Section 2
Original German
§ 2.
Von der Erkenntnißart, die allein metaphysisch heißen kann.
a)
Von dem Unterschiede synthetischer und analytischer Urtheile überhaupt.
Metaphysische Erkenntniß muß lauter Urtheile a priori enthalten, das erfordert das Eigenthümliche ihrer Quellen. Allein Urtheile mögen nun einen Ursprung haben, welchen sie wollen, oder auch ihrer logischen Form nach beschaffen sein, wie sie wollen, so giebt es doch einen Unterschied derselben dem Inhalte nach, vermöge dessen sie entweder blos erläuternd sind und zum Inhalte der Erkenntniß nichts hinzuthun, oder erweiternd und die gegebene Erkenntniß vergrößern; die erstern werden analytische, die zweiten synthetische Urtheile genannt werden können.
Analytische Urtheile sagen im Prädicate nichts als das, was im Begriffe des Subjects schon wirklich, obgleich nicht so klar und mit gleichem Bewußtsein gedacht war. Wenn ich sage: alle Körper sind ausgedehnt, so habe ich meinen Begriff vom Körper nicht im mindesten erweitert, sondern ihn nur aufgelöset, indem die Ausdehnung von jenem Begriffe schon vor dem Urtheile, obgleich nicht ausdrücklich gesagt, dennoch wirklich gedacht war; das Urtheil ist also analytisch. Dagegen enthält der Satz: einige Körper sind schwer, etwas im Prädicate, was in dem allgemeinen Begriffe vom Körper nicht wirklich gedacht wird; er vergrößert also meine Erkenntniß, indem er zu meinem Begriffe etwas hinzuthut, und muß daher ein synthetisches Urtheil heißen.
b)
Das gemeinschaftliche Prinzip aller analytischen Urtheile ist der Satz des Widerspruchs.
Alle analytische Urtheile beruhen gänzlich auf dem Satze des Widerspruchs und sind ihrer Natur nach Erkenntnisse a priori, die Begriffe, die ihnen zur Materie dienen, mögen empirisch sein, oder nicht. Denn weil das Prädicat eines bejahenden analytischen Urtheils schon vorher im Begriffe des Subjects gedacht wird, so kann es von ihm ohne Widerspruch nicht verneint werden; ebenso wird sein Gegentheil in einem analytischen, aber verneinenden Urtheile nothwendig von dem Subject verneint und zwar auch zufolge dem Satze des Widerspruchs. So ist es mit den Sätzen: Jeder Körper ist ausgedehnt, und: Kein Körper ist unausgedehnt (einfach), beschaffen.
Eben darum sind auch alle analytische Sätze Urtheile a priori, wenn gleich ihre Begriffe empirisch sind, z. B. Gold ist ein gelbes Metall; denn um dieses zu wissen, brauche ich keiner weitern Erfahrung außer meinem Begriffe vom Golde, der enthielte, daß dieser Körper gelb und Metall sei: denn dieses machte eben meinen Begriff aus, und ich durfte nichts thun als diesen zergliedern, ohne mich außer demselben wornach anders umzusehen.
c)
Synthetische Urtheile bedürfen ein anderes Prinzip als den Satz des Widerspruchs.
Es giebt synthetische Urtheile a posteriori, deren Ursprung empirisch ist; aber es giebt auch deren, die a priori gewiß sind, und die aus reinem Verstande und Vernunft entspringen. Beide kommen aber darin überein, daß sie nach dem Grundsatze der Analysis, nämlich dem Satze des Widerspruchs, allein nimmermehr entspringen können; sie erfordern noch ein ganz anderes Prinzip, ob sie zwar aus jedem Grundsatze, welcher er auch sei, jederzeit dem Satze des Widerspruchs gemäß abgeleitet werden müssen, denn nichts darf diesem Grundsatze zuwider sein, obgleich eben nicht alles daraus abgeleitet werden kann. Ich will die synthetischen Urtheile zuvor unter Classen bringen.
1) Erfahrungsurtheile sind jederzeit synthetisch. Denn es wäre ungereimt, ein analytisches Urtheil auf Erfahrung zu gründen, da ich doch aus meinem Begriffe gar nicht hinausgehen darf, um das Urtheil abzufassen, und also kein Zeugniß der Erfahrung dazu nötig habe. Daß ein Körper ausgedehnt sei, ist ein Satz, der a priori feststeht, und kein Erfahrungsurtheil. Denn ehe ich zur Erfahrung gehe, habe ich alle Bedingungen zu meinem Urtheile schon in dem Begriffe, aus welchem ich das Prädicat nach dem Satze des Widerspruchs nur herausziehen und dadurch zugleich der Nothwendigkeit des Urtheils bewußt werden kann, welche mir Erfahrung nicht einmal lehren würde.
