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Kant's a priori methods for recognizing necessary truths

J. A. Brook

In the second edition, Kant summarized the question behind the Critique of Pure Reason this way: "How are a priori synthetic judgments possible?" (B19(1)). We can easily understand his interest in synthetic judgments; he thought that analytic ones could not tell us anything new (A5-6=B9). There are only two ways to get judgments that are analytic, by drawing out what is contained in our concepts and by combining the resulting propositions inferentially into arguments. Neither could ever tell us anything not already "thought in [the concepts we have used], though confusedly" (A7=B10/11), and even if they could, they could not give us anything against which to test them for truth or falsity. "In the mere concept of a thing no mark of its existence is to be found" (A225=B272; cf. Bxvii-xviii). In the search for knowledge, analytic judgments get us nowhere. That is why analysis is useless for establishing propositions of traditional metaphysics about God, freedom and immortality, as Kant argues in the Dialectic. Few would disagree with this verdict on analytic judgments, or criticize a preference for synthetic ones. But why was he so interested in those synthetic judgments that are a priori? As we will soon see, the answer to this question quickly leads us to Kant's interest in necessary truth, and to some peculiarities in what he offers us, in both his theory of necessity and his theory of the a priori.

I think the most immediate answer to our question is that, as Kant saw it, both the kinds of synthetic judgment that most interested him are a priori, the judgments that make up mathematics and physics and those that would have to make up metaphysics if it contains knowledge at all. The former are a priori because the propositions they express are necessarily and universally true, not contingently true or true of only part of a domain. Except for the ones Kant rescues, the latter do not contain real knowledge, but if they did, this knowledge would have to be obtained a priori. We have no sensible awareness of what these propositions describe, neither to God, freedom (uncaused choices) and immortality, or to the things Kant rescues.

In addition to tying the two sides of his project together in this way, Kant might have had other reasons to be especially interested in the a priori, though here we can only speculate. In his time, philosophers (and not just philosophers) were interested in what the mind could do unaided by sensible observation. Hume had recently launched a stinging attack on the very possibility of the mind coming to know anything new in any way other than by sensible observation. And, of course, theology, morality and psychology were conducted almost entirely by non-empirical analysis, the latter in the form of rational psychology.(2)

By 'a priori', Kant means: 'known independently of experience' (B2), where by 'experience' Kant means and from now on I will mean 'sensible awareness'. (At least that is what he meant here. He used 'a priori' in other ways, too, a point to which we will return.) By 'independently of experience', Kant means that some aspect of our belief in a judgment is based on something other than what we observe or could observe, and yet is not based on inference either. As we will see later, the 'something other' is a certain exercise of our cognitive or conceptual imagination.

Privileging Necessity

If the above partly explains why Kant started with a priori synthetic knowledge, it was still a peculiar place to start. Why should it matter whether a judgment is believed or known a priori or not? Surely what matters is whether it is true and we are justified in believing it -- whether it is knowledge. Kant, however, had a more restricted view of what counts as knowledge, or, perhaps a better way to view it, he placed stringent conditions on justification. I think this puts us on the trail of his deepest reason for privileging a priori knowledge. For Kant, we do not have genuine knowledge about something until we know not just its contingent features but things about it that are necessarily so (here we do not need to worry about what kind of necessity he might have had in mind). But all knowledge is expressed in judgments, propositions, etc., and we can know that a judgment, proposition, etc., is necessarily true only by a priori means. "Experience teaches us that a thing is so and so, but not that it cannot be otherwise" (B3). Thus only propositions, judgments, etc., known a priori, at least in part, could contain genuine knowledge. And that is why Kant emphasized synthetic judgments that are a priori. What mattered about a judgment being a priori is that this reflected its necessity.

In addition, Kant took it as evident that we have such knowledge. As was true of most philosophers in his time, Kant took mathematics to be the exemplar of knowledge, and it seemed obvious to him that the propositions of mathematics are necessarily true (cf. B4-5, B17, B20-21; we will return to these passages).(3) Kant had in mind mainly arithmetic and Euclidean geometry, though later he included algebra (A717= B745ff.) and must have held the same view of calculus and analytic geometry. It seemed similarly evident to him that the propositions of physics (Newtonian physics) have the same necessity. Thus it seemed evident to him that we already have knowledge that consists of necessary truth. Since all knowledge that goes further than merely spelling out what is already built into concepts is also synthetic, and, as Kant thought, not only physics but also mathematics goes further than that, he also saw these two bodies of knowledge as synthetic. He then takes as his task to show how these types of synthetic a priori knowledge are possible and where similar knowledge is not possible (in metaphysics). Indeed, carrying out this two-sided task was the single broad goal of the first Critique (B19).(4)

The synthetic, necessary and therefore a priori knowledge that Kant had in mind is knowledge of general propositions, propositions which say something about all members of a class or kind. Even the best possible empirical support for a general proposition, Kant thought, would never ensure that it is universally true (B3-4). Presumably his thought was that if a proposition is not necessary, it is not guaranteed to be true of all members of the class or kind it is about, only of those members so far observed; it is thus vulnerable to refutation. When a proposition is necessarily true, this limitation is overcome. Thus, it is superior to a contingent one; unlike the latter, the former is not merely approximative, limited to as much of the world as we have observed or to merely inductive generalizations, rules of thumb. For us to know (and not just believe) that a general proposition is true, we must know that it is necessarily true. It describes what will always be the case in our world as we observe it, indeed not just in our world but in any world, and with guaranteed accuracy, complete invulnerability to refutation. That is to say, it yields certainty. (If so, it is also proof against some important forms of scepticism. However, scepticism did not become a concern for Kant until the second edition, as a result, it is believed, of misunderstandings of the first edition broadcast by Garve and Feder in their famous review.(5)) Kant was clearly uneasy about privileging the necessary, or perhaps about his lack of arguments for doing so ("reason ... is so insistent upon this kind of knowledge", he says (A2-3), seeming to imply, '... but I am not sure why'). But he seems never to have questioned it. In fact, merely being necessary is not enough; to have the superior sort of knowledge of something, we must also know that the propositions describing it are necessary. Otherwise, even if the proposition contains superior knowledge, we do not. To know that a general proposition contains knowledge, then, two conditions must obtain: (1) the properties it ascribes to the object or state of affairs it describes must have "true universality and strict necessity" (A2, B2), and (2) we must know that they do (A2 and B2 imply this, too).

In addition to the arguments I have just reconstructed, doubtless other considerations pushed Kant in the direction of believing that genuine knowledge must consist of necessary and universal truths. Perhaps the most important additional one was his long pre-critical romance with rationalism. Indeed, until recently most philosophers have believed that necessary truths have special epistemic virtues.

The Focus on A Prioricity rather than Necessity

Whatever his reasons, Kant's decision to start with a priori judgments and their possibility disguised important differences between the two kinds of a priori judgment mentioned earlier. What struck him about mathematics and science was not so much that we know a feature of the judgments contained in them a priori so much as that these judgments are necessarily true. At B17, for example, he considers the judgments, "that in all changes of the material world the quantity of matter remains unchanged; and that in all communication of motion, action and reaction must always be equal", and says, "both propositions, it is evident, are ... necessary ...". The connection to the a priori is that to gain the knowledge that these judgments are necessarily true, we have to use a priori means (cf. B3 again). By contrast, what struck him about metaphysics was that we do not have sensible access to the various objects of its inquiries. Thus, if we are aware of them at all, it must be by a priori means. However, not only are these judgments not necessarily true, they are not true at all, not as far as mere mortals can know, at any rate. (Perhaps Kant thought that if they were true, then they would be necessarily true, but I will not investigate that strange, Kripkean modal construction.) So whereas the a priori-related feature of metaphysical judgments is our lack of sensible access to their objects, the a priori-related feature of mathematics and science is that the propositions that make them up are necessarily true.

