Methodological Considerations:

Application on Metaphysics:

Kant does not explicitly discuss the first difference (the one of the subject-matter = object), probably because at this stage he still does not want to discriminate between General Metaphysics in the traditional sense and Critical Metaphysics in the sense of his renovation project. The former, divided into Ontology, Rational Cosmology, Psychology and Theology, deals with transcendent objects, which Kant has never entirely renounced but now regards as a heavy baggage, while the latter investigates the boundaries of knowledge from principles and represents the knowledge Kant is presently seeking. The point is that in this latter sense, Metaphysics does not have any special subject-matter apart from abstract concepts.
Therefore Kant starts his demarcation with the second regard: the sources of metaphysical knowledge. "Its very concept implies that they cannot be empirical." Its notions "must never be derived from experience". In other words, metaphysical knowledge is not "physical", regardless whether the physical refers to the external reality (natural science) or internal reality (empirical psychology). As the very name "Metaphysics" suggests, its knowledge is "meta-physical" (after/above the physical), that is to say it is non-empirical. Kant explicates the meaning of the word "metaphysical" as a unique determination, but at a closer scrutiny what he says comes down to two different ideas that reveal the above ambiguity:

1. Traditional:

Metaphysics is "the knowledge lying beyond experience", that is to say, its principles and concepts surpass experience both by their source and by their scope.

The first connotation is not identical with the second. Both metaphysics are  non-empirical, but they entail two very different positions with regard to experience. The first speaks about the knowledge "lying beyond experience" whereas the second indicates that metaphysical knowledge is to be understood as an a priori knowledge "underlying experience".
It is not the same to acquire knowledge "after experience" (hereafter, afterlife) and to gain knowledge about the forms coming "before experience" (forms of intuitions and categories). Although Kant does not denounce this ambiguity here (he speaks rather as if there is a continuum between the traditional, popular and scientific connotations) it is clear that his formulations point to two different and perhaps opposed ideas of Metaphysics.

Note: Both the traditional and the Kantian Metaphysics are non-empirical, but only the first goes beyond possible experience. The gulf between these two gets only strengthened by equating the popular notion of Metaphysics with the realm of the supernatural. In contrast, Kant's metaphysics is laid on the firm ground of legislative principles.

The internal tension between the two senses of "non-empirical" ('after' and 'before experience', 'lying beyond' and 'underlying', transcendent and transcendental) leads to the following division within the concept of Metaphysics:

In the first part of his discussion about the "sources of Metaphysics" Kant still preserves the continuity with the traditional (transcendent) nature of metaphysical knowledge. Later on he even recognizes that transcendent metaphysical objects represent "the noblest" objects of human concerns ("the knowledge of a highest being and of a future existence" corresponds to the two main objectives of Cartesian Metaphysics: "the existence of God and the immortality of the soul"). But in the second part of the same paragraph Kant obviously refers only to the transcendental concept of Metaphysics, which is evident from the way how he delineates Metaphysics from two other theoretical sciences: Physics and Mathematics (demarcation between these three disciplines is an old Aristotelian problem). By comparatively applying all three Kantian criteria of knowledge one gets the following diagram:

Object

Concepts
Adequate

Nature
"Deep science"

Concepts
Constructs/Exact

Source

Pure Reason

Experience

Pure Reason

Kind

A priori in abstracto

Empirical

A priori in concreto

We should be able now to understandable why Kant could not have drawn a clear cut demarcation line between Metaphysics and other theoretical sciences by simply defining their specific objects. Since Mathematics is also a non-empirical science and since its subject-matter is also abstract and a priori (concepts of reason), a further differentiation between it and metaphysical concepts is needed to distinguish between the two. This differentiation is carried out in the Critique of Pure Reason to which Kant here refers. The distinction addresses the nature of metaphysical concepts as opposed to mathematical.