2) Mathematische Urtheile sind insgesammt synthetisch. Dieser Satz scheint den Bemerkungen der Zergliederer der menschlichen Vernunft bisher ganz entgangen, ja allen ihren Vermutungen gerade entgegengesetzt zu sein, ob er gleich unwidersprechlich gewiß und in der Folge sehr wichtig ist. Denn weil man fand, daß die Schlüsse der Mathematiker alle nach dem Satze des Widerspruches fortgehen (welches die Natur einer jeden apodiktischen Gewißheit erfordert), so überredete man sich, daß auch die Grundsätze aus dem Satze des Widerspruchs erkannt würden, worin sie sich sehr irrten; denn ein synthetischer Satz kann allerdings nach dem Satze des Widerspruchs eingesehen werden, aber nur so, daß ein anderer synthetischer Satz vorausgesetzt wird, aus dem er gefolgert werden kann, niemals aber an sich selbst.
Zuvörderst muß bemerkt werden: daß eigentliche mathematische Sätze jederzeit Urtheile a priori und nicht empirisch sind, weil sie Nothwendigkeit bei sich führen, welche aus Erfahrung nicht abgenommen werden kann. Will man mir aber dieses nicht einräumen, wohlan so schränke ich meinen Satz auf die reine Mathematik ein, deren Begriff es schon mit sich bringt, daß sie nicht empirische, sondern blos reine Erkenntniß a priori enthalte.
Man sollte anfänglich wohl denken, daß der Satz 7 + 5 = 12 ein blos analytischer Satz sei, der aus dem Begriffe einer Summe von Sieben und Fünf nach dem Satze des Widerspruches erfolge. Allein wenn man es näher betrachtet, so findet man, daß der Begriff der Summe von 7 und 5 nichts weiter enthalte, als die Vereinigung beider Zahlen in eine einzige, wodurch ganz und gar nicht gedacht wird, welches diese einzige Zahl sei, die beide zusammenfaßt. Der Begriff von Zwölf ist keinesweges dadurch schon gedacht, daß ich mir blos jene Vereinigung von Sieben und Fünf denke; und ich mag meinen Begriff von einer solchen möglichen Summe noch so lange zergliedern, so werde ich doch darin die Zwölf nicht antreffen. Man muß über diese Begriffe hinausgehen, indem man die Anschauung zu Hülfe nimmt, die einem von beiden correspondirt, etwa seine fünf Finger oder (wie Segner in seiner Arithmetik) fünf Punkte, und so nach und nach die Einheiten der in der Anschauung gegebenen Fünf zu dem Begriffe der Sieben hinzuthut. Man erweitert also wirklich seinen Begriff durch diesen Satz 7 + 5 = 12 und thut zu dem ersteren Begriff einen neuen hinzu, der in jenem gar nicht gedacht war, d. i. der arithmetische Satz ist jederzeit synthetisch, welches man desto deutlicher inne wird, wenn man etwas größere Zahlen nimmt; da es denn klar einleuchtet, daß, wir möchten unsern Begriff drehen und wenden, wie wir wollen, wir, ohne die Anschauung zu Hülfe zu nehmen, vermittelst der bloßen Zergliederung unserer Begriffe, die Summe niemals finden könnten.
Eben so wenig ist irgend ein Grundsatz der reinen Geometrie analytisch. Daß die gerade Linie zwischen zwei Punkten die kürzeste sei, ist ein synthetischer Satz. Denn mein Begriff vom Geraden enthält nichts von Größe, sondern nur eine Qualität. Der Begriff des Kürzesten kommt also gänzlich hinzu und kann durch keine Zergliederung aus dem Begriffe der geraden Linie gezogen werden. Anschauung muß also hier zu Hülfe genommen werden, vermittelst deren allein die Synthesis möglich ist.