Rather than focusing on this necessity, however, Kant choose to focus on the a prioricity of our knowledge of it. One result of this is that his views on both necessity and the relationship of necessity to a prioricity are seriously underdeveloped. For him, necessity is the prior notion, and he uses it to construct a criterion, in fact the only criterion he offers in the Introduction, of a prioricity -- "if we have a proposition which in being thought is thought as necessary, it is an a priori judgment ..." (B3). (Since necessary judgments are also universal, universality is an equally good criterion, one moreover that no inductive judgment can ever satisfy (B4).) However, first, necessity is not a terribly good criterion because it does not apply in any straightforward way to metaphysical judgments, if it applies to them at all, and secondly and far more importantly, Kant does not tell us how we manage to recognize the necessity of judgments or discriminate the necessarily true from the merely true. On B4, he says "it is easy to show that there actually are in human knowledge judgments which are necessary and in the strictest sense universal", but he makes no attempt to do so, not in the Introduction at any rate.

Why did Kant think that the judgments of mathematics and at least some of the judgments of physics are necessarily true? He simply thought that this is evident. Recall the remark from B17 quoted just above. Or consider the footnote to B21, where he says that "we have only ... to consider the propositions at the beginning [ie. at the foundation] of (empirical) physics ... in order to be soon convinced that they constitute a physica pura, or rationalis ... " (B21), that is to say, a body whose 'beginning' propositions at least are necessarily true. For a somewhat more complicated example, consider Kant's response to Hume on causality. Kant represents Hume as urging that "an a priori proposition" with respect to the connection of an effect with its cause "is entirely impossible" (B19-20), and responds as follows:

If he [Hume] had envisaged [the] problem in all its universality, he would never have been guilty of this statement, so destructive of all pure philosophy. For he would then have recognized that, according to his own argument, pure mathematics, as certainly containing a priori synthetic propositions, would also not be possible; and from such an assertion his good sense would have saved him [B20, my emphasis].

One wonders what Quine would say about that! The important point, however, is that Hume did not attack the a prioricity of causal propositions, not directly at least, he attacked their necessity. To be sure, as one of his arguments he did urge that we could never know the truth of such propositions a priori. But he made this point to show that these propositions are not necessarily true, not because it is intrinsically important.(6) And when Kant says in response that mathematics at least is certainly a priori, his basis for saying this, I think, was his certainty that its judgments are necessarily true.

In short, in the Introduction at least, Kant seems to have simply taken it for granted that the judgments of mathematics and science are necessary, and then used this concept of necessity to anchor his notion of the a priori.(7) Like cognitive science two hundred years later, Kant tended to take the credentials of knowledge-claims at face value and concentrate instead on the conditions under which we make them, how we could be aware of their various features, and how they hang together. He then focused his attention on the consequent claim that necessity could only be recognized a priori. How is it possible to make or recognize the connection between subject and predicate in such judgments a priori and recognize that it is necessary a priori (A9=B13)? (Moreover, Kant did not even stick with this topic, as we will see.)

We are suddenly faced with a host of questions. Why was Kant so sure that the proposition of mathematics and at least the 'beginning' propositions of physics are necessarily true? Did he have something better than his sense of the evident to back him up, something to justify his boast that "it is easy to show that there ... are in human knowledge judgments which are necessary and in the strictest sense universal" (B4)? What kind of necessity did he have in mind? It is not at all clear why Kant was certain that the truths of mathematics and physics are necessary, or what he thought would lend support to this certainty.

Something else is far from clear, too. If only an a priori cognitive capacity could construct or recognize the necessity of a necessary truth, what is this capacity like? What lets us construct and recognize necessary truths? In the case of mathematics, Kant's answer to this question is clearer, in general terms at least, than how he would have responded to the questions about necessity. We come to recognize the necessity of a mathematical judgment by constructing an instance of the state of affairs it describes in something called pure intuition. Notice that this argument has nothing to do with necessary conditions of experience, Kant's most characteristic way of approaching the a priori. We will examine the significance of that later. Maybe in clarifying how Kant thought we can recognize mathematical truths a priori, we can also find something to help us resolve our puzzle about other kinds of necessary truth and about how he could be so sure that certain truths were necessary. Before we embark on these explorations, however, we must first make some distinctions explicit that have only been implicit so far.

The Scope of the A Priori: Capacities and Propositions, Origin and Way of Knowing

As will be clear from the above, Kant's theory of the a priori has two fundamental parts, one to do with judgments (propositions, principles, etc.), and one to do with cognitive capacities. Note that this distinction, fundamental to his whole epistemology and theory of mind, has nothing directly to do with the division of a priori judgments and propositions into analytic ones and synthetic ones. Both sides of this latter division are found within the judgment part of the division I am discussing. Unlike contemporary philosophers, who usually restrict the term 'a priori' to propositions (their modal status or epistemic grounding; Kripke for example(8)), Kant also applied the term to certain capacities, the cognitive capacities we use to gain knowledge.

A priori cognitive capacities do two very different jobs in Kant (at least two). One is the job of ordering sensible input into judgmentally-unified representations. The other is the job of proving necessary propositions and recognizing their necessity. Kant's views on the former task are well-known; the capacities used in it include the forms of intuition -- the capacity to form representations of space and time and to locate things spatially and temporally --- and the forms of judgment -- the capacity to make judgments using the concepts and inference-patterns of Aristotelian logic, mainly in the specific form of the categories. I will focus, however, on the less-studied task of using the capacities to recognize and prove necessity.

A priori propositions also come in two kinds. There are those that do no more than spell out what is contained in a subject-term. These are the analytic a priori. And there are those that connect the subject-term to a predicate not contained within it, yet where no sensible awareness can determine either the soundness of the connection (metaphysics) or its necessity (mathematics and physics). These are the synthetic a priori.

A priori cognitive capacities and a priori propositions connect in various ways. Perhaps the most important is that a capacity to use a priori propositions, principles, etc., is a major component of the capacity to make judgments. That Kant thought that synthetic a priori propositions, principles, etc., play a big role in our judgment-making is well-known. However, he also thought that analytic a priori propositions, principles, etc., play an important role. Our capacity to judge is structured by the forms of judgment. These forms of judgment are themselves a priori, not acquired from experience. In order to use them, the concepts and principles of inference that make them up must be spelled out, in the categories and the inference-principles derived from the categories. The propositions that spell out these concepts and forms of judgment, that describe the marks [merkmalen] of each concept or form that are "thought in [it]" (A8), "breaking it up into those constituent concepts that have always been thought in it, though confusedly" (A7=B11), will be analytic. All this points up something important: certain analytic propositions, namely those "breaking up" a priori concepts and forms of judgment, play a far larger role in Kant's picture of cognition, and also in Kant's picture of the mind, than is often noticed.(9)

As well as two different things that can be a priori, there are two different ways in which something can be a priori. It can be knowable independently of experience or its origins can be in something other than experience. Kitcher has coined the useful terms a priorio (for 'a priori origins') and a priorik (for 'known a priori') to capture this distinction.(10) Given the literal meaning of 'a priori', those two ought to exhaust the meanings of the term. Some commentators, Kitcher included, think Kant used the term in a third sense, too, as a rough synonym for 'necessary', as in 'necessary and universal' truths. Kitcher has coined an additional term for this sense. She calls it a prioril (for 'logically a priori'). Some contemporary philosophers certainly use the term 'a priori' in this sense, Putnam for example.(11) I am not sure, however, that Kant did. Even when he referred to propositions that he took to be necessary as a priori propositions, as in 'synthetic a priori propositions', he calls them a priori, I think, because they are necessary, not as an alternative term for necessity. Kant was clearly able to distinguish necessity and a prioricity in the Introduction. I see little reason to think that he suddenly lost this ability later. Indeed, he speaks specifically of necessity throughout the work (cf. A93=B126, A104, A106 and B219f. for just a few of the available examples) and explicitly separates necessity and a prioricity at A93=B126, B219 and other places. In short, I think he used the term 'a priori' in only the two ways Kitcher captures in her terms 'a priorik' and 'a priorio'. (He does explicate the notion(s) in a variety of ways, however; for one curious one, cf. A148=B188).

Let us now connect this distinction between the two ways of being a priori, being knowable non-sensibly and originating non-sensibly, to the distinction between the two kinds of thing that can be a priori, propositions, etc., and capacities. The two distinctions go together, but not in perfect tandem. What makes capacities a priori is their origin. Thus they are a priorio. We could not acquire them from experience because we must already have them to have experience (sense-dependent, judgmentally-unified experience at any rate). Moreover, since the forms of intuition and the forms of judgment are not propositions but means of acquiring knowledge, they do not have modal status and therefore the question of the knowability of their status, a priorik or otherwise, does not even arise. Not being objects of knowledge, they are neither necessary nor contingent. The a priori capacities can be described in propositions, of course, and these propositions would then be either necessary or contingent and therefore either a priorik or a posteriori. But the forms of intuitions and the forms of judgment themselves do not have modality.