In a nutshell, metaphysical concepts are universal concepts in abstracto, whereas mathematical concepts require intuition which considers them in concreto. Metaphysics has to expose its concepts as adequate to the objects, whereas mathematical concepts are exempt from that obligation because they stem "from the construction of concepts". This means that Metaphysics must conform to the rules of concept synthesis if it is ever to attain the truth (its knowledge being gained not from axioms but from concepts by means of reason). Conversely, Mathematics strives only for validity of theorems derived from whatever axioms we may choose (mathematical definitions can never be faulty or wrong). Since the validity of Mathematics does not depend on the veracity of its axioms, its superb exactness goes hand in hand with the utmost arbitrariness (at the beginning). By contrast, in philosophy definitions come only at the end, after we ascertain the true nature of the object, not at the beginning of the process.

Transcendent, from the Latin transcendere = climb across, go beyond; something superior or surpassing (for instance, God), something that goes beyond that what is given to our experience (metaphysical knowledge in the traditional sense). According to Kant, however, there can be no knowledge of anything transcendent. Search for it is a "transcendental illusion".  

Transcendental, from the Latin transcendere = climb across, go beyond; originally, that which surpasses the most general categories (transcendentalia); according to Kant, that which precedes experience and determines its content beforehand (a priori); also the a priori analysis of pure reason which elucidates the conditions of possibility of any experience; the transcendental investigative procedure is "occupied not so much with objects as with the mode of our knowledge of objects in so far as this mode of knowledge is to be possible a priori".

Waldo Emerson's "Transcendentalism" and the so called "Transcendental Meditation" of Macharishi Macheshi do not have anything in common with the transcendental in this sense apart from the sheer homonymy (the similarity of sound); their real connotation is rather "transcendent"; indeed in non-terminological parlance these two words are very often used interchangeably - a remnant of the medieval equation between transcendentalia and transcendentia.

Having determined both the object and the faculty of metaphysical knowledge Kant moves on to the question what "kind of knowledge…can alone be called metaphysical" (§ 2 of the Preamble). His answer is that Metaphysics must consist of "a priori judgments".
Note: The word "judgment" does not have here the popular meaning of an evaluative statement. Kant uses it in the manner of the traditional theory of logic according to which judging relates a predicate term to a subject term within an assertion. The underlying thought of every proposition is "I judge that S is P". So conceived, every judgment is a function of unity of our representations expressed in different conjunctions of the subject and the predicate term (general form: 'S is P'). From this point of view all judgments are synthetical in the sense of connecting a subject with a predicate by means of a copula ('is' or 'is not'). However, with regard to "their content" ("relations of ideas" or empirical connections) and the ground of their veracity (a concept of a subject or an empirical object) the status of various judgments could be very different and this difference is what Kant now addresses.

Explicative vs. Expansive:

If the predicate only explicates the subject "adding nothing to the content" of the proposition the resulting judgment is explicative or analytic. If, on the other hand, it adds something to the subject (expands on it) the judgment is expansive or synthetic (in the specific sense of putting together different representations). In the first case, the constitutive terms of the judgment (S and P) fully coincide ("stand in the relation of logical identity"), which secures upfront the analytic status of the judgment, whereas, in the second, the connection between the terms remains synthetic and the veracity of the judgment cannot be determined logically, but needs to be tested empirically.

Note 1: Although these distinctions largely coincide with each other their respective principles are not identical. The a priori - a posteriori division is centered around the epistemic source of our propositions, while the analytic - synthetic distinction points to the semantic reference of respective propositions. The designations necessary or contingent are supposed to reveal the logical status of propositions under scrutiny.

Note 2: All these distinctions have been challenged by W.V. O. Quine who has shown that the so called necessary truths (= a priori, analytic) presuppose the equality of meaning among synonyms no matter whether their basis is logical, conventional or practical. This presupposition is so problematic that it negates the special status of analytic vis-a-vis a posteriori propositions and renders redundant the very distinction. The distinction between a priori and a posteriori rests on an ambiguous notion of meaning - it is just a convention without real ground. According to Quine, there are no distinct types of reality in the world that would require different types of judgments.

