
How to Become a Pure Mathematician (or Statistician):
a List of Undergraduate and
Basic Graduate Textbooks and Lecture Notes
怎樣研習純數學﹝或統計學﹞：
本科與基礎研究課程參考書目
Print friendly version: here [.pdf]
Last Revised: 8/1/2009
Stage 1: 
Elementary Stuff, Intro. Discrete
Mathematics, Intro. Algebra, Intro.
Calculus 
Stage 2: 
Linear Algebra,
Intro. Higher Algebra, Calculus,
Complex Variables,
Differential Equation, Probability & Statistics 
Stage 3: 
Intro.
Analysis, Abstract
Algebra, Intro. Number Theory, Intro.
Topology,
Differential
Geometry,
Optional: Math. Modelling, Stat. Inference, Probability
& Stochastic Processes, Stat. Computing 
Stage 4: 
Foundations and Discrete Mathematics,
Analysis, Algebra,
Number Theory, Geometry,
Topology,
Optional: Further
Calculus, Mathematical Physics, Probability,
Statistics, Biostatistics 


It's like if you want to
be a good pianist,
you have to do a lot of scales and a lot of practice,
and a lot of that is kind of boring, it's work.
But you need to do that before you can really be very expressive and
really play beautiful music.
You have to go through that phase of practice and drill.
 Terry Tao
About this article:
 What for?
OK, there is a way to become
a good theoretical physicist. Here is a guide to study pure
mathematics, or even more. This list is written for those who want to learn mathematics
but have no idea how to start. Yup, a list for beginners. I don't claim that this list makes you a good pure
mathematician, since I belong to the complement of good pure
mathematician. I make no attempt to define what pure mathematics is, but
hopefully it will be clear as you proceed. I also highlighted several
books that you would really like to keep in your own library. You probably like
to read those books again and again in your life. Free material
excluded. Note the highlighted
list does NOT indicate those books are good for
beginners. I shall try to keep
this list up to date whenever I exist.
 Assumed knowledge.
I assumed you have high school mathematics background (i.e. basic trigonometry, Euclidean geometry,
etc). The
aim of this page is to introduce what different branches of mathematics
are; and recommended a few notes or texts.
Scientists in other fields and engineers may skip first or second
stages and begin at later stages, according to their prior knowledge.
 Time.
It takes approximately one year for each stage (except for stage 4, I
list more material in each field for more advanced studies), for a full time
student. Part time students may double the time. But its better for anyone to understand most parts of stage
n before proceeding to n+1, for some integer n in {1, 2, 3, 4}. If you decided to attend a class, don't expect the professor can teach, it always happen, especially in higher level courses.
What's the order of courses to study within a stage doesn't really
matter, usually. One doesn't need to read every listed book within a
subject to master the subject. I listed more than enough so that you
can scout around to find one that you feel comfortable with. Some
people like to consult a few books, beware of the symbols from
different books in such cases.
Moreover, it often happens that you couldn't solve a problem
within an hour. It's not surprise to spend a week or more to tackle
one problem. Things may come to your head suddenly. Shouting eureka is
the high point of a mathematician.
 "Axiom of choice".
My selection will not be bounded by any publication press, author's nationality
or religion. It relies on two factors:
well written or cheap. These two factors are not mutually exclusive. I
treat "free" as an element in "cheap". Note the price
factor may by irrelevant, sometimes I get a HK $2xx book and Amazon
says its US $1xx (~HK $7xx)...... with the only difference is,
perhaps, I got the international edition. Moreover, some Chinese press
in mainland China published photocopied of English text with a
relatively cheap price.
Bear
in mind that, just because one is a good mathematician doesn't imply he's a good author or educator. Perhaps Terry Tao is an exceptional
case. To study science, reading the
classics (the Elements, Dialogo sopra i due massimi sistemi del mondo,
the Principia, Disquisitiones Arithmeticae, Principia Mathematica,
etc) is optional. While for literature or philosophy, I wonder if any
well educated student has never study Shakespeare or Plato.
To view [.pdf] get
Adobe Reader, to view [.ps] download this
and this,
or visit this page,
to view [.djvu] get this.
Get WinRAR for [.rar] files.
 Comments.
