Is Addition Synthetic?

All right, now I'm starting to figure it out. I suppose. At least I have the source material, as baffling as it is. Forgive the long blockquotes in what follows, but I thought clarity more important than elegance, at least for this post.

I'm shocked that I haven't made a blog post about the difference between analytic and synthetic judgments, and between a priori and a posteriori judgments, but a search reveals nothing, so here I go. For Kant, all judgments (statements of the form "A is B") can be described in those four terms, consisting of two dichotomies, as follows:

1. Analytic judgments are those in which the predicate is contained implicitly in the subject, so that the judgment expresses a definition and, ultimately, a tautology. As an example: "A bachelor is an unmarried male human."

2. Synthetic judgments are those where the predicate goes beyond what is barely expressed in the subject, and adds two things together. As an example: "That dog is black."

3. A priori judgments are universal and necessary. The negation of the judgment could not possibly be without contradiction. As an example: "A circle is round."

4. A posteriori judgments express a truth that is specific, not universal and necessary. Although expressing the opposite may (if the judgment is true) express a falsehood, it does not express a logical impossibility; the negation of an a posteriori judgment is not contradictory. As an example: "This book is heavy."

Analytic a priori judgments obviously exist as definitions. On the other hand, synthetic a posteriori judgments exist whenever a contingent fact is expressed. Analytic a posteriori judgments cannot exist, because for something to be true by definition, yet capable of being negated without contradiction, is impossible. The final potential combination is a synthetic a priori judgment. As Kant asks, "How are synthetic judgments possible a priori?" Much of the first Critique is fixed on finding an answer to that question.

Kant brings up an example, our old friend "7 + 5 = 12", in the introduction to the B edition of the Critique of Pure Reason:

It is true that one might at first think that the proposition 7 + 5 = 12 is a merely analytic one that follows, by the principle of contradiction, from the concept of a sum of seven and five. Yet if we look more closely, we find that the concept of the sum of 7 and 5 contains nothing more than the union of the two numbers into one; but in [thinking] that union we are not thinking in any way at all what that single number is that unites the two. In thinking merely that union of seven and five, I have by no means already thought the concept of twelve; and no matter how long I dissect my concept of such a possible sum, still I shall never find in it that twelve. We must go beyond these concepts and avail ourselves of the intuition corresponding to one of the two: e.g., our five fingers, or (as Segner does in his Arithmetic) five dots. In this way we must gradually add, to the concept of seven, the units of the five given in intuition. For I start by taking the number 7. Then, for the concept of the 5, I avail myself of the fingers of my hand as intuition. Thus, in that image of mine, I gradually add to the number 7 the units that I previously gathered together in order to make up the number 5. In this way I see the number 12. That 5 were to be added to 7, this I had indeed already though in the concept of a sum = 7+5, but not that this sum is equal to the number 12. Arithmetic propositions are therefore always synthetic. We become aware of this all the more distinctly if we take larger numbers. For then it is very evident that, no matter how much we twist and turn our concepts, we can never find the [number of the] sum by merely dissecting our concepts, i.e., without availing ourselves of intuition.

Just as little are any principles of pure geometry analytic. That the straight line between two points is the shortest is a synthetic proposition. For my concept of straight contains nothing about magnitude, but contains only a quality. Therefore the concept of shortest is entirely added to the concept of a straight line and cannot be extracted from it by any dissection. Hence we must here avail ourselves of intuition; only by means of it is the synthesis possible.

The technical term "intuition" needs to be defined. Intuition is that by which the raw data of experience are organized in such a way that a coherent experience can be made out of the chaos. That is, when sense data reaches our minds, it is just a mass of data of various types, containing within itself no principle of organization. Although we never have any experience of this type, because it would be incomprehensible and thus escape our understanding entirely, I think an illustration helpful. Imagine the experience of seeing a single tree in a field of grass, at noon on a cloudless day. Visual sense-impressions are coming from all over - blue sky, yellow sun, green grass, brown bark, &c.; but even describing those impressions with such language is misleading, because the data consist of a variety of colors, with (as yet) no organizing principle assigning the blue sense-impressions to the concept "sky" or even to any segregated area of the experience. Similarly, auditory sense-impressions will impinge the mind through the ears, but nothing in the raw experience itself exists to allow us to conclude that this particular sound is the wind rustling the leaves, that the same sound a second ago arose from the same source and thus the two sounds constitute a continuous, single sound.

Intuition is the first organizing principle to be applied to this raw data. Intuition has two forms: space and time. Space is the form of outer intuition; it presents things in experience as outside self, as extended and separated from the self and from each other. Time presents the mind's impressions as occurring in temporal succession, so that not experience is presented as occurring over a duration instead of at a single instant. Taken together, space and time make experience coherent by organizing things spatially ("there is an object of that size, in that direction, in that position") and temporally ("that bird was on that branch, then flew down to the ground, and now is pecking along the ground"). Out of a jumble comes a real experience. Besides intuition, concepts are necessary fully to understand what is going on in an experience, but intuition alone at least makes experience comprehensible.

