I recently marked over $100$ assignments for a multivariable calculus course. One question which a lot of people did poorly was proving a given set was open. Aside from issues relating to rigour and logic (or lack thereof), I noticed an issue that I wasn't as aware of before this experience. A lot of students used English words incorrectly (from a mathematical point of view) when writing sentences within their proof.

Some of the words/phrases I am referring to are:

  • As
  • Assume
  • Let
  • Suppose
  • Hence
  • Therefore
  • Such that
  • There exists

Is anyone aware of a reference which explains how to use these words and phrases (and others that frequently occur) in a mathematical context?

  • 2
    $\begingroup$ Great question! $\endgroup$
    – user42912
    Apr 18, 2013 at 11:06

3 Answers 3


Charles Wells, The Handbook of Mathematical Discourse; it’s available as a PDF here. His site abstractmath.org may also be useful.

  • $\begingroup$ Thanks Brian, I've only had a quick look but this seems like an excellent resource. $\endgroup$ Apr 18, 2013 at 7:36
  • $\begingroup$ @Michael: My pleasure. I’ve never actually used it, but it comes closer to what you describe than anything else that I’ve seen. $\endgroup$ Apr 18, 2013 at 7:37
  • 2
    $\begingroup$ The link don't work for me :( $\endgroup$
    – Paul
    Apr 18, 2013 at 10:13
  • 1
    $\begingroup$ @Paul: Does the link still not work? $\endgroup$ Oct 5, 2013 at 2:46
  • 1
    $\begingroup$ New link over here. I would edit the answer but the edit would be too short. $\endgroup$ Nov 9, 2014 at 1:54

Again ('cos I've recommended it before) I can very warmly recommend getting your students to beg/borrow/buy and then read the excellent

Daniel J. Velleman, How to Prove it: A Structured Approach (CUP, 1994 and much reprinted, and now into a second edition).

From the blurb: "Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs." And in doing that, the most important thing, Velleman's book should get students to understand the lingo properly as they better understand what they are doing when they make a supposition here, draw an inference there, use a quantifier, etc.

  • $\begingroup$ I have seen this book mentioned before but I completely forgot about it. My university library doesn't have a copy but I will definitely encourage them to get one. $\endgroup$ Apr 18, 2013 at 13:02
  • $\begingroup$ His website has some interesting tidbits as well. In particular, the Proof Designer looks like it could be useful. $\endgroup$ Apr 18, 2013 at 16:06

I'm using

Franco Vivaldi Mathematical Writing (Springer Undergraduate Mathematics Series)

which I find very useful. It contains many definitions of basic mathematical objects (sets, maps...) as well as examples of proves.


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