Analytic Judgments:

Analytic judgments with empirical concepts

Kant agrees with his predecessors that analytic judgments are a priori ("are in their nature a priori cognitions"). They objectively precede and ground our knowledge even when they come later in the temporal acquisition of knowledge. But they perform that role simply by making more "distinctly" or fully "conscious" what has been "already actually thought in the concept of the subject". Thus, strictly speaking, they are only clarifying judgments. The predicate does not enlarge the scope of our knowledge of the concept, but only analyzes it (this is the reason why Kant aptly chooses the label "analytic" as their designation). For instance, the analytic judgment "All bodies are extended" does not amplify the concept of body. It only states what is already included in it: "extension was really thought to belong to (the) concept of body before the judgment was made, though it was not expressed". In other words, the analytic judgment takes the object (= body) according to its concept (the subject) and does nothing else but asserts its self-sameness. This is the reason, on the other hand, why an analytic judgment must be necessarily and a priori true, true by virtue of the formal identity between the subject and the predicate. 

Note: Kant claims that the connection of the subject and the predicate could be analytical even when both concepts are empirical provided that it is a priori, but his example "Gold is a yellow metal" can hardly convince anybody. In § 3 he refers to Locke's Essay Concerning Human Understanding (Bk., 4, Ch. III, § 9) as an anticipation and corroboration of his view, but in the very same paragraph Locke explicitly treats the gold-predications like having certain weight, being yellow, being malleable and fusible as non-analytic. By all standards, Kant's example should be regarded rather as a synthetic proposition based on experience. The assertion of gold's yellowness is not any different from the statement "gold melts at 1,064.4C°". Moreover, it could be false - when beaten into thin foils gold appears green. Kant must have known this but he obviously did not envision gold as an element in the sense of the periodic table of elements. He spoke about gold only as "a yellow metal" in the sense of ordinary language, which usage, once accepted as the common concept, by definition entails yellowness as an essential characteristics of gold.    

Kant's predecessors have applied the law of (non) contradiction to all judgments indiscriminately simply as the negative criterion of their viability. He, however, thought that one had to distinguish between a negative and a positive application of the law. In the first case we simply establish formal conditions for a judgment to be viable, in the latter the principle itself secures the validity of the judgment. Kant contends that "all analytic judgments depend wholly on the law of contradiction" in the latter sense. Of course, all judgments should be free of self-contradiction no matter whether they are true or not, but analytic judgments are true simply by virtue of being non-contradictory. And the other way round, if there is a real possibility that they contradict themselves in terms of accepting the subject and denying the predicate they cannot be analytic. But if an "affirmative analytical" judgment is true based on this law, then its predicate "cannot be denied (of the subject) without contradiction" because the negation would then violate the self-identity of the concept (the predicated connotation being part of the concept). To use the same example as Kant, if we said "All bodies are unextended" we would contradict the subject concept by claiming basically that "No body is extended". To avoid it we have only two choices: either to drop the negation of the predicate (which, by reverting to the complementary class, restitutes the initial judgment: "All bodies are extended") or to change the subject term into its compliment ("No bodies are un-extended."). As we glean from the second case, the "opposite is necessarily denied of the subject in an analytic but negative judgment". This corroborates the claim that in analytic judgments both the subject and the predicate are locked up in the relation of identity whose subsistence is sanctioned by the "law of contradiction".

Synthetic judgments are products of the synthetical work of our cognitive faculties insofar as they deal with experience. However, experience is not a mere bundle of representations, but "a continued synthesis of perceptions". We know something only if we can represent it in its connections with other representations. These connections are not self-evident and therefore require stepping out of the subject concept toward the "testimony of experience" ("passing beyond the concept of the subject"). The knowledge of facts is in this sense based on the synthesis of the object in its manifold relations. In other words, we need to represent the object itself by means of an additional synthesis which goes beyond the (minimal) concept of the subject. Expansive predication has the function of establishing this linkage of additional representations: it supplies the predicates that are not already included in the subject. Thus "judgments of experience are always synthetic." Unlike analytic judgments, which are by the rule universal or general (focusing on the core concept), synthetic judgments are typically particular (speaking about some instances or some features). Surprisingly, Kant gives as example a judgment which is universal: "All bodies have weight."

