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

9 thoughts on “Is 1+1=2 Intuited?

  1. Hi Tanasije,

    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.

  2. Hi Pete,

    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.

  3. Tanasije,

    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.

  4. Pete,

    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?

  5. Hi Tanasije,

    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.

  6. Hi Pete,

    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.

  7. I think we are probably a long way from knowing, with any sort of certainty, whether or not 1+1=2 is an a priori piece of knowledge. The concept is obviously extremely simple, but it might be possible to raise an infant under carefully controlled conditioning so as to create the axiomatic basis that 1+1=3 or =0. That is, the knowledge that 1+1=2 is a justified true belief because, in the context of daily life, it is repeatedly shown to be true over and over again. The concept is simple and is mathematically pure, so it is no surprise that even young infants have already integrated it into reasoning. Frustration or surprise at exposure to the idea that 1+1≠2 is, I believe, the result of discord with previously applied knowledge of one’s environment rather than a contradiction to some sort of intuition. Conditioning someone to accept 1+1≠2 would require careful (perhaps impossibly careful?) control of the environment precisely for this reason. An infant cannot define 1+1=2 in the mathematical language we use for such transactions, but I believe he already knows it to be true in application from experience, not from intuition absent of experience. It is also possible that this sort of knowledge is developed internally before birth, which, again, would not necessarily make it intuition, but would create difficulties in setting up the hypothetical conditioning to 1+1≠2.

  8. Hi Curtis,

    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.

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