The Ergonomic Proof
Very often I find proofs difficult to read. But it was hard to pin down why, until I came across a patently obvious proof that nevertheless managed to set off the same mental triggers in me that more difficult proofs normally do.
I found it in the 'Solutions to Exercises' of the Open University's Chapter D4 of maths course MS221, in a proof that the square of an even number is even itself:
An even number is of the form 2n where n is an integer. Its square is
(2n)^2 = 4(n^2) = 2*2(n^2).
Now m=2(n^2) is an integer, so the square (2n)^2 is of the form 2m, where m is an integer, and so (2n)^2 is even.
Here is my preferred proof:
- All even numbers have a factor 2.
- Therefore all even numbers can be expressed as 2n, where n is an integer.
- Therefore the square of any even number can be expressed as (2n)^2 = 4(n^2).
- Therefore the square of any even number has a factor 4.
- Therefore the square of any even number has a factor 2.
- Therefore the square of any even number is itself even.
Comparison of proofs
- Proof A requires you to hold the whole proof in mind at once, it refers to itself here there and everywhere. In comparison, proof B only requires you to make jumps from one statement to another. - Minimize mental overhead.
- Proof A makes no explicit reference to the nub of the proof: that we are aiming to find a factor of 2 in our target. In comparison, this idea is made explicit (in step 2) as well as being central to Proof B. - Make explicit central ideas and motivations.
- Proof A freely mixes notation and natural language, forcing the reader to make frequent jumps from one symbolic domain to the other for no real purpose. In comparison, Proof B is mainly in natural language, with a crucial transformation step in notation. There is no need to switch domains frequently. - Language mixing carries a signficant comprehension overhead, so use it sparingly.
- Proof A makes frequent, extraneous reference to the fact that various expressions are in fact integers. It wouldn't occur to most people to question this, and those that would would know that all even numbers are integers, and that all the operations used in the proof are closed in the integers anyway. - Keep to the point.
- Proof A introduces a needless letter 'm', Proof B doesn't. Proof A has two equalities, Proof B only has one (we don't need to show that 4 is even...). - Occam's Razor.
- Proof A hinges on the manipulation of symbols. Proof B also does (the equation in step 3), but has a greater emphasis on the manipulation of concepts. Conceptual manipulation is more intuitive and satisfying than symbolic manipulation, and more conducive to understanding. Proof B would be improved if we could show why the equation in step 3 is true. Probably a geometric diagram would help here. - Conceptual manipulation is preferable to symbolic manipulation.
Proof A and Proof B are essentially the same, but I know which one I'd rather be confronted with.
Rhetorical awareness is as important in mathematical proof as in any other kind of argument. At root, we are not trying to be automated theorem solvers, although this ability implies various useful skills. However, we are trying to persuade ourselves and each other of various things. We should remember this when we do maths.