Note: The German original, printed by J.F. Hartknoch in Riga (1783), reads here "Some bodies are heavy." At first glance, this is in conformity with the (commonsense?) idea that not all bodies have weight. This assumption receives experimental corroboration through the fact that bodies in outer space become weightless. But Kant could not have known this. It is not an outright implication of the gravity theory either. Therefore, from the point of view of the eighteenth century knowledge it is an understatement and does not put into a sharp contrast the difference with analytic judgments. The statement "All bodies have weight" offers in that respect a much better reading. It is formulated as a universal judgment stating something which is seemingly very difficult to distinguish from the concept of body. As a matter of fact, this proposition was traditionally regarded as analytic, but Kant agrees with Hume that it, on the contrary, represents a piece of synthetic knowledge a posteriori. The point is that weight is not contained in the concept of body the way extension is. We need to add it to the idea of body as "something not actually thought in the universal concept of body". Its negation does not produce a contradiction, even if it is associated with body "insofar we can think of". All this suggests syntheticity which Kant explicitly attests ("must therefore be called synthetic").

Kant disagrees that all synthetic judgments are empirical (although synthesis is a suitable designation for all judgments of this kind, it is not tantamount to empirical linkage of a predicate to the subject). To be sure, synthetic judgments could be derived from experience tout court ("of empirical origin"), that is to say, they could be simply a posteriori judgments, as, for example, the following observations: "This piece of gold is 1 inch long" or "Kant was a bachelor."
Thus, in contrast to analytic judgments, which are always non-empirical even when they operate with empirical concepts ("it would be absurd to base an analytic judgment on experience"), empirical judgments "are always synthetic" although not all synthetic judgments are empirical.

Kant believes that there exists yet another class or sub-class of synthetic judgments which are "certain a priori and which spring from pure understanding and reason".
This means that synthetic judgments could be either

The very idea of a priori synthetic judgments appears as something self-contradictory, particularly if we equate syntheticity with the notion of a posteriori (like A. Ayer). But this identification would be, for Kant, entirely unwarranted. If I affirm a predicate B as necessarily and universally (though not logically) linked to the subject A, then B is an a priori synthetic representation of A, regardless whether the concept of A is a priori or not. A priori synthesis attributes a necessary property to the subject which is not contained in it.

Note: The logical table of judgments provides an exhaustive list of all possible judgments divided into four logical types along with a parallel table of the concepts of pure understanding "which contain the a priori conditions of all synthetic and necessary judgments" (§ 23).

Since they are synthetic synthetic a priori judgments do not "spring from the law of contradiction". They "must depend upon other principles than the law of contradiction", which they also, of course, must respect. But they cannot be deduced from it alone. The law of contradiction is not the principle of Metaphysics but of pure Logic. The knowledge of facts, however, is based on the principle of causation rather than on Logic. As Kant demonstrates in the central part of the Prolegomena, the general law of all synthetic a priori judgments stems from the insight that the conditions of the possibility of experience are the conditions of the possibility of the objects of experience. Synthetic a priori principles cannot be, therefore, derived from mere thinking (Logic). But they cannot be deduced from sheer intuitions (Experience) either. The concept of cause or the principle of sufficient reason cannot be represented intuitively - precisely because they are the principles of objects, or better to say, of the objectivity of objects. They "must express what experience in general… contains" and they do that by adding some necessary components "to the sensuous intuition and its logical connection in a judgment". These conceptual components define the a priori rules for possible generation of synthetic a priori judgments. Synthetic judgments a priori are amplifying in the sense of giving more than mere explication, but necessary in giving more certainty than regular experiential judgments.

Note: Richard Rorty surmises that the whole "story about synthesis" in terms of our active subjectivity was invented to explain the possibility of synthetic a priori judgments. The two doctrines are indeed mutually intertwined, but the general idea that our cognitive faculties operate synthetically was for Kant more fundamental even if it was not developed prior to the discovery of synthetic a priori judgments.

