First, the text itself:
Mathematical judgments are one and all synthetic. Although this proposition is incontestably certain and has very important consequences, it seems thus far to have escaped the notice of those who have analyzed human reason; indeed, it seems to be directly opposed to all their conjectures. For they found that all the inferences made by mathematicians proceed (as the nature of all apodeictic certainty requires) according to the principle of contradiction; and thus they came to be persuaded that the principle of contradiction is also the basis on which we cognize the principles [of mathematics]. In this they were mistaken. For though we can indeed gain insight into a synthetic proposition according to the principle of contradiction, we can never do so [by considering] that proposition by itself, but can do so only by presupposing another synthetic proposition from which it can be deduced.Critique of Pure Reason, B14-B17, Pluhar translation. Brackets indicate translator's insertions, all emphasis original.
We must note, first of all, that mathematical propositions, properly so called, are always a priori judgments rather than empirical ones; for they carry with them necessity, which we could never glean from experience. But if anyone refuses to grant that all such propositions are a priori - all right: then I restrict my assertion to pure mathematics, in the very concept of which is implied that it contains not empirical but only pure a priori cognition.
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 arise. That 5 were to be added to 7, this I had indeed already thought 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.
It is true that a few propositions presupposed by geometricians are actually analytic and based on the principle of contradiction. But, like identical propositions, they serve not as principles but only [as links in] the chain of method. Examples are a = a; the whole is equal to itself; or (a+b)>a, i.e., the whole is greater than its part. And yet even these principles, although they hold according to mere concepts, are admitted in mathematics only because they can be exhibited in intuition. [As for mathematics generally,] what commonly leads us to believe that the predicate of its apodeictic judgments is contained in our very concept, and that the judgment is therefore analytic, is merely the ambiguity with which we express ourselves. For we say that we are to add in thought a certain predicate to a given concept, and this necessity adheres indeed to the very concepts. But here the question is not what we are to add in thought to the given concept, but what we actually think in the concept, even if only obscurely; and there we find that, although the predicate does indeed adhere necessarily to such concepts, yet it does so not as something thought in the concept itself, but by means of an intuition that must be added to the concept.
That judgments in geometry depend on the immersion of pure concepts in space seems unobjectionable. Indeed, Kant's strongest argument that mathematical judgments are synthetic proceeds by analogy with his claim that "a straight line is the shortest distance between two points" is comprehensible only by assuming the further condition that the concept of a straight line is first to be immersed in Euclidean space. To attempt to elucidate spatial relations without assuming that space exists (even if merely assumed to exist as a necessary condition for making a geometrical judgment in the first place) would be a futile effort, and pure geometrical concepts like that of a straight line would be comprehensible only through the relatively unproductive dissection of concepts analytically, by the principle of noncontradiction. Immersion of pure geometric concepts in space allows an explication of what those concepts entail when considered, not merely as abstract concepts of the mind, but as simultaneously being determinations of a singular intuitive representation (pure space). Because space is an a priori method of representing objects, the pure concepts of geometry lose none of their universal and necessary character for being displayed as determinations of space. For that reason, immersion in space does not destroy the a priori character of geometric concepts; it merely allows them to connect with each other and go beyond mere concepts in geometric judgments. What would be sterile, unproductive definitions of concepts become rich explications of the nature of extension when those concepts are applied to actual space.
This interpretation of Kant's basic claim, that geometric concepts gain their full richness and interrelatedness only when immersed in space, concedes the point far too easily, but for now, I want to postpone criticism and accept it arguendo. Assuming Kant is right, then, the principles of pure geometry are all synthetic a priori. The analogy to principles of pure arithmetic is more difficult. First, it is not clear what singular presentation is necessary to make the principles of arithmetic possible synthetically. There are two, and only two, singular presentations that comprise the full range of pure human intuition - space and time. That geometric principles, considered as possible determinations of the infinite given magnitude that is space, gain full relevance by being considered simultaneously with that space is uncontroversial, even trivial. If the analogy holds for arithmetic, however, then arithmetic principles, such as the simple act of adding seven and five to create twelve as their sum, will have to be determinations either of space or of time.
Addition as a determination of space has some intuitive appeal. Objects, such as the fingers of one's hand (as in Kant's example and in simplistic counting), can be mentally gathered together, first one group, then another, until a new group, the sum of the first two, is formed. Describing this procedure as a determination of space is problematic. For one thing, as Kant says in the Transcendental Aesthetic, all the parts of space are simultaneous. Thus, if "7 + 5 = 12" were regarded as a determination of space, the sum and the addition of the addends would be simultaneous. The mediation that intuition is supposed to provide for this arithmetic procedure would be missing. Contra the case with geometric principles, where properties that are the result of extension become added to pure geometric principles through the medium of space, with addition, space adds nothing.
