Mathematics

My understanding is that we can use systems like Peano Arithmetic to prove the seemingly basic truth that 1+1=2. Do such proofs actually give us reasons to believe that 1+1=2 that we didn't have before? Are they more fundamental or compelling than whatever justification a mathematically-naive person would have to believe that 1+1=2?

Comments

There are genuine philosophers of math on the Panel, but while we wait for them to respond I'll take a stab at your questions, which are epistemological as much as they're mathematical. I think we can answer yes to the first question without having to answer yes to the second question, but the answer to both questions may be yes.

As I understand the Peano Proof that 1 + 1 = 2, the gist is that the definitions of 'successor', 'addition', and '2' imply that 1 + 1 = 2. The successor of 1 is defined as 2, and addition is defined so that the result of adding 1 to any number is the successor of that number. Therefore, the result of adding 1 to 1 is 2. If the Peano Proof constitutes a reason to believe that 1 + 1 = 2, then it's surely a reason we didn't have before we had the Peano Proof. So I (somewhat tentatively) answer yes to your first question, regardless of the answer to your second question.

Even if we grant the infallibility of the deductive inferences in the Peano Proof, the Proof depends on Peano's postulates. So the answer to your second question depends on whether those postulates are 'more fundamental and compelling than whatever justification a mathematically naive person would have to believe that 1 + 1 = 2'. That, in turn, depends on what the mathematically naive person's justification is. In my answer to Question 5387, I argued that actual or possible processes of physical counting and aggregation are only fallible grounds for believing arithmetical claims such as 1 + 1 = 2. If those are the mathematically naive person's grounds, then they aren't as compelling as infallible grounds would be.

Are the Peano postulates infallible grounds? Some would regard that question as ill-posed because, they say, a postulate is just a stipulation and therefore not something that can be objectively true or false. But I'm not so sure. Euclid's Parallel Postulate helps to define Euclidean geometry, and in that sense one can regard it as merely a stipulation. But the Parallel Postulate is false if construed as a claim about any possible space and may actually be false as a claim about the physical space of our universe; in that sense it has an objective truth-value. So too might the Peano posulates have objective truth-values, in which case it makes sense to ask if they might be false. For what it's worth, I can't see how they could be false. But, as I said earlier, I can see how physical counting and aggregation might fail to confirm the claim that 1 + 1 = 2. So I tentatively answer yes to your second question as well.

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