I don't think you've responded to my linked comment. But OK, looking up a result in a math book could count as an experiment, as could any method by which you might learn about dyslexia or whatever you suspect might be confusing you. If you don't believe anything like that could happen to you, either you made that judgement based on experience and science or you are very badly misguided.

Ahem. I can think of many ways that some broadly defined "experimental method" could come into play there.

The point was less about the physical world applications of 2+2=4, and more about the fact that any belief you have is ultimately based on the evidence you've encountered. In the case of purely theoretical proofs, it's still based on your subjective experience of having read and understood the proofs.

Humans are sometimes literally insane (for example, not being able to tell that they're missing an arm). Also, even the best of us sometimes misunderstand or misremember things. So you need to leave probability mass for having misunderstood the proof in the first place.

(The followup to this post is this one: http://lesswrong.com/lw/mo/infinite_certainty/ which addresses this in some more detail)

Hi, I am a mathematician and I guess most mathematicians would not agree with this. I am quite new here and I am looking forward to reactions of rationalists :-)

I, personally, distinguish "real world" and "mathematical world". In real world, I could be persuaded that 2+2=3 by experience. There is no way to persuade me that 2+2=3 in mathematical world unless somebody shows me a proof of it. But I already have a proof of 2+2=4, so it would lead into great reform of mathematics, similar to the reform after Russel paradox. Just empirical experience would definitely not suffice. The example of 2+2=4 looks weird because the statement holds in both "worlds" but there are other paradoxes which demonstrate the difference better.

For example, there is so called Banach-Tarski paradox, (see Wikipedia). It is proven (by set theory) that a solid ball can be divided into finitely many parts and then two another balls of the same size as the original one can be composed from the pieces. It is a physical nonsense, mass is not preserved. Yet, there is a proof... What can we do with that? Do we say that physics is right and mathematics is wrong?

Reasonable explanation: The physical interpretation of the mathematical theorem is just oversimplified. This part of mathematics does not fit to this part of physics. The false statement about physics is just different from the true mathematical statement.

But the Banach-Tarski paradox has no physical equivalent. We can not test it empirically, we can just believe the proof. This is probably what I would think if my experiences showed me that 2+2=3. It would appear that in our real mysterious world just 2+2=3 but in mathematical world, which was designed to be simple and reasonable, still 2+2=4.

Similarly, we can guess whether and how the physical universe is curved, yet the Euclidean space will be straight and infinite by definition, no matter what we will experience.

Sure, it can be argued that if mathematics does not reflect the real world then it is useless. Well, set theory is a base for almost all math fields. Even though the particular result called Banach-Tarski paradox have no practical use, more complicated objects in the mathematical universe are used in physics well. Restriction to just "empirically testable" objects in mathematics is a counter-intuitive useless obstacle. In such view, there is no sixth Ackermann number or the twin prime conjecture has no meaning. I can barely imagine such mathematics.

I understand that you may want a simple way to handle theists but abandoning abstract mathematics (or calling it "false") is definitely not a wise one.

If we are the only civilisation to exist in the history of the Earth, then we will probably become extinct not in mild way, but rather in a way which will prevent any other civilisation from appearing. There is higher probability of future (man-made) catastrophes which will not only end human civilisation, but also prevent any existence of any other civilisations on Earth.

I don't believe that this follows. It is surprising that we are apparently the only civilization to so far appear on Earth, but if we accept that we are, we should not assume that we have accomplished this by destroying the future.

However, while I feel strongly that this is this case, I do not feel confidant that I can express it in a way that would be understood by someone who does not agree with me.

If someone could explain clearly why I am right, or alternatively, why I am wrong, I would greatly appreciate it.

(For context, here is what I would write given my currently semi-formed understanding: "while it makes sense to compare ourselves to a time-line independent view of the world to test the probability that our assumptions about the world are correct, it does not make sense to assume that our assumptions about the world will guide the future.")