Thanks for the interesting comment. Let me say first that it is amazing for me of how much people are extreme empiricists. I would never have known that a priori.
I think your comment about conditioning where the infants won’t get surprised by the impossible outcome is important, and shows that the example I gave is not very big argument in the discussion of the intuitive belief that 1+1=2 (e.g. probably if you repeat the “impossible outcome” lot of times, infants will eventually get more surprised by the “possible outcome”). And it might be even more easier to teach children to pronounce that 1+1≠2.
I still think though that 1+1=2 is a priori. I will try to explain my reasoning in the next post.

1.I’m not sure how the experiment with 234 figurines would work. Do such experiment on me, and remove 10 of them while the screen is up – and I won’t be surprised when the screen is dropped. Most of the people won’t be. Now, it might be possible that I count the things while they are carried one by one behind the screen, and then count them after the screen is dropped, but that would necessarily include language (as in the counting I would pronounce the numbers in my head).
Now, I accept your point that it is weird to say that those babies had belief that 1+1=2 without having the abstract notions (equals, addition), and it is puzzling to me too, but here is argument that the intuition shown in those children that when you get one thing, and add one more thing, there should be two things, is general…
One can take different kinds of figurines, fruits, and things, and repeat the experiment with the infants, or do some other configuration of experiments, and one should get the same result. Now, what is our understanding that 1+1=2, if not this understanding that in this kind of generalized case we would expect every time what the infants are shown to expect also? We can just express it in abstract notions. That’s why I wonder if there is some non-linguistic behavior which might help us distinguish our belief that 1+1=2, from whatever expectation that babies have and which makes them surprised.

2.What I’m interested in is this – if some person didn’t learn by being taught, how would one learn it? You gave two examples of how something can come to be without being taught. One is inventing (like inventing the chess game), and the other is learning empirical truth (“there was a nail sticking out of the wall”). But I can’t imagine how would any of those be applicable of the untaught learning of 1+1=2. Would be like in the chess case, that “originator” of 1+1=2 would decide that 1+1=2 (could someone else decide that 1+1=3) ? Or would it be like in the empirical case, where someone is noticing that in all cases where you put together one and one more, you get two? But in this second case, that seems already present in every infant. I don’t think you wanted to say that it is either of those cases, but I’m just explaining what I’m interested in.

1. I already gave a hint in an earlier comment: do the baby experiement with an arbitray number of figurines, like 234. That would help show, without linguistic report, that the person had beliefs about addition as such.

2. I don’t know what “empircal” or “concrete” has to do with it. I gave a definition of intuition (over at Brain Hammer) in terms of knowing not based on learning. More on this below.

3. I can’t conceive of a chess game that involves three players and three colors. Such a game might be chess-like, but it wouldn’t be chess. How do I know what the essence of chess is? Because I learned it from people who learned it from people who invented it. At no point does anything worth calling intuition come into play, even though lots of what I can or can’t conceive does. Therefore, your inability to conceive of 1+1 equalling anything other than 2 shows nothing relevant to the question at hand.

1.If it isn’t about the words (or symbols), can you think of any other kind of behavioral non-linguistic dispositions which would show that a person believes that 1+1=2?

2.I can easily see how one can empirically come to belief that there was a nail sticking out of the wall, but how does one come to learn that 1+1=2(without being taught)? Maybe the concrete cases on which one can learn that 1+1=2 can point to behavioral tests for the belief that 1+1=2. (from point 1)

3.While I can conceive that there wasn’t a nail sticking out of the wall, I can’t conceive that one and one is not equal to two. Doesn’t that point to the intuitive base of my understanding of 1+1=2?

1. No, this isn’t about words at all. It is about the general competence that needs to be exhibited in grasping
number, addition, and equality. If someone’s knowledge is insufficently general to grasp for some arbitrary number what happens when you add one to it, then they don’t grasp what adding is.

2. Learning doesn’t require being taught. Another way of learning something is by discovering it yourself. I learned that there was a nail sticking out of the wall just by running into it. I wish some one else taught me to do otherwise, but unfortunately I had to learn it the hard way.

1.Are you saying that someone needs to know how to express in words what they believe, in order for them to be counted as believing something? Do you think that is the case in general, or particularly in this case? I’m inclined to say that someone can believe something, even without being able to express it using words. So, I would say that learning the language is necessary requirement for children to be able to express the belief, but is not necessary requirement for having the belief.

2.I agree that 1+1=2 can be learned, but if that’s the whole story about where the belief that 1+1=2 comes from, wouldn’t you need to assume infinity of teachers backwards in time.

I think that experiments like these do show that infants perceive and remember quantities. However, I don’t think they show that those infants know or even believe that one plus one equals two.

My daughter is 4 1/2 and I’m in the process of teaching her some basic arithmetic. She can count up to 20, and when I ask her, if “you have one and then another one, how many do you have” she has some difficulty but sometimes gets the right answer. If I ask her “what does one plus one equal?” she says “four” as often as she says “two” and “I don’t know”. I’m inclined to say, then, that she neither knows nor believes that 1+1=2 and further, when she eventually does know it, it will be because she learned it.

When does someone know that 1+1=2? When they grasp a sufficient amount of information to know what “plus” and “equals” means, and that further requires that they be able to figure out not only what 1+1 equals, but also what 234+1 equals.

I think it is clear that the infant experiement you cite wouldn’t work if you started with 234 figurines and then added one more figurine behind the screen.