Below is a list of notes on mathematical proofs. The notes are directed at beginners who want to learn how to write mathematical proofs. PROOF TECHNIQUES 1) Introduction to mathematical arguments (by Michael Hutchings) http://math.berkeley.edu/~hutching/teach/113/proofs.pdf 2) How to Write Proofs - A short tutorial on the basics of mathematical proof writing (by Larry W. Cusick) http://zimmer.csufresno.edu/~larryc/proofs/proofs.html 3) How to write proofs: a quick guide (by Eugenia Cheng) Department of Mathematics, University of Chicago http://math.unice.fr/~eugenia/proofguide/ 4) Notes on Methods of Proof by Peter Williams http://www.math.csusb.edu/notes/proofs/pfnot/pfnot.html 5) A brief guide to writing proofs (Polytechnic university) http://www.math.poly.edu/courses/ma2312/WritingProofs.pdf 6) A few words about proof (Berkeley Math Circle) http://mathcircle.berkeley.edu/proof.pdf 7) Understanding Mathematical Induction (Idris Hsi) http://www.cc.gatech.edu/people/home/idris/AlgorithmsProject/ProofMethods/index.html http://www.cc.gatech.edu/people/hom...Methods/Induction/UnderstandingInduction.html 8) Basic proof methods (David Marker) MATH 215, Introduction to Advanced Mathematics, Fall 2006 http://www.math.uic.edu/~marker/math215/methods.pdf --------------------------------------------------------------- GUIDELINES FOR MATHEMATICAL PROOFS 1) Guidelines for Writing Mathematical Proofs (Jessica K. Sklar) http://www.plu.edu/~sklarjk/499f06/499proofguidelines.pdf 2) Introduction to Mathematical Reasoning (John M. Lee) Conventions for Writing Mathematical Proofs (Math 310, Spring 2006) http://www.math.washington.edu/~lee/Courses/310-2006/writing-proofs.pdf 3) How to do math proofs (wikiHow) http://www.wikihow.com/Do-Math-Proofs 4) Some hints on mathematical proof by David Goss http://www.math.ohio-state.edu/~goss/style.html 5) Proof-Writing Tips (Ezra N. Miller) Math 5707, Spring 2004 http://www.math.umn.edu/~ezra/5707/tips.html HOW TO WRITE MATHEMATICS BADLY 6) How to write mathematics badly (Entry in the Mathematics Weblog) Part 1: http://www.sixthform.info/maths/?p=147 Part 2: http://www.sixthform.info/maths/?p=148 Part 3: http://www.sixthform.info/maths/?p=149 If you know more links, feel free to post them here.
Hi rococophysics, I don't have a list of books, but I found some books on amazon.com. This book here looks good, though I haven't read it. At the bottom of the page you will find other books ("Customers Who Bought This Item Also Bought"). If you click on the image of the book "The Nuts and Bolts of Proofs" ("Search Inside" function) and go to the page after page 13 you will find a list of books.
More on proof writing: 1) Notes on Math Proof (by Bruce Ikenaga) http://marauder.millersville.edu/~bikenaga/mathproof/mathproofnotes.html Covers many topics. 2) Math 23b Proofs (by Kiyoshi Igusa http://people.brandeis.edu/~igusa/Math23bF07/Math23b.htm An introductory course on math proofs. 3) A Guide to Proof-Writing (by Ron Morash, University of Michigan-Dearborn) http://www.csd.abdn.ac.uk/~kvdeemte/teaching/CS3511/lectures/slides/proofwriting.pdf 4) Writing Proofs (by Tim Hsu) http://www.math.sjsu.edu/~hsu/courses/generic/proof.pdf 44 pages long 5) ∀ Proof Writing ∃ This Reference Book A Student’s Guide to Intermediate Mathematical Proofs (by Kiddo Kidolezi, David Molk, Maurice Opara, Dan Shea and Priscilla S. Bremser) http://community.middlebury.edu/~bremser/MA091_HANDBOOK.pdf A document written by students. 6) Proof Writing and Presentation Tips (by Erika L.C. King) http://math.hws.edu/eking/pandptips Tips for writing good proofs.