The situation is a little more complicated with respect to a priori propositions, principles, etc. They can be a priori in both ways, both a priorik and a priorio. For Kant, the propositions, principles, etc., required for experience are clearly a priorik; they express necessary truths and we can know this only a priorik. However, they are also a priorio. Because they are required for experience, they could not be acquired from it. Nevertheless, when Kant calls a proposition, principle or judgment a priori, he generally means that it is a priorik, and when he calls a cognitive capacity a priori, he almost always means that it is a priorio.

A Slide in Kant's Articulation of the Problem

We left two issues hanging. One concerned Kant's basis for his certainty that mathematics and physics contain necessary truth, the other how he thought we could recognize such necessity a priori, that is to say, by what we can now call a priorik means. Let us begin with how Kant himself sets up these issues in the Introduction. He begins with analytic propositions, which he defines as those "in which the connection of the predicate with the subject is thought though identity", and glosses 'though identity' as:

[the predicate] adding nothing to the concept of the subject, but merely breaking it up into those constitutive concepts which all along have been thought in it, though confusedly ... [A7=B10-11]

Synthetic propositions are all the rest, where the understanding must rely on "something else (X) ... if it is to know that a predicate, not contained in [the concept of the subject], nevertheless belongs to it" (A8). Kant offers the following example of the two processes of gaining knowledge:

By ... analysis I can apprehend the concept of body through the marks [merkmale] of extension, impenetrability, figure, etc., all of which are thought in this concept. To extend my knowledge, I then look back to the experience from which I have derived this concept of body, and find that weight is always connected with the above marks [merkmalen]. Experience is thus the X which lies outside the concept A, and on which rests the possibility of the synthesis of the predicate 'weight' (B) with the concept (A) [A8].(12)

He then turns to the question of what this X could be in the case of a synthetic a priori (i.e. synthetic necessary) proposition, and says,

It cannot be experience, because [the concepts in a synthetic a priori proposition are connected], not only with greater universality, but also with the character of necessity, and therefore completely a priori and on the basis of mere concepts [A9=B13].

In the first edition, he then says,

A certain mystery lies here concealed; and only upon its solution can the advance into the limitless field of the knowledge yielded by pure understanding be made sure and trustworthy. What we must do is to discover ... the ground of the possibility of a priori synthetic judgments, [etc.] [A10]

-- and stops!

Kant's treatment of analytic propositions is not without its problems.(13) What interests me here, however, is how little he says about what allows us to connect a predicate to a subject in such a way that the result is both synthetic and necessary. He tells us that it "cannot be experience", and that it is "on the basis of mere concepts" (cf. the quote from A9=B13 just above), but that is all. Neither claim is very helpful, and the second one also contradicts his characterization on A8 of syntheticity and risks reducing synthetic necessity to a form of analyticity. In the second edition, Kant provided some additional material. He deleted the final remark just quoted, and added two new sections. Though they contain a full summary of the approach to mathematics, natural science (physics) and metaphysics as bodies of synthetic a priori propositions laid out in the Prolegomena, written in the meantime, they too say little about how propositions can be both synthetic and a priori (necessary) or what kind of X could allow us to recognize a priorik that a synthetic proposition is necessary.

The new material of the second edition does do something else, however. It helps us spot something going wrong. In the transition from the just-quoted passages of the first edition to the new sections of the second, Kant is making a slide from one topic to another one, presumably without noticing it. The new topic then occupies not just the new sections written for the second edition but most of the rest of the book! The topic he starts with is necessity and a prioricity. In the new material, he then introduces the question: "How are a priori synthetic judgments possible" (B19)?(14) So far so good. However, this question is ambiguous. It could mean:

1. How is it possible for these judgments to be necessary, and/or to be a priorik?

or it could mean:

2. How is it possible for us to make such judgments, have the propositions that result?

Kant now slides from the first question to the second -- and does not touch the first again in the entire Doctrine of Elements, i.e., for four-fifths of the book!

Kant's answer to 2. is, roughly, that it is possible for us to make these judgments and have these propositions because it is necessary that we make and have them if we are to have experience (of certain kinds). This argument-strategy then becomes the central strategy of the whole Doctrine of Elements, both the Analytic and the Dialectic; i.e. for the next four-fifths of the book. I will refer to it as the central strategy. It is very important to see why it does not address the question in interpretation 1., Kant's official topic in the Introduction, but only the one in 2. Before I attempt this task, however, I want to clarify get the ambiguity itself a bit clearer.

To help us see how 1. is different from 2., consider a passage where Patricia Kitcher gets caught by the same ambiguity. She is attempting to reconstruct the central strategy:

Kant's general argumentative strategy can be framed in terms of the three senses of "a priori" [presented earlier]. He will justify our ability to know certain a prioril [i.e. necessary] propositions, by showing through an a priorik argument, that these propositions contain elements that are a priorio.(15)

Leave aside the question of the merits of this reconstruction for the moment and instead ask, what does Kitcher think Kant wants the argument to do? Does he want it to justify our conviction that certain propositions are necessary and/or a priorik (1. above)? Or does he want it to 'justify our ability' to have and use such propositions (2. above)? Once again, the ambiguity lies in the question. Kitcher's "justify our ability to know certain a priori propositions" perfectly reproduces the ambiguity in Kant's "How are a priori synthetic judgments possible?" (B19). The strategy Kitcher describes could not even touch the first question, it would seem. That a proposition contains elements that are a priorio does nothing to show that it is necessary or that this necessity is knowable a priorik. The central strategy is an a priorik argument, so the whole argument that certain propositions and principles are required for experience and therefore are a priorio is an a priorik argument. But from that it does not follow that there is anything a priorik in our knowledge of these propositions and principles themselves. Indeed, in the Introduction, Kant argued in precisely the opposite direction: from necessity to a prioricity!(16) These last observations introduce our next topic: the relation of the central strategy to necessity and a prioricity.

Necessity and The Central Strategy

The central strategy argues that propositions and principles of mathematics and physics, some of them at least, are a priori by arguing that, because we must use these principles and propositions to have experience, we could not acquire them from experience. This is an a priorik argument for an a priorio conclusion. Many people take this strategy to be Kant's main or even only way of arguing for the necessity of certain propositions and principles. That, however, is not how Kant himself presents it. In the remarks introducing the Transcendental Deduction chapter, he presents it as a way of justifying our application of these propositions and principles to objects of sensible experience (A85=B117). We are justified in applying them because we must apply them; using them is a necessary condition of having experience (or experience of certain sorts). Thus it is at least not obvious that either (a) Kant's central project or (b) the central strategy he uses to pursue it have anything much to do with the necessity ascribed to propositions and principles in the Introduction.

Some might want to object that Kant was not concerned with any necessity other than being a necessary condition, even in the Introduction. Kitcher might be an example. "[Kant] employs an unusual sense of 'necessity'. Something is necessary if it is true in all worlds we could experience constituted as we are".(17) If something must be the case in all worlds we could experience, that is necessity enough. If this objection is sound, most of what I say in this paper is misguided. But is it sound? Arguments of the central strategy have the form:

3. Necessarily (If experience, then S is P (or we must judge that S is P, or whatever)).

It would seem on the face of it that 3. is very different from 4.:

4. Necessarily (If experience, then necessarily (S is P) (or we must judge that necessarily (S is P), or whatever)),

though the semantics of the difference have been the object of some debate. It would seem, on the one hand, that the central strategy can only generate arguments having the form of 3., but, on the other, that in the Introduction Kant views the propositions of mathematics and physics as having the form of the consequent of 4., as being necessarily and universally the case in themselves. What they describe "must necessarily be so" (A1) and "cannot be otherwise" (B3). If it would take an argument of the form of 4. to demonstrate this necessity, not one of the form of 3., arguments of the central strategy do not touch the question of why the propositions of mathematics and physics are necessary in themselves, or how they are a priorik knowable.