Judgment, from the Latin jus = law; 1. popular meaning: evaluating something; 2. in traditional (Aristotelian) logic, asserting or denying a predicate of some subject; 3. for Kant, who follows this old usage, any judgment is a relationship of concepts, the subject and the predicate, within a proposition. This explains why he freely alternates the term "judgment" with the term "proposition". However, his theory of judgments includes also a grasp of the object as well as of the "I" ("the objective unity of apperception") and this moment moves him away from a purely logical conception of judgments.

Subject, from the Latin subjectum (i.e., Greek hypokeimenon) = something underlying, under-carriage, substrate; 1. medieval philosophy, grammar, logic: the bearer of a property or an attribution (i.e., of a predicate), the subject-term, substance; 2. modern philosophy: the self, the mind, the "I", the consciousness; ordinary language: (a) object, topic, (b) the subordinate (person). In logic the subject-term is a singular term which names an entity.

Predicate, from the Latin praedicatus = proclaimed; something asserted about something else, that which is affirmed or denied about a subject. The predicate-term is a general term which is true (or not) of an entity.

Analytic Judgments, from the Greek analytikos = pertaining to taking apart; the propositions (judgments) which are necessarily true due to the relation between their concepts (logically true: "A = A") or true by virtue of the meanings involved (true by definition): "All bachelors are unmarried" or "a bicycle has two wheels".

Synthetic Judgments, from the Greek synthetikos = put together; the judgments in which what is asserted by the predicate has not been thought in the subject; for instance, "all bachelors are happy" or "this bicycle has an elongated seat". Synthetic judgments are not logically true nor true by definition.

Principle of (Non) Contradiction, one of the laws of the traditional Aristotelian logic which says that an affirmative proposition and its negation cannot be both true at the same time in the same respect. For instance, "X is white" and "X is not white" cannot be both true at the same time (or false). Kant omits the phrase "at the same time" in his phrasing of the law, because he wants to avoid all temporal (empirical) references in the formulation of general principles. This principle is for him a universal although negative condition of admissibility of all our judgments but specifically the test of veracity for analytical judgments (if not contradictory they are true).

Principle of Sufficient Reason, the principle of universal causation ("nothing happens without a cause"); was introduced by Leibniz as the fourth law of logic and the governing principle of the truths of fact.

Kant is convinced that precisely the judgments which express a priori cognitions of synthetical kind make up the body of Metaphysics, Mathematics and pure (theoretical) Physics. In order to demonstrate more cogently their existence he first undertakes a careful analysis of mathematical propositions. The greatest novelty (and at the same time the most controversial point) of Kantian analysis is the contention that mathematical propositions are not analytic (as has been always thought) but synthetic, although synthetic a priori: "mathematical judgments are all synthetic". Kant acknowledges that this thesis goes directly against all accepted notions, but he claims that "it is incontestably certain and most important in its consequences". Unfortunately, it has been overlooked.
"This fact seems hitherto to have altogether escaped the observation of those who have analyzed human reason." This oversight was caused by an incorrect understanding of the role the law of contradiction plays in Mathematics. Mathematical conclusions all proceed in accord with the law of contradiction, "as is demanded by apodictic (mathematical) certainty", but they are not known or derived solely from it. Mathematical propositions can be established by the law of contradiction indeed, but never by that law alone. They also need some intuition and synthesis in time. Of course, they are not derived from experience either. "All strictly mathematical judgments are a priori, and not empirical, because they carry with them necessity, which cannot be obtained from experience." But they are not analytic, for mathematics would not be able to furnish additional knowledge if its statements were just a matter of definition or stipulation.

Note: The prevalent view in the philosophy of mathematics has been and still is that mathematical knowledge is a priori. Both formalism and conventionalism seem to agree on this. Only some empiricists contend that mathematical truths are very general empirical propositions. But a growing number of philosophers in the aftermath of Goedel's and Quine's insights challenge the mainstream view thus providing new support for Kant's theory. Leonard Nelson and Irving Copi assert syntheticity for geometry and arithmetic respectively.

