Abstract
Kant’s theory of arithmetic is not only a central element in his theoretical philosophy but also an important contribution to the philosophy of arithmetic as such. However, modern mathematics, especially non-Euclidean geometry, has placed much pressure on Kant’s theory of mathematics. But objections against his theory of geometry do not necessarily correspond to arguments against his theory of arithmetic and algebra. The goