Mathematics is built out of arguments. These are normally logical arguments, sequences of things which we say are true. We know they’re true because either they start from something we assume to be true or because they follow from logical deduction from things we assumed were true. Even calculations are a string of arguments. We start out with an expression we’re interested in, and do things which change the way it looks but which we can prove don’t change whether it’s true.
A fallacy is an argument that isn’t deductively sound. By deductively sound we mean that the premises we start with are true, and the reasoning we follow obeys the rules of deductive logic (omitted for clarity). if we’ve done that, then the conclusion at the end of the reasoning is — and must be — true.
A deductively sound argument is a wonderful thing. It tells us something we know must be true. That something will be true not just here and now — the way we might know “today is the 5th of June” is true — but always. It would be true anytime, anywhere in the universe. For that matter it would be true in other universes, if they existed. If nothing existed then … well, it’s very hard to say what would be true if nothing existed. Let’s step cautiously back from talking about “nothing”.
If we start out from a premise that isn’t true, or if we use a reason that isn’t part of the rules of deductive logic, then the argument is fallacious. It’s simple as that.
That isn’t to say the conclusion is wrong, by the way. If I present to you the argument, “I am a competent mathematician and I tell you that 53 is a prime number. Therefore 53 is a prime number”, I am making a fallacious argument. The conclusion is correct, but the fact that I’m a competent mathematician doesn’t prove that. Among other things you don’t know that I actually am a competent mathematician and not just a madman with a blog. You also don’t know that I’m not making a mistake or a joke or playing a prank. (This is a good point to look up “Grothendieck’s Prime”.) I promise you I am at least competent, and not making a mistake or a joke, and that 53 is prime. (I suppose I should double-check that before publishing). But I’ve given only a fallacious argument to prove it to you.
A fallacious argument can still be convincing. If you write to the United States Department of State and ask, and they tell you that Samuel Tilden was never President of the United States, that would convince nearly everyone that Tilden never was President. Formally speaking, that’s a fallacious argument. It’s even the same kind of fallacy I made with 53 up there. But you’d be perverse to insist they might be wrong about this.
Now … any interesting mathematical conclusion has a lot of reasoning in it. Nobody ever identifies every single bit of logical deduction, because that requires breaking the argument down into so many steps, each making so little progress, that the argument would be unreadable. We write out instead an argument that we are confident can be proved logically sound. How do we know there isn’t a fallacious step somewhere in the argument?
Normally, we know because we build interesting mathematical arguments out of smaller, simpler arguments that we proved already. Sometimes we proved them as mathematics majors or as graduate students. Sometimes we proved them by finding a book where someone who wasn’t us actually proved it. Or we start from an argument that is very much like one of these smaller, simpler arguments already proved, and trust that however we varied things isn’t going to change the soundness of the argument.
Much of the time this works out. Sometimes we’re wrong, and erratas or withdrawn proofs follow, chased down by a lot of sulking and fuming. Sometimes we’re even lucky and the reason that our proof was wrong turns out to be interesting, and to lead to new discoveries. That happens most rarely, but it almost makes up for the pain of finding the fallacy.