One set of texts that present a viable alternative to the ‘Discipline’ as Kant’s central mathematical text is the Kant-Schultz correspondence and Schultz’s resulting commentary on the KrV.131 The correspondence contains one of Kant’s clearest
statements on the role of time in the construction of mathematical concepts and also gives us some interesting material to work with concerning the role of axioms in arithmetic. There are four reasons to think that Kant’s communication with Schultz and Schultz’s commentary are an appropriate starting place for an examination of Kant’s mathematics: First, The texts contain an extremely interesting discussion of axioms, and axioms play a pivotal role in contemporary mathematics. In fact, axioms tend to play an important role in any sort of systematic thought. It is possible that understanding axioms is the key not only to Kant’s mathematics but his philosophy as a whole. Second, Schultz’s provides us with the only authorized commentary of the KrV. There is good reason to think that his opinions reflect Kant’s own. Finally, the interpretation of Kant’s texts that most readily
131 Though not a section of the KrV I feel that the discussion of the Kant-Schultz and Schultz’s commentary best fit here as both the correspondence and Schultz’s commentary were directly addressed to the KrV.
presents itself if one focuses on these passages conforms most easily to Kant’s earlier writings (in particular the Prize Essay). If one wants to emphasize the continuity of Kant’s thought and the Leibnizian influence, focusing on Schultz’s commentary and the related correspondence is extremely helpful.
For those concerned about the role of intuition in mathematics the Kant-Schultz correspondence is an excellent place to start. The clearest statement on the role of time in mathematical construction made by Kant himself is found in a letter to Schultz.
As you [Schultz] quite rightly note, time has no influence on the properties of numbers (as pure determinations of magnitudes), yet it may [influence] the property of every alteration (as of a
quantum) which itself is possible only relative to a specific
condition of inner sense, and its form (time). Yet notwithstanding succession, which every construction of magnitude requires, the science of number is a pure intellectual synthesis, which we represent in thoughts. But in so far as magnitudes (quanta) are to be determined according to this [synthesis], they must be given to us so that we can apprehend their intuition successively, and thus this apprehension is subjected to the condition of time.132
Time has no influence on the properties of numbers. I take this to mean that a number thought of at one time will not have different properties if thought of at a different time. Though it could also mean that time is not a necessary pre-condition for thinking of a
specific number or for the existence of the schema of number (i.e. we could still have number even if time was not a condition for the possibility of inner intuition. In such a world numbers would only be represented spatially). However, any alteration of numbers (adding, subtracting or counting) must take place in time, and the intuition of magnitudes must be apprehended successively. If this is the entire role time plays in the construction of numbers, time seems to have no further sway over mathematics then it does over any other intuition. Kant seems to attribute a special task in mathematical construction to time, but what that task is remains to be seen.
The confusion over time’s role in arithmetic for Kant was first expressed by Johann August Eberhard. Eberhard’s criticism came in response to Schultz’s Prüfung. Eberhard knew of Schultz’s proposed First Axiom of Arithmetic which states, “The
quantity of the sum is the same whether one adds the second to the first or the first to the second, i.e., it is always the case that a + b = b + a.”133 Eberhard suggests that the
symmetry of arithmetic equations which Schultz proposes as an axiom of arithmetic serves to negate the role of time in arithmetical construction. “So the truth of a
proposition of arithmetic does not depend on time, i.e., on the order in which the parts of the sum have been thought…I ask anyone whether that does not mean that the truth of a proposition does not depend on time at all.”134
Eberhard, much like Hankel, also comments on the controversial finger-counting passage, and seems to suggest that such counting makes addition a merely empirical process.
133 Schultz quoted in Martin, Arithmetic and Combinatorics, 103. 134 Eberhard quoted in Martin Arithmetic and Combinatorics, 102.
The reply to Eberhard’s objection comes not from Kant, but from Schultz, and is worth quoting at length.
What Mr. Eberhard has to say concerning the apodictic certainty of
arithmetic and analysis is based chiefly on the consideration that
the pure intuition of time does not lie in the concept of number itself as its object, but only in the limits of our power of
representation. But if this is the case, then Mr. Eberhard himself must admit that the concept which we have of a number is merely sensible, and thereby actually holds the intuition of time within itself. This is because, according to his system, the limits of the power of representation are precisely the source of sensibility. He therefore unavoidably contradicts himself, because counting units without adding them successively to one another is an obvious contradiction.135
Schultz claims that Eberhard’s complaint is based on a misunderstanding of the role of time in the production of number concepts. According to (Schultz’s reading of) Eberhard the only reason one needs time for the process of enumeration is that all of our cognition, regardless of its content, must take place in time. If adding or counting is a process, then time must be part of our intuition of that process.