Einige andere Grundsätze, welche die Geometer voraussetzen, sind zwar wirklich analytisch und beruhen auf dem Satze des Widerspruchs; sie dienen aber nur, wie identische Sätze, zur Kette der Methode und nicht als Principien, z. B. a = a, das Ganze ist sich selber gleich, oder (a + b) > a, d. i. das Ganze ist größer als sein Theil. Und doch auch diese selbst, ob sie gleich nach bloßen Begriffen gelten, werden in der Mathematik nur darum zugelassen, weil sie in der Anschauung können dargestellt werden. Was uns hier gemeiniglich glauben macht, als läge das Prädicat solcher apodiktischen Urtheile schon in unserm Begriffe, und das Urtheil sei also analytisch, ist blos die Zweideutigkeit des Ausdrucks. Wir sollen nämlich zu einem gegebenen Begriffe ein gewisses Prädicat hinzudenken, und diese Nothwendigkeit haftet schon an den Begriffen. Aber die Frage ist nicht, was wir zu dem gegebenen Begriffe hinzu denken sollen, sondern was wir wirklich in ihm, obzwar nur dunkel, denken; und da zeigt sich, daß das Prädicat jenem Begriffe zwar nothwendig, aber nicht unmittelbar, sondern vermittelst einer Anschauung, die hinzukommen muß, anhänge.
Musean translation
Mouseia’s complete machine-assisted Musean translation, made directly from the complete German of the Prolegomena (AA IV 253–383, all §§1–60 and the Appendix), for fidelity, the steps of Kant’s argument, clarity and structure.
§ 2.
On the Kind of Cognition That Alone Can Be Called Metaphysical.
a)
On the Distinction between Synthetic and Analytic Judgments in General.
Metaphysical cognition must contain nothing but judgments a priori: the distinctive character of its sources requires this. Yet whatever their origin or logical form, judgments differ in content in a way that makes them either merely explicative, adding nothing to the content of cognition, or ampliative, increasing the given cognition. The former may be called analytic, the latter synthetic judgments.
Analytic judgments say nothing in the predicate except what was already actually thought in the concept of the subject, though not so clearly or with the same awareness. If I say, “All bodies are extended,” I have not enlarged my concept of body in the least, but have merely analyzed it: even before the judgment, extension was actually thought in that concept, though not expressly stated. The judgment is therefore analytic. By contrast, the proposition “Some bodies are heavy” contains something in its predicate that is not actually thought in the general concept of body. It therefore increases my cognition by adding something to my concept, and must accordingly be called a synthetic judgment.
b)
The Common Principle of All Analytic Judgments Is the Principle of Contradiction.
All analytic judgments rest entirely on the principle of contradiction and are by their nature cognitions a priori, whether or not the concepts serving as their material are empirical. For since the predicate of an affirmative analytic judgment is already thought in the concept of the subject, it cannot be denied of that subject without contradiction. Likewise, in an analytic but negative judgment, its opposite is necessarily denied of the subject, again by the principle of contradiction. This is how matters stand with the propositions “Every body is extended” and “No body is unextended (simple).”
For this very reason all analytic propositions are also judgments a priori, even when their concepts are empirical: for example, “Gold is a yellow metal.” To know this I need no further experience beyond my concept of gold, which contained the thought that this body is yellow and is a metal. This constituted my very concept, and I needed to do nothing but analyze it, without looking beyond it for anything else.
c)
Synthetic Judgments Require a Principle Other Than the Principle of Contradiction.
There are synthetic judgments a posteriori, whose origin is empirical; but there are also synthetic judgments that are certain a priori and spring from pure understanding and reason. Both agree, however, in never being able to arise from the principle of analysis—the principle of contradiction—alone. They require yet another, quite different principle, although, from whatever principle they are derived, they must always be derived in accordance with the principle of contradiction. For nothing may contradict this principle, although not everything can be derived from it. I shall first divide synthetic judgments into classes.
1) Judgments of experience are always synthetic. It would be absurd to base an analytic judgment on experience, since I need not go beyond my concept at all to make the judgment, and therefore need no testimony from experience. That a body is extended is a proposition established a priori, not a judgment of experience. Before I turn to experience, I already have all the conditions for my judgment in the concept, from which I need only extract the predicate according to the principle of contradiction. In doing so I can also become conscious of the judgment’s necessity, which experience could not even teach me.