Geometry is synthetic, as Kant indicates, and it's pretty clear why that must be so. The principles of geometry are just explications of the various relations different parts of space have to each other. Because space is not an objectively real thing, i.e., it does not exist as a thing in itself or as a relation among things in themselves, but rather exists only because of our capacity to perceive things outside ourselves, the study of the parts of space is a study of intuition, and, ultimately, of synthetic a priori judgments. The judgments are synthetic because geometric concepts do not consist of tautologies, because, in addition to the subject, the intuition of space must be added in order to reach the predicate, so that the predicate is not implicitly contained in the subject. Those same judgments are a priori because they could not be any other way; i.e., we could not possibly think a concept of space that would contradict the principles of geometry, so those principles are universal and necessary.

As a side note, what non-Euclidean geometry does for this conception and what relativity does for the ideality of space and time is unclear. I would contend that they do not actually refute the assertion that space and time are mere forms of intuition, though they do make intuition more abstract than Kant posited.

Back to my example in the previous post on this topic. "A triangle is a three-sided figure" is analytic, because it consists merely in a definition of "triangle." This is not as evident as I first thought, but it's still true. It is not evident because "three," "side," and "figure" are terms laden with intuitive implications. If those terms are meaningless without intuition's being applied to them, then "a triangle is a three-sided figure" turns out to be synthetic after all. I do not believe this to be a real problem. Purely as defining characteristics, and not characteristics that go into applying space to the concept of a triangle, those terms are essentially non-spatial in signification. The pure concept of a triangle as a three-sided figure says nothing about extension in space of the sides, that a figure must be extended, or that the three sides must be in different regions of space from each other. In order to use the concept in a meaningful way, of course, we need to attach spatial relations to those parts and to immerse the triangle in space. Once we have done that, then various qualities not contained in the definition become easy to find, and the synthetic a priori judgment "the sum of the interior angles of a triangle is equal to two right angles" is true.

What intuition makes "7 + 5 = 12" true? Kant rather unfortunately uses as examples some concrete instances of intuition in the quoted passage (fingers on one's hand and dots) to explicate the synthetic nature of the judgment, but those examples do not serve to locate the pure form of intuition involved in the judgment, which is our purpose here. However, Kant claims that we analogize time in spatial terms in order to understand it, and that the product of that analogy is seeing time as a line:

And precisely because inner intuition gives us no shape, do we try to make up for this deficiency by means of analogies. We present time sequence by a line progressing ad infinitum, a line in which the manifold constitutes a series of only one dimension.

What else is represented by a straight line in common experience? The integers, on the "number line." Given that time cannot be (or simply is not, in most people's understanding) represented except by resort to a spatial model of temporal succession, then anything we present in intuition will be represented spatially. Although space and time are different, time is understood in spatial terms, at least analogously. This may clarify Kant's fingers-or-dots explanation of addition. Although fingers and dots are symbols with spatial elements, in the act of addition, they are not being used qua extended objects. Instead, the process of counting five fingers, or five dots, to add to the seven already suspended in the mind is (I argue) a spatial analogy for time. Five follows four follows three follows two follows one follows zero, and by means of a temporal succession (most in evidence when we are doing the simple act of counting), five is thought, not merely as a unit (as seven was thought when suspended in the mind) but as the final stage in a succession of earlier stages. This five is added, one unit at a time, to that seven, in order to produce the sum, twelve. "7 + 5" contained the idea of addition without any indication of the intuition needed to make that addition into a further concept ("12"). "7" is a unit; "5" is a unit; "+" is an essentially contentless operator; "12" is a unit.

Of course, in actual experience, we rarely actually do addition on the fingers of our hands. It's simply not very efficient, and we have developed procedures for doing complex arithmetic without resorting to simple analogies; but, what Kant is saying, is that before those procedures can be invented, when humans first become aware of what the concept of addition of two numbers means, this concept makes no sense without some simple act of intuition. That intuition is no less important because it is so implicit; indeed, being implicit, lying at the foundation of experience and making experience itself possible, is what makes intuition indispensable. The use of shortcuts that go beyond what intuition presents in the first place is what allows us to engage in efficient uses of the intellect, but we should not forget the basic components of experience, vital both for initial comprehension and for continuing coherence at all levels of complexity.

Analytic judgments are static. They explicate nothing but what is contained in the concept. Synthetic judgments, on the other hand, are dynamic, and synthetic a priori judgments show something changing, but changing according to a fixed law. "7 + 5 = 12" is just such an example - the addition of the two numbers creates a new number, but according to universal and necessary laws of thought.

Arithmetic is the study of time; geometry is the study of space. Therefore, "7 + 5 = 12" is synthetic a priori because it requires immersion in time in order to make sense, for one concept to imply the other.

QED