The post Are these questions unambiguous? in the blog Explaining Mathematics concerns the funny way mathematicians use the number “two” (Note [3]).  This is discussed in, based on usage quotations (see Note [1]) in the Handbook of Mathematical Discourse. They are citations  54, 119, 220, 229, 260, 322, 323 and 338.  The list is in the online version of the Handbook (see Note [2]) which takes forever to load.  (There is a separate file for users of the paperback book but it is currently trashed.)

The usage quirk concerning “two” is exemplified by statements such as these:

  1. The sum of any two even integers is even.
  2. Courant gives Leibniz’ rule for finding the Nth derivative of the product of two functions.  (This is from Citation 323.)
  3. Are there two positive integers m and n, both greater than 1, satisfying mn=9? (This is from Explaining Mathematics.)

Statements 1 and 2 are of course true.  They are still true if the “two” things are the same.  Mathematicians generally assume that such a statement includes the case where the two things are the same.  If the case that they are the same is excluded, the statement becomes an unnecessarily weak assertion.

Statement 3, in my opinion, is badly written.  If the two positive integers have to be distinct, the answer is “no”.   I think any competent mathematical writer would write something like, “There are not two distinct integers m and n both greater than 1 for which mn = 9″.

It is fair to say that when mathematicians refer to “two integers” in statements like these, they are allowed to be the same.  If they can’t be the same for the sentence to remain true, they will (or at least should) insert a word such as “distinct”.

Of course, in some sentences the two integers can’t be the same because of some condition imposed in the context.  That doesn’t happen in the citations I have listed.  Maybe someone can contribute an example.


[1] In the Handbook, usage quotations are called “citations”.  It appears to me that the commonest name for citations among lexicographers is “usage quotations”, so I will start calling them that.

[2] I created the online version of the Handbook hastily in 2006.  It needs work, since it has TeX mistakes (which may irritate you but should not interfere with readability) and omits the quotations, illustrations, and some backlinks, including backlinks for the citations.  Some Day When I Get A Round Tuit…

[3] This funny property of “two” was discussed many years ago by Steenrod or Knuth or someone, and is mentioned in a paper by Susanna Epp, but I don’t currently have access to any of the references.


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