Schultz’s reply highlights the fact that if time is necessary for arithmetical operations only in the same sense it is necessary for all cognition, then number becomes
135 Schultz quoted in Henry Allison, The Kant-Eberhard Controvery (Baltimore: Johns Hopkins University Press, 1973), 174.
something empirical. All empirical intuition must occur in time, and while pure cognition utilizes time (time is the pure form of inner sense) it does not hold time as its object. Mathematics must use time in a special sense because in ordinary intuition time is not in the object intuited but only in the mind which intuits it. If numbers are empirical
intuitions then there is no time in them, only time added to them by the mind which intuits them. As quoted above, Kant claims that “number is nothing other than the unity of the synthesis of the manifold of a homogeneous intuition in general, because I generate time itself in the apprehension of intuition.”136
Kant does not here grant to number the status of an empirical intuition. Time is generated in the apprehension of intuition. Number is not that which is apprehended, but that which provides unity to the synthesis of the manifold of a homogenous intuition in general. It is the generality of numbers (what Kemp Smith refers to as their abstract nature137) that separates them from empirical intuitions. As with all the schemata, number provides unity to our intuitions by allowing them to be brought under the categories. On Schultz’s reading of Eberhard, the intuition of time must both be and not be in the concept of number. Since this contradiction cannot be the case, time must not lie in number concepts in so far as they are concepts, but in so far as these number concepts are counted as individual numbers. When we count we count successively and time is a necessary component of the counting. The role of time in the formation of numbers is unique not because of how time is utilized, but, rather
136 KrV, A142-143/B182. 137
Kemp Smith, Commentary, 132. Kemp Smith claims that Kant refers untenably to mathematical objects as having “a dual mode of existence,” being both abstracted and constructed. I assume he is referring to Kant’s description of the construction of a triangle at KrV, A713-714/B741-742, but this passage (nor any other I can find) does not seem to support Kemp Smith’s view.
because time does not apply to numbers in the same way in which it applies to intuitions of empirical objects. Numbers, like time, are added to the objects.
It is the Kant-Schultz correspondence which gives Gottfried Martin a basis for his theory that Kant granted axioms an important role in arithmetic.Martin draws on
Schultz’s discussion of axioms in his commentary to the KrV as well as Leibniz’s
writings on mathematics. While Schultz, and many of Kant’s other students, believed that arithmetic must have axioms (reflexivity, for example) Kant himself explicitly states that there are no axioms of arithmetic. Martin argues that the agreement of Kant’s students on this issue combines with other evidence to form a convincing case for Kant eventually abandoning this view sometime after the publication of the KrV. 138
We can already begin to see that taking the Kant-Schultz correspondence (and related texts) as most central creates more problems than it solves. While the reasons for considering the passages as crucial remain true we also see, first, that the notion of what an axiom is has shifted somewhat from Kant’s time to ours. It is not clear how much of what he has to say concerning axioms maps on to current usages of the concept. It is true that axioms generally are an important part of any system, unless that system explicitly lacks axioms. It is extremely hard to get around Kant’s clear statement that arithmetic has no axioms and one must wonder what implications this has for metaphysical systems as well. Second, while Schultz’s commentary was authorized by Kant it was never read by Kant in its published form. There is no way of knowing what changes were made
138 For more on the differences between Schultz and Kant concerning the possibility of axioms for arithmetic see Martin, Arithmetic and Combinatorics. For the opposing view point see Parsons, “Kant’s Philosophy of Arithmetic.”
between the version Kant saw and the published version or how Kant would have
responded to the final version. Finally, while interpretations based on the Schultz texts do show a Kant who bears a strong resemblance to his pre-critical self, this is not necessarily a desirable trait. There were rather dramatic shifts in Kant’s thinking between the Prize Essay and the KrV, and a robust theory of Kant’s mathematics needs to account for these as well.139 In the end it is not clear that what Kant had to say about axioms is central to his thought, and the claim that arithmetic has none may be far less interesting than it originally seemed.