Is 1+1=2 Intuited?

Over at Brain Hammer, Pete Mandik as a part of his post Abusing the Being-Knowing Distinction proposed that a following principle can be valuable in attacking claims of intuited metaphysical truths:

It is intuited that P. Philosophical theory T explains P. But philosophical theory T is inconsistent with the fact that P is intuited. Too bad for T!

In the comments of the post I wondered if the following example would fall also under that principle:

  1. It is intuited that 1+1=2
  2. Russell and Whitehead gave theory of why 1+1=2 in Principia Mathematica
  3. If 1+1=2 was true *because* of the logic in Principia, nobody could’ve possibly intuit its truth.

Pete raised the issue if 1+1=2 is intuited at all. I said I believe it is intuited, and Pete said that he believes that it is learned.
As I don’t want to abuse the comment-space over at Brain Hammer for discussing a question which is just marginally connected to the central topic of the post, I will put my thoughts on it here.

The question of what is a belief, and the issue of when we can say that one beliefs something is surely problematical.
When we say that a person P believes that X, we don’t e.g. expect that P continuously contemplates on the truth of X. Probably we will require only that in certain cases, in the right conditions, his belief that P to be actualized somehow, affecting what the person says, what the person does and so on. Based on that we can talk abut dispositional vs. occurrent belief. If that is so, then we can say that the person doesn’t have actually to claim that 1+1=2, in order to count as believing as 1+1=2. But if the person doesn’t claim that 1+1=2 (or sincerely report that he believes that 1+1=2), how do we get to know that person believes that 1+1=2?
One behavioral sign besides actually P claiming X, might be that P is surprised if he/she observes a situation in which X is shown false. While of course, this behavioral act doesn’t have to mean that P believes that X (generalizing beliefs to dispositions to act would fall into some kind of behaviorism), it is clear that it can be taken as a good indicator.

The question if this kind of behavior can be found in human infants has been tested by experiments. Wynn K.(1990)1 did experiments where he presented the following sequence to four and half month olds.


After this, two outcomes were presented, which are marked in the following picture as a)possible outcome and b)impossible outcome, and the time infants kept looking at the outcome was measured in each case.


In psychology, the sufficiently longer time in those cases is taken to represent a surprise. And that’s what Wynn got for the impossible outcome. The infants looked longer at it.

While I’m not very  knowledgeable about the field of developmental psychology, it seems to me that if those results are true and if the interpretation of the results is right, it isn’t very unreasonable to talk about infants having belief that 1+1=2. However because this is shown for very early pre-linguistic age, it makes it problematic to speak of a learned theory.

1Wynn, K. (1992) Addition and subtraction by human infants. Nature, 358: 749-750, reference taken from Lakoff&Nunez (2000) Where mathematics comes from p.16-17