MATHE MATI CS 220—DISCRETE MATH EMAT ICS —S EP TE MB ER 26, 2006
COMMENTS ON WRITING PROO FS
A GUIDING META PH OR
Most mathematical writing narrates an intellectual journey. Mathematics is generally be-
lieved to be the study of that which can be proved: that is, deduced by valid reasoning from
self-evident axioms and precise definitions. Will our intellectual journey be an ascetic one?
Will we carry all our supplies on our backs, walk barefoot, hunt and gather food, and navigate
by the stars?
We will not—and very few mathematicians ever do, in the sense of breaking all concepts
down to the smallest possible building blocks. Some terrain is familiar—such as the natural
numbers or ordinary algebra—so both the writer and the reader can easily fill in a few small
details of reasoning. Some terrain (such as a full analysis of valid reasoning with quantified
statements) is too difficult to attack without heavy equipment.
The less familiar the material is, though, the more detailed guidance the writer must provide
the reader. As we begin writing proofs, the conventions for structuring proofs are themselves
unfamiliar. Thus we must be very careful in watching the reasoning underneath even simple
proofs: are we progressing from hypothesis to conclusion? What is “given” information, and
what must be demonstrated?
TIPS F OR WRITING PR OOF S
Voice. The author can address the reader in several ways:
•imperative (“Turn left”),
•cooperative (“We now turn left”), or
•passive (“A left turn is now made”).
The narrative is always in the present tense, as the mathematical journey occurs both as the
author is writing (when it takes place in the author’s mind) and as the reader is reading (when
the journey is re-enacted in the reader’s mind).
Definitions. In a well-planned mathematical excursion, all terms appearing in a theorem should
be defined before the theorem is stated. In a book or article, these definitions might appear
earlier in the same work. In Math 220, we are all traveling together. Definitions stated in
lecture, in our textbook, or in handouts may be taken to be common knowledge and need not
be restated in your homework.
All our current definitions are imperative: “Define an integer nto be wiggly if...”. When
using the imperative in mathematical writing, the writer can assume that the reader follows all
orders as soon as they are given. Thus citing a definition is recalling a known fact. Appropriate
wordings for restating a definition during a proof include “Recall that an integer nis defined to
be wiggly when...,” or “By the definition of wiggly, nis...”
Variables. The letters we use to refer to mathematical objects whose specific identities we do
not know are tremendously important. Here are some ground rules:
(1) Every variable appearing in a proof must be described in some way. Perhaps it appears
in the statement of the theorem. Perhaps it arises from a universal or existential in-
stantiation. But a proof must be self-contained, in the sense that we must know where
every variable has come from.
(2) Each variable can have at most one definition, however. When you existentially instan-
tiate multiple times, you must give the resulting variables different names. There’s no
reason to assume that the values obtained from separate instantiations are equal!
