Munafo's Laws of Mathematics
1. Once a mathematician learns something, they do not repeat it.
To give some specific examples:
- Once you know the definition of "abelian group", you must not actually give the definition in any of your papers.
- If some other author, or you (in a previous paper) showed why some function is limited from above by N1.2x, you cannot repeat the proof.
- If you and someone else both gave proofs of the N1.2x result (which, by the First Law, must be different) neither of you may repeat either of them.
In all cases, readers are expected to either know the omitted information, or go find the source and read it. In general, once any result is known (regardless of who showed that result), it cannot be restated.
This practice applies mostly to academic articles, but is also common on web pages, in discussion forums, and very often in private emails.
Richard Feynman states in [1] that mathematicians consider any theorem to be "trivial" once there is a proof. Thus, all (true) mathematical propositions can be classified into the trivial and the unproven.
Scott Aaronson has a blog article [2] that begins by stating that he often receives submissions claiming to be proofs of something known to be rather difficult, then goes on to list signs that a received mathematical "proof" is probably wrong. His list includes "[wasting] lots of space on standard material". Accordingly, any explanation that is below the level of the author is not considered helpful to the reader, but instead is considered damaging to the credibility of the author — and authors therefore avoid including explanation.
Main Corollary:
1. a. I am not a mathematician.
Munafo's First Law, applied to me, states that if I were a
mathematician, I would not repeat things I have learned. However, I
often explain things I have learned in great detail, thus I cannot be
a mathematician.
The Inverse of the First Law would imply that Since I am not a mathematician, I am free to say whatever I want, including things I have learned. This does not necessarily follow from the First Law, but happens to be true in my case: if I actually were a (professional) mathematician I would probably feel compelled to obey the Laws.
Humorous Corollaries:
1. b. No mathematician says anything they did not work out themselves.
Thus the common joke wherein a mathematician answers a question
like "how do I make soup?" by describing one step then stating that
the problem has now been reduced to a previously-solved problem.
1. c. No writers of mathematics textbooks are mathematicians.
This is clearly not true, thus the First Law is technically false.
The First Law could be "fixed" by adding a qualifier, but I'm not
sure which qualification would be the best.
1. d. Any literary work (such as a popular song) that contains repeated words, or which repeats words found in another literary work, is not a work of mathematics.
2. Statements must be made in the most abstract manner possible.
If your findings can be expressed in two different ways, both equally provable, choose the more abstract expression.
Even if it can be described more concretely without loss of generality, abstraction somehow makes it better.
3. Examples are worthless.
Examples prevent one's expression from being abstract; they draw the reader's attention to specific cases, possibly suggesting that the general statement does not hold; and they are always redundent inasmuch as the general statement implies all possible examples.
This article is expanded from the older article Munafo's Law of Mathematical Discourse.
Old versions of the Laws:
1. Never repeat anything you have learned from another mathematician.
1. Once a mathematician learns something, they must not repeat it.
1. Once a mathematician learns something, they are not allowed to repeat it.
1. Once a mathematician learns something, they are not allowed to teach it to anyone else.
1. Once a mathematician learns something, they are not allowed to explain it to anyone else.
[1] Richard Feynman, "A Different Set of Tools" (in 'Surely You're Joking, Mr. Feynman!': Adventures of a Curious Character (New York: W. W. Norton), pp. 69-72, (1997)
[2] Scott Aaronson, Ten signs a claimed mathematical proof is wrong, blog article (2008)
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