Links to Amazon for most of the listed book are included, so that you
have an easy access to other users' comments. Note that the comments
are sometimes quite extreme: for the same book, one rated it with 5 star (with dozens of
people supporting) and at the same time another rated 1 star (with dozens of
people supporting again), especially for introductory discrete
mathematics, probability and statistics
books. It seemed to me that lots of people study these subjects
because they need to, they want to apply mathematics. Large proportion
of these readers are lack of mathematical maturity. If they can't pass
the exam, you know... In contrast, most pure mathematics students study
because they like the subject and enjoy it. So, ask yourself, why do
you study?
 Other resources.
Although I'm not into reading books online, I should remind you
that MIT's
open recourse, the Archimedeans
and Wikibooks
provide another great sources of materials. These are excluded in the
following list. The list below
aimed to recommend books or (usually) printable notes. Google
books allow you to preview sections from a book. Schaum's
Outlines series are cheap, but I seldom include them, you may
search the relevant if you like.
 Me.
A product of School
of Mathematics and Statistics, UNSW,
Sydney, Australia. I've taken all undergraduate core pure mathematics
and statistics courses there, with all pure mathematics courses in higher
level (perhaps equivalent to Honors Courses in the US which focus on
theory), whenever they exist.
Also, I was Terry Tao's teacher's student, Michael Artin's student's student,
Gottfried Leibniz's student Jacob Bernoulli's student Johann Bernoulli's
student Leonhard Euler's student Joseph Lagrange's student Simeon Poisson's
student's student's student's student's student's student's student's
student's student, Max Planck's student's student's student's student and Thomas Kuhn's
student's student. Just feel like I'm an idiot.
 Disclaimer.
I'm not responsible for any external link.
 Comments/suggestion for a book, etc. Either make
comment in the blog http://hbpms.blogspot.com/
or email me, contact can be found in the main page.
For many reasons, high school mathematics is taught in an informal way,
we'll fix things up here. The first stage contains the basic material that is required for most scientists
and engineers, discrete mathematics is particularly meant for
computer scientists.
Elementary Stuff:
Here is a list that helps you to refresh and enrich your high school mathematics.
Introductory Discrete Mathematics:
You will study logic and set theory in an introductory discrete mathematics. The
concepts you learnt here will play a key role in your later studies
and improve your thinking skill. You will also study proofs, functions (injective,
surjective, bijective inverse),
introductory graph theory and number theory, Euclid's algorithm, discrete probability
(counting, nCr, nPr) etc in this course.
Only for Proof, Logic and Set Theory:
In general:
Introductory Algebra:
Complex number, polynomials, matrix, system of equation, Gaussian elimination,
vector space, linear transformation, etc. T.M. Aposotol's Calculus Vol.
1 in the calculus sections also introduce topics that is relevant to
this area. A few Americans (exchanged to Sydney) in my third year abstract algebra class told us they
couldn't follow because they haven't learnt much complex number. In order to
have fun with
linear algebra and complex variables in stage 2, it's better to learn complex number now.
 Anton H. and Rorres C. Elementary
Linear Algebra  The application version introduced how linear
algebra can be applied to Markov Chains, Graph Theory, Games of
Strategy, Computer Graphics, Fractals, Chaos, Cryptography, Genetics,
etc.
 Axler S. Linear Algebra Done Right
 See Axler's page for this.
 Chen W.W.L. Linear
Algebra (FREE!)  Read Chapter n's,
for all positive integer n<8.
 Chen W.W.L. Miscellaneous Topics in First Year Mathematics
(FREE!)
 Dawkins P. Algebra
(FREE!)
 Gardner R.B. Linear
Algebra (FREE!)  Leave Chapter 6 and
7 to the next stage.
 Joyce D.E. A Short Course on Complex Numbers
(FREE!)
 Lang S. Undergraduate
Algebra  He's a member of the Bourbaki and wrote many books.
Not friendly for beginners, but you should get use to it.
 Lay D.C. Linear Algebra and Its Applications
 Matthews K. Elementary
Linear Algebra
(FREE!)
 Santos D. Linear Algebra Notes
[.pdf] (FREE!)
 Strang G. Linear Algebra and Its Applications
 Wedderburn J.H.M. Lectures
on Matrices
(FREE!)
 山东大学数学学院：秦静教授线性代数课件
[.rar] (FREE!)
