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:

Worlds Out of Nothing by Jeremy Gray

Non-Eucliden Geometry by H. S. M. Coxeter

There’s much more out there, but hopefully, these two will serve as a good place to start.

One can think of Pythagoras theorem as a physical theory (empirically verifiable), that for small scales enables you to “predict” the length of the hypotenuse of a triangle, using previously measured lengths of it sides. Using bits of string, plumblines you can create three physical points of a right angle triangle and what pythagorases “theorem” predicts is that the square root of the lengths of the two shorter sides (an abstract mathematical operation on two numbers) just happens amazingly (to me anyway) to give the same number as a direct measurment of the length of the hypothenuse using (say) a piece of string. To me the various “geometrical proof” beg the question by assuming implicitly some property of geometrical space that is characterized by its metric being pythegorean in the first place. Please, please tell me where I am wrong because a lot of people I have spoken to about this seem to just nod their heads but seem not to understand?

If one is able to define a physical notion of length and angle, then one can certainly hypothesize that the 3 lengths given by the sides of such a physical right triangle are related by the Pythagorean relation. For clarity of communication, this thought shouldn’t be referred to as the “Pythagorean theorem,” but instead as, perhaps, the “Pythagorean hypothesis” or the “Pythagorean law.” That is, thinking about whether the lengths of the sides of some physical right triangle obeys the Pythagorean relationship versus thinking about the mathematical assertion that is the Pythagorean theorem, are quite different. It would create confusion if the two distinct ideas are conflated. (Notice that if space turned out to be non-Euclidean, that would not disprove the Pythagorean theorem!)

By the way, your suggested Pythagorean hypothesis is a local one (since you restrict to “small scales”) and I think that there is an important distinction between local and global, so the specific thought you describe in your first sentence could be referred to as the “local Pythagorean hypothesis.” Also, defining a physical notion of length and angle is already a challenge. You use string to measure length, though the current SI definition of the meter is in terms of light travel (which brings in the complication that light is influenced by gravity). The book “Worlds Out of Nothing” has a lot to say about this.

Regarding your second to last sentence, I’m not exactly sure to what question you are referring in “beg the question.” If the question you are referring to is whether physical space is locally Euclidean, then I can certainly understand why the Pythagorean theorem might compel one to wonder about this question, and even wonder about it irrespective of how the Pythagorean theorem is proven.

You are correct that in an axiomatic system where the Pythagorean theorem is true, there is the implicit assumption that the geometric space is Euclidean. All mathematical theorems are tautologies (at least, if the theory is sound). One of the fascinations of math, I think, is that there are so many tautologies that are surprising, and there are surprising sets of axioms that seem consistent (such as the ones that give rise to non-Euclidean geometries).