Now it will be objected that I am begging the question against Kitcher. If she is right, the necessity of A1 and B3 just is whatever necessity arguments of the form of 3. show propositions and principles, and also the forms of intuition, to have. By arguments of this form, we can infer that a proposition or principle obtains, or a form of intuition applies, in all worlds that we could experience. Is it so clear that Kant wanted any stronger necessity than this? It is hard to be certain, but it is reasonably clear that he did. Kant had in mind propositions and principles describing what cannot be otherwise. But all arguments of the form of 3. support is the idea that if the events or states of affairs in question were to be otherwise, they could not be experienced by us. That is a very different thing from saying that they could not be otherwise, must necessarily be as they are (A1 and B3). In short, as well as being necessary for experience, Kant had in mind that the propositions and principles of mathematics and physics are themselves necessary and universal. Clearly he continued to subscribe to the same view of their necessity throughout, as the Second Analogy and the Postulates of Empirical Thought make clear. Indeed, Kant's discussion of the third Postulate makes reference to both ways of being necessary.(18)

Among Kant commentators who have seen that he had more than necessity relative to experience in mind, Dryer, Harper and perhaps Allison come to mind.(19) Harper points out that in the Postulates Kant even distinguishes between something like formal or logical necessity (in connection with possibility, the first Postulate) and material or causal necessity (third Postulate) (A218-34=B265-87).(20) At least by implication, Allison also has Kant advocating necessity other than necessary conditions, because he sees that the necessity of mathematical propositions is what Kant had in mind when he talked about the necessity of mathematics and physics.(21) Dryer, however, makes the most important point.

Dryer begins by pointing out that for Kant the necessity of mathematics and physics is no weaker than the necessity of an analytic proposition. He then observes that with all three, the necessity consists in the fact that we see that the concept of the one is "inseparable" from the concept of the other. I think what Dryer is on to something, but I would put it this way. The necessity consists not in the impossibility of experiencing something without something else, but in the fact that we can find no way to think of the one without the other.(22) When an alternative to a pattern of events E or a state of affairs SA cannot be conceived, we have grounds for holding that E or SA could not be otherwise (this, of course, is not the only way to support a claim of necessity). Merely being necessary for experience provides nothing comparable. Or to put the point another way, the proposition 'If we have experience, then E (SA) obtains' may have more than the necessity of a necessary condition; we may not be able to conceive of ourselves having experience without E or SA obtaining. But from this it does not follow that E or SA are necessary in the same way.

If so, the central strategy can provide no argument at all that propositions and principles are themselves necessary or for the claim that this necessity is knowable only a priorik, their centrality in the overall structure of the Critique notwithstanding. Though themselves a priorik, as I have said, these arguments only show that the things they are about are a priorio. Contrary to Kitcher, "an a priorik argument that propositions contain elements that are a priorio" has no power to show that such propositions are necessarily true (her a prioril). For all that such an argument could show, a proposition that we require in order to have experience (a priorio) could even be contingent! From another direction, notice that arguments of the central strategy establish the same necessity for the forms of intuition, space and time, as they do for any proposition or principle. However, it does not even make sense to consider whether the forms of intuition "must necessarily be so" (A1) or not; they are not even propositional. Put yet one more way, if the argument of the Transcendental Deduction is right, a priorio elements will be a part of every proposition, and so of contingent ones as much as necessary ones. That a proposition is a priorio or has a priorio elements tells us nothing about its modal status or how we know that status.

A Better Strategy

It is far from clear that Kant ever thought through the relation of the central strategy to propositional necessity far enough to see all this. Indeed, he may even have believed at times that the central strategy could justify his conviction that what the propositions of mathematics and physics describe "necessarily must be" as they are (A1). It is hard to say. However, he did have a second strategy, and it may have been his considered one. To approach it, recall the one passage in the Introduction that does seem to be directly relevant to establishing the necessity of propositions and how we recognize this necessity a priorik. I have in mind the discussion of construction in mathematics on B15-16. Whatever we may think of this discussion, it does at least delineate a strategy. The strategy is to form an image of what a proposition describes and then construct a procedure for demonstrating it.(23) To work, we must do this in pure, not empirical intuition -- the non-empirical, a priorio intuition of space (geometry) and time (arithmetic) that we all have (A713=B741), and the faculty we use to do this is productive imagination (though Kant does not formally introduce the term until A118 and first gives it this role only on A157=B196, The Highest Principle of All Synthetic Judgments; A165=B205 does so more clearly.) Here we do have a strategy with prima facie potential to demonstrate that the propositions of mathematics at least are necessary, without appeal to sensible experience or analysis of concepts. At any rate, it has more potential than the central strategy. Unfortunately, Kant offers nothing similar at this stage for the propositions of physics.

Indeed, he does not even tell us how the strategy is supposed to work. How could constructing geometric figures, patterns of successive units, proof procedures, etc., in pure intuition, whatever that is, allow us to recognize that mathematical propositions are necessary? Perhaps the reason for this is that Kant introduced the strategy not to show that mathematics is necessary or a priorik knowable, but to show that it is synthetic! Whatever, he does not mention the strategy again in the whole Doctrine of Elements, except for a few allusions on occasions when he finds himself talking about mathematics (A162=B203ff.; A221-24=B268-72; A240=B299). There is also an interesting remark on A157=B196, in The Highest Principle of All Synthetic Judgments:

Although we know a priori in synthetic judgments a great deal regarding space in general and the figures which productive imagination describes in it [an allusion to the method of construction?], ... yet even this knowledge would be nothing but a playing with a mere figment of the brain, were it not that space has to be regarded as a condition of the appearances which constitute the material for outer experience [an application of the central strategy].

'Mere figment' may be dramatic license; what is important about this passage is that Kant is distinguishing something, something to do with the internal structure of spatial relationships, from the central strategy, their application to 'appearances'.

Kant also says nothing more about the faculty that allows us to recognize propositional necessity a priorik. Even if the central strategy could prove that some propositions are necessary, what would allow us to recognize this feature of them? Yet, because it was vital to Kant that he show that the propositions of mathematics and at least some propositions of physics are necessarily true, he had to show they are knowable a priorik, at least in part. Given the difficulties facing the central strategy in this regard, it would be remarkable if he had nothing else to offer.

Sure enough, if we look far enough, we do find something else. The issues we are examining are fundamentally methodological: how do we do the things that we do? Thus, we might expect to find them addressed in the often-neglected last part of the book, the Transcendental Doctrine of Method, and that is what we find.(24) The first of the two chapters of this part of the book is called The Discipline of Pure Reason. In the first Section of this chapter, The Discipline of Pure Reason in its Dogmatic Employment, and the fourth, The Discipline of Pure Reason in respect of its Proofs, Kant lays out just the account that we have been seeking. He tells us how we can know that propositions are necessary, he tells us how we can recognize their necessity a priorik, and, indeed, he tells us how to do the former by doing the latter. The account is far from pellucid. But at least it is there. The basic idea is that exploring our conception of "an object [of experience] in general" (A788=B816) can do the same job for necessary truths about objects of experience as constructions in pure intuition do for geometry and arithmetic (or that 'symbolic constructions' do for algebra).

Moreover, this technique is of more than historical interest. If we remove the idea that it is exploring propositions describing what 'could not be otherwise' necessity, as I think we should do anyway, and view it as a method for exploring very general constraints on relations and systems in the imagination, it becomes something very like a method at the heart of contemporary cognitive science, the method of exploring general constraints on systems with thought-experiments. Cognitive science is exploring constraints on systems able to perform cognitive tasks whereas Kant is exploring constraints on the propositions we can construct in mathematics and physics, but otherwise the parallel is close. Indeed, Kant himself explored constraints on cognitive systems using the same method in other places, the subjective part of the Transcendental Deduction being the most notable example. Here is what Kant has to say about the technique.

How to Discover A Priori that A Synthetic Proposition is Necessary

Few philosophers have ever tried to maintain that any knowledge, let alone the most important kind of knowledge, is both synthetic and necessary. Until recently, most philosophers accepted that some propositions are necessarily true. Both Kant's immediate rationalist predecessors and virtually all post-Humean empiricists have also maintained, however, that such propositions are analytic. Since 1950, on the other hand, many philosophers have come to believe that much of what was called analytic truth is really synthetic. But these philosophers also maintain that such truths are neither necessary nor knowable a priorik. A great deal has been written on these issues in the last few decades. All I will try to show here is how Kant thought we could recognize propositions that avoid this fork, propositions that are necessary yet still synthetic.

