Note from the Translators

The text of this translation is based on the Akademie text contained in volume 20 of Kant's Gesammelte Schriften, ed. Königlich Preussischen, later Deutschen Akademie der Wissenschaften zu Berlin (Berlin: de Gruyter, 1900–). We used bold type where in the original spaced type (Sperrdruck) is used. For easy cross-reference, the Akademie page numbers have been inserted in square brackets in the text below. All other insertions in square brackets are ours. For clarification we sometimes give the original German word or phrase in parentheses. We have also consulted the translated excerpts in Allison (Reference Allison1973), the Italian translation by Claudio La Rocca (Kant Reference Kant1994), and the French translation by Michel Fichant (Kant Reference Kant1997).

We want to thank Claudio La Rocca for providing us with the Italian translation, contained in his edition of Kant's essay against Eberhard (Kant Reference Kant1994). We also thank Piter van Tuinen and Philip Westbroek for their assistance with the translation of the text fragment from Raphson's De spatio reali and an anonymous referee for Kantian Review for helpful comments on the main text. In addition, we thank Wolfgang Ertl and Thomas Land for their comments on the translation, and Wolfgang also for clarifying the meaning of the scholastic concept of ‘res’.

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[AA 20: 410] Pieces from a Kästner or KlügelFootnote ¹ can provide value to any collection, without their exactly having the intention to reveal the truth in matters where others would have been in error. The three treatisesFootnote ² by Councillor Kästner in this second volumeFootnote ³ concern the manner in which the geometer can meet the demands which can be made on him because of [the issue of] the possibility of his object, its determination and the unprovable principles governing it, and are wholly limited to mathematics, which is not at all favourable to the assertions of Mr Eberhard; since precisely the contrast of its ability to the inability of metaphysics to meet these demands in any way (provided this happens with the certainty that one can request from all putative rational cognition) lets the latter appear in so much more a disadvantageous light.

On p. 391 of the second volume, it is correctly said that ‘Euclid assumes the possibility of drawing a straight line and describing a circle [411] without proving it’,Footnote ⁴ which amounts to saying: without proving this possibility by means of inferences; for the description, which occurs a priori by means of the imagination in accordance with a rule, and is called construction, is itself the proof of the possibility of the object. The mechanical drawing (Zeichnung) (p. 393) which presupposes it [i.e. the construction] as its model is hereby not at all taken into consideration. That, however, the possibility of a straight line and a circle cannot be demonstrated mediately through inferences, but only immediately through the construction of these concepts (which is not at all empirical), is due to the fact that among all constructions (exhibitions which are determined according to a rule in intuition a priori) some must be the first, such as the drawing (Ziehen) or the describing (in thought) of a straight line and the rotation thereof around a fixed point, where neither the latter can be derived from the former, nor these from any other construction of the concept of a magnitude. The constructions of other concepts of this sort in space are all derived in geometry, and this derivation Mr Kästner calls the demonstration of their possibility. Against this manner of assuming the possibility of that in regard to which one is immediately conscious of the ability to construct its concept, the Critique has nothing in the slightest to say. Rather it cites this as an [412] example for dogmatic metaphysics, in order to do the same for its own concepts, whereby it is noted: that, if no exhibition in the intuition (whether, as is the case with concepts of geometry, this be possible [merely] a priori, or also, as with those of physics, empirically) were added to the concept, then we could not make out by means of mere concepts that such a thing, as is thought under the concept of magnitude, or which corresponds to the concept of cause, is possible. This reservation and the demand, based upon it, that metaphysics provide for all its concepts a corresponding intuition (for which it already suffices when one connects according to a rule of combination which can also be exhibited in intuition, that which is given in some intuition), is here of utmost importance. For with all due respect for the principle of [non-]contradiction and without in the slightest offending against it, metaphysics can initially introduce a priori concepts, which can be formed in pure intuition (as in geometry); then such concepts as can at least be formed in experience (such as the concept of cause); and further such concepts which, to be sure, cannot without contradiction be formed in any conceivable example, [413] but in many other respects (e.g. practical) are very reasonable; lastly however let all enthusiastic delusion and alleged philosophical insight into which one in fact has no insight at all, creep in.Footnote ⁵ For all barriers to the freedom of waxing lyrical are removed, as soon as one frees the subtle reasoner (Vernünftler) from the obligation to prove the objective reality of his concepts of things, of which he claims to have theoretical knowledge, by means of intuition (which though is not a seeing,Footnote ⁶ but a representation of the particular insofar as it is not merely thought, but given for thought) without which guarantee he goes into rapturesFootnote ⁷ among mere thought entities.—Very wisely but of little consolation for Mr Eberhard, Councillor Kästner says therefore (p. 402): ‘I leave it undecided whether outside geometry the possibility of exhibiting a thing a priori can be set forth in such a way that one thereby shows that there is no contradiction in its concept.’ He adds, in a very apposite and illuminating way: ‘Euclid would demand of Wolff (in respect of the possibility of a most perfect being)Footnote ⁸ [414] that he make (machen) a most perfect being; namely in precisely the sense in which Euclid makesFootnote ⁹ the icosahedron, in the understanding.’Footnote ¹⁰ The latter cannot mean that this solid figure is in the intellect, but it only means that according to a rule, which the understanding thinks, that concept is given a corresponding intuition a priori (in the imagination).

