We easily recognize a familiar face in a group of people.
We recognize her face in a moment, without analyzing the details of the face. In fact, if someone has asked us to describe features of that face, or to answer questions about it while not looking at it, there is a good chance that most of us would fail to do so. I’m thinking here about features like color of the eyes, the form of the lips, the length of nose and so on.
I would say that as far as we can answer such question or provide descriptions, it is because either we can bring that familiar face from the memory before our “inner eye”, or because we had explicitly attended to that feature in the past (we might have had intentionally focused on it, or it attracted our attention).
When we see a person’s face, we don’t see it as an aggregate of multiple features, but we see it and later recognize it as a whole, as a gestalt. But while we can remember and recognize things as a whole without learning about specific features, still in any case when the face is before us, it is there as analyzable. When we look at it, the features are there – open to our possibility to focus on them, open to our skills for analysis which may be “triggered” either spontaneously or by some kind of reflex (e.g. when something attracts our attention).
And in those cases we can attend to different things, we can attend to one eye, or to the other eye, or to both eyes at once, we can attend to something which we previously didn’t notice – for example we can attend to a small part of the curve of the left eye, or to the relative brightness of a specific place vs. another, or to the number of speckles on the right cheek, etc…
But while we attend to one of the things, our consciousness is not limited to it. As when at the start the gestalt was there as gestalt presenting the possibility for wealth of different abstractions; now when we attend to a specific abstraction (feature), it is there as an abstraction – as a part of the whole which is still there to return to.
So, let me now turn from familiar faces and gestalts to a priori truths.
In a previous post I was wondering if some tests in developmental psychology might be taken as a hint that the infants believe that 1+1=2 based on intuition. While it might be taken as a hint, as it was pointed in the comments by Pete and Curtis those experimental results hardly show anything decisive – one can explain the results in different ways which wouldn’t include any mention of intuitive truths.
But let me look at the issue in the context of the previous discussion…
When we have in front of us two things, we can see them as a gestalt, as “two things”, something which we can see and recognize. But the same gestalt, which we can name as “two”, is also analyzable – it is implicit in that gestalt that I can attend to the one or to the other of those two things. And while attending to one of them, as in the case with the face, the other one is not disappearing – it is still there. And I can spontaneously change the way I attend to the whole, I can focus on the other one. So, in that whole of two things, I can switch my attention, look at the both things as a whole (two), or I can attend to each of them separately.
In this way, we are presented with the a situation, in which the same gestalt is analyzable either as one two, or as two ones. And that goes for any gestalt which I can imagine – as long the gestalt is two, it will present possibility for attending to two separate ones (among which we can switch the attention).
This a priori relation of possibilities is there, be it if what we characterize as two is in front of us, or if as in the case with the experiment done with the kids mentioned in the other post, one or both of the things are tracked (as hidden behind the screen), or even if we have imaginary gestalt.
This is, I think, what is behind our intuitive understanding of what we express by 1+1=2. The equation shouldn’t be taken as identifying two separate sides, namely a)1+1 and b)2 , but as expressing that the whole, if it is characterized as two, can be also characterized as one and another one. The identity is in the whole, and the equality is expressing the necessity that in every possible world the whole which is 2, is also 1+1 and vice versa.
And at the end, let me finish with a doubt I have…
While I’m pretty convinced that I have grasp of the a priori truth of the equation that 1+1=2 (meaning what has been just described), I’m not very sure what to say about the issue if the truth of 1+1=2 is analytical.
On one side, it seems to me that the potential to analyze the whole into separate things (i.e. to focus on the one and the other) is what is required for something to be named as two, but on other side having on mind the case with the face, I’m thinking that a whole might be recognized as two even without explicitly or implicitly being aware of those possibilities.
A brood comb 
Tanasije,
This might be a bit off topic, but I’m fascinated by your comments on gestalt and its relation to the a priori. So, I hope these comments help and not hinder your thoughts on the matter.