Kant begins his discussion in the Discipline of Pure Reason by reminding us that the propositions of geometry and arithmetic, though synthetic and a priori, can be demonstrated, as he puts it a bit later, "intuitively through the construction of the[ir] concept" in pure intuition (B748).(25) (Throughout this part of the Critique, when Kant speaks of concepts he seems to mean propositions or representations that are expressed in statements and so have propositional structure, not concepts in a strict sense. I will speak of propositions.) To construct here "means to exhibit a priori the intuition which corresponds to the concept" (B742). So we demonstrate that a mathematical proposition is necessarily true and recognize its necessity by constructing a non-empirical instance of the state of affairs it describes, complete with the spatial and/or temporal positions and relations such a state of affairs would display. We can do this empirically (though then we would never get necessity), but when we do it non-empirically, we do it "by imagination alone" (B741), that is to say, by the 'middle' of the three faculties of sensibility, imagination and understanding that we require to represent (intentional (A104)) objects (cf. Transcendental Deduction, A101, A118, A120).(26) To construct and explore the spatial and temporal features of such an instance by imagination alone, we must have a non-empirical ('pure') representation of space and time in what Kant calls pure intuition.

Generally, Kant gives few examples. However, he does give some examples of this process. Recall the one in the Introduction. How do we determine that 7 + 5 = 12, he asks (B15)? "I may analyze my concept of such a possible sum as long as I please", he says,

still I shall never find the 12 in it. Instead, we have to go outside these concepts and call in the aid of the intuition which corresponds to one of them, our five fingers, for instance, ... adding to the concept of 7, unit by unit, the five given in intuition [B15].

The example he now takes up is the triangle, and "what relation the sum of its angles bears to a right angle" (B744). He makes the important point that we could never determine this relationship by examining our concept of a triangle. Then he describes how a geometrician would proceed, by the method of construction familiar from high school. Similarly with any fundamental proposition of geometry. To see that 'shortest distance between two points' and 'straight line' together make up a necessary proposition, we must imagine drawing a straight line. We then see that only such a line could mark out the shortest distance between two points. That is to say, we see that any other way of constructing a line will produce one that is longer. Next making the claim that algebra also uses a process of construction, but out of symbols rather than lines and circles (a claim that some might find less convincing), Kant concludes that what gives mathematics its decisive superiority over the philosophers' examination of concepts is this method of construction (B749).

This story about construction and its relation to demonstration, which I have sketched in only the broadest outline, is fairly well known; in note 24 I cited a sample of the authors who have explored it. Kant clearly has nothing but respect for construction as a method of demonstration. He urges that there is no more need for a critique of reason in this employment than in its purely empirical employment (B739). Here what interests me is not this story but what Kant contrasts with it. By comparison to mathematics, the employment of reason in philosophy is unreliable and prone to delusions of grandeur; and constructions are not available to help us out (a point to which I will return). Nevertheless, Kant thinks that he can sketch a method that is just as good. His discussion has two parts. One is found in Section 1 of the chapter we are discussing, the other in Section IV. In Section I, Kant tells us what an adequate method of proof in philosophy cannot be like; 'mediat[ing] on concepts' gets nowhere, yet it is impossible to construct a priori instances of the concepts that interest us. In Section 4, he tells us about the only way that will work. Even though Kant does not say so explicitly, if the method he describes there works, given the propositions in philosophy credited to it, it will also have provided the justification Kant sought in the Introduction for our conviction (or at any rate, his conviction) that, like mathematics, the fundamental propositions of physics also contain necessary and universal truths.(27) This method has received little independent attention, perhaps because it looks a lot like the central strategy and perhaps because Kant develops it in even less detail than he develops the method of construction. However, not only is the method not just a repetition of the central strategy, it goes beyond the latter in one crucial respect. As a result, Kant claims for it what cannot be claimed for the central strategy: the capacity both to prove(28) that certain propositions are necessarily true a priorik and to let us recognize this necessity a priorik.

Why are imagined constructions not available for the synthetic propositions of philosophy and physics? These are propositions about causes and what the other categories describe (B752). Only quantities allow of being constructed; we can obtain knowledge of qualities "only through concepts" (B743). The reason is that we have no pure intuition of what quality concepts name: "the only intuition that is given a priori is that of the forms of appearance, space and time" (B748). Thus, we can "cannot represent [such a concept] in intuition ... except in an example supplied by experience" (B743). (By contrast, both Dryer and Allison suppose that proving the necessity of scientific and philosophical propositions requires the representation of imagined objects in pure intuition, just as in mathematics.(29)) As becomes clear a bit later, what Kant means is that qualitative concepts are a device for synthesizing other, empirical intuitions, and do not come equipped with an intuition of themselves (B747). This is not exactly a transparent distinction. I think Kant means that simply by having for example the concept of a triangle, we can imagine what a triangle would look like (for Kant, this would require constructing an image of a triangle (A120)), but from the concept of a cause alone, we cannot imagine what a cause would look like. For that, we also need to become aware of real events that are related to one another.

If we cannot proceed by way of construction, then how can we proceed to prove that the propositions of metaphysics and physics are necessary and recognize their necessity, both a priorik? In Section I, Kant offers a highly original idea, though he puts it obscurely. The propositions and concepts in question are devices for synthesizing empirical intuitions, and they come complete with constraints on how they can be used. If we want to examine these constraints for necessity, the only way to do so, Kant suggests, is to examine the most general concept of the objects to which we apply them, "the concept of a thing in general" (B748) or, as he puts it later, "an object in general" (B816). Now ask, how are the two related? It is vital to get this clear. Is Kant just asking -- the central strategy again -- what the conditions for experiencing an object in general are? I think not. I think he is urging that exactly the reverse relationship is the important one. By studying what a thing in general must be like, we can explain the constraints on our a priorio propositions and concepts. So let us form a representation of an object in general and find out what such an object must be like. And that is all we find in Section 1. (Why is this notion not a representation in pure intuition? A good question. The answer may again be that from its concept alone, we could not imagine what such a thing would look like.)

Or rather, that is almost all we find in Section 1. After an interesting discussion of why philosophy does not have available to it definitions, axioms or demonstrations (the foundations of physics, too?), Kant concludes the section as follows:

pure reason does, indeed, establish secure principles, not however from concepts alone, but always only indirectly through relation of these concepts to something altogether contingent, namely, possible experience [B765].

This assertion immediately gives rise to a question. Was the earlier reference to a thing in general really introducing something new, or is Kant simply reintroducing the central strategy, confusing the a priorio and necessary conditions of experience with the a priorik and necessary truth? We do not get a full answer to these questions until Section 4. However, Kant brings up the object in general again in the very next sentence: "When such experience (that is, something as object of possible experiences) is presupposed, these principles are indeed apodeictically certain [true necessarily]" (my emphasis). That is, Kant is again separating the idea of an object of possible experience out for specific attention, precisely distinguishing it from the conditions of possible experience. He seems to think he can get something out of the former that he cannot get out of the latter. It is time to turn to Section 4.

The first paragraph of that section once more raises the question whether Kant is doing anything more than reintroducing the central strategy. It also once again distinguishes between the conditions of experience and of an object of experience:

The proof proceeds by showing that experience itself, and therefore the object of experience, would be impossible without such a connection [B811].

Kant will soon reverse this inference, back to the direction we just saw in B765.

The passage in which he gives his definitive statement of the method of proof in metaphysics begins on B815 with the curious claim that there can be only one proof for each transcendental proposition (by 'transcendental' here, he seems to mean not only necessary for experience but also necessarily true). The reason seems to be that all such proofs "can contain nothing more than the determination of an object in general" (B816) and we only have a concept of one object in general. To show how such a proof might go, Kant takes as his example the principle of universal causality, the principle that necessarily every event has a cause. In the Transcendental Analytic, he says,

we derived the principle that everything which happens has a cause, from the condition under which alone a concept of happening in general is objectively possible -- namely, by showing that the determination of an event in time, and therefore the event as belonging to experience, would be impossible save as standing under such a dynamical rule [B816].

