### 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.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.

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.

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 progressingWhat 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.ad infinitum, a line in which the manifold constitutes a series of only one dimension.

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

## 5 Comments:

Was just puzzling over this very question and a web search found this blog.

I don't see how the statement 2+5=7 differs from 'a bachelor = unmarried man'. If in judging the first we have to perform the mental operation of counting, one could equally say in evaluating the second we have to perform the mental operations of substitution/recognition or language processing. In both cases we then examine the result and see if it is an identity. If the former is 'synthetic' then so is the latter, no? Why is 'counting' any different from 'language processing'?

Put another way...

If we train ourselves to recognise 1+1 as 2 and 1+1+1+1+1 as 5 and 1+1+1+1+1+1+1 as 7 without consciously counting, does it then become analytic? Or is the counting still there, unconsciously, in which case its no different from our unconscious translation of words to concepts.

It all baffles me.

Me again. Yup, I'd stick my neck right out and venture that Kant might be wrong.

2+5=7 requires us to engage in a mental process (a process which we have to be taught to do) to confirm its truth or falsity, and 'a bachelor is a married man' does as well, there is no difference in the two cases that I can see, other than the first involves 'counting' and the second 'logic/language processing'.

Perhaps 'synthetic a priori' is really just that our brains are structured so that we have an ability to learn to count, to use language and logic. All those skills nevertheless have to be taught to us so they aren't entirely a priori at all.

Or maybe I've misunderstood entirely.

I don't see how the statement 2+5=7 differs from 'a bachelor = unmarried man'. If in judging the first we have to perform the mental operation of counting, one could equally say in evaluating the second we have to perform the mental operations of substitution/recognition or language processing. In both cases we then examine the result and see if it is an identity. If the former is 'synthetic' then so is the latter, no? Why is 'counting' any different from 'language processing'?Mechanically, even in analytic judgments, we have to perform some act in order to recognize that the predicate is contained in the subject. Otherwise, the judgment would not be a judgment; that is, the subject would be recognized in an instant to contain all the predicates contained in its definition. But this is sort of what occurs in analytic judgments. "A bachelor is an unmarried man" is equivalent to "A bachelor is a bachelor;" all analytic judgments are true by the principle of identity, so that all the predicate does is make clear what may be hidden in the subject, without explicating anything beyond the subject.

In the process of addition (following Kant), the idea of addition of two numbers does not implicitly contain the number of the sum at all. "7 + 5" is the concept of a sum; "12" is the concept of a single number that is the end result of a process of counting, one-by-one. Think of it this way - producing each side of the equation requires a different activity. On the left side, what is performed is a recognition of the number 7 and then a step-by-step addition of the units making up the number 5. On the right side, only the concept of the number 12, as a whole, not as a sum, is found.

That may be (almost certainly is) obscure. I wish I could explain it better. If you accept Kant's transcendental idealism, and the concomitant acceptance of space and time as forms of intuition, then "7" "5" "7 + 5" and "12" are all concepts that cannot be understood without the employment of intuition. None of them makes sense until the step-by-step process is done to construct the concept from pure intuition. But what prevents "bachelor" from being recognized immediately as "unmarried man"? The concept does not have to be built stepwise; all that can be predicated of the subject (its comprehensive definition) is contained in it without any assistance from intuition.

Some confusion may arise because, when considering what a bachelor is, we may conceive of an image of a bachelor in our minds to assist reason. But this image is convenient (and, as Kant starts out the Critique saying, all knowledge starts with experience) and not necessary, as in the case of arithmetic.

If we train ourselves to recognise 1+1 as 2 and 1+1+1+1+1 as 5 and 1+1+1+1+1+1+1 as 7 without consciously counting, does it then become analytic? Or is the counting still there, unconsciously, in which case its no different from our unconscious translation of words to concepts.This is exactly what I had considered as a roadblock to understand (and agreeing with) Kant. We almost never actually count on our fingers when doing arithmetic. We almost never count at all, really, but simply recognize that 7 + 5 = 12, and so, for instance, 37 + 65 = 12 + 90 = 102. That, at least, is how I do arithmetic; sometimes I can eliminate the middle step, depending on how swiftly my mind feels like working. In any case, the laborious task of counting out 65 units to add to 37 is never actually done. But the ways our minds actually do specific tasks are irrelevant for how those tasks are justified by reason as true. Judgments involving addition are true because they can be broken down into just the sort of slow, inefficient, intuition-based counting I described. Consider the case of calculus. When you differentiate, you aren’t considering whether the theorems of calculus can be derived validly from axioms. But the validity of calculus depends on such logical relations. Similarly, the validity of arithmetic judgments depends on the capacity for the human mind to count.