2) Mathematical judgments are all synthetic. This proposition seems hitherto to have wholly escaped the notice of those who analyze human reason, and indeed to run directly counter to all their conjectures, although it is indisputably certain and of great importance for what follows. Because it was found that all the inferences of mathematicians proceed according to the principle of contradiction—as the nature of all apodictic certainty requires—people persuaded themselves that the fundamental propositions too were known from the principle of contradiction. In this they were greatly mistaken. A synthetic proposition can indeed be discerned according to the principle of contradiction, but only if another synthetic proposition is presupposed from which it can be inferred; never in itself.
First we must note that properly mathematical propositions are always judgments a priori, not empirical, because they carry necessity with them, and this cannot be drawn from experience. If anyone refuses to grant me this, then I shall restrict my assertion to pure mathematics, whose very concept entails that it contains not empirical cognition but only pure cognition a priori.
At first one might well think that the proposition 7 + 5 = 12 is merely analytic, following from the concept of a sum of seven and five according to the principle of contradiction. On closer examination, however, we find that the concept of the sum of 7 and 5 contains nothing more than the union of the two numbers into a single one; it does not in the least specify what this single number embracing both is. The concept of twelve is by no means already thought when I merely think of that union of seven and five. However long I analyze my concept of such a possible sum, I shall never find twelve in it. We must go beyond these concepts by calling on the intuition corresponding to one of the two—say, our five fingers, or five points, as Segner does in his arithmetic—and thus gradually add the units of the five given in intuition to the concept of seven. We therefore really enlarge our concept through the proposition 7 + 5 = 12, adding to the first concept a new one not thought in it at all. That is, an arithmetical proposition is always synthetic. This becomes all the clearer if we take somewhat larger numbers: it is then evident that, however we turn and twist our concept, we could never find the sum by merely analyzing our concepts without calling on intuition.
Nor is any fundamental proposition of pure geometry analytic. That the straight line between two points is the shortest is a synthetic proposition. For my concept of straightness contains nothing of magnitude, but only a quality. The concept of the shortest is therefore added in its entirety, and cannot be drawn from the concept of the straight line by any analysis. Here, then, we must call on intuition, through which alone the synthesis is possible.
Some other fundamental propositions presupposed by geometers are indeed analytic and rest on the principle of contradiction; but, like identical propositions, they serve only as links in the method, not as principles: for example, a = a, the whole is equal to itself, or (a + b) > a, that is, the whole is greater than its part. Yet even these, although valid according to mere concepts, are admitted in mathematics only because they can be exhibited in intuition. What commonly makes us believe here that the predicate of such apodictic judgments already lies in our concept, and that the judgment is therefore analytic, is merely an ambiguity of expression. We ought, namely, to add a certain predicate in thought to a given concept, and this necessity already attaches to the concepts. But the question is not what we ought to think in addition to the given concept, but what we actually, even if only obscurely, think within it. Here it becomes clear that the predicate attaches to that concept necessarily indeed, but not immediately: rather, through an intuition that must be added.
Plain English translation
Mouseia’s complete Plain English edition, made independently and directly from the same complete German text.
§ 2.
The Only Kind of Cognition That Can Be Called Metaphysical.
a)
The Difference between Synthetic and Analytic Judgments in General.
Metaphysical cognition must contain nothing but a priori judgments. The distinctive character of its sources requires this. But whatever origin judgments may have, and whatever their logical form, there is a difference in their content. They are either merely clarifying, adding nothing to the content of cognition, or extending, increasing the cognition given. The first can be called analytic judgments, the second synthetic judgments.
In their predicate, analytic judgments say only what was already actually thought in the concept of the subject, though not as clearly or with equal awareness. If I say that all bodies are extended, I have not increased my concept of body in the slightest. I have only broken it down into its elements. Extension was already actually thought in that concept before the judgment, even if not explicitly stated. The judgment is therefore analytic. By contrast, the proposition that some bodies are heavy contains something in its predicate that is not actually thought in the general concept of body. It therefore increases my cognition by adding something to my concept, and must be called a synthetic judgment.
b)
The Principle Common to All Analytic Judgments Is the Principle of Contradiction.
All analytic judgments rest entirely on the principle of contradiction and are by their nature a priori cognitions, whether or not the concepts that supply their content are empirical. The predicate of an affirmative analytic judgment is already thought in the concept of the subject. It therefore cannot be denied of the subject without contradiction. Similarly, in an analytic but negative judgment, its opposite is necessarily denied of the subject, again according to the principle of contradiction. This is how it stands with the propositions: every body is extended, and no body is unextended (simple).
