Learntofish's Blog

How to write Math Proofs

Posted by Ed on October 6, 2009

Proof is the idol before whom the pure mathematician tortures himself.
Sir Arthur Eddington (1882 – 1944), The Nature of the Physical World

The very essence of mathematics is the concept of a proof. But writing one is not so easy. Below is a compilation of notes that describe how to write proofs in mathematics. They are intended for beginners.

PROOF TECHNIQUES

1) Introduction to mathematical arguments (by Michael Hutchings)
http://math.berkeley.edu/~hutching/teach/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)
http://cheng.staff.shef.ac.uk/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) Basic proof methods (by David Marker)
MATH 215, Introduction to Advanced Mathematics, Fall 2006
http://www.math.uic.edu/~marker/math215/methods.pdf

7) Notes on Math Proof (by Bruce Ikenaga)
http://marauder.millersville.edu/~bikenaga/mathproof/mathproofnotes.html
Covers many topics.

8 ) Math 23b Proofs (by Kiyoshi Igusa)
http://people.brandeis.edu/~igusa/Math23bF07/Math23b.htm
An introductory course on math proofs.

9) 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

10) Writing Proofs (by Tim Hsu)
http://www.math.sjsu.edu/~hsu/courses/generic/proof.pdf
44 pages long

11) ∀ 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.

12) Proof Writing and Presentation Tips (by Erika L.C. King)
http://math.hws.edu/eking/pandptips
Tips for writing good proofs.

13) How to write mathematics (by Martin Erickson), 5 pages
http://www2.truman.edu/~erickson/manual6.pdf

14) Errors in mathematical writing (by Keith Conrad), 5 pages
This one is good because Keith shows an example for good and bad mathematical writing.

15) 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!

GUIDELINES FOR MATHEMATICAL PROOFS

1) 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

2) How to do math proofs (wikiHow)
http://www.wikihow.com/Do-Math-Proofs

3) Some hints on mathematical proof (by David Goss)
http://www.math.ohio-state.edu/~goss/style.html

How to write mathematics badly (Entry in the Mathematics Weblog)
Part 1: http://www.sixthform.info/maths/?p=147
New Link for Serre’s talk: http://modular.fas.harvard.edu/edu/basic/serre/
Part 2: http://www.sixthform.info/maths/?p=148
Part 3: http://www.sixthform.info/maths/?p=149