The alternative, then, is to regard arithmetic principles as requiring immersion in time to become comprehensible. Indeed, the quoted passage, describing the procedure for adding two numbers, is rich in temporal language ("start" "then" "gradually" "previously"). Thus it seems plausible that Kant's describing the procedure as a temporal progression is not accidental, but is meant to reveal the underlying resort to intuition to make even simple arithmetical concepts understandable. To cognize means, indeed, to submit something to a priori synthesis in order to bring it within the range of objects that properly can be considered by the human mind.
The key problem with this interpretation of arithmetic is that it confuses the quasi-temporal procedure of conducting arithmetic operations with a necessary precondition for the very comprehensibility of arithmetic at all. It is natural to regard the binary operation symbolized by "7 + 5" as a command to "regard 7 as given, and add one unit at a time until the number of units added to 7 is equal to 5." That the result of this operation will be the same number as that more succinctly symbolized by "12" is obvious from the laws of mathematics. The question, though, is not whether a successive procedure can produce a single sum identical to a number with an existence independent of that procedure, but whether this identity of the two sides of the equation rests on an a priori, time-based synthesis. Kant's claim could be extended beyond arithmetic to logical operators. Thus, the antecedent in a conditional would be conceived of as given, and the relation between antecedent and consequent would depend on an a priori synthesis (because, presumably, the mind contemplates a unidirectional dependence between them). But a logical tautology, such as "If either p or p then p" is surely analytic. That the conception of a one-way temporal relation (antecedent preceding consequent) can be thought as connected with the concept of implication does not mean that implication is unknowable without being brought under temporal relations. In fact, as those familiar with Kant will know, when time is brought into the pure conceptual relation of a hypothetical syllogism, the relation of causation and dependence, one of the basic concepts of physics, appears. If logic can be implemented in separation from temporal relations, then it is not clear why arithmetic would be different.
The supposed synthetic nature of arithmetic breaks down further when the "7 + 5 = 12" example is examined in all its parts. Leaving aside the relations between numbers (addition and identity), take the numbers in themselves. Take a single number by itself. Because "12" can be broken down into a succession of additions, one unit after another until all twelve are gathered together into the single number, it seems that literally nothing involving numbers is possible without some sort of resort to intuition. Kant himself does not seem to have embraced this expansive view, if his emphasis in the quoted passage on the presence of intuition in the steps of the process of addition can be understood as an implicit limitation. Whatever the case may have been with Kant, it is difficult to see how a single number could be understood as being incomprehensible without resort to the singular presentation of time.
It is possible, however, that space is, after all, if not fundamentally at the basis of arithmetic concepts, at least important to make clear the properties of numbers. Numbers are often visualized as existing on a number line; similarly, time is represented as a one-dimensional progression, the time line. Kant makes the point himself in the Transcendental Aesthetic (granted, the comparison is made to show that, because time can be represented by means of spatial relations, then it has the same character as space, which Kant had just proven was a pure a priori intuition). This does not seem to avail, however. If time is inherently merely an intuition with one dimension in the same way that space is a three-dimensional intuition, then time is merely a kind of space, and only space is the real intuition. Obviously, though, the analogy is merely that, a way of understanding the progression of time by imagining that it occurs along a one-dimensional spatial continuum, not meant to imply that it actually has the same properties as space. And the analogy is not complete. Time proceeds in one direction, inexorably moving through successive inner presentations, whereas space simply exists in its entirety, the whole given at once. One can move, either actually or in imagination, back and forth in space. Time does not work this way. Thus the time line has a further limitation that motion is at a constant rate, in only one direction. Thus, visualizing time as motion along a line is useful when considering relations between events, but it does little to make explicable the nature of time itself. Still less does it serve to make comprehensible the nature of numbers and arithmetic relations. Why the position of the number twelve along the number line should have any of the same features as the position of an event on the time line is unknown. Events occur at different times because otherwise experience would not be comprehensible as a succession of events, and physics as a succession of causes related to effects. Numbers occupy different positions along the number line because numbers describe different quantities.
That last sentence ought to give us pause. It was rather glib. It could express two different views about the reality of mathematical truths. Either different numbers express different quantities by definition, or different numbers correspond to different independently existing objects of nature. Kant, of course, held to the view that objects are what they are because they are for minds and are the product of organizing functions' having been applied to reality to create experience. But there could be a difference between pure mathematics and applied mathematics that would locate at least some of the principles of mathematics in general logic, logic divorced from an intuition. If, however, "mathematical judgments are one and all synthetic," then they must be determinations of intuition (or discursive concepts, but that possibility can be rejected).