Yup, I'd stick my neck right out and venture that Kant might be wrong.You would not be the first.

2+5=7 requires us to engage in a mental process (a process which we have to be taught to do) to confirm its truth or falsity, and 'a bachelor is a married man' does as well, there is no difference in the two cases that I can see, other than the first involves 'counting' and the second 'logic/language processing'.Do we have to be taught to engage in that mental process? Sure; but when we are taught it, does not something in us recognize that it was productive of truth? If you’ve ever read Plato’s

Menothen this difficulty is already apparent to you. The truth or falsity of that equation cannot depend fully on what we have been taught, because we cannot be taught the ultimate criterion of truth. Either we derive truth from repeated occurrence, which leads only to inductive generalization and no universality, or the truth of at least some judgments is already known to us, and we compare them as we experience them to the ideal already within us. Neither of those was satisfactory to Kant and he rejected them by claiming that, although we do not possess true judgments before experience, we possess valid forms of judging innately. We cannot tell, a priori, whether the content of a judgment is true, but we know whether a judgment is valid as regards form.Further, you’ve been pointing out that the bachelor judgment involves “language processing.” But analytic judgments are definitions; thus analytic judgments are inherently linguistic because they involve dissection of terms to discover what they signify. Pure logic is different from counting. Counting requires construction of a concept; pure logic deals only with the form of judgments. “A bachelor is an unmarried man” is true because it an instance of “A is A” which is a valid judgment as regards form. “7 + 5 = 12” is not true simply because of its form, but the content of the judgment has to be added to it. That’s what makes it synthetic, but because “7 + 5 = 12” is always true, it is also a priori.

Perhaps 'synthetic a priori' is really just that our brains are structured so that we have an ability to learn to count, to use language and logic. All those skills nevertheless have to be taught to us so they aren't entirely a priori at all.Our brains are certainly structured to receive knowledge, to learn to count, use language, and use logic. Kantian epistemology goes beyond the brain’s/mind’s capacity to absorb knowledge and posits that the brain/mind contains certain forms it recognizes as valid and to which it will subsume all experience. The skill of doing addition well is learned; I don’t think the same can be said for the capacity to recognize that addition is true of the world as we know it. You would be right if a judgment of addition is right “because our teachers told us so, and we’ve all come to a collective agreement about the principles of addition.” Instead, addition is right because our minds are not capable of thinking in any way that contradicts or rejects addition.

Skills that we have to acquire are not a priori – otherwise, who would have to learn them? Everyone would have those skills. But the judgments whose proper determination those skills cultivate can be about a priori subject matter. We have to have some innate forms of understanding the world, or else we could never be sure of anything (or we could rely on a rationalist mysticism, which I am sure is even less attractive today than in Kant’s age). I keep thinking back to

Meno.From the Introduction to B: "There can be no doubt that all our cognition begins with experience...But even though all our cognition starts

withexperience, that does not mean that all of it arisesfromexperience."Does this help at all?

Thank you for responding. I tried reading Critique of Pure Reason years ago and found myself arguing with it too much to proceed (its hard arguing with someone long dead). I am trying again. I have saved this entire page to print and read and think about. May reappear here when I have got it straight. The key then is that there's a crucial difference between counting and logical processing of concepts? Will have to ponder that a bit more.

Thanks for your explication.

I had an idea for a condensed explanation.

The bachelor sentence is analytic because it's self-evident by pure concepts.

Now try to answer this question with pure concepts: "What number comes after 1?"

It's impossible, because "after" implies a temporal relation, which brings intution into it already.

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