"The essential and distinguishing feature of pure mathematical knowledge among all other a priori knowledge is that it cannot at all proceed from concepts, but only by means of the construction of concepts." To construct a concept means to represent it in an intuition a priori. As mathematical necessity is not purely logical but constructivist, mathematical propositions cannot be generated only "by dissection of the concept". "Pure mathematics, and especially pure geometry, can have objective reality only on condition that they refer merely to objects of sense." This way of proceeding requires synthetical judgments which go beyond concepts toward that which is represented in intuition. According to Kant, this comes down to judgments of the third type, namely to wit, synthetic a priori judgments. He shows that this holds true both of Arithmetic and Geometry at least as far as they are branches of pure Mathematics.

Note: Great mathematicians like Frege and Hilbert admit that at least geometry is founded on intuition but they stop short of accepting syntheticity as the right designation for mathematics in general.

Previous analyses of arithmetical propositions were mislead by the simplicity of their examples (typically, Descartes and Hume were considering very simple operations like "2 + 2 equal 4" or "2 + 3 = 5" which create the impression that the result is analytically deduced from the concepts of additives). To avoid the same delusion of analyticity these thinkers fell prey to, Kant purposely takes much less obvious propositions. His formula of choice is: "7 + 5 = 12". He contends that one misunderstands this proposition if it is viewed as an analytical statement supported by the law of contradiction. He is aware that people reason like this: to say that the sum of 7 + 5 is anything other than 12 would be contradictory, therefore this is an analytic proposition. But he insists that the concept of the sum of 7 and 5 contains "merely their union in a single number", not the particular number 12. An analytical version of the said equation would be "7 + 5 = the sum of 7 and 5". But this is not what the mathematical proposition "7 + 5 = 12" states. It says that "the sum of 7 and 5" equals the particular number 12. To replace "the sum of 7 and 5" with the number 12, which is not "thought by merely thinking of the combination of seven and five", we "must go beyond these concepts, by calling to aid some intuition", for instance 5 fingers or five points and to add them "successively". In other words, we need to get engaged in counting which clearly indicates that the judgment "7 + 5 = 12" truly amplifies our knowledge of the subject concept "7 and 5". Since we cannot figure out the sum of 7 and 5 without resorting to these calculative aids, the conclusion is inevitable that "arithmetical judgments" are in fact "synthetic". The synthetic character of arithmetical propositions becomes even more manifest when we perform mathematical operations with bigger numbers and magnitudes.

Note: The most questionable point in Kant's argument is the contention that "the sum of 7 and 5" does not "in any way" already include 12. What about the sum of "twenty and five"? Can one say that "twenty and five" does not include "twenty five"? Kant could point out that "twenty and five" is only a formulation of a task, whereas "twenty five" is the solution (Max Apel), but the relationship seems to be clearly analytic. However, the same cannot be asserted for all equations. Kant is certainly right in claiming that the sum of "twenty and five" does not include "five times five" analytically. To be sure, the propositions of arithmetic are rather satisfiable than necessarily valid - some could be denied without entailing any contradiction.

Kant attests the same a priori syntheticity for geometrical propositions. This time his example is: "Straight line is the shortest path between two points." He claims that the concept of straight line establishes only the quality of the line (being straight as opposed to being curved or zigzag), not its quantity (distance). Straight line is the line produced by moving a point into one and the same direction. The concept of direction contains a priori the notion of the straight, but not the longitude. Therefore, the aforementioned proposition is synthetic, though a priori. The concept of the "shortest" is entirely "additional" and cannot be "obtained by any analysis of the concept 'straight line'". To be able to reach the conclusion that the straight line is the "shortest path" we must resort to intuition (indirect way) and use it as a representational aid in the matter.
Kant admits that upon reflection one must closely link the "shortest distance" with the concept of the "straight line", but his point is that distance was not actually thought in the concept itself. The reason for fusing these two things into one and for the ensuing transformation of syntheticity into supposed analyticity is the ambiguity of the phrase "straight line" (Kant speaks about "the duplicity of the expression"). It could denote the nature of a line as well as shortest distance between two points.
Note: By the same token Kant would regard the following statement as synthetical a priori: Circle is a plane curved figure that encloses bigger space than any other with the same perimeter.

