Suggestions for writing Proofs

569 April 16, 2020
  • Be sure you are clear in your own mind about are assuming as opposed to what you need to prove. Don’t use language that claims things are true before you have actually proven they are true. e.g. you could start a proof of the Pythagorean Theorem with “We want to show that a2 + b2 = c2,” but don’t start with. “Therefore square of the hypotenuse equals the sum of the squares of the other two sides.” (It seems silly to warn you about that, but you would be surprised at how often students write such things.)
  • Be sure your argument accounts for all possibilities that the hypotheses allow. Don’t add extra assumptions beyond the hypotheses, unless you are considering cases which, when taken together, exhaust all the possibilities that the hypotheses allow.
  •  Don’t confuse an example, which illustrates the assertion in a special instance, with a proof that accounts for the full scope of the hypotheses.
  •  Be wary of calculations that start at the conclusion and work backwards. These are useful in discovering a line of reasoning connecting hypotheses and conclusion, but often are logically the converse of what you want to show. Such an argument needs to be rewritten in the correct logical order, so that it proceeds from hypotheses to conclusion, not the other way around.
  • Don’t just string formulas together. Your proof is not a mathematical diary, in which you write a record of what you did for your own sake. It is an explanation that will guide another reader through a logical argument. Explain and narrate. (And don’t include things you thought about but ended up not using.)
  •  Don’t use or talk about objects or variables you haven’t introduced to the reader. Introduce them with “let” or “define” so the reader knows what they are when they are first encountered. On the
    other hand, don’t define things long before the reader needs to know about them (or worse, things that aren’t needed at all).
  •  Keep notation consistent, and no more complicated than necessary.
  •  When you think you are done, reread your proof skeptically, looking for possibilities that your argument overlooked. Perhaps get someone else (who understands proofs) to read it. Make it your responsibility to find and fix flaws in your reasoning and writing before you turn it in.
  •  One thing that some find awkward at first is the traditional use of first person plural “we . . . ” instead of “I” in mathematical writing. The use of  “I” is fine for personal communications, talking about
    yourself. But when you are describing a proof it is not a personal statement or testimony. The “we” refers to you and all your readers – you are leading all of them with you through your reasoning. Write so as to take them along with you through your arguments. By writing ‘we” you are saying that this is what everyone in your logical tour group should be thinking, not just your personal perspective.
  • Separate symbols by words. e.g. “Consider q2, where q
    ∈

    Q” is good. “Consider q2, q

    ∈

    Q” is vague.

  •  Don’t start a sentence with a symbol.
  • A sentence starting with “if” needs to include “then.”
  •  Use words accurately. An inequality is not an equation, for instance.