Introductory Calculus:
As you may have seen, calculus is crucial. It is not only important in
advanced mathematics but also can be applied to many other fields. If
you learn first year calculus in a college, your suggested text is
probably J. Stewart's Calculus. This is a standard modern text,
it is colorful and quite detailed.
If you prefer it, that's fine. This book is thick, you would like
to leave a few section, say vector calculus, double, triple, line and
surface integrals, to the next stage. I would suggest a few nonstandard
texts and notes.
The first thing we need to fix up is the definition of limit, learn the
epsilon and delta proof. Other key concepts are the mean value theorem,
fundamental theorem of calculus, logarithm and exponential functions,
inverse trigonometric functions, hyperbolic functions, basic differential
equations, limit of sequences, indeterminate form, infinite series, basic
idea of several variables.

Mathematics program in the University are constructed in the way that
courses are offered in favor of other schools (Physics, Chemistry, Biology, Engineering, Economic,
etc). For example, Physics students have to take linear algebra, several
variable calculus, mathematical analysis and differential equation. As a
consequence, textbooks are written and catalog like the following way.
Several authors focus on the application and often do calculation without
justifying, beware of it.
Linear Algebra:
Vectors, vector spaces, linear transformations, multilinear map, inner
product spaces, norms, orthogonality, GramSchmidt algorithm,
QRfactorisation, least square, Householder algorithm, normal matrices,
Jordan canonical forms, CayleyHamilton theorem, minimal and
characteristic polynomials, direct sum decompositions, generalised
eigenspaces, functions of matrices, exponentials of matrices, etc will be studied in this course. Such material can be applied to linear programming, computer graphics,
fractals, and many areas in natural sciences and social sciences.
Introductory Higher Algebra:
I've taken this course entitled "Finite Mathematics", a
course for computer scientists, software engineers and pure
mathematicians (optional but useful). Assuming you have learnt the very basic of number theory in the
discrete mathematics section, you are ready to get a taste of higher
algebra here. Things like prime numbers, tests for primality, Fundamental Theorem of
Arithmetic, Fermat's little theorem, Gauss' lemma, Euler's theorem, Chinese remainder theorem and their applications
(coding, RSA etc.) are concerned in this section. This is a bridge that
connects first stage basic algebra, discrete mathematics and the third stage abstract algebra.
Calculus (Introductory Real Analysis, Several
Variables Calculus, Vector Calculus, etc):
Learn the very basic concept of real analysis, like open set, close set,
boundary point, closure, limit point, bounded set, connected set, compact
set, BolzanoWeierstrass theorem. To get yourself ready for the stage 3
analysis, make sure you understand those stuff. Next, the Lagrange multiplier, inverse and
implicit function theorems can also be studied. For the several variables
section, gradients, double, triple and surface integrals, cylindrical coordinates,
Green's theorem and divergence theorem are the key things. Also learn
basic Fourier series.
Complex Variables (Introductory Complex
Analysis):
Another beautiful branch of mathematics. While you're in high school, you may wonder what's the point of
introducing imaginary number i, does it really exists? Is it really
useful? In an introductory complex analysis course, you will see the
beauty of this construction. Gauss, Riemann, Weierstrass and Cauchy
are key figures in this area. Key concepts to learn are: analytic
functions, CauchyRiemann equations, contour integral, CauchyGoursat
theorem, residues and poles.
Differential Equation:
Isaac Newton originally studies behaviour of dynamical systems using
differential equations. Mathematical models are often described with
differential equations and they are widely used in many fields,
including physics, chemistry, biology, economics etc. Mathematical
methods used in solving differential equation also play an important
in advanced studies. Key topics to learn are: Wronskians, series
solutions, reduction of order, variation of parameters, Frobenius normal
form, Bessel's equation, Legendre's equation, two points boundary problem,
Fredholm alternative, Green's function, complete orthogonal system, SturmLiouville
problems (inhomogeneous, singular), basic partial differential equation,
elliptic eigenproblems, heat equation and wave equation.
Probability and Statistics:
Well, being a well educated mathematician, you should have basic knowledge of
statistics.
Things to learn: basic probability theory (independent events,
conditional probability, Bayes' Theorem), random variables (r.v.),
expectation, convergence of r.v., maximum likelihood estimator, basic
hypothesis testing, pvalues are the basic. If you want to go a bit
further, check out linear regression, linear model, residual, categorical
predictors, logistic regression, ANOVA (analysis of variance) etc.