Note 1: In the original text the paragraph with this explanatory remark is obviously misplaced during the printing of the manuscript. It does not make sense immediately after the passage discussing downright analytic propositions. The original edition apparently suffers from some mix-up that has gone unnoticed either in the manuscript or in the galley proofs.

Note 2: We can grant that 'A straight line is the shortest distance between two points' is a synthetic proposition unless we phrase it as a proposition of a Euclidean geometry - there it is true by implication. As to the assertion of syntheticity for geometrical propositions in general, one should bear in mind common intuitive aspects of geometrical demonstrations. At the time of Kant it was customary to establish theorems by means of the construction of figures and this fact could have convinced Kant that he was correct in claiming that geometrical propositions are a priori though synthetic. Furthemore, Kant did not know for the possibility of non-Euclidean Geometry. On the contrary, he was convinced that Euclidean Geometry was the only possible Geometry, and consequently the true Geometry of space. Obviously, if Kant thought that the only possible geometry was Euclidean (three dimensional) he was wrong and this mistake is often adduced against him. But he was right in implying that the space of human experience is Euclidean. Our intuition of space conforms to the space of Eucliedan geometry. In fact, the very possibility of non-Euclidean geometries could be interpreted as an indication that geometry is not analytic.

Analyticity in Mathematics:

Note: Definitory propositions like "triangle is a plane figure with three sides" are analytic in this sense. The same holds true of geometrical demonstrations which are based on intuitive axioms. The problem, however, is that this type of proof is not entirely adequate because it grounds mathematical demonstration on spatial perceptions. Anyway, the fact is that a proposition is analytic with regard or within one language. The problem is how to make it analytic for all languages. This was Kant's dream, but Quine has shown that to gauge analyticity or syntheticity of isolated propositions without placing them in a particular system or language must be always misleading.

Many judgments that pertain to Metaphysics are in truth analytic (as are those that pertain to Mathematics). Metaphysics has been always using analytic propositions the way Mathematics uses them: "We can be shown indeed many propositions, demonstrably certain and never questioned; but these are all analytic…" However, they have only a supportive role within Metaphysics. They "rather concern the material and the scaffolding for metaphysics than the extension of (metaphysical) knowledge, which is our proper object in studying it."
The analytic judgments of the aforementioned kind are very instrumental in explicating certain operative concepts of Metaphysics, but they do not really belong to its body of knowledge. Only their constitutive concepts could be regarded as immanent to metaphysical considerations.

Judgments

a priori

a posteriori

Analytic
Explicative

Apodictic Logic
Pertaining to Metaphysic

Synthetic
Expansive

Empir. Natural Science
Empirical Psychology
Assertoric, Problematic

In raising the question of synthetic a priori judgments we should resist the temptation to conceive them simply as a priori products of "metaphysical imagination". Metaphysics, as envisioned by Kant, is not a self-generated art or discipline, but the science of how we conceive objects a priori. The source of metaphysical judgments is not metaphysical imagination, but "the laws of pure reason". Although not necessary in themselves (logically), they are necessary for experience which is organized through intuition and concepts. "But the generation of a priori knowledge by intuition as well as by concepts, in fine, of synthetic propositions priori, especially in philosophical knowledge, constitutes the essential subject of metaphysics."

Note: The Kantian pair of intuitions and concepts obviously corresponds to the Humean pair of impressions and ideas. The duality of intuitions and concepts encompasses both types of representations and both sides of our subjectivity, the receptive and the active one. Intuitions are immediate representations of immediately present and particular objects whereas thinking provides mediate representations of something in general. Through intuition our organs are "approached" by appearances, through thought we bestow a priori forms of intuition to perceptions. Real knowledge is unification of intuitions and concepts. A merely intuited or conceived object is not an object at all; the latter is just a thought, the former just an incomprehensible perception.

