Recently at the Girls’ Angle club, a girl asked: Has the Pythagorean theorem been proven?
That was a terrific question! Unfortunately, I didn’t have time to discuss it as well or as fully as I would have liked. A lot of math hovers around it, so I decided to amplify my in-club remarks in a blog post.
First, there’s a common misconception that I’d like to expose. The misconception is that the Pythagorean theorem is a statement about the relationship between the lengths of the sides of right triangles found in the real world. It is not. It is a statement about the relationship between the lengths of the sides of a mathematical concept known as a right triangle.
If this comment strikes you as off, it could mean that you have not understood the distinction between mathematics and science. Mathematics is a universe of abstractions, not physical objects. People often confuse mathematical abstractions for actual things because natural observations inspired so much mathematics. Perhaps when one thinks of a plane, one imagines a table top or a tennis court or a sheet of glass. But we know that the physical objects are imperfect models of the pristine, abstract concept of a plane, infinitely thin, absolutely flat, and extending without end. Perhaps the surface of a frozen lake inspired the mathematical concept of the plane, but the concept is an abstraction endowed with idealized properties. Mathematical theorems are about these abstracted objects. Science and applied math model actual things with these abstracted objects. And the Pythagorean theorem is a mathematical theorem, not a scientific hypothesis.
When a land surveyor applies the Pythagorean theorem to deduce some length, that surveyor is assuming that distances in the real world obey the relationship given by the theorem. This assumption is extra-mathematical and is part of what distinguishes math from science. In order to use math in science, scientists have to make some assumptions that connect the mathematical world to the natural world. But while science concerns itself with the “real world,” mathematics, strictly speaking, does not.
And yet, time and time again, people improperly call real world illustrations of the Pythagorean theorem, “proofs” of the Pythagorean theorem. No doubt you’ve seen the so-called “water-based ‘proof’ of the Pythagorean theorem.” (Such as this one or this one. Both are nice demos, but the first title is, strictly speaking, inappropriate.) If you understand this misconception, then you realize that such demonstrations are not proofs. (By the way, of these demonstrations, one can also note that many illustrate only a single instance of the Pythagorean theorem because they involve a very specific right triangle, whereas the Pythagorean theorem is a statement about all right triangles. That is a valid observation, but that is not the reason for the misconception I am now addressing. This problem of being too specific to capture the full generality of the theorem can be a problem with mathematical proofs too, such as if one were to prove the Pythagorean theorem in the specific case of an isosceles right triangle, then erroneously conclude that, therefore, the Pythagorean theorem is valid for all right triangles.)
The mathematical proof of the Pythagorean theorem does not depend on the geometry of the space of the world that we inhabit. Instead, it is about whether a geometric statement follows logically from some set of axioms.
The question of whether distances in the “real world” obey the Pythagorean theorem is a scientific hypothesis and can be checked experimentally. You could regard the “water-based proof” as just such an experiment. Like all scientific hypotheses, this one could be disproved some day, but it will never be proven.
If the water-based “proof” of the Pythagorean theorem isn’t really a proof, what is?
When a mathematical statement is proven, it is only proven relative to some set of axioms, which are statements made without any proof at all. Axioms are assumptions upon which a theory is woven.
Euclid’s Elements contains a valiant attempt to provide such a deductive proof by clearly stating axioms that are accepted as true and then showing a way to logically deduce, step by step, the Pythagorean theorem. Although subsequent analysis of the Elements has shown that not all assumptions used were explicitly stated, the procedure used is the right mathematical one: Axioms are stated, logical implications of these axioms are explored. And any statement that is logically deduced has been proven, relative to those axioms.
In the case of the Pythagorean theorem, many proofs have been supplied. Alexander Bogomolny has collected 99 of them on his extensive website Cut The Knot. Now, most of these proofs do not supply detailed logical deductions all the way down to foundational axioms of some geometry. Doing so would make reading the page as unpleasant and frustrating as swimming through peanut butter. Instead, they bridge the theorem logically to other mathematical results. The presumption is that the reader either accepts that these other results have been proven (and proven without using the Pythagorean theorem), or can prove or find a proof of them.
The important point is that no theorem can be proven true or false in the absence of axioms.
The next time you’re asked to prove something in math, perhaps the first thing you should say is, “what am I allowed to assume?”
And this leads to the following modification of the student’s question:
Are there any interesting collections of axioms from which the Pythagorean theorem can be proven to be false?
The short answer is “Yes!” The long answer is the story of non-Euclidean geometry.
There’s so much more to say, and others say it so much better than I. So here I’ll refer interested parties to two books:
There’s much more out there, but hopefully, these two will serve as a good place to start.