If you've taken proofs classes from different professors, you know that we share some basic beliefs about writing, but we also have our own quirks and preferences. (If you haven't yet taken proofs classes from different professors... get ready! We share some basic beliefs, but we also have our own quirks and preferences). What you'll see below are the guidelines that I consider the standards for submitting to the Gibbons Journal of Mathematics (by which I mean, sending in a writing assignment to be assessed in a proofs class that you're taking with me).
To me, mathematical writing breaks into three (entangled) pieces: Logic, Mathematics, and Style.
Logic.
The authors
- state claims (propositions, theorems, lemmas, etc) as logical statements;
- state claims appropriate to the assigned problem;
- write proofs with first and last sentences that indicate the proof technique they are using;
- this criterion relaxes a bit after the first proofs course; if you're proving a theorem directly, you can just get to the proof without repeating the hypotheses at the beginning of the proof. But if you're in your first proofs course, please include the first and last sentences that lay out your hypotheses and conclusions!
- define notation and symbols before using them;
- employ a valid proof technique for the statement as written;
- employ a valid proof technique for the statement appropriate to the problem;
- include all necessary cases in the proof;
- write examples with a clear topic sentence;
- write prose using evidence to support claims;
- employ sound logic even in mathematical prose.
Mathematics.
The authors
- use definitions, theorems, and other results from the appropriate chapter(s);
- cite definitions, theorems, and other results from the appropriate chapter(s);
- cite named definitions, theorems, and results;
- employ correct calculations;
- do not re-prove prior results.
Style.
The authors
- use the appropriate LaTeX template for the course;
- adhere to mathematical style conventions, such as:
- using math mode for math symbols,
- writing sentences that do not start in math mode,
- not writing “two-column” proofs (more here: Paragraph Proofs),
- using aligned equations when a string of equations is too long for a page,
- using inline equations for minor mathematical manipulations;
- using appropriate LaTeX environments (proof, prop*, example*, etc);
- discerning when to write an example instead of a proposition and proof;
- adhere to the conventions of grammar;
- write clearly and concisely in a way that does not obscure the logic or mathematics.
Here's a note about the difference between a proposition in math (versus formal logic). In math, we understand propositions (and theorems, lemmas, corollaries, etc) to be statements that are true (and proved/provable) so that we can cite them when we need to use them. You can see the confusion that would ensue if a paper contained "Proposition 1: All even integers are prime" under that assumption, even if followed immediately by a counterexample! So, if you are showing that a statement is not true as part of a writing assignment, here's what that might look like:
Example. It is not the case that every even integer is prime. Notice that the integer 8 is even. Although 8 divides 24 = (2)(12), it divides neither 2 nor 12. Therefore, by (definition of prime/proposition about prime numbers/whatever it is), 8 is not prime.
Example. It is not the case that every even integer is prime. Notice that the integer 8 is even. Although 8 divides 24 = (2)(12), it divides neither 2 nor 12. Therefore, by (definition of prime/proposition about prime numbers/whatever it is), 8 is not prime.
To break that down into more of a template:
Example. [Topic sentence: what's the point of this example?] [Supporting mathematical evidence, less formal than a proof, but still following the stylistic conventions of good mathematical writing!] [Concluding sentence.]
