and consequently insufficient reason to explain why God situated bodies in these points and not in others. Similarly, if time were independent from things, there ←53 | 54→would be no reason why things happen in one moment rather than in another. But, then, what are time and space? Leibniz proposes not considering them as absolute positions, but as relations: space is an order of coexistences, while time is one of successions. As he affirms they are no-things nor attributes, but rather idealitas, in the sense that they consist of relations, orders, abstracted from material objects and then provided with universality and necessity in opposition to the overflowing matter of senses. In this sense, they are defined as idealitas and not objects, res. The problem is: if space and time derive from sensibility, then the sciences based on them (physics and mathematics) depend on sensible objects, which are contingent, and cannot be universal or necessary. Is the claim of mathematics and physics to be considered as universal and necessary only an illusion, or can it be justified?
Kant’s 1770 conception of space and time as forms might be interpreted as a first, successful attempt to provide an original solution to this question. If some sciences (such as geometry and arithmetic) are based on space and time and if Kant’s forms of sensibility do not derive from senses, then it is possible to justify their universal and objective value. However, Kant’s doctrine still has some limitations.
Although Kant’s conception of space and time distinguishes him from previous traditions, he is still very influenced by the Wolffian division between inferior and superior faculties, receptivity and spontaneity, thus generating incoherences, or at least ambiguities in his doctrine - for instance, regarding his account of sensibility and the distinction between phaenomena and noumena -. More specifically, Kant defines sensibility as receptivitas, the faculty through which the subject can be affected by the object, while the understanding is conceived as the faculty of representing what is not present in the senses. Whilst the former deals with things as they appear (uti apparent), the latter focuses on things as they are (sicuti sunt). This division reechoes the traditional separation between primary qualities, which belong to the things in themselves and are grasped through the understanding62, and secondary qualities that depend on the subject and its sensibility. But, as already stressed, sensibility cannot be defined as passive, since it is ←54 | 55→characterised by the pure forms of intuitions, through which impressions do not affect the subject, as it were, automatically, but according to an order.
Then, if on the one hand the doctrine of space and time exposed in the Dissertation opens the way to a new approach to the problem of knowledge, on the other hand it contains limitations and incoherences, as underlined by Kant himself. As he states in the famous letter to Herz from 21st February 1772, the most important philosophical question focuses on the link between representations and objects. What causes this link? While it might be easy to explain it in reference to sensibility (as it could be possible to affirm that representations reflect objects as they are produced by their affection) on the other hand the relation between the intellectual action and objects is harder to justify. In the letter to Herz, Kant provides only a negative definition of the understanding’s activity: it is the faculty of representing things we are not affected by, it is not an abstraction from the senses, and neither is it a production such as the efficient causality of an intuitive understanding. Unfortunately, Kant does not delve into detail with this kind of action, nor does he explain whether, and if so how, such activity possesses an objective significance: how can it not simply be a product of the imagination? How is its reference to actual things (sicuti sunt) justified? Are there forms and schemata similar to the ones of sensibility also for the understanding? If the reference to space and time as schemata allows Kant to explain the possibility of the objective value of sciences and of our sensible experience as well, why should not a similar solution be valid also for the activity of the understanding? As Kant himself puts it in the letter to Marcus Herz from 21st February 1772:
“In my Dissertation I was content to explain the nature of intellectual representations in a merely negative way, namely, to state that they were not modifications of the soul brought about by the object. However, I silently passed over the further question of how a representation that refers to an object without being in any way affected by it can be possible. I had said The sensuous representations present them as they are. But by what means are these things given to us, if not by the way in which they affect us? And if such intellectual representations depend on our inner activity, whence comes the agreement that they are supposed to have with objects […]?” (AA X, p. 130–31)63←55 | 56→
Moreover, another limitation of the Dissertation can be seen in the fact that, although it states that the difference between understanding and sensibility is not a qualitative one, it is difficult to avoid the thought that the knowledge concerning things uti apparent is somehow inferior to that of the things sicuti sunt (Cassirer 1918). It seems that there are two worlds and that the intelligible one is absolutely separated and opposed to the sensible one, kind of a Platonic ideal world, which can be grasped only when the understanding is freed from the bounds of senses, thus revealing itself as a traditional, metaphysical, perfect realm. As Cassirer states (Cassirer 1918, p. 137) in the Dissertation, Kant assumes a distinction between a purely creative understanding and a purely receptive one. But since our understanding falls not under these two kinds, a new concept of understanding needs to be elaborated by Kant. The overcoming of the separation between these two worlds and accounts of the understanding will be one of the findings of the Critique of Pure Reason.←56 | 57→
47 I do not intend to delve into the question concerning the validity of Kant’s account of mathematics, but just to touch upon the criticism made by Koriako. According to Koriako, Kant presents mathematics and its constructive method “with philosophical eyes” (Koriako 1999, p. 3), thus presenting a misleading view of mathematics. Consequently, if Koriako is right, the particularity of the mathematical construction should no longer be a valid solution to the problem concerning the method to apply pure concepts to experience. Even if this is the case, as it seems, that Kant’s account of mathematics has philosophical grounds, my historical claim, that links mathematical construction and schematism, should not suffer. Within Kant’s perspective, if mathematics deals with the application of pure concepts, then, mathematical construction should be regarded as a procedure comparable to that of schematism. The question concerning the actual mathematical validity of such a constructive method is not my concern.
48 “Man kann zu einem jeden allgemeinen Begriffe auf zweierlei Wege kommen, entweder durch die willkürliche Verbindung der Begriffe, oder durch Absonderung von demjenigen Erkenntnisse, welches durch Zergliederung ist deutlich gemacht worden. Die Mathematik faßt niemals anders Definitionen ab als auf die erstere Art. Man gedenkt sich z. E. willkürlich vier gerade Linien, die eine Ebene einschließen, so daß die entgegenstehende Seiten nicht parallel sind, und nennt diese Figur ein Trapezium. Der Begriff, den ich erkläre, ist nicht vor der Definition gegeben, sondern er entspringt allererst durch dieselbe. Ein Kegel mag sonst bedeuten, was er wolle; in der Mathematik entsteht er aus der willkürlichen Vorstellung eines rechtwinkligen Triangles, der sich um eine Seite dreht. Die Erklärung entspringt hier und in allen andern Fällen offenbar durch die Synthese. Mit den Definitionen der Weltweisheit ist es ganz anders bewandt. Es ist hier der Begriff von einem Dinge schon gegeben, aber verworren oder nicht genügsam bestimmt. Ich muß ihn zergliedern, die abgesonderte Merkmale zusammen mit dem gegebenen Begriffe in allerlei Fällen vergleichen und diesen abstrakten Gedanken ausführlich und bestimmt machen.”
49 On the one hand, it is true that both philosophy and mathematics deal with the problem of construction or exhibition of pure concepts in intuition, but on the other hand