Apodictic, from the Greek apodeixis = pointing out, proof, demonstration; an apodictic proposition states what must be the case, in contrast to an assertoric one, which states what is the case, and a problematic one, which states what can be the case.

Experience, from the Latin experientia = trial, experience; 1. that what is happening to the subject, events, feelings, states of consciousness; 2. that what stems from the acts of the subject; for Aristotle, a general knowledge derived from many memories of the same thing; for Kant, the realm of possible knowledge, practically identical with nature in the sense of the Newtonian physics: "Experience consists in the synthetic connection of phenomena (perceptions) in consciousness, so far as this connection is necessary."

Intuition, The English rendition of the German Anschauung = mode of visual perception (comes from the Latin in + tueor = to see, look, gaze); in Kant, the faculty of apprehending the concrete singularity of things as they are given in space and time but not yet subsumed under abstract concepts; it is both something conditioned (receptive aspect of experience with regards to objects) and conditioning (pure forms of sensible intuition, space and time, precede and determine all actual impressions of objects).

In accord with the general idea of transcendental approach, transcendental philosophy purports to provide "the complete solution of the problem" how synthetic a priori judgments are possible. The metaphysical a priori becomes the transcendental with a synthetic power. The elaboration of this position is an inquiry "into purely rational knowledge". It explains in which sense the cognition a priori "lies at the basis" of Metaphysics. As theory of knowledge, transcendental philosophy secures the foundations of theoretical sciences along with Metaphysics. Consequently, "transcendental philosophy" is the investigation of those conditions that are necessary conditions of all cognitions a priori. Therefore, "transcendental philosophy" must precede all Metaphysics (including the "scientific" one) by securing its possibility.

Note: These formulations suggest that transcendental philosophy coincides with the critique of pure reason. It is certainly not psychology of knowledge. But the whole tenor of the Prolegomena as well as particular passages favor a clear separation between the two, while equating the system of transcendental philosophy with Metaphysics as a science (=system of a priori principles). The Critique retains a preparatory function with regard to that objective. Even though Kant likes to call the preliminary part of his project "the Metaphysics of Critical Idealism", it should not be confused with the Metaphysics as a science about those "noble objects". Metaphysics as theory of knowledge (transcendental) is different from the dogmatic (transcendent) Metaphysics which goes beyond experience.

In keeping with the adopted analytical method, Kant first scrutinizes two sciences of "theoretical knowledge", pure Mathematic and pure Physics, which "exhibit to us objects in intuition". Having demonstrated that these theoretical sciences possess cognitions from pure reason, he then shows how they achieve the conformity of their a priori cognitions with concrete objects (this approach in concreto is opposed to the procedure of the Critique of Pure Reason that starts from concepts in abstracto). The explanation how the a priori judgments of Mathematics and Physics are valid is at the same time a proof of their validity. Finally, he turns to the metaphysical science of a priori principles in general (metaphysica generalis) and explains how Metaphysics as science is possible. Thus the transcendental examination of the conditions of such science advances through four stages:

1. How is pure mathematics possible? (§§ 6-13)
2. How is pure natural science possible? (§§ 14-39)
3. How is metaphysics in general possible? (§§ 40-60)
4. How is metaphysics possible as science?

Mathematics, Physics and Metaphysics all make universal claims as exemplified in synthetic a priori judgments ("Straight line is always the shortest distance between two points.", "All bodies attract each other with gravitational force.", "All events have cause."). Truly, the possibility of equally valid metaphysical judgments is still not recognized, but the very existence of Mathematics and Physics provides the required proof. From the possibility of the given knowledge a priori in Mathematics and Physics follows the possibility of metaphysical a priori knowledge.
However, the possibility of scientific Metaphysics must be prepared by an appropriate discussion of Metaphysics as a "natural disposition", which explains the necessity of the fourfold (instead of threefold) articulation of the problem.