Is 1 + 1 = 2 subject to proof, or is it a matter of definition?

[Liberal]

@Orbifold

However, since I have said in this very thread both “I agree that it’s entirely possible, logically speaking, that one day gravity will reverse and the Earth will spontaneously explode”, and “Yes, a scientific hypothesis can be overturned at any time by new data”, why are you addressing this comment to me?

At this point, I don’t even know!

Let me just ask you : do you believe that a scientific test of any kind can determine the truth — TRUTH, mind you! — of the proposition that 1 + 1 will always equal 2?

[Polerius]

@SentientMeat

One can only define one object next to another as two objects. You don’t discover it by observing the consequences.

Did you just, ahem…, say that we can only define that one object plus another object equals two objects?

[Liberal]

@Polerius

Did you just, ahem…, say that we can only define that one object plus another object equals two objects?

Yes, he did. But no one is saying that 2 isn’t defined. It is defined as the successor of 1. Definitions don’t have truth value, and don’t prove anything. What the Peano proof shows is that, if 2 were the value defined, and if addition were what it is defined as, then 1 + 1 would equal 2. EVERY word has a definition. That doesn’t mean that definitions are invoked as inferences.

[Orbifold]

@eagle

OK, Orbifold, let’s settle this between us: I can agree with that, as long as you agree that scientifically prove 1+1=2 through an experiment for example with pebbles not actually proves 1+1=2 (which can only be proved mathematically) but instead it proves the scientific hypothesis, that pebbles as cardinalities follow the ‘law’ of 1+1=2 (or more general of a+b=c, which can be further verified with more experiments).

Yes, I agree with that, although since there’s been some debate over the meaning of the word “proof” in a scientific context perhaps we should settle for “supports” or “provides supporting evidence for”.

@liberal

Let me just ask you : do you believe that a scientific test of any kind can determine the truth — TRUTH, mind you! — of the proposition that 1 + 1 will always equal 2?

No, I do not. Since I have quibbled with you over the meaning of “proof” in this thread, I feel the need to be precise and say that no scientific test will ever establish with mathematical and logical certainty that 1+1 will always equal 2 in the real world. Which seems to be equivalent to the question you are asking, as you mean it.

If you had asked a different question, namely “Do you believe that repeated scientific tests can establish the proposition that 1+1=2 to within the standard of evidence required by the principles of science?” then I would answer yes. But that is, of course, a different question, and one on which I think we agree as well.

[Liberal]

@Orbifold

No, I do not. Since I have quibbled with you over the meaning of “proof” in this thread, I feel the need to be precise and say that no scientific test will ever establish with mathematical and logical certainty that 1+1 will always equal 2 in the real world. Which seems to be equivalent to the question you are asking, as you mean it.

If you had asked a different question, namely “Do you believe that repeated scientific tests can establish the proposition that 1+1=2 to within the standard of evidence required by the principles of science?” then I would answer yes. But that is, of course, a different question, and one on which I think we agree as well.

Yes, we do agree. Completely.

Now, I’m going to throw a monkey wrench into it. All deduction (which is what proves that 1 + 1 = 2 is true) begins with induction. That’s what the premises (or postulates or axioms) are — logical propositions (or assertions or statements) that are offered without proof. Seldom are axioms controversial because they’re usually things like A=A, or other propositions that seem, to most people, to be self-evidently true. How are these axioms usually derived? Well, by definition they’re not deduced from anything else. Rather, they’re statements that we believe fit with our observations.

So…

The proof of 1 + 1 = 2 requires Peano’s fifth axiom, also called the Induction Axiom. It basically states that every natural number has a successor. The proof of that? None was offered. It’s an axiom. But it was derived pretty much the same way as you say above. After sufficient trial, and based on simple common sense, it is induced that no matter what natural number you name, you can increase it by one. So, although the proof is a deductive proof, it — like all deductive proofs — stands on the foundatoin of induction.