I do not know what Kant is referring to in the Analytic. Certainly he did nothing like what he sketches here in the proof of the Second Analogy, which is about the relation of causality to our ability to experience events, not to fixing events themselves in time. Perhaps Kant thinks he did more there than he did, did more than apply the central strategy when that is all he did. At any rate, notice two things. First, Kant distinguishes between conditions of being an event, and conditions of it belonging to experience. This shadows the earlier distinction between conditions of being an object and conditions of experiencing an object. Secondly, Kant returns the direction of the inference to the order of B765. We again establish that something could belong to experience (satisfies the conditions of experience) by showing that it could be.

In these distinctions we can find, at last, the method of proof we have been looking for. Granting that a state of affairs SA being necessary for us to experience an object O or event E does not show that this or any link between SA and O or E 'could not be otherwise' (cf. B3), the same limitation would not seem to apply if SA is necessary for O or E to exist. Here a statement of the form,

5. If O [or E], then SA,

would indeed state something that "must necessarily be so" (A1). Put in terms of the distinction we took over from Dryer earlier, here SA being true of O is not just a condition of us experiencing O, it is a condition of us conceiving of O. In the present example, "the determination of an event in time" (my emphasis), not just "the event ... belonging to experience", would be impossible if the event were not caused. If, however, we cannot fix a time for an event, then not only can we not conceive of experiencing the event, we cannot conceive of the event even existing. (For present purposes, I do not need to examine whether Kant's argument is sound.)

Kant drew a distinction between conditions of experience and conditions of being a possible object of experience and cognate distinctions repeatedly in the Discipline chapter. It seems to me likely that he did so precisely because examining the latter opens the way to arguments like the one just sketched, but examining the former does not. That is to say, examining the conditions of being an object or an event has a potential to establish that certain features of objects "must necessarily be so" (A1), but examining the conditions of experiencing the event or object does not.(30) Thus we at last have the method of proof that Kant set out in the Introduction to seek but from which he soon got diverted (for the next nearly 800 pages!) by switching to the question of how we can have propositions that are synthetic but a priori. Contrary to what Kant implies in the passage from B816 quoted just above, I have not found any place in the Analytic where he used anything like this method, and he certainly did not discuss it earlier. He mentioned the method of construction a few times, as we noted earlier, but never this method of examining an object in general.

The interesting question now becomes this. How, that is, by the use of what capacity, do we recognize that a proposition like the causal principle is necessary? What plays the role in this sort of proof that construction in pure intuition plays in mathematical demonstration? Here Kant offers us even less than he offered concerning construction. He says a little. Such proofs are conceptual yet ostensive (ostensiv), ostensive yet non-intuitional. That is to say, though based on concepts and specifically the concept of an object in general, not intuitions, such proofs are ostensive, not just acroamatic (B763) -- sentential -- and apagogical -- syllogistic (B817-8). But that is not much. 'Ostensive' -- what does that mean? (Kant uses the term in only one other place in the whole Critique, as part of a claim that the concept of a highest intelligence does not refer to anything that exists (A671=B699).)

Here I think we have to allow that there is a lacunae in Kant's system, one that Kant seems not to have noticed. The problem begins with pure intuition. Kant wants to restrict pure intuition to representations of the spatial and temporal. When we explore the concept of an object in general, we are exploring its qualities and relations, not the structure of its shape or its iterative (for Kant, temporal) properties. Thus we cannot be exploring the object in general in pure intuition. Kant had some reasons for wanting to restrict pure intuition, but he did not need to restrict it this severely. His criteria of empirical reality do not need the restriction and would continue to block the ontological extravaganzas of his predecessors without it. Moreover, restricting pure intuition this tightly creates some real problems for him. Aside from the specific problem of accounting for our awareness of the necessity of necessary truths, perhaps the most intense problem it creates for him is that there is suddenly no place for productive imagination to operate with respect to such truths. Yet if forming a notion of an object in general is not an exercise of the imagination, it is hard to see what it could be. It is not an exercise of reproductive imagination, which is the process of carting previously-experienced representations up into the present, so that leaves only productive imagination. Kant has a problem! Indeed, Kant should have seen that he had a problem even within the boundaries of mathematics. Suppose his account in terms of pure intuition works for demonstration in geometry and arithmetic. How could it provide an account of calculation in algebra, or calculus, or analytic geometry, all of which use what he himself calls symbolic construction?

In my view, Kant could go either of two ways. He could free the image-building activity of productive imagination, the activity of forming imagined instances of concepts and propositions, from its servitude to pure intuition. Or he could expand pure intuition to include more than imagined spatial and temporal phenomena. We could do the former at less cost to Kant's overall system than the latter. Since Kant himself gives us little to help us build such an account, we have to invent it for ourselves. Go back to the idea that proof in philosophy is ostensive. Here I think Kant means the following. To prove the causal principle, we use not only our concept of causality (which would get us nowhere by itself) but also a representation of an object in general. That is why this kind of proof "combines with the conviction of its truth insight into the sources of its truth" (B817). This representation is a "singular representation", a representation of a single instance of an object stripped of all its contingent features.(31) Since the object need not be represented as having spatial or perhaps even as having temporal features, the representation of it could be said not to be a representation in pure intuition, in the strict sense of the term. However, in other respects, the representation would be like one in pure intuition.(32)

Now the meaning of Kant's allusions to ostension becomes clear. Reference to this imagined object would be a kind of ostension. Opening up the notion of productive imagination as we have, we might now call it ostension in the productive imagination!(33) If we adopt the above suggestion, proof in philosophy would be as ostensive as in geometry and more ostensive than in algebra (at B745, Kant explicitly contrasts geometry as ostensive with algebra as symbolic). That seems right. If so, Kant's claim that such proofs are proofs obtained 'from concepts' (B752) might have to be modified, but this claim might have to be modified anyway. Even without our additions, it is difficult to see how we could reconcile this claim with the claim that such proofs are ostensive. Anyway, how could a form of proof that is not analysis of concepts and must involve more than discursive reasoning be purely conceptual?

The sort of activity we are discussing is close to unique. It is not conceptual analysis, yet it is not empirical either. It is research in the imagination. It is easy to see why Kant thought of it as a priorik, yet aimed at synthetic propositions. Clearly, it does not aim to generate analytic truths. Yet it is also a priorik, independent of experience, not a posteriori. One of Kant's most important ideas is that in order to experience, we must have time-fixing, space-fixing, and conceptual capacities available to us. Whatever order or divisions there may be in things in the world, the perturbations of the sensitive surfaces of our body (retinas, ear drums, finger tips), which are the basis of sensible representation, do not come pre-ordered, certainly not with the order of the world. Thus, we need abilities that are independent of that order if we are to recognize it, abilities that "our own faculty of knowledge supplies from itself (sensible impressions serving merely as the occasion)" (B3).(34) As Kant put it in a letter to Beck late in his life, "We must synthesize if we are to recognize anything as synthesized (even space and time)."(35) Curiously enough, we can use these capacities to explore the structure of imagined objects just as well as empirically-encountered ones. Indeed, Kant thinks that investigating imagined objects has an advantage over real ones; it can determine whether the objects' properties and relationships must be as they are. Thus, such explorations are the a priorik method Kant recommends for investigating whether synthetic propositions are necessary.

Links to Current Work

As will be clear by now, I conceive of Kant's notion of proof in philosophy as very much like our contemporary notion of a thought-experiment. However undeveloped it might be, in one respect Kant's notion takes us further than most contemporary discussions. If proof in philosophy is an exercise of productive imagination, generating and exploring a representation of an object in general, the next question is: using such a representation, how could we generate a proof? If our account of this form of proof above needed an analogue of representing lines and figures in pure intuition, I think it now needs an analogue of the activity of construction. Notice that to prove the causal principle, we imagine what we would have to know to do something. Specifically, we imagine ourselves fixing an event in time; then we explore what we would have to know to do so. This feature of the method gives it an efficacy that no exploration of the content of a concept could ever have. If we are imagining the doing of something, we are imagining ourselves working through a causal process: what we need to do it and where the absence of something would stop us from doing it. As a result, the findings of this sort of thought-experiment have a clarity and a certainty that, for example, neither analysis of a concept nor trying to spot the essential properties of an imagined object could ever have.(36) Compare the thought-experiment we give to undergraduates of trying to imagine a leaf in Siberia suddenly turning blue for no cause. This experiment is notoriously inconclusive. What makes it so, I think, is that we have no idea what the underlying mechanism might be like.(37)

Though the explorations in the imagination that we now call thought-experiments are the heart and soul of both cognitive science and a lot of contemporary philosophy, we know little about how the mind is able to do them. Dennett has made a start with his exploration of intentional objects and of the idea of a 'notional world', but the topic is radically under-examined.(38) (His notional world strikes me as very much like Kant's intuition with a population of representations.) Indeed, imagination and therefore thought-experiments play a role in all science. Even if science starts from observables, as soon as it postulates unobservables to explain the observed, it leaves the realm of the experienced actual and enters the realm of the imagination.

