¹ ‘I cannot say, therefore, that the world is infinite in space … Any such concept of magnitude, as being that of a given infinitude, is empirically impossible, and therefore, in reference to the world as an object of the senses, also absolutely impossible’ (A520/B549).

Note: Except as indicated, I will use the Smith, Kemp (1929) translation of the Critique of Pure Reason (New York: St. Martin's)Google Scholar.

² ‘Now we have the cosmic whole only in concept, never, as whole in intuition’ (A519–520/B546–7).

³ ‘Consequently, the original representation of space is an a priori intuition, not a concept’ (A25/B39–40).

⁴ See van Dalen (1990) for an account of the political side of this confrontation.

⁵ Hilbert (1922).

⁶ Brouwer (1928).

⁷ ‘It is the nature of an individual substance or complete being to have a concept so complete that it is sufficient to make us understand and deduce from it all the predicates of the subject to which the concept is attributed’, Leibniz (1686a), §8. Indeed, in this context, Leibniz arrogates the medieval notion of haecceity to the complete concept.

⁸ ‘… when everything which enters into a definition or distinct knowledge is known distinctly, down to the primitive concepts, I call such knowledge adequate. And when my mind grasps all the primitive ingredients of a concept at once and distinctly, it possesses an intuitive knowledge. This is very rare, since for the most part human knowledge is merely either confused or suppositive’. Leibniz (1686a), §24.

‘In this way all adequate definitions contain primitive truths of reason and consequently intuitive knowledge. It can be said in general that all primitive truths of reason are immediate with respect to an immediateness of ideas’. Leibniz (1704), IV, ii, 1.

⁹ ‘Let us be content with looking for truth in the correspondence between the propositions which are in the mind and the things which they are about’, ibid., IV, v, iii.

‘It would be better to assign truth to the relationship amongst the objects of the ideas by virtue of which one idea is or is not included in the other’, ibid.

¹⁰ ‘…an objective cognition is knowledge (cognitio). This is either intuition or concept (intuitus vel conceptus). The former relates immediately to the object and is singular, the latter refers to it mediately by means of a feature which several things may have in common’. (A320/B376–7). Translation by CP.

¹¹ ‘In whatever manner and by whatever means a mode of knowledge may relate to objects, intuition is that through which it is in immediate relation to them, and to which all thought as a means is directed’. (A19/B33).

¹² This, in turn, is the essence of his Copernican revolution. When he says in that famous passage

Kant is saying that our human intuitions play the ontological role of Leibnizian complete concepts: they tell all there is about their objects. Thus, Kant has fashioned a ‘humanized’ version of Leibniz's notion of haecceity.

¹³ ‘But there is no lowest concept (conceptus infimus) or lowest species in the series of species and genera under which not yet another would be contained, because it is impossible to determine such a concept. .... In respect to the determination of the concepts of genera and species, the following general law is valid: There is a genus that can no longer be a species; but there is no species that can no longer be a genus’, ‘Lectures on Logic’, I, 1, fn. 11, in Hartman and Schwarz (1974).

¹⁴ See Posy (2000) for an elaboration of this point.

¹⁵ Famously, Kant argues in the Amphiboly that just seeing the spatial distance between a pair of objects suffices to tell them apart. This is his refutation of the identity of indiscernibles:

‘Thus in the case of two drops of water we can abstract altogether from all internal difference (of quality and of quantity), and the mere fact that they have been intuited simultaneously in different spatial positions is sufficient justification for holding them to be numerically different. … For one part of space, although completely similar and equal to another part is still outside the other, and for this very reason is a different part, which when added to it constitutes with it a greater space. The same must be true of all things which exist simultaneously in the different spatial positions, however similar and equal they may otherwise be’ (A264/B320), emphases added.

¹⁶ ‘But all thought must, directly or indirectly, by way of certain characters, relate ultimately to intuitions, and therefore, with us, to sensibility, because in no other way can an object be given to us’ (A19/B33).

¹⁷ See A58/B82 and A191/B236.

¹⁸ ‘In whatever manner and by whatever means a mode of knowledge may relate to objects, intuition is that through which it is in immediate relation to them, and to which all thought as a means is directed. But intuition takes place only in so far as the object is given to us. This again is only possible, to man at least, in so far as the mind is affected in a certain way. The capacity (receptivity) for receiving representations through the mode in which we are affected by objects, is entitled sensibility’ (A19/B33).

¹⁹ ‘The postulate bearing on the knowledge of things as actual does not, indeed, demand immediate perception (and, therefore, sensation of which we are conscious) of the object whose existence is to be known. What we do, however, require is the connection of the object with some actual perception, in accordance with the analogies of experience, which define all real connection in an experience in general’ (A225/B272).

²⁰ ‘In the mere concept of a thing no mark of its existence is to be found’, ibid.

²¹ See Kripke (1972), and Putnam (1975). Famously, this theory allows us to refer successfully to someone or something even though we might have an incomplete or even incorrect concept of the object of reference. And Kant correctly sees that the causal notion of empirical reference, which he now champions, does allow for fallibility and revisability:

²² ‘And indeed what useful purpose could be served by defining an empirical concept, such, for instance, as that of water? When we speak of water and its properties, we do not stop short as what is thought in the word, water, but proceed to experiments. The word, with the few characteristics which we attach to it, is more properly to be regarded as merely a designation than as a concept of the thing, the so-called definition is nothing more than a determining of the word’, ibid.

