The role of proofs in mathematical writing

This post outlines the way that proofs are used in mathematical writing. I have been revising the chapter on Proofs in abstractmath.org, and I felt that giving an overview would keep my mind organized when I was enmeshed in writing up complicated details.

Proofs are the sole method for ensuring that a math statement is correct.

  • Evidence that something is true gooses us into trying to prove it, but as all research mathematicians know, evidence only means that some instances are true, nothing else.
  • Intuition, metaphors, analogies may lead us to come up with conjectures. If the gods are smiling that day, they may even suggest a method of proof. And that method may even (miracle) work. Sometimes. If it does, we get a theorem, but not a Fields medal.
  • Students may not know these facts about proof. Indeed, students at the very beginning probably don’t know what a proof actually is: “Proof” in math is not at all the same as “proof” in science or “proof” in law.

A proof has two faces: Its logical structure and its presentation.

The logical structure of a proof consists of methods of compounding and quantifying assertions and methods of deduction.

  • The logical structure is usually expressed as a mathematical object.
  • The most familiar such math objects are the predicate calculus and type theory.
  • Mathematical logic does not have standard terminology (see Math reasoning.) Because of that, the chapter on Proofs uses English words, for example “or” instead of symbols such as $P\lor Q$ or $P+Q$ or $P||Q$.
  • For beginning students, throwing large chunks of mathematical logic at them doesn’t work. The expressions and the rules of deduction need to be introduced to them in context, and in my opinion using few or no logical symbols.
  • Students vary widely in their ability to grasp foreign languages, and symbolic logic in any of its forms is a foreign language. (So is algebra; see my rant.)
  • The rules of deduction do not come naturally to the students, and yet they need to have the rules operate automatically and subconsciously. They should know the names of the nonobvious rules, like “proof by contradiction” and “induction”, but teaching them to be fluent with logical notation is probably a waste of time, since they would have to learn the rules of deduction and a new foreign language at the same time.
  • I hasten to add, a waste of time for beginning students. There are good reasons for students aiming at certain careers to be proficient in type theory, and maybe even for predicate calculus.

Presentation of proofs

  • Proofs are usually written in narrative form
  • A major source of difficulties is that the presentation of a proof (the way it is written in narrative form) omits the reasons that most of the proof steps follow from preceding ones.
  • Some of the omitted reasons may depend on knowledge the reader does not have. “Let $S\subset\mathbb{Q}\times\mathbb{Q}$. Let $i:S\to\mathbb{N}$ be a bijection…” Note: I am not criticizing someone who writes an argument like this, I am just saying that it is a problem for many beginning students.
  • Some reasons are given for some of the steps, presumably ones that the writer thinks might not be obvious to the reader.
  • Sometimes the narrative form gives a clue to the form of proof to be used. Example: “Prove that the length $C$ of the hypotenuse of a right triangle is less than the sum of the other two sides $A$ and $B$. Proof: Assume $C\geq A+B$…” So you immediately know that this is going to be a proof by contradiction. But you have to teach the student to recognize this.
  • Another example: in proving $P$ implies $Q$, the author will assume that $Q$ is false implies $P$ is false without further comment. The reader is suppose to recognize the proof by contrapositive.

Translation problem

  • The Translation problem is the problem of translating a narrative proof into the logical reasoning needed to see that it really is a proof.
  • Many experienced professional mathematicians say it is so hard for them to read a narrative proof that they read the theorem and the try to recreate the proof by thinking about it and glancing at the written proof for hints from time to time. That is a sign of how difficult the translation problem really is.
  • Nevertheless, the students need to learn the unfamiliar proof techniques such as contrapositive and contradiction and the wording tricks that communicate proof methodology. Learning this is hard work. It helps for teachers to be more explicit about the techniques and tricks with students who are beginning math major courses.

References

Added 2014-12-19

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