My background is in mathematics* and now I teach mathematics for a living. As any other math teacher can attest, this means that I teach students how to do proofs.
*Two thoughts here:
1. I cannot fathom a more disinteresting five words to open a blog.
2. My background is actually in video games, sarcasm, and an obscene amount of coffee. (I highly recommend Ethiopian Yirgecheffe if you get the chance.) I once read somewhere someone’s definition of mathematician as “a machine for turning coffee into theorems.” Seems a bit too on the nose…
Doing so inevitably leads to the following conversation:
Student: “Sir? What actually is a proof?”
Me: “Y’know something? That’s a really good question. Why don’t you wait until we’ve done a couple first. That way, the definition will make a bit more sense. OK?”
Every parent out there knows that this is the official answer for “I don’t have the foggiest idea.” So all I need to do is stall them long enough for them to forget their original question. And, fortunately, goldfish have longer attention spans than students.
Thus, I don’t really ever need to know what a proof actually is.
…
…
…
At least, I didn’t need to know…
A few years ago, I needed to do some research about geometry teachers.*
*And there it is, friends: the single most depressing sentence in history.
In particular, I needed to research their opinions about using proofs in the classroom.*
*This sentence was also in the running for “most depressing sentence,” but only took the bronze. By the way, second place was awarded to the group of sentences that begin with “As the parent of a junior high boy…”
Have you ever heard the quip, “Opinions are like noses; everyone has one.” Well, definitions of proof among mathematicians are the exact opposite: no one has one.
It’s almost like we math teachers have formed this educational cabal in which the First Rule of Defining Proof is “never talk about defining proof.”
Thus, I had to turn to mathematical philosophers for some ideas about the definition of proof.*
*Good grief… Just look at these sentences. This is turning into the 2020 of blog posts.
Well, believe it or not, someone actually did try to give a definition.
The title of Reuben Hersh’s paper kind of gives away the answer: “Proving Is Convincing and Explaining.” I say kind of because “convincing and explaining” are more what (good) proofs do, rather than what they are.
*Reuben Hersh. “Proving Is Convincing and Explaining,” Educational Studies in Mathematics 24 (1993): 389-399.
Fellow nerds can read the whole thing at https://moodle.tau.ac.il/2018/pluginfile.php/316018/mod_resource/content/1/Hersh%201993%20ESM.pdf
But by page three of the article, it becomes a bit depressing (common theme, eh?) to find out that Hersh has three definitions of proof. Well, I guess having three definitions to sort through is a lot better than the “none” that I had before.
His first definition is rooted in the etymology (big Greek words!) of the word prove. The Latin probus means “good”; the subsequent Latin probare means “to test” or “to approve” or “to demonstrate.” The resulting English word prove is a combination of these two ideas: essentially, “to demonstrate the goodness” (or “soundness” in this context).
Hersh summarizes this with his first definition of proof:
(1) Test, try out, determine the true state of affairs.
This use of the words usually applies more to things than it does to people or ideas–and almost in a passive sense. For example, the maxim “The proof is in the pudding” is actually a misquote of a longer statement: “The proof of the pudding is in the tasting.” In other words, the quality of the pudding (a thing) is demonstrated (is proved) by being tasted.
Well, that first definition is fine, I guess. But I usually like to browbeat people with something like “Prove it to me!” And the definition above just doesn’t work in this case. (Though I will certainly accept gifts of pudding at any and all times.)
Hersh’s THIRD definition is the type of monstrosity that could only be cobbled together by something equally monstrous…mathematicians. And a cobbled-together Frankenstein it is. For though I said earlier that no mathematician has a genuine definition of proof, Hersh was able to take the fleshy, sub-human concepts that mathematicians have about proof and put together this hideous eyesore:
(3) A sequence of transformations of formal sentences, carried out according
to the rules of the predicate calculus.
The less said of this, the better.*
*Perhaps we should just call this third definition “The Definition That Shall Not Be Named.”
But the SECOND definition really cuts to the core of what is usually meant by the word proof:
(2) An argument that convinces qualified judges.
The more I look at this definition, the more I just love it. Of course, I’m looking at this as a definition that I can give in the math classroom, and this definition perfectly describes what we are trying to do in the mathematical community. Yes, the Definition That Shall Not Be Named is perhaps more precise (at least, to the two people on the planet who know what the words in that definition mean) in a mathematical sense, but every mathematician still thinks that certain proofs are more…well…”beautiful” than others.
For example, there are dozens of proofs of the famed Pythagorean Theorem. The website linked here has–and I kid you not–118 proofs of that theorem. Some of them are agonizingly stale,* but some of them are just more aesthetically pleasing (at least to mathematicians). I, for one, am a little bit partial to version #3. Version #5 is also pretty interesting–especially once you realize that it was first proposed by President James Garfield in the late 1800s.
*OK…most people would say that all 118 are agonizingly stale.
But what makes a proof more…uh…”beautiful” than another? Well, beauty certainly is subjective (“…in the eye of the beholder”), but certain proofs do more than just convince: they explain. Proof #3, for example, not only demonstrates that the Pythagorean Theorem is true, it helps the reader visualize why the pieces fit together to make it true.
This is comparable to one mechanic who tells you that a certain sound indicates a certain problem versus another mechanic who actually shows you the two parts grinding together. In each case, you might know that the car has a particular problem, but only in the second case would you truly understand why.
Thus, an argument that explains while convincing just resonates more than an argument that only convinces.
Likewise, I prefer the “an argument that convinces qualified judges” definition of proof FAR more than either of the other two. Why? It accomplishes the two-fold goal of first stating what a proof is, but also stating what a proof does.
Now, you may disagree with the definition, but I think this gives us a good test case: I will try to prove to you that this definition of proof is sound.*
*Those of you with some experience in logic know that “proving a definition” is a logical impossibility. But what I’m doing here is demonstrating (i.e., “proving”) why one definition is preferable to another.
Unfortunately, I’m already over 1000 words and my “proof of prove” has at least three parts. Thus, I will pick up with this next time.
But let me finish by trying to put all of this in context. In my last post, I mentioned about the Scarecrow from The Wizard of Oz receiving a mere piece of paper…a diploma…a degree (the delightfully ridiculous “Doctor of Thinkology”).
The last part of the definition of proof is “qualified judges.” What makes one a “qualified judge”? A piece of paper?
Quiet Contempt 