I bet there is an alternative – one could read that word as ‘odd’ – a priori interpretation of the two objects. The alternative rendering could show that there are actually 4 (or possibly more) objects.
Let’s call one object p and the other object q. p is one object, and q is another object. Together, they are two objects.
But there is also a third object. That is, there is an object composed of p, q, and the two objects p and q.
Also, there might be a fourth object. Suppose that p is to the left of q. That means there is “p is to the left of q”, “q is to the right of p”, p, and q. Thus, there is a fourth object.
Based upon that there might be a fifth object. The fifth object is composed of “p is to the left of q”, “q is to the right of p”, p, q, and “p and q.” Thus, there is a fifth object.
You’re on to something here when you say that 1+1=2 is not analytic in the way we might say that “all bachelors are unmarried males” is analytic.
(I’m pretty sure that Hilary Putnam has an example like this in his work, but I can’t remember where it’s located. I remember it as him talking about 3 marbles in a bag and asking how many objects were in the bag. He suggested 7, which is based on a similar analysis as the one I offered above about p and q.)
Hi Joe,
Thanks for your comment and nice words.
I would agree that there are multiple ways to attend to a given whole (a pair), which points to different a priori relations between possibilities for different abstractions (interpretations).
Let me connect to what you say, and mention one issue I thought to also discuss in the post, but decided for the sake of shortness and clarity not to.
Namely, I mentioned that one can attend to the whole as a pair (and in that case we would have the right side of the 1+1=2 equation, i.e. ‘2’), but also that there are two separate ways to attend to the parts. One can attend to p, where p is “the one”, and q is “the other one”; and also one can attend to q, where q is “the one”, and p is “the other one”.
Now in concrete situation it is true that there is lot of determinations for p and q, e.g. p is to the left of q, q is to the left of p, p is bigger (e.g. apple), q is older (person), and so on.
However in the case of 1+1, we do abstract from this possibility for other determinations, and in such attending the two ones are not determined in any other way except by “being another to each-other”. In such abstract equality it is same for our spontaneous decision which we would take as “the one”, and which would be “the other”. So, abstracting from other determinations, there is a symmetry in which they are reduced to same abstractions from the pair.
While this might not very interesting in the case of 1+1, in the case of 2+1=3, there are few ways (3 to more exact) to attend to the whole, and which might be described by 2+1. However because of the abstract symmetry of the ones among each other, they are all covered by the formalism – 2+1.
One other thing to point to, is that when we attend to “the one” of the whole and thus abstract from “what is left (a pair)”, we can also attend to “the other (pair)” and abstract from “the one”. In that way we have symmetry 3=2+1=1+2.
If you are trying to connect the mechanisms by which people recognize faces and by which they determine the validity of arithmetic propositions, it seems to me that you are on tangent from what is generally known about facial recognition. This seems to be an ability specific to our species, by which I mean to say that it is humans only that seem to be able to discriminate among human faces with the amazing precision we do, and that this ability is something that is hard-wired and can be lost by pathologies affecting brains. A tumor in just the right place can affect and even destroy this ability. I think Oliver Sachs has reported this in “The Man WHo Mistook His Wife for a Hat.”
The mechanisms? The cogs and wheels? Of course not! I don’t even believe that “mechanism” is good word to describe whatever is happening, but anyway the thinking here goes in another direction.
The faces were taken merely as an example in which we can focus on the “whole thing”, and then focus on some part of the face. I could’ve discussed a complicated texture and focusing attention on some part of the texture, seeing a piece of furniture and focusing attention on some specific thing about the furniture, or, which is closer to the case of math, seeing a bunch of ants and focusing attention on individual ant.
On the point that we have innate abilities, I fully agree. Abilities to focus on wholes or their parts, or focus attention to multitudes or to individuals which are part of those multitudes, to recognize different kinds of things (like faces), to recognize what other people look at, to track things which go behind other things, and so on…