The long nights of Swedish winter tend to send one a little insane. This year’s stroll down lunacy lane led me to explore some of the curious pre-installed software on my Mac, more specifically, Garage Band. Garage Band seems to have been designed with the musically inclined in mind, rather than the academic, but by the end I was having so much fun that I had to produce a video. Hence the arrival of the unashamedly silly, Epic Pythagoras. But I also had a somewhat educational message that I wanted to share. And worried that this might perhaps (maybe) have been lost in the endless lightning strikes and bizarre sound effects, I’d like to tell you that side of the story today!
Epic Pythagoras is a video about the Pythagorean theorem: given a triangle with side lengths \(x\), \(y\) and \(z\) (where \(z\) is larger than \(x\) and \(y\)), if the triangle is right angled, then
\[x^2+y^2=z^2.\]
In fact, as is slightly less well known, the converse is also true. Namely given a triangle with side lengths \(x\), \(y\) and \(z\), if \(x^2+y^2=z^2\), then the triangle must be right angled. There are famously many proofs of the Pythagorean theorem (though some obfuscate this converse statement), and in the video I have animated my personal favourite.
So far so good, but why bother? Sure, it is nice to emphasise the converse statement, but the Pythagorean theorem has been so extensively covered, if education was my goal, wouldn’t I have been better off following a path less trodden? To such criticisms, my mind immediately leaps to the following quote:
During the 1877 England-Scotland game the Honourable Alfred Lyttleton, when challenged about his failure to pass the ball, remarked to his team-mate but social inferior Bill Mosforth, “I am playing purely for my own pleasure, Sir!”
The Ball is Round: A Global History of Football – David Goldblatt
I am certainly not trying to claim any social superiority here, but no matter how I try to spin it, I doubt I could convince you that I made this video for any other reason than my own pleasure. However if I were to lift out an educational message, it would be on the role of proof by contraposition, rather than anything to do with Pythagoras (or the theorem that for some reason bears his name). Proof by contraposition is a technique I use all the time. Certainly it lacks the glamour of its more fashionable cousin proof by contradiction. But it so often gets the job done (at least for engineers pretending to be mathematicians), and so many proofs by contradiction are really just (poorly phrased) proofs by contraposition. In my opinion it deserves greater emphasis, at least for someone learning about the construction of logical proofs.
To understand how proof by contraposition works, let’s focus again on the converse statement in the Pythagorean theorem, and suppose that we want to prove that given a triangle, if \(x^2+y^2=z^2\), then the triangle must have a right angle in it. To construct our proof, let’s focus on the two statements about triangles that can be either true or false:
A: \(x^2+y^2=z^2\)
B: the triangle has a right angle in it
Our goal is to show that given any triangle, truth of A implies truth of B. Proof by contraposition simply asserts that our goal is equivalent to showing that falsity of B implies falsity of A. We make use of this in the video by showing that for triangles without right angles, \(x^2+y^2\neq{}z^2\), establishing the sought contraposition, and hence the converse statement in the Pythagorean theorem.
To reveal what is going on, let’s reframe this argument in terms of the set of all triangles. In this context, our original objective is to show that the set of all triangles for which A is true is a subset of the set of all triangles for which B is true (i.e. if A is true for a triangle, B cannot be false, and so truth of A implies truth of B). Similarly, proving the contrapositive demonstrates that the set of all triangles for which B is false is a subset of the set of all triangles for which A is false. Therefore there can be no triangles for which B is false and A is true, and so truth of A implies truth of B. You can find a nice illustration of this on a Euler diagram on the Wikipedia page.
As a concluding remark, observe that every implication that can be proved by contraposition can also be proved by contradiction. To see this, suppose that we have a proof of the contrapositive step. That is, we know how to prove that falsity of B implies falsity of A. Following the usual patter of proof by contradiction, we now assume the truth of A and falsity of B. Our contrapositive proof then asserts that since B is false, so is A, which is a contradiction. So in some sense proof by contradiction has earned its more illustrious billing (proof by contradiction is also not limited to the proof of implications). However proof by contradiction does not require that our contradiction is the simultaneous truth and falsity of A. Rather the simultaneous truth and falsity of any statement that can be deduced from the assumption of truth of A and falsity of B will suffice. Understanding this distinction better illuminates the steps involved in logical proofs, what is being sought, and the tactics that can be used in their construction!