Probability:
Statistics:
Second course in Statistics:

At this stage, you begin to learn modern Pure Mathematics. Yup, this is
the beginning. Focus on the proof. Don't expect you can solve a
problem by plugging numbers into formulae. It often happens in advanced text
that the author skips some steps in a proof or calculation, while
elementary text gives detail explanations. But as G.F. Simmons has said,
'[t]he serious student will train himself to look for gaps in proofs, and
should regard them as tacit invitations to do a little thinking on his
own.' Moreover, '[i]t is a basic principle in the study of mathematics,
and one too seldom emphasized, that a proof is not really understood until
the stage is reached at which one can grasp it as a whole and see it as a
single idea. In achieving this end, much more is necessary than merely
following the individual steps in the reasoning. This is only the
beginning. A proof should be chewed, swallowed, and digested, and this
process of assimilation should not be abandoned until it yields a full
comprehension of the overall pattern of though.'
Also get maturity and learn how to write mathematics, by reading good books (?). I don't think Gauss or Galois can get full
mark in mathematics assignments these days, for 1) something obvious to
them may not be obvious to everyone else, and 2) they don't bother to
explain to others. The point is, every steps in a proof should be
logically related and everyone can go easily from the previous step to the
next. This is not the case for nonpure mathematics. If you insist, I can
show you a copy of full mark statistics assignment consists work of mere
calculation and computation. (Of course not done by me!)
I had tried to list more books so
that you can compare and choose one or two that fit you. Despite that
most of following courses are 'pure', they can be applied to other
graduated level science subjects. In the analysis class, Ian
Doust even showed us an article in the Econometrica (Hildenbrad
W. and Metrens J.F. Upper Hemicontinuity of the Equilibriumset
Correspondence for Pure Exchange Economies, Vol. 40, No. 1) talks about
liminf, limsup, measure, weak topology and stuffs like that. If one
insists that the pure stuff is useless, one is just ignorant, I don't
bother to argue with those people anymore.
Introductory Analysis:
At this stage you should have learnt limits, continuity and convergence,
over the reals. These are central concepts of calculus in both one and several
variables. These can be generalised. You will study
the idea of several spaces (metric space, topological space, Banach, Hilbert),
compactness, connectedness, linear operators, elementary Lebesgue
theory and measure theory, etc.
Contents of
analysis courses varies almost everywhere (a.e.). As you may see, the term
analysis appear in the first and second year calculus to forth year
analysis. Kolmogorov and Fomin's
text fits this stage well, they cover most of the above topics. Apostol,
Marsden and Hoffman, and Simmons make good references. Several books
entitled analysis are indeed advanced calculus (stage two calculus) or not aimed for mathematicians.
Stay away from Mathematical analysis for business and finance or the
like.
Only for metric spaces and topological spaces (also consult the list of Topology
below):
Also deals with Lebesgue theory, Banach spaces and Hilbert spaces, etc:
Abstract Algebra:
One of the beautiful branches of mathematics. You will see how we study symmetry. Topics to be covered are
basic group, ring and field theory.
Only discuss Groups:
Rings and Fields:
In general:
 Anderson M. and Feil T. A First Course in Abstract Algebra:
Rings, Groups, and Fields
 Arapura D. Abstract
Algebra Done Correctly [.pdf] (FREE!)
 Artin M. Algebra
 Not aimed at this level, but this is the one you like to keep in
your own library. As M. Artin states in "A Note for the Teacher"
that several sections would make a coherent course, you may simply
follow his list at this stage. My lecturer recommended this partly
because M. Artin was his supervisor, I recommend this but I have no
direct relationship with M. Artin.
 Beachy J.A. and Blair W.D. Abstract Algebra Online Study Guide
(FREE!)
 Birkhoff G. and Mac Lane S. A Survey of Modern Algebra
 Blomqvist C. Algebraic Systems
(FREE!)
 Chan D. Higher
Algebra Lecture Notes [.pdf] (FREE!)
 Diagram omitted, it's best to attend Daniel's lecture.
 Clark A. Elements
of Abstract Algebra  Kind of handbook.
 Connell E.H. Elements of Abstract and Linear Algebra
(FREE!)
