If you have a middle-school-aged child, you’ve probably endured countless conversations where you think you’ve clearly explained your point, but it is always answered with a “Yes but”, and a further rationalization. Recently I was in such a situation, trying to convince my daughter to scoop the cat litter, and descending down an infinite regress of excuses. It occurred to me that this conversation was very similar to one that Achilles had with the Tortoise in Lewis Carroll’s famous 1895 dialogue, “What the Tortoise Said to Achilles”. I was actually surprised to realize that I hadn’t yet recorded a Math Mutation episode on this classic pseudo-paradox. So here we are.
This dialogue involves the two characters from Zeno’s famous paradoxes of motion, the Tortoise and Achilles, though it is on a totally different topic. Achilles presents a pair of propositions, which we can call A and B, as he and his pet discuss an isosceles triangle they are looking at. Proposition A is “Things that are equal to the same are equal to each other.” Proposition B is “The two sides of this Triangle are things that are equal to the same.” Achilles believes that he has now established another proposition, proposition Z: “The two sides of this Triangle are equal to each other.” But the Tortoise is not convinced: as he states, “I accept A and B as true, but I don't accept the Hypothetical”.
Achilles tries to convince the Tortoise, but he believes there is an unstated proposition here, which we will call Proposition C: “If A and B are true, then Z must be true.” Surely if we believe propositions A, B, and C, then we must believe Proposition Z. But the Tortoise isn’t convinced so easily: after all, the claim that you can infer the truth of Proposition Z from A, B, and C is yet another unstated rule. So Achilles needs to introduce proposition D: “If A, B, and C are true, then Z must be true.” And so he continues, down this infinite rabbit-hole of logic.
On first reading this, I concluded that the Tortoise was just being stubborn. If we have made an if-then statement, and the ‘if’ conditions are true, how can we refuse to accept the ‘then’ part? Here we are making use of the modus ponens, a basic element of logic: if we say P implies Q, and P is true, then Q is true. The problem is that to even be able to do basic logical deductions, you have to already accept this basic inference rule: you can’t convince someone of the truth value of basic logic from within the system, if they don’t accept some primitive notions to start with.
One basic way to try to resolve this is to redefine “If A then B” in terms of simple logical AND, OR, and NOT operators: “If A then B” is equivalent to “B or NOT A”. But this doesn’t really solve the problem— now we have to somehow come across basic definitions of the AND, OR, and NOT operators. You can try to describe the definitions purely symbolically, but that doesn’t give you semantic information about whether a statement about the world is ultimately true or false. Logicians and philosophers take the issue very seriously, and there are many long-winded explanations linked from the Wikipedia page.
I personally like to resolve this pseudo-paradox by thinking about the fact that ultimately, the modus ponens is really just a way of saying that your statements need to be consistent. For any reasonable definition of “implies” and “true”, if you say P implies Q, and claim P is true, then Q must be true. You might nitpick that I haven’t defined “implies” and “true” in terms of more primitive notions… but I think this is just an instance of the general problem of the circularity of language. After all, *any* word you look up in the dictionary is defined in terms of other words, and to be able to exist in this world without going insane, you must accept some truths and definitions as basic building blocks, without having to be convinced of them. Hardcore philosophers might object that by accepting so simple an explanation and blindly using modus ponens willy-nilly, I’m being as stubborn as a Tortoise. But I’m OK with that.
And this has been your math mutation for today.