What if mathematical judgments are determinations of intuition by analogy? Consider the complex plane, or the mapping of functions on the Cartesian coordinate system. Arithmetic, it turns out, is actually geometry! Now if what Kant had said about geometry, the "easy case" of mathematical judgments' being true only under the assumption of intuition, was correct, then perhaps all mathematical judgments are simply determinations of space. What relation time and number have would be unclear. Despite the appeal of accepting that all mathematical judgments are judgments about spatial relations, this resolution seems merely to promote a mapping relation of mathematical concepts to geometric relations to an expression of ontological significance.
The rot spreads. That geometrical judgments can be made to correspond to determinations of space does not mean that space is a prerequisite for their having content. Twice in the quoted passage, Kant ought to have caught the mistake. First, that counting can be done on the fingers of one's hand does not mean that it necessarily involves intuition. Second, that what Kant admits are purely logical concepts ("a = a; the whole is equal to itself; or (a+b)>a, i.e., the whole is greater than its part") can be analogized with intuition (one can imagine a whole thing occupying a larger space than one of its own parts) means merely that humans rely on intuition to guide thought. But the question "Is addition synthetic?" does not ask whether it is helpful to use intuition to perform additive tasks, but whether addition has meaning only as a determination of one or both forms of human intuition. That all math may be like general logic and merely be easier to deal with in thought when assisted by intuition, rather than depending on intuition to exist in the first place, is not so easily dismissed.
The nagging background question through this discussion is what precisely makes mathematical judgments true. The facility with which I assumed that the Introduction could be understood without dealing with the nature of mathematics was obviously baseless. Without answering that question, the Kantian approach to the issue of how mathematics is synthetic crumbles, as evidenced by the previous paragraph. An examination of that background question is in order.
"7 + 5 = 12" is either analytic or synthetic. If analytic, it is true by definition, because the concept of "7 + 5" includes "12" implicitly. Kant clearly disagrees; but if the truths of mathematics are essentially arbitrary, then they are merely tautologies. Kant's transcendental idealism would deny any necessary correspondence between these tautologies and an experience in the creation of which they did not participate. If this is a scandal, it is a scandal for applied math, not pure math, which can be as analytic as it wants. The possible lack of absolute certainty would not be in the principles of mathematics themselves, but in their universal and necessary application to concrete events in the world. "7 + 5 = 12" can be analytic even if "7 apples added to 5 apples results in a 12-apple collection" depends on a synthesis. Indeed, perhaps this admittedly simplistic distinction between "pure" and "applied" mathematics points the way forward. "7 of object A and 5 of object A makes 12 of object A" is not universal and necessary in the world of sense. The combination of drops of water produces, not many drops of water, but a single new quantity of water. Without knowledge of the cohesive behavior of water, a person would be unable to make any sense of the result. The additional assistance of experience solidifies what would otherwise be merely a contingent connection between "this many drops and that many drops" and "such-and-such quantity of water." Of course, in the water example, reliance on experience and not on an a priori condition for the possibility of experience means that the synthesis involved holds a posteriori. For the synthesis required to apply pure math to experience to hold a priori, it must be a determination of intuition or discursive concepts (the two classes of organizing principles that bring the manifold under a determinate, cognizable structure).
Taking space and time as Kant takes them, and without getting into a discussion either of spatial dimensions in excess of three or of spacetime, it is not immediately clear that mathematics applies to the world of sense through a determination of intuition. To be sure, if space and time were not a priori ways of organizing sense data, then experience would be presented in a haphazard, probably incoherent way (assuming that things in themselves have no inherent qualities that would otherwise make organization possible). Space and time have quasi-subjective qualities (space is outer intuition, time inner) that are not attached to mathematical principles. Of course, if mathematics is merely a determination of space, time, or both, then it is not contradictory to assume that the intuitions contain more relations than mathematics, which consists of determinations of them. The difficulty that causes hesitation in identifying intuition as the act that imbues mathematical significance on experience is the indeterminacy of the "stage" in which data go from being mathematically incoherent to capable of being discussed in mathematical terms. Things in space can be described using geometry; does geometry begin to describe their relations simultaneously with their being considered in space, or does an already-existing necessary relation become additionally interpretable as spatial at that moment? Keep this question in mind, while for now merely recognizing that the supposed a priori synthesis we have been searching for may occur at some other stage in cognition than the intuition-applying stage.
