38 2. Background
A proof is an object of convincing. It should be an explicit,
specific, logically sound argument that walks step by step from the
hypotheses to the conclusions. Avoid vagueness and leaps of deduc-
tion, and strip out irrelevant statements. Be careful to state what
you are trying to prove in such a way that it does not appear you are
asserting its truth prior to proving it. More broadly, make sure your
steps are in the right order. Often, a good way to figure out how to
prove something is to work backwards, in steps of “I get this as a con-
clusion if this other property holds; this property holds whenever the
object is of this kind; and oh, everything that meets my hypothesis is
an object of that kind!” However, the final proof should be written
from hypothesis to kind of object to special property to conclusion.
True mathematical proofs are very verbal, bearing little to no re-
semblance to the two-column proofs of high school geometry. A proof
which is just strings of symbols with only a few words is unlikely to
be a good (or even understandable) proof. However, it can also be
clumsy and expand proofs out of readability to avoid symbols alto-
gether. For example, it is important for specificity to assign symbolic
names to (arbitrary) numbers and other objects to which you will
want to refer. Striking the symbol/word balance is a big step on the
way to learning to write good proofs.
Make your proof self-contained except for explicit reference to
definitions or previous results (i.e., don’t assume your reader is so
familiar with the theorems that you may use them without comment;
instead say “by Theorem 2.5, . . .”). Be clear; sentence fragments and
tortured grammar have no place in mathematical proofs. If a sentence
seems strained, try rearranging it, possibly involving the neighboring
sentences. Do not fear to edit: the goal is a readable proof that does
not require too much back-and-forth to understand. There is a place
for words like would, could, should, might, and ought in proofs, but
they should be kept to a minimum. Most of the time the appropriate
words are has, will, does, and is. This is especially important in proofs
by contradiction. Since in such a proof you are assuming something
that is not true, it may feel more natural to use the subjunctive, but
that can make things unclear. You assume some hypothesis; given
that hypothesis other statements are or are not true. Be bold and let
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