Thus the concept of ‘decahedron’ does not contain a contradiction, but the mathematician does not hold it to be valid that, just because this concept is possible, its object is possible, but demands that one [415] exhibit it in intuition, as it is then shown that this concept does indeed not contradict itself but does contradict the conditions of the construction of a regular solid.Footnote ¹¹

The demand placed on the metaphysician would therefore be this one: he must represent (vorstellig machen) by means of some example what he means by reality, i.e. the absolutely positiveFootnote ¹² of things; but he can only get [this example] from objects of experience, of which everything which one can call real in them, is according to its essential constitution dependent on conditions, delimited and inseparably connected with negations, so that one cannot leave these out of the concept of reality without at the same time cancelling it out, hence no example (corresponding intuition) can be found for the concept of pure reality, still less for the idea of the connection in one entity of all, however heterogeneous reality; this demand therefore forces the metaphysician to admit that in this case as for the concept of a supersensible being in general, its possibility ([i.e.] the objective reality of its concept) can simply not be proven.

[416] The expression of Mr Kästner is therefore, if somewhat striking, meaningful and good, and the Critique can always grant him that in order to prove the possibility of a thing it is not enough that one find no contradiction in its concept, but one must be able to construct (machen) its object in the understanding, either, as in geometry, by means of pure intuition (in the construction of the concept), or, as in natural science, from the matter and in conformity with the rules that experience furnishes us.

That which Councillor Kästner, on pp. 403–19, expounds on the representation of space, is entirely meant for the mathematician, in order to determine the legitimate use that the latter makes of that representation, and is just as disadvantageous for Mr Eberhard as the preceding, as it is said on p. 405: ‘Whatever one wants to call this concept of geometric space, either pictorial or non-pictorial, I leave that to whoever determines the meaning of these words. For me, it is abstracted from sensible representations.’Footnote ¹³ Mr Eberhard's whole exposition of space, however, revolves around those expressions, and it would be impossible for him to determine their meaning.

When Mr Kästner says: for him, as mathematician, the concept of [417] space is abstracted from sensible representations, this can likewise hold for the metaphysician; for without application of our sensible capacity of representation to actual objects of the senses, even that which may be contained a priori in these would not become known to us. This should however not be understood as if that representation of space were first to originate from and be generated through the sensible representation, which would contradict the properties of space, which in mathematical propositions are grasped a priori, (p. 406) ‘not demonstrated through observation, measurement and weighing up (but a priori)’.Footnote ¹⁴

Since that which is expounded from p. 407 to [p.] 419 merely concerns the use of the concept of the infinite in geometry, it lies outside the scope of this review. However, since it might seem to Mr Eberhard and others that this should equally have been a refutation of the infinity of space, of which the Critique says that it is inseparably attached to this representation, it is appropriate for a review of a journal which has made metaphysics its main [418] subject to point out the different uses of the concept of infinity in both sciences.