If Kant was on to a method that has continued, in one form or another, to play a large role in philosophy and science, it does not follow, of course, that it could do what he thought it could do. Can demonstrations in the imagination establish that propositions are necessarily true? This is a big question and one that has been hotly debated the last two decades. On the one hand, people like Kripke have argued with great vigour that it can.(39) On the other hand, people like Dennett and the Churchlands argue that thought-experiments yield results no different in modal status from 'real' experiments, and, moreover, suffer the additional liability of not being under the discipline of reality. Though my sympathies are with the latter view, I will not try to adjudicate that dispute here. Just describing the dispute allows us to do one useful thing. It allows us to distinguish between research into a special kind of truth, and a special kind of research. Whether or not Kant was researching a special kind of necessary truth, I would submit that he most certainly laid out the rudiments of a special kind of research.

In fact, the important question concerning the propositions he claimed to be true necessarily and universally is not whether they are necessary but whether they are true. Indeed, it is hard to see any good reason why Kant himself insisted that the propositions in question are necessary. If these propositions turn out to be just contingent statements of abstract and high-level constraints on what objects or objects of certain kinds could be like, why should this matter to him? Kant himself, of course, had a strong predilection for necessary truth. If we can find no good reason for this predilection, however, that in itself would make him significantly more accessible and palatable to contemporary theorists.

ENDNOTES

1. References to the Critique of Pure Reason will be given in the text using the standard 'A' and 'B' notation. `A' refers to the pagination of the first edition (1781) and `B' to the second edition (1787). (Of the numerous editions of the Critique, Kant prepared only these two himself.) I will use Norman Kemp Smith's 1927 translation, Immanuel Kant's Critique of Pure Reason (London: Macmillan Co. Ltd., 1963), sometimes with minor changes. Except in one part of the paper noted later, a reference only to one edition means that the passage in question does not appear in the other one.

2. Kant's attack on rational psychology in the chapter on the paralogism is, to my mind, the most successful discussion in the whole of the Critique of Pure Reason. Kant's attempt to "deny knowledge" in various domains "in order to make room for faith" (Bxxx) was at the heart of his reasons for writing the Critique of Pure Reason, though this has sometimes been overlooked.

3. This of course was the general view at the time. Descartes took geometry as his exemplar of certain knowledge. As a very different example, we might recall that in 1763 the Berlin Academy proposed as a topic, 'Are the Metaphysical Sciences Amenable to the Same Certainty as the Mathematical?'.

4. This is how Kitcher sees Kant's central task, too (Kant's Transcendental Psychology (New York: Oxford University Press, 1990), p. 15).

5. Paul Guyer says that Feder was the villain. Apparently the passages that upset Kant were mostly added by Feder to Garve's original draft (Kant and the Claims of Knowledge (New York: Cambridge University Press, 1987), p. 434).

6. Hume uses the denial of a priori knowledge argument in the Enquiry Concerning Human Understanding (1748), ed. Charles W. Hendel (Indianapolis: Library of the Liberal Arts, 1955), p. 42. In A Treatise on Human Nature (1736), ed. L. A. Selby-Bigge (Oxford: Clarendon Press, lst ed. 1888) he does not refer to a prioricity in the relevant section at all (cf. pp. 78-84).

7. Kant made the same assumption about the necessity of the propositions of mathematics and physics in Prolegomena to Any Future Metaphysics (1783), revision of Carus trans. by Lewis White Beck (Indianapolis: Library of Liberal Arts, 1950)).

8. Saul Kripke, "Naming and Necessity", in: G. Harman and D. Davidson, eds., Semantics of Natural Languages (Dordrecht: D. Reidel Co., 1972).

9. Two secondary issues arise in connection with this discussion of the analytic a priori. The first is that we should acknowledge that Kant accepted that we have and use analytic propositions linking concepts derived from experience, too (B3). I do not need to consider this kind of analytic proposition here. The second concerns his view of the relationship between analyticity and definition.

It would be natural to think that when Kant talks about "breaking [a concept] up into those constituent concepts that have always been thought in it, though confusedly" (A7=B11), he is talking about defining the concept. However, in the only section of the first Critique devoted explicitly and single-mindedly to definition, Kant raises very Putnam-like objections to the very possibility of defining concepts:

... an empirical concept cannot be defined at all, but only made explicit [perhaps something like being explicated in Carnap's sense]. For since we find in it only a few characteristics of a certain species of sensible object, it is never certain that we are not using the word, in denoting one and the same object, sometimes so as to stand for more, and sometimes so as to stand for fewer characteristics [1. Definitions, A727=B755-A733=B760].

Kant then illustrates the problem, as did Putnam two hundred years later, by the example of gold.

The relation of this scepticism about definition, which comes very late in the Critique, to Kant's ready acceptance of analyticity in the Introduction, deserves more exploration than I can give it here. Unlike our contemporaries, Kant did not base analyticity upon definitions (cf. L. W. Beck, "Kant's Theory of Definition" (Studies in The Philosophy of Kant (Indianapolis: Bobbs-Merrill Publishers, 1963), pp. 61-73)), and D. P. Dryer, Kant's Solution for Verification in Metaphysics (London: George Allen & Unwin Ltd, 1966), p. 324).

10. Op. cit., p. 15.

11. 11. See, for example, his "There is at least One A Priori Truth", reprinted in Philosophical Papers, Vol. 3: Realism and Reason (Cambridge: Cambridge University Press, 1983). The truth he has in mind is not every statement is both true and false. He means that this statement is necessary truth, could not be false, not that we know its truth independent of experience (though he might believe that, too).

12. 12. Kemp Smith translates 'merkmale' and 'merkmalen' as 'characters'. I think that term misses both the simplicity and the special technical sense of 'merkmal', a term whose literal means is 'mark', 'sign', 'indication', etc., but which often means something like 'differentiating feature' (differentiae) in Kant's work. (What exactly the term means, however, is controversial.)

13. 13. Henry Allison gives a good account of them ((Kant's Transcendental Idealism (New Haven, CN: Yale University Press, 1983), Ch. 4) and they have been explored by many other commentators. Because Jonathan Bennett argues that, in the end, so-called synthetic a priori propositions are really just unobviously analytic ones, he in particular devotes a lot of attention to the problems in Kant's account of analyticity ((Kant's Analytic (Cambridge: Cambridge University Press, 1966), Ch's 1 and 3). Kitcher makes some interesting remarks about the resemblances between Kant's remarks on analytic connections and Quine's attack on the very idea (op. cit., p. 27).

14. 14. That Kant had this question in mind in the first edition, too, is indicated by way we end up with a parallel question if we change the equivalent part of the deleted first-edition passage into a question: "What ... is ... the ground of the possibility of a priori synthetic judgments[?]" (A10).

15. 15. Op. cit., p. 18.

16. 16. One commentator who gets the direction of Kant's argument right is William Harper ("Kant on the A Priori and Material Necessity", in Robert E. Butts, ed. Kant's Philosophy of Physical Science (Dordrecht: D. Reidel, 1986), pp. 239-272). Though he does not make the point explicitly, the whole structure of his paper presents Kant as arguing from necessity to a prioricity.

17. 17. Op. cit., pp. 23-24.

18. 18. A further complication is that Kant also had a notion of necessary existence. Indeed it is in terms of this notion that he first explicates the schematized form of category of necessity, strangely enough (A145=B184), even though he explicates the category itself in other terms, in the second edition at least (B111).

