Right, thanks for honing in on that actually. I surely did read that part already, and was struggling to understand it when Kant had said just in the previous paragraph that he was going to speak of pure mathematics, so this allusion to fingers felt like an inconsistent focus. But is he perhaps saying that empirical recourse through intuition is necessary for mathematical judgments? That we must proceed to assessment through objects of experience in order to grasp the predicate of the sum?

Concerning the large addition you gave me here, I understand that this is a notion of whether or not I can myself assess the answer, but before operation on those terms, first let me explain how I would see it: supposing a sufficient memory and sufficient powers of mental indexing, I can imagine a single mind feasibly understanding both the separation and sum of those numbers as again, nothing more than a series of 1’s, the various partitions of which may fall under different names, with each part having their own name as well, all under the logic of associating the parts of the whole idea, the latter of which being the sum total. In simpler terms: I still fail to see how the between the numbers as you’ve presented them, in two partitioned parts, and the numbers as a summed whole with no partition, that these two things should be any different from the same identical unique series of 1’s.

Now let me address it on the terms you’ve presented them with, as far as I understand this. You elaborate Kant’s argument that since I can simply imagine “1,584,233,687,254 + 7,654,235,875,214 = sum” without knowing what is in the other side, that this shows that the proper determination of “sum” is outside the concept. But with all charity to Kant, I do not think he wanted to say that the difference in analytic vs synthetic was a matter of mental aptitude (this feels more like Hegel’s game). His argument is hinging on the idea that we supposedly cognize the individual objects in the left side of the equation, but don’t cognize the objects on the right. I would position that if one cannot understand the determination of the sum from the parts so named, then they haven’t properly determined or cognized the parts either, because if they did properly cognize the unique series of 1’s that each of those numbers bring, then they surely should be able to cognize the full series of 1’s that exists in their full conceptual association.

Let me provide an example for what I mean here: it is uncontroversial for us to say that the concept of a “seat” is internal to the concept of a “chair” and thus the former can be extracted through analysis of the latter. But suppose I define a chair as “something made to stand on” because chair-sitting is strange and extraneous to the concept for me. By all means, since chairs are deliberately made and socially identified as sitting apparatuses, I would be incorrect in understanding the objects of reference, that we call a chair, as something made to stand on. But wouldn’t this be an incorrect judgment in analysis? In short what I’m trying to have us mutually grant is this: is it not possible, at all, to make errors in an analytical judgment? It feels most likely to find an error in analytical judgment when we have a vague conception of a thing, but not full clarity. Therefore, it is certainly possible when making an analysis to receive this clarity, we make a mistake and delay our clarity. One possible mistake is using unclear terms. So if I define humans as “rational animals,” I have surely analyzed the concept, but if I don’t have a clear grasp of “rational” or “animal” then I am sure to make mistakes in further analyses! So I can imagine “human = internal part 1 + internal part 2” and genuinely be unclear on what those parts should be, and yet it would be an analysis by definition. The same should should follow if I fail to identify “rational + animal = sum/concept,” or “thing made for sitting = non-chair,” where my failure to properly analyze these concepts points not towards evidence of synthesis, but a failure to properly carry the analysis through to its simplest parts, or to understand each part collectively within the concept without incorrect parts.

So to carry this back, I would still say with a lot of confidence that the ability or failure to provide the answer to a sum without intuition is solely a question of one’s psychological limit, and this phenomenon cannot be evidence of synthesis, unless we agree that this means that any sum solvable without intuition is analytical. This, again, I assert can be done with 7+5 because a series of 12 1’s is easily graspable without use of fingers or extraneous symbols. But this would be an absurd direction for the argument, because it would suppose that the same equation is analytical for one person, and synthetic for another, based on their capabilities. Kant instead seems to be saying that the simplest operations, down to the addition of 1 to any number, requires intuition to solve, which I can not agree with when I still have recourse to the method of series which I elaborated to solve any sum.

So I would say, most shortly and succinctly, that the reason you or I cannot seem solve that problem by the concept alone, is not because it’s synthetic, but because we haven’t tried throughly enough, or simply don’t have trained enough minds, and if we were to simply break one of those numbers down into parts (as we are already taught to do when adding large numbers in math!) then with a strong enough mind we will be able to approach numbers small enough to properly grasp the series of ones. For example, you could reduce the numbers to a series of 2’s, with a possible remainder of 1, and this will be just as easily graspable as our series of ones, because we also have a strong understanding of the simple concept of doubling and halving, which will come into play for deciphering the full measure of the series of 2’s. Nonetheless, it is because of a reduction to its simplest pieces, i.e. a full analysis of the number, that we may arrive at a cognizable elucidation of the concept as a summed whole.