2.6. Some Notes on Proofs and Abstraction 39

the whole contraption go up in flames when it runs into the statement

it contradicts.

Your audience is a person who is familiar with the underlying

definitions used in the statement being proved, but not the statement

itself. For instance, it could be yourself after you learned the defi-

nitions, but before you had begun work on the proof. You do not

have to put every tiny painful step in the write-up, but be careful

about what you assume of the reader’s ability to fill in gaps. Your

goal is to convince the reader of the truth of the statement, and that

requires the reader to understand the proof. Along those lines, it is

often helpful to insert small statements (I call it “foreshadowing” or

“telegraphing”) that let the reader know why you are doing what you

are currently doing, and where you intend to go with it. In particular,

when working by contradiction or induction, it is important to let the

reader know at the beginning. More complicated proofs, the kinds

that take several pages to complete, often benefit from an expository

section at the beginning, that outlines the proof with a focus on the

“why” of each step. A more technical portion that fills in all the

details comes afterward.

You must keep to the definitions and other statements as they are

written. In fact, a good strategy for finding a proof of a statement is

first to unwrap the definitions involved. Be especially wary of men-

tally adding words like only, for all, for every, or for some which are

not actually there. If you are asked to prove an implication it is likely

the converse does not hold, so if you “prove” equivalence you will be

in error. Statements that claim existence of an object satisfying cer-

tain hypotheses may be proved by producing an example, but if you

are asked to prove something holds of all objects of some type, you

cannot do so via a specific example. Instead, give a symbolic name

to an arbitrary object and prove the property holds using only facts

that are true for all objects of the given type. Similarly, the term

without loss of generality, or WLOG, appears from time to time in

proofs to indicate a simplifying but not restrictive assumption. If you

use the term make sure the assumption truly does not restrict the

cases. For example, one may assume without loss of generality that

the coeﬃcient of x in the equation of a plane is nonnegative, since if