[419] Metaphysics must show how one can have the representation of space, geometry however teaches how one can describe a space, viz., exhibit one in the representation a priori (not by drawing). In the former, space is considered in the way it is given, before all determination of it in conformity with a certain concept of object. In the latter, one [i.e. a space] is constructed (gemacht). In the former it is original and only one (unitary) space, in the latter it is derived and hence there are (many) spaces, of which the geometer however, in accord with the metaphysician, must admit as a consequence of the foundational representation of space, that they can only be thought as parts of the unitary original space. Now one cannot name a magnitude, in comparison with which each assignable [unit] of the same type is only equal to a part of it, anything other than infinite. Thus, the geometer, as well as the metaphysician, represents the original space as infinite, in fact as infinitely given. For this is in itself specific to the representation of space (and in addition that of time), such as can be found in no other concept: that all spaces are only possible and thinkable as parts of one single space. Now when the geometer [420] says that a line could always be extended no matter how far one has drawn it, then this does not mean what is said in arithmetic about numbers, namely that one can always, and endlessly, increase them through the addition of other units or numbers (for the added numbers and magnitudes, which are expressed through it, are for themselves possible, without them having to belong, together with the previous ones, as parts, to one magnitude); rather, that a line can be extended to infinity means so much as: the space in which I describe the line, is greater than every one line which I may describe in it; and thus the geometer grounds the possibility of his task of increasing a space (of which there are many) to infinity on the original representation of a unitary, infinite, subjectively given space. Now that the geometrically and objectively given space is always finite agrees completely with this; for it is only given through its being constructed (gemacht). That, however, the metaphysically, i.e. originally, nonetheless merely subjectively given space, which (because there is no plurality thereof) cannot be brought under any concept which would be constructible, but to be sure contains the ground of the construction of all possible geometrical concepts, is infinite [421] only indicates that it consists in the pure form of the sensible mode of representation of the subject, as a priori intuition; hence in this, as singular representation, the possibility of all spaces, which goes to infinity, is given. With this also agrees entirely what Raphson,Footnote ¹⁵ according to Councillor Kästner's quotation on p. 418, says, [namely] that the mathematician is always only concerned with an infinito potentiali [a potential infinite], and [that] actu infinitum (the metaphysically given [infinite]) non datur a parte rei, sed a parte cogitantis [an infinite in actualityFootnote ¹⁶ is not given on the side of the thing, but on the side of the thinker];Footnote ¹⁷ this latter mode of representation is however not thereby fabricated and false, but rather lies at the foundation of the infinitely progressing constructions of geometrical concepts, and leads metaphysics to the subjective ground of the possibility, i.e. the ideality, of space, with which and the [422] debate on this doctrine the geometer is absolutely not concerned; for he would have to get involved in a dispute with the metaphysician about how the difficulty that space, and everything that fills it, is infinitely divisible and nevertheless does not consist of infinitely many parts, is to be resolved.

In all this the reviewer finds Councillor Kästner fully in accord with the Critique of Pure Reason, also there where, on p. 419, he says of geometrical doctrines: ‘Never does one infer from the image, but from that which the understanding thinks when considering the image (beym Bilde denkt).’ He undoubtedly understands by the first the empirical drawing, by the second the pure intuition that accords with a concept, i.e. a rule of the understanding, namely the construction of it [i.e. of the concept], which is not an empirical exhibition of the concept. When, however, he cites the Philosophisches Magazin, as if herewith he had hit on and confirmed Mr Eberhard's opinion of the pictorial (Bildlichen) in contrast to the intelligible, then he is much mistaken. For he [i.e. Eberhard] understands by pictorial (Bildlichen) not something like the figure in space as geometry would view it, but space itself (although it is hard to understand, how one could form an image (Bild) of something external to oneself [423] without presupposing space); and his intelligible (Intelligibeles) is not for instance the concept of a possible object of the senses, but of something that the understanding must represent, not in space, but as its ground on the basis of which it [i.e. space] can first be explained. But he will readily be excused for this misunderstanding by anyone who has felt the difficulty of connecting a self-consistent concept with this expression that is used so diversely by Mr Eberhard.