The importance of mastering the art of writing valid proofs that do not make careless unstated assumptions or unproven assertions can not be understated. Oftentimes when you view some statement as initially obvious, it will turn out to be either dead wrong or at the very least hold most of the meat of the proof in proving it. At their core, basic proofs are really easy and frequently just a matter of unwrapping the definition and following your nose, but getting into the right mindset for them might take the neophyte some practice in order to see them that way.
Therefore you should work through a book or two on proofs before moving onto advanced mathematics and then blaming those books for being written badly because you lacked the prerequisite mathematical maturity from skipping this step. Some good books to learn proofs are:. If you find yourself struggling with proofs, then the following books provide more hand holding on the subject but at the cost of excluding some additional material :. After this, set theory and mathematical logic are the logical continuation of this material and reading books on them will deepen your understand of what sets and proofs really are as well as mathematics as a whole with meta-mathematics.
They also make excellent next steps in getting better at proofs and abstract mathematics in general before moving on to the much more difficult subjects like algebra and analysis. Combinatorics, graph theory , linear algebra involving vector spaces , and number theory textbooks would then be the next level to practice on and are fairly easy to read at this stage of mathematical maturity.
Since you will likely find yourself revising your proofs quite often, now would be an ideal time to finally learn LaTeX pronounced "lay-tech" to typeset your proofs and future papers in.
This is the formal study of the Foundations of Mathematics using mathematics, particularly on Set Theory which much of mathematics is built on and Mathematical Logic which studies what proofs are and the limits of what can be done. When starting in this subject the question of where to start pops up. Ideally, you would want to know a some logic while studying set theory and know some set theory while studying logic leading to a bit of a dilemma. This is solved by most introductory books giving just enough material on the other subject so you don't get lost but once you move on to intermediate and beyond books, you are assumed to have already studied both set theory and logic at least at the introductory level.
Enderton is a gentle, clear, and easy to read textbook that's perfect for someone just finishing a book or course on proofs and looking for the next step to improve their math skills further. He will construct the real numbers from ZF axioms in the first five chapters. These books would be better for someone who already has a few proof based math courses under their belt. They're a notch harder than Enderton and go into a few more advanced topics too. See the universal recommendations on Statistics. See the universal recommendations on Design of Experiments.
These 2 books are aimed at high school students with knowledge of elementary algebra to give them a taste of pure mathematics. Abstract Algebra also called Modern Algebra or just Algebra is the study of mathematical structures that consist of a set with algebraic rules defined on the set's elements.
This enables us to prove general results that depend only on the particular rules the structures have and not a particular example structure like the rationals, reals, quaternions, polynomials, matrices, or integers modulo n we have in mind that satisfies those rules.
Abstract Algebra is not to be mistaken with College Algebra as that refers to the Elementary Algebra that is typically done in grade school. If you're looking for resources on that, see the Precalculus section above. Herstein's Topics takes a fairly conventional approach to the subject while Artin's book does things in a rather unique and geometrical way. While both are well written texts on their own, but pairing them is very useful.
With the piecemeal development of calculus by Cavalieri , Pascal , Fermat , Descarte , Leibniz , Euler , Lagrange , Fourier and many others, calculus gradually showed itself to be a powerful yet deeply troubled tool. These were not just pointless trifles that could be brushed off as more philosophy than math but of increasing practical importance. As time went on, many counterexamples and not just pathological ones where the naive application of the methods of calculus would produce erroneous results cast a shadow on the validity of all other results of calculus.
Many critics wanted to end its study altogether and relentlessly mocked the concept of infinitesimals as "the ghosts of departed quantities". Since the triumphs for calculus were both numerous and far reaching, mathematicians strongly sought to make the results of calculus proven rather than discarding the subject all together. This sparked off a massive revolution in mathematics and the field of analysis quickly exploded into various distinct but interconnected directions.
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Students just finishing the study of calculus and basic proofs often fail to realize the sheer importance of careful work in analysis and scoff the whole subject off as merely "intellectual or autistic masturbation". This mentality comes from being coddled with the toy-problems you see in calculus that are selected to hide any possible nastiness that comes from complicated situations that frequently arise in science and engineering. Even if the student is aware of the importance of being careful, they are often insulted when forced to work through "obvious" theorems.
The problem here is that many results in analysis that seem obvious are frankly very difficult to prove for example see the Jordan curve theorem or even dead wrong. In order to gain the ability to prove important and powerful theorems hidden away in analysis, students need plenty of practice working through basic problems to gain familiarity and mathematical maturity to move on to difficult work even if this means you need to spend time proving that "every open ball is open".
A good reference to keep with you and refer to often is "Counterexamples in Analysis" by Gelbaum and Olmsted published by Dover Books. A lot of the exercises in analysis often boil down to spamming the triangle inequality until you get the result you want. If you haven't done much work with inequalities since grade school, practicing them can make the subject seem vastly easier.
This is where the results of single variable calculus are finally made both rigorous and generalized. The gold standard for the subject is the first 8 chapters of Rudin's "Principles of Mathematical Analysis" whose slick proofs and challenging exercises can't be beat. Chapters 9 and 10 of Rudin on multivariable analysis are bit sparse to learn from so you're better off moving fuller treatment on analysis on manifolds see below to learn from. Apostol's "Mathematical Analysis" goes through a bit more material than Rudin, gives more worked out proofs, and has relatively easier problems.
If you're struggling with Rudin, give Apostol a try. The price paid is that his books is quite longer than Rudin and larger time investment. This is the study of analysis on multidimensional spaces making multivariate and vector calculus rigorous and pushing the subject further.
Dover Books on Mathematics | Awards | LibraryThing
A good grounding in linear algebra is required. Math Textbook Recommendations. MAA Book Reviews.
Chicago undergraduate mathematics bibliography. Amazon's "So you'd like to Axiomatic Set Theory by Patrick Suppes. Boolean Algebra and Its Applications by J. Eldon Whitesitt. A bridge to advanced mathematics by Dennis Sentilles. Building Models by Games by Wilfrid Hodges.
Calculus of finite differences by George Boole. Calculus of Variations by Robert Weinstock. Calculus of variations with applications by George M.
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- The history of the calculus and its conceptual development | Open Library.
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Calculus Refresher by A. Dennis Lawrence. Challenging Problems in Algebra by Alfred S. Character theory of finite groups by I. Martin Isaacs. Combinatorics for Computer Science by S. Gill Williamson. Combinatorics of Finite Sets by Ian Anderson. Complex Analysis with Applications by Richard A. Complex Variables by Stephen D.
Complex Variables by Francis J. Computational Methods of Linear Algebra by V. A Concept of Limits by Donald W. Concepts of Probability Theory by Paul E. Conformal Mapping by Zeev Nehari. Coordinate Geometry by Luther Pfahler Eisenhart. A Course in Mathematical Analysis, Vol. Curvature and Homology by Samuel I. Curvature in Mathematics and Physics by Shlomo Sternberg. Differential Forms by Henri Cartan. Differential Geometry by Erwin Kreyszig.
The History of the Calculus and its Conceptual Development
Differential Geometry by Heinrich W. Differential Geometry by William C. Differential Manifolds by Antoni A. The Divine Proportion by H. Dynamical Systems by Shlomo Sternberg. Elementary Concepts of Topology by Paul Alexandroff. Elementary Decision Theory by Herman Chernoff.
Elementary Mathematics from an Advanced Standpoint, Vol. Elementary statistics with applications in medicine by Frederick E. Elementary Topology by Michael C.