Another stage may be the stage at which discursive concepts make intelligible sense data. It would be wrong to suggest that this stage occurs "after" intuition is applied, but it seems at least plausible to imagine intuition without concepts. No intelligibility, but bare existence in experience, would be possible of such an intuition. Mathematics does not appear comprehensible as a determination of these concepts, in any case. For the objects of mathematics are not things that are otherwise given whose relations must be made clear by external concepts, but things that are created in the act of thinking them. A triangle is a product of imagination, whereas the objects of physics are things actually given. If discursive concepts were necessary to make sense of mathematical objects, then those objects would have to be already-existing, intuited things acquiring conceptual character through thought. This is impossible.
Questions and criticisms abound; it is time for me to say something positive about the issue. I think pure mathematics must be analytic. Mathematics is a branch of logic, or vice versa, and logic is tautological. I do not see what was so apparent to Kant, that something has to be added to the concept of the subject of a mathematical judgment in order to make the predicate inhere in it necessarily. The puzzle remains: why do these tautologies say anything about external reality? For one of two reasons:
First, perhaps Kant was on to something after all. Even if mathematics consists of tautologies, it requires some sort of synthesis to bring those tautologies into experience and make them apply to the manufactured reality of human minds. But it is not insofar as things are thought in space, or in time, or as having certain properties that they can be described by mathematics. Instead, the manifold, even before it is intuited, is susceptible of being described mathematically. The transcendental affinity of the manifold would include, among its very basic features of organization, and agreeableness to mathematics. The transcendental affinity is difficult to understand, and I don't want to get distracted with it. But it is plausible that, whatever original synthesis this affinity accomplishes for the mind's subsequent work, it includes a way of organizing completely random data into mathematically-rich relationships. Thus, intuition finds spatial relationships in the data because of an affinity of the relations the manifold already has to spatial relations. An analogy between pure geometry and geometry applied to space is thus a product of the mind's very first way of organizing experience.
Second, perhaps Kant was simply wrong. Mathematical relations hold among things in themselves, and math is true because it is an inherent feature of things. This is a mathematical realist position. Indeed, if, as Gödel seemed to indicate (I will prudently avoid a tangent on this issue, mostly because I want to treat it in much more depth at a later date!), mathematical judgments are true even if their truth cannot be evaluated by the formal conditions of thought, reality might be mathematical even independent of human minds. One of the very few positive things we could say about noumena (besides that they must exist for experience to exist at all) would be that there are mathematical relations among them. This would establish why mathematics is true - things are just that way - but would fail to respond to Hume's objections about a prioricity.
The danger should be obvious. If Kant is wrong that mathematical judgments are synthetic a priori, then the sharp phenomenal/noumenal distinction collapses. A feature of noumena would be known; moreover, we could not be certain of the universality and necessity of mathematical judgments because they would be products of induction. Additionally, mathematics would really be a kind of metaphysics, a discipline dealing with the nature of things in themselves, so that the sequestration of speculative metaphysics accomplished in the Critique would come to nothing. To say the least, this is a shame.
One remaining interpretation saves transcendental idealism and avoids the murky waters of mathematical realism. Consider Gödel's insight and its apparent significance. It says, essentially, that as we currently understand the mind, it is not possible to determine the truth of mathematics simply by the operation of axioms and rules of inference. Either this is because math is true independently of our minds, or we are fundamentally ignorant about a key aspect of cognition. Under this second interpretation, to say that mathematical judgments are true merely because we pour our own assumptions into them could still be correct. Transcendental idealism can breathe easier, although reconciling this thought with Kant himself is another matter. Kant thought that a proper critique of the power of reason would have to outline all the functions of the mind, broadly indicating what the mind contributes to the formation and understanding of experience. Though specific judgments under the broad sections of the outline would not have to be cataloged in the critique itself, they at least would have to be species of one of the exhaustively-enumerated genera in the critique. This discussion has been attempting to locate mathematical judgments within all the genera, to no avail, and this solution suggests that the enumeration was incomplete. If so, the Critique of Pure Reason was not a total success. In fact, it left inexplicable a rather important type of cognition.
My own view is that mathematics does not depend on human minds for its existence or truth. There are certain things that simply are true, and though this does not seem philosophically satisfactory, a proper skepticism about the reach of the human mind leads to this leap of faith. As with Aristotelian first principles, math is simply true; no further basis on which to rest mathematical truth can be found. Constant reconsideration of this act of faith is what philosophical rigor demands, and if a better-supported position presents itself, I fully expect to change my mind. Given the inherently interesting nature of the discussion, I expect this consideration will not be my last.