For this same reason, all analytic propositions are a priori judgments even when their concepts are empirical. For example: gold is a yellow metal. To know this I need no further experience beyond my concept of gold, which would contain the fact that this body is yellow and is a metal. That constituted my concept itself. I only had to analyze it, without looking beyond it for anything else.
c)
Synthetic Judgments Need a Principle Other Than the Principle of Contradiction.
There are a posteriori synthetic judgments, whose origin is empirical. But there are also synthetic judgments that are certain a priori and arise from pure understanding and reason. Both kinds agree in this: they can never arise from the principle of analysis, namely the principle of contradiction, alone. They require a quite different principle as well. Still, from whatever basic principle they are derived, they must always be derived in agreement with the principle of contradiction. Nothing may conflict with this principle, even though not everything can be derived from it. I will first divide synthetic judgments into classes.
1) Judgments of experience are always synthetic. It would be absurd to base an analytic judgment on experience. I do not have to go beyond my concept at all to make the judgment, and therefore need no testimony from experience for it. That a body is extended is a proposition established a priori, not a judgment of experience. Before I turn to experience, I already have all the conditions for my judgment in the concept. I need only draw the predicate from it according to the principle of contradiction. In doing so, I can also become aware of the judgment's necessity, which experience could not even teach me.
2) Mathematical judgments are all synthetic. This proposition seems until now to have entirely escaped the notice of those who analyze human reason. Indeed, it seems directly opposed to all their conjectures, though it is indisputably certain and very important in its consequences. People found that all the inferences of mathematicians proceed according to the principle of contradiction, as the nature of all apodictic certainty requires. They therefore persuaded themselves that the basic propositions too were known through the principle of contradiction. In this they were badly mistaken. A synthetic proposition can indeed be understood according to the principle of contradiction, but only by presupposing another synthetic proposition from which it can be inferred, never on its own.
First, we must note that properly mathematical propositions are always a priori judgments, not empirical ones, because they carry necessity with them, and necessity cannot be taken from experience. But if anyone will not grant me this, then I will restrict my claim to pure mathematics. Its very concept implies that it contains no empirical cognition, but only pure a priori cognition.
At first one might think the proposition 7 + 5 = 12 is merely analytic, following from the concept of a sum of seven and five according to the principle of contradiction. But closer inspection shows that the concept of the sum of 7 and 5 contains nothing more than the union of the two numbers into a single one. This does not at all tell us which single number contains them both. The concept of twelve is not already thought when I merely think of that union of seven and five. However long I analyze my concept of such a possible sum, I will not find twelve in it. We must go beyond these concepts by calling on the intuition that corresponds to one of them: for instance, our five fingers, or five points, as Segner does in his Arithmetic. Then we add the units of the five given in intuition, one by one, to the concept of seven. So the proposition 7 + 5 = 12 really does extend our concept. It adds a new concept to the first, one not thought in it at all. That is, an arithmetical proposition is always synthetic. We become all the more clearly aware of this if we take somewhat larger numbers. It is then obvious that, however we turn our concept this way and that, we could never find the sum merely by analyzing our concepts without calling on intuition.
No basic proposition of pure geometry is analytic either. That the straight line between two points is the shortest is a synthetic proposition. My concept of straightness contains nothing about magnitude, only a quality. The concept of the shortest is therefore an entirely new addition. No analysis can draw it from the concept of a straight line. So here we must call on intuition, which alone makes the synthesis possible.
Some other basic propositions that geometers presuppose are indeed analytic and rest on the principle of contradiction. But like propositions of identity, they serve only as links in the method, not as principles. For example: a = a, the whole is equal to itself; or (a + b) > a, that is, the whole is greater than its part. Yet even these, although valid through concepts alone, are admitted in mathematics only because they can be presented in intuition. What usually makes us believe here that the predicate of such apodictic judgments already lies in our concept, so that the judgment is analytic, is simply an ambiguity in the expression. We must add a certain predicate in thought to a given concept, and this necessity already belongs to the concepts. But the question is not what we must think in addition to the given concept. It is what we actually, even if only obscurely, think in it. And here we find that the predicate does indeed belong necessarily to that concept, but not directly. It belongs through an intuition that must be added.
About this section
Section 2 is part of Prolegomena to Any Future Metaphysics by Immanuel Kant. This page sets the German original beside 2 direct-source translations, so any line can be checked against the source. It is section 5 of 72 in the work, which is kept in the European tradition.
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