’s Principia Mathematica

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.

Probability Theory: the Logic of Science.

Or anything by Edwin T Jaynes.

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.

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.

Visual Complex Analysis by Tristam Needham.

I always like to see mathematical problems in pictures whenever I can, and this one pushes the ‘keep it visual’ approach to the limits.

Needham won an award for some of the work in there.

Nicolas Bourbaki’s Éléments de mathématique (specifically Topologie Générale and Algèbre).

the man who loved only numbers, innumeracy, a beautiful mind. these three books have shaped my thinking and love of mathematics…books on math..not exactly a lot of math in them however.

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.

dummit and foote’s abstract algebra

it taught me, more than anything, how to be precise.

disclaimer: i’m computer science not math

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.

god created the integers by Stephen Hawking is the best book…… for mathematics.

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.

I would read a book about Perelman’s proof of the Poincaré conjecture (or even the papers themselves). Oh, you mean the book had to be written when I was starting?

- Prove $_2F_1\left(\frac13,\frac13;\frac56;-27\right)\stackrel{\color{#808080}?}=\frac47$
- How do we show that the function which is its own derivative is exponential?
- Finite sum of reciprocal odd integers
- Prove that the union of three subspaces of $V$ is a subspace iff one of the subspaces contains the other two.
- Uniform convergence and integration
- Vector spaces with fractional dimension
- Why is the set of integers in $\mathbb{R}$ closed?
- How to obtain a closed form for summation over polynomial ($\sum_{x=1}^n x^m$)?
- Using permutation matrix to get LU-Factorization with $A=UL$
- Try not to get the last piece, but with a twist!
- Proof that $t^8+2t^6+4t^4+t^2+1$ is reducible in $\mathbb{F}_p$
- Limit problem: $\sqrt{x+1} – \sqrt{x}$ as $x$ approaches infinity
- What is the motivation of Measure Theory when there is probability theory?
- If we accept a false statement, can we prove anything?
- Calculating the Zeroes of the Riemann-Zeta function