²³ I stressed the minimal distance here because in the Antinomy Kant is concerned with the sequential progress of our knowledge. Thus, the question as to whether an infinite sequence of regions might converge to a finite limit is simply not to the point.

²⁴ ‘Thus the first and negative answer to the cosmological problem regarding the magnitude of the world is that the world has no (…) outermost limit in space’ (A520/B548).

²⁵ See for instance Dummett (1978), (1991).

²⁶ Indications of an underlying proto-assertabilism are scattered throughout Leibniz's work. One of the most explicit texts is the ‘General Inquiries about the Analysis of Concepts and Truths’ (1686b) where in §130 he says: ‘That is true, therefore, which can be proved, i.e. of which a reason can be given by analysis …’, English translation by Parkinson (1966). I explore this point more fully in Posy (2003).

²⁷ See Posy (1992).

²⁸ See Putnam (1981). See also Posy (2000), (2003).

²⁹ Similarly, at each checkpoint, k, with positive outcome, our grasp cannot determine whether the actual universe is identical to a contemplated one in which there are exactly k occupied region.

³⁰ ‘In natural science, (…) there is endless conjectures, and certainty is not to be counted upon’ (A480/B508).

³¹ ‘We cannot assume that every part of an organized whole is itself again so organized that, in the analysis of the parts to infinity, still other organized parts are always to be met with; in a word, that whole is organized to infinity. This is not a thinkable hypothesis’ (A526/B554).

³² Moreover, there is no causal link to the ‘next organizing concept’, so we cannot allow ourselves to claim the existence of the k + 1 ^st division already at the k ^th checkpoint.

³³ We might ask, for instance, whether there will ever be a conceptual system with exactly 12 fundamental quantities.

³⁴ This is clear in the first edition version of the Aesthetic where in speaking of the infinity of mathematical space, Kant says ‘If there were no limitlessness in the progression of intuition, no concept of relations could yield a principle of their infinitude’ (A25).

³⁵ See in particular B16–18, B40–41.

³⁶ ‘We cannot think a line without drawing it in thought, or a circle without describing it’ (B155).

³⁷ ‘Space and time are quanta continua, because no part of them can be given save as enclosed between limits (points or instants), and therefore only in such fashion that this part is itself again a space or a time. (…) Such magnitudes may also be called flowing, since the synthesis of productive imagination involved in their production is a progression in time, and the continuity of time is ordinarily designated by the term flowing or flowing away’ (A169–70/B211–12).

³⁸ ‘The various similar parts of the curve are therefore not connected with each other by any law of continuity, and it is only by the description that they are joined together. For this reason it is impossible that all of this curve should be included in any equation’, Euler (1755) in Stüssi and Favre (eds.) (1947).

³⁹ ‘Therefore all concepts, and with them all principles, even such as are possible a priori, relate to empirical intuitions, that is, to the data for a possible experience. Apart from this relation they have no objective validity, and in respect of their representations are a mere play of imagination or of understanding. (…) The mathematician meets this demand by the construction of a figure, which, although produced a priori is an appearance present to the senses. (…) The concept itself is always a priori in origin, and so likewise are the synthetic principles or formulas derived from such concepts; but their employment and their relation to their professed objects can in the end be sought nowhere but in experience, of whose possibility they contain the formal conditions’ (A239–40/B298–9).

⁴⁰ This is a constant theme in Brouwer's thought. See in particular Brouwer (1928).

⁴¹ See op. cit.

⁴² Brouwer's student, Arend Heyting, gave one of the initial formulations of inductively defined truth conditions for assertabilism. And he did this as part of an attempt to formalize the reasoning that underlay Brouwer's intuitionistic mathematics. Dummett's book on intuitionism (1977) centers on this aspect of Brouwer's thought.

⁴³ Doxiadis (2000).

⁴⁴ A note for aficionados: those Kantian paths in Figures 1 and 2 can represent Brouwerian choice sequences. In earlier work, I have used diagrams such as these to represent Kripke models, and then have used the intuitionistic reading of the logical particles to provide a precise account of some Kantian arguments and concepts. In particular, with an appropriate interpretation of the predicate B, Figure 1 will validate the formula ∀x~ ~∃y B(x, y), (i.e., given any x, you cannot deny that there will be a y such that B(x, y) holds) while Figure 4 will validate ∀x∃y B(x, y) (i.e., given any x, you can affirmatively assert that there will be a y such that B(x, y) holds). The difference between these two formulae (read intuitionistically) nicely summarizes the difference between regulative force, on the one hand, and constitutive claims, on the other. See for instance Posy (1992). Figure 1 here gives a more fine tuned account of Kant's reasoning than the diagrams I used in this and some other papers.

⁴⁵ Indeed we can say definitely that r^# ≠ 1/2, though still we are not entitled to say r^# < 1/2. Brouwer (1925) actually has an elaborate analysis of the ‘splitting of mathematical concepts’.

⁴⁶ Hilbert (1926).

⁴⁷ I owe special thanks to Michela Massimi for her support and patience.