to write a proof: begin by stating what you want to prove, precisely. then make sure you know what all the words in that statement mean. then identify the hypotheses of your statement, and start using them to head towards your desired conclusion. look at the desired conclusion and see if you know any other theorems that have that as a conclusion, then try to see if their hypotheses can be verified in your setting. another approach is to negate the desired statement, and try to deduce a known false statement.
learn how to use basic language correctly. e.g. learn the distinction between the converse of a statement (whose truth is unrelated to that of the original statement), and the contrapositive, whose truth is equivalent. E.g. if the statement has form "P implies Q", the converse is "Q implies P", and the contrapositive is "notQ implies notP". e.g. Every good boy does fine is equivalent to "if X is a good boy, the X does fine". the converse is: "If X does fine, then X is a good boy". the contrapositive is: "If X does not do fine, then X is not a good boy". Or, if f is continuous on [0,1] then f is bounded on [0,1] (true). converse: if f is bounded on [0,1] then f is continuous on [0,1] (false). contrapositive: If f is not bounded on [0,1] then f is not continuous on [0,1] (true).
The video link, "Serre: Writing Mathematics," (aka How to write mathematics badly) was broken as well, but can be found here: http://modular.fas.harvard.edu/edu/basic/serre/ I also recommend the book "An Introduction to Mathematical Reasoning" by Peter J. Eccles Its pretty basic, but useful for someone like me who had never been exposed to formal math or proofs.
Thanks m00npirate for the link! More notes: 1) How to write mathematics (by Martin Erickson), 5 pages http://www2.truman.edu/~erickson/manual6.pdf 2) Errors in mathematical writing (by Keith Conrad), 5 pages http://www.math.uconn.edu/~kconrad/math216/mathwriting.pdf This one is good because Keith shows an example for good and bad mathematical writing. 3) An Introduction to Proofs and the Mathematical Vernacular (by Martin V. Day), 123 pages http://www.math.vt.edu/people/day/ProofsBook/ This one is actually a book with 123 pages!
Just to add my two cents on this I just started this whole proof writing business and learning proof writing from the wrong book can just confuse you more. So I'd recommend if you want to take up proof writing to learn it from more then one source especially if you don't have a teacher. I'd Recommend: How To Prove It: A Structured Approach by Daniel J. Velleman (It is great at explaining some basic concepts and the majority of the common proof writing concepts however it isn't as heavy in math as most books.) For that I'd recommend The Nuts and Bolts of Proof Writing ( I forget Who its by) but its heavy in example.) If you're really adventurous and want a challenge with almost no explanation my college uses "A Transition To Advanced Mathematics" ( Its a hard book for beginners if you don't have teacher since it has few examples and little to no explanation )
More material: 1) Why do we have to learn proofs Joshua Cooper, Associate Professor Department of Mathematics, University of South Carolina 2) Book of Proof A complete book on mathematical proof writing by Richard Hammack, Associate Professor, Virginia Commonwealth University, Department of Mathematics and Applied Mathematics You can download the book as PDF for free. Or you buy it for 13$ or 11€ on amazon which I'd say is a very good investment. 3) Writing Mathematics Ethan Bloch, Professor of Mathematics, Bard College 4) Video lectures by Dr Joel Feinstein How and why we do mathematical proofs Definitions, Proofs and Examples
Thanks a lot. Very useful. I shouldn't have bought Velleman or Solow. These notes are just enough, no need to buy a book...
Your first link is dead. Here's what is should be: http://www.millersville.edu/~bikenaga/math-proof/math-proof-notes.html
Approved textbooks by The American Institute of Mathematics (the books are open-access): 1) A Gentle Introduction to the Art of Mathematics by Joe Fields 2) Mathematical Reasoning: Writing and Proof by Ted Sundstrom