 Dummit D.S. and Foote R.M. Abstract
Algebra  If you could afford this. Standard recommended text
in the U.S. Read sections recommended
by the authors in the preface, leave others to the next stage. See Foote's
web for errata.
 Fraleigh J.B. A
First Course in Abstract Algebra  Fraleigh made it, he gives a good
introduction abstract algebra for newbie. Ideal for a first coures. See also Afra's incomplete
notes [.pdf].
 Garrett P. Intro
to Abstract Algebra [.pdf] (FREE!)
 Gilbert W.J. Modern Algebra with Applications
 Gillian J.A. Contemporary Abstract Algebra
 As the author have said, this book includes 'lines from popular
songs, poems, quotations, biographies, historical notes, hundreds of
figures, dozens of photographs, and numerous tables and charts."
If you prefer those things. See the author's
website for resources.
 Goodman F.M. Algebra: Abstract and Concrete (FREE!)
 Herstein I.H. Abstract Algebra
 Ikenaga B. Abstract Algebra (FREE!)
 Jacobson N. Basic
Algebra I 
New edition of his Lectures in Abstract Algebra.
 Knapp A.W. Basic Algebra
 See the
web for correction.
 Rotman J.J. A First Course in Abstract Algebra 
Check out Rotman's page for
errata.
 Shahriari S. Lectures
on Algebra I [.pdf] (FREE!)
 李華介：大學基礎代數 (FREE!)
 The best free material on introductory abstract algebra I have seen so far,
provided you can read Chinese.
Introductory Number Theory:
Gauss once said "mathematics is the queen of sciences and number theory the queen of mathematics." G..H. Hardy once said in A Mathematician's Apology that, "If the theory of numbers could be employed for any practical and
obviously honourable purpose, if it could be turned directly to the furtherance of human happiness or the relief of human suffering,
as physiology and even chemistry can, then surely neither Gauss nor any other mathematician would have been so foolish as to
decry or regret such applications. But science works for evil as well as for good (and particularly, of course, in time of war); and
both Gauss and less mathematicians may be justified in rejoicing that there is one science at any rate, and that their own, whose
very remoteness from ordinary human activities should keep it gentle and clean."
He wrote it on November 1940. It turned out coding theory, where algebra
and number theory are applied, played a key
role in the World War II.
Introductory Topology:
Topology is an extension of geometry and built on set
theory, it's about continuity. There is a joke that topologists are those who cannot distinguish between a
doughnut and a coffee or tea cup. Wikipedia
has a [.gif] to demonstrate this. You may have studied open sets, open
neighbourhood, interior, closure, boundary, basis, continuity, compactness
and connectedness etc. in the analysis course, move on to homemorphism,
homotopy and fundamental group.
As in abstract algebra and analysis, several selected
books or notes cover more than enough. You may leave a few sections to
study at the next stage.
 Armstrong M.A. Basic Topology
 Another nice introductory text by Armstrong, he succeeded in
developing reader's geometrical insight.
 Crossley M.D. Essential Topology
 Undergraduate introductory level, detailed explanation.
 Davis S.W. Topology 
It's like lecture notes, many proofs are left as exercises and many exercises
are proofs.
 Dugundji J. Topology
 Another topology text with quality.
 Engelking R. General Topology
 Gemignani M.C. Elementary Topology
 Henle M. A
Combinatorial Introduction to Topology
 Plenty of illustrations. Indeed many concepts in this subject can
and should be visualised.
 Ikenaga B. Notes on Topology (FREE!)
 McCluskey A. and McMaster B. Topology Course Lecture Notes (FREE!)
 Mendelson B. Introduction to Topology 
Very nice introduction to the metric spaces, topological spaces,
connectedness and compactness. Recommended for first reading.
 Moller J.M. General
Topology [.pdf] (FREE!)
 Morris S.A. Topology Without Tears (FREE!)
 Munkres J.R. Topology

Another nice introduction. Go further than Mendelson's one. Also cover
topics like the Tychonoff theorem, metrization theorems, complete
metric spaces and function spaces, the fundamental group and covering
spaces.
 Singer I.M. and Thorpe J.A. Lecture Notes on Elementary Topology and Geometry
 Steen L.A. and Seebach J.A. Counterexamples in Topology
 Strickland N. Topology Notes (FREE!)