19. 19. In a criticism that has attained some notoriety, Strawson charges that Kant's argument for the necessity of causal propositions (the Second Analogy) runs the two together (The Bounds of Sense (London: Methuen, 1967), Part Two: III).

20. 20. Op. cit., pp. 245-248.

21. 21. Op. cit., p. 80.

22. 22. Op. cit., pp. 99ff. and the many other references to necessity. Both Dryer's distinction and my recasting of it smack of psychologism -- a problem for another paper.

23. 23. In an important paper on what Kant really had in mind by construction in geometry, Friedman convincingly argues that the important thing we construct in pure intuition is not an image, an imagined instance, of the object of a mathematical proposition but a procedure for demonstrating it: a sequence of Euclidean constructions, or a sequence of calculations, that is to say, symbolic manipulations ("Kant's Theory of Geometry", The Philosophical Review XCIV (1985), pp. 455-506).

24. 24. D. P. Dryer is one commentator who does not neglect these sections of the Doctrine of Method (op. cit., especially Chapter 7). My view of Kant's theory of a priorik proof owes a lot to his, though I disagree with him on a fundamental point, as we will see. Among recent English-speaking commentators, Patricia Kitcher makes some references to these passages (op. cit., pp. 14-17), and Michael Friedman discussion the ones that discuss mathematical demonstration (op. cit.). There is also an brief but interesting footnote on the method of construction in Ted B. Humphrey, "The Historical and Conceptual Relations between Kant's Metaphysics of Space and Philosophy of Geometry", Journal of the History of Philosophy 11 (1973), pp. 483-512 and L. W. Beck discusses various parts of the Methodenlehre in a number of works. As a rule, however, it is hardly mentioned; Strawson, Bennett, Allison and Guyer are among the well-known examples.

An exception to this general rule is found in students of Kant's philosophy of mathematics. They pay a lot of attention to the remarks in the Methodenlehre about construction in geometry and, to a lesser extent, those about what Kant calls 'symbolic construction' in algebra; cf. M. Thompson, "Singular Terms and Intuitions in Kant's Epistemology", Review of Metaphysics 26 (1972-73), pp. 314-343; C. Parsons, "Infinity and Kant's Conception of the 'Possibility of Experience'" and "Kant's Philosophy of Arithmetic", both collected in Mathematics in Philosophy (Ithaca, NY: Cornell, 1983); and Friedman, op. cit.

25. 25. Since the pages are exactly the same except for the page-numbers in the two editions in the Method, I will cite only the page numbers in B from now on.

26. 26. The most interesting study of the role of the imagination in Kant's model in general is Wayne Waxman's Kant's Model of the Mind (Oxford: Oxford University Press, 1991). Waxman argues that for Kant the imagination plays a large role, not just in the sort of imaginative exercise I am exploring, but also in ordinary sensible representation. Dryer emphasizes the role it plays, too.

27. 27. That Kant does not make this point explicitly is remarkable. Yet he never distinguishes physics from philosophy nor treats the former separately anywhere in the chapter. We come to realize that if his method justifies the statements in philosophy that Kant credits to it, it also justifies his conviction that some of the propositions of physics are also necessary and universal only from the examples he uses. I have in mind in particular the example of causality that figures so centrally in the definitive statement of the method in Section 4 (A788=B816), but the way he discusses the other categories in Section 1 points in the same direction (A724=B752). I should note, if only to say that I will not discuss, the complicated question of the relation of a priorik and a posteriori pursuits in physics. That Kant clearly recognized the central role of the latter comes through more clearly in the Prolegomena and The Metaphysical Foundations of Natural Science than in the first Critique. Very roughly, I think a posteriori investigation give us grounds for thinking that propositions of physics are true, while a priorik investigations establish that what they assert could not be otherwise.

28. 28. 'Prove' (beweise) is the word Kant uses (A783=B811ff.). Though the method cannot demonstrate the propositions of philosophy, neither construction nor demonstration being available in philosophy (A726-7=B754-5), it can prove them. I will leave the sorting out of this distinction to another occasion.

29. 29. Dryer, op. cit., Ch. 7; Allison, op. cit., p. 78-80. Thus there has to be something wrong with both accounts. To be fair, Dryer does recognize that Kant says what I just quoted him as saying (p. 285). He just thinks Kant did not mean quite what he says there. In favour of both commentators, I should also acknowledge that, in later works, Kant himself says on occasion that pure intuition is required to prove synthetic a priori propositions (e.g., What Real Progress Has Metaphysics Made in Germany Since the Time of Leibniz and Wolff? ed. and trans. Ted Humphrey (New York: Abaris Books, 1983), Ak. XX, p. 266, quoted by Allison on p. 78, and the passages quoted by Dryer on pp. 284-5.). However, there is some question as to whether the works in question are a reliable indication of Kant's views. They are certainly completely inconsistent with the view expressed in the first Critique. It should additionally be noted that this issue may be smaller than it looks. Virtually anyone looking at what Kant says about the object in general would arrive at the view that it is some kind of singular representation, therefore something like a representation in pure intuition. I return to this issue briefly below.

It seems to be a little unclear when What Real Progress ... was written. Humphrey suggests 1793 as the probable date; a competition on the same topic set was being held by the physics class (!) of the Academy of Berlin (see the translator's introduction, ibid., p. 13). De Vleeschauwer says that Kant mailed three manuscripts subsequently edited into this work to a friend in 1800, apparently viewing them as ready for publication (The Development of Kant's Thought (1939), trans. A. R. C. Duncan (London: Thomas Nelson and Sons Ltd., 1962). Whatever, it is a late work (Kant died in 1804, just short of his 80th birthday), not even published during Kant's lifetime, and it is flatly inconsistent with the first Critique on the point in question.

30. 30. It is over this distinction that Dryer and I part company. Dryer does not give the kind of weight I give to the distinction between conditions of having experience and conditions of being an object (op. cit., Ch. 7). He also thinks that one can somehow establish that propositions are necessarily true by finding out that having and using them is a necessary condition of experiencing. Thus he does not see that the direction of proof in the method of proof in philosophy of the Discipline chapter is from the conditions of being an object to the conditions of experiencing an object, not the reverse. That is to say, he does not see the method of proof Kant describes in this chapter as anything more than a repetition of the central strategy.

31. 31. Lectures on Logic, 1, Ak. IX, p. 91. As Allison points out (op. cit., p. 80), we need a representation of an instance of some sort to prove synthetic propositions, and this sort of singular representation does not suffer from the limitations of inductive generalization.

32. 32. That is why I said earlier that the issue between Dryer and Allison and me on the issue of pure intuition may be smaller than it looks.

33. 33. The lack of enthusiasm Wittgensteinians might feel for this latter idea can easily be imagined(!).

34. 34. Kitcher provides a fine account of this strain of a priorio constructivism in Kant's thought (op. cit., Ch's 3 and 4, particularly pp. 72, 77, and 80). As she also notes, it allows us to answer the charge that Kant's whole doctrine of synthesis is built upon an archaic, atomistic theory of sensible stimulation. Her kind of account also captures a lot of what is living in transcendental idealism. In recent cognitive science, the general idea behind it has really taken off, indeed in a number of different directions and under a number of different names.

35. 35. Letter of July 1, 1797 (Ak. XI:514).

36. 36. The view I am developing here would fit very naturally with Friedman's view that the job of construction in mathematics is to invent and work through a demonstration or calculation in the imagination (op. cit., pp. 496ff.). The virtues of causal-mechanism explanations have been explored recently by Wesley Salmon (Scientific Explanation and the Causal Structure of the World (Princeton: Princeton University Press, 1984)) and Richard Miller (Fact and Method: Explanation, Confirmation and Reality in the Natural and Social Sciences (Princeton: Princeton University Press, 1987)).

37. 37. Asking for a mechanism comes pretty close to begging the question in this case, but I will let that difficulty pass.

38. 38. Intentional objects in "Two Aspects of Mental Images" (in Brainstorms (Montgomery, VT: Bradford Books, 1978), notional worlds in "Beyond Belief" (in The Intentional Stance (Cambridge, MA: Bradford Books/MIT Press, 1987).

39. 39. "Naming and Necessity", op. cit.