 Thurston W.P. The Geometry and Topology of ThreeManifolds (FREE!)
 Viro O.Y., Ivanov O.A., Netsvetaev N.Y. and Kharlamov V.M. Elementary Topology: Problem Textbook (FREE!)
 Over 400 pages of free material.
 Ward T.B. Topology
Lecture Notes [.pdf] (FREE!)
 Wilkins D.R. Topology
(FREE!)
Differential Geometry:
You may wonder, geometry takes up a large portion in
high school mathematics, why isn't there any geometry course in the first
two stages? In fact, geometry is kind of imbedded in stage two calculus (several variables) and linear algebra
courses, they are usually
assumed and will be used for this course. Here differential calculus is used to study geometry.
Key things to study: multilinear algebra, curvature and torsion, SerretFrenet equation, fundamental
theorem of curves, Poincaré Index theorem (plane and surface),
exterior calculus, Gauss' theorema
egregium, geodesics, GaussBonnet theorem.
Moving frame, due to Élie Cartan, is an approach to
geometry of surface. According to my lecturer John
Steele, it is "computationally the easiest, notationally the
neatest, aesthetically the best, makes the definitions more natural and
the proofs of the two major theorems easier. The moving frame method also
points the way towards several important ideas in modern differential
geometry and theoretical physics. The downside (if there is one) is the
reliance on exterior calculus of differential forms." O'Neill, for
example, uses this approach and he manages to prove Gauss' theorema
egregium in half page, see p.281.
In general:
Tensor Analysis and Manifolds:
Differential Forms:
Mathematical Modelling (optional):
"All models are wrong, some are useful." George Box.
Using mathematics to model real world phenomena is useful, but not the main
concern of pure mathematician.
Statistical Inference (optional):
Build upon stage 2 statistics, you will learn here the CramerRao bound, uniform minimum variance unbiased estimators, NeymanPearson theory, Bayesian
inference, basic bootstrap and robustness, introductory nonparametric
(including the sign test, Wilcoxon signed rank test, McNemar's chisquare
test, WaldWolfowitz runs test, MannWhitney U test, KolmogorovSmirnov
twosamples test, KruskalWallis analysis of ranks, Spearman's R and
Kendal's Tau), etc. Note a few texts say they are aimed at graduated
level, partly because people from other fields only learn this material in
graduated schools, as a result the texts have to set up basic probability
and statistics (you have done it, stage 2 stuff) for them.
Probability and Stochastic (Random) Processes
(optional):
Conditional expectation, Poisson process, Markov
chains, renewal theory, queueing theory, reliability theory, Brownian motion and stationary processes. This topic is particularly
useful for electrical and computer engineers, actuarial study, finance or things like that.
See also left material from stage 2 probability. If you want to learn this with measure, see stage 4 probability list.
Statistical Computing (optional):
These days, you can't do statistics without computer,
in practice. R is a freeware, SPLUS uses the same language, so I
recommend books of this kind. Also, see UCLA's Statistical Computing page for online resources.

Equivalent to the (fourth) honours year in Australia, or the basic graduated
year in other places. I shall outline the key topics and give short
lists of references, you may scout around to see what makes you interest. I'm far from being an expert,
let me know if you
have a suggestion! Note that at this stage, the levels or difficulties of books
within a certain topic may vary a lot. A few subfields may be highly
related, e.g. further reading in Number Theory and algebraic geometry in Geometry.
Moreover, further reading should belong to stage 5.
Foundations and Discrete Mathematics:
Foundation:
Logic (Mathematical, Symbolic, Proof Theory, etc.):
According
to Wikipedia, Enderton, Hamilton, and Mendelson are meant for undergraduate; while Andrews, Barwise (ed.) and Shoenfield are meant for graduate.
 Aigner M. and Ziegler G.M. Proofs from THE BOOK
 Andrews P.B. Introduction to Mathematical Logic and Type Theory
 Ash C.J., Crossely J.N., Brickhill C.J., Stillwell J.C. and Williams
N.H.
What is Mathematical Logic?
 Barwise J. (ed.) Handbook of Mathematical Logic
 Not for beginners, mind you.
 Bilaniuk S. A Problem Course in Mathematical Logic (FREE!)
 Blackburn P., de Rijke M. and Venema Y. Modal Logic
 Blackburn P., van Benthem J. and Wolter F. Handbook of Modal Logic (FREE!)
 Bornat R. Proof and Disproof in Formal Logic: An Introduction for Programmers
 Oxford Texts in Logic 2.
 Carney J.D. Introduction to Symbolic Logic
 Chiswell I. and Hodges W. Mathematical Logic  Oxford Texts in Logic 3.
 Church A. Introduction to Mathematical Logic
 Copi I.M. Symbolic Logic
 Enderton H.B. A Mathematical Introduction to Logic
 Hamiltion A.G. Logic for Mathematicians
 Hedman S. A First Course in Logic: An Introduction to Model Theory, Proof Theory, Computability, and Complexity
 Oxford Texts in Logic 1.
 Hilbert D. and Ackermann W. Principles of Mathematical Logic
 Hunter G. Metalogic
 Kleene S.C. Mathematical Logic
 Kunen K. The Foundations of Mathematics
[.ps] (FREE!)  Includes set
theory, model theory and proof theory. Readers are assumed to
feel comfortable with abstract mathematical structures (groups
and fields) and know basics calculus.
 Mac Lane S. and Moerdijk I. Sheaves in Geometry and Logic: A First Introduction to Topos Theory
 Manin Y.I. A Course in Mathematical Logic
 Mendelson E. Introduction to Mathematical Logic
 Rautenberg W. A Concise Introduction to Mathematical Logic
 Shoenfield J.R. Mathematical Logic
 Simpson S.G. Mathematical
Logic [.pdf] (FREE!)
 Srivastava S.M. A Course on Mathematical Logic
 Takeuti G. Proof Theory
 Troelstra A.S. and Schwichtenberg H. Basic Proof Theory
 Walicki M. Introduction to Logic
[.pdf] (FREE!)
Set Theory:
Graph Theory:
Combinatorics:
Cryptography, Coding and Information Theory (not so pure):
First stage introductory discrete mathematics and basic
linear algebra are usual assumed knowledge.
 Ash R.B. Information Theory 
An introductory text.
 Bellare M. and Rogaway P. Introduction to Modern
Cryptography (FREE!)
 Berlekamp E.R. Algebraic Coding Theory
 Coutinho S.C. The Mathematics of Ciphers: Number Theory and RSA Cryptography
 Goldreich O. The Foundations of Cryptography (FREE!)
 Goldwasser S. and Bellare M. Lecture Notes on Cryptography (FREE!)
 Hall J.I. Notes
on Coding Theory (FREE!)
 Hamming R.W. Coding and Information Theory
 If you are serious about this field (I mean you are not merely
interested in cracking ciphers like those the Zodiac Killer used), you
must learn Hamming error correcting codes, this is the origin.
 Hill R. A First Course in Coding Theory
 Humphreys J.F. and Prest M.Y. Numbers, Groups and Codes 
Friendlier than Roman.
 Johnson O. Information Theory and the Central Limit Theorem
 Koblitz N. A Course in Number Theory and
Cryptography
 Lysyanskaya A. Cryptography and Cryptanalysis (FREE!)
 MacKay D. Information Theory, Inference, and Learning Algorithms (FREE!)
 Take a look, and give me a reason why you shouldn't download it if
you are studying this subject.
 Menezes A.J., van Oorschot P.C. and Vanstone S.A. Handbook of Applied Cryptography (FREE!)
 Mollin R.A. An Introduction to
Cryptography
 Pless V. Introduction to the Theory of ErrorCorrecting Codes
 Pretzel O. ErrorCorrecting Codes and Finite Fields
 Roman S. Coding and Information Theory 
More advanced text, basic abstract algebra required.
 Salomaa A. PublicKey Cryptography
 Schneier B. Applied Cryptography: Protocols, Algorithms, and Source Code in C
 See author's
site.
 Smith L.S. Cryptography: The Science of Secret Writing
 Elementary, suitable for nonmathematicians.
 van Lint J.H. Introduction to Coding Theory
 Algebra and probability required.
 Vaudenay S. A Classical Introduction to Cryptography: Applications for Communications Security
 See the web for extra
material.
Analysis:
Materials listed in stage
3, which have not been covered there, may be studied here.
Functional Analysis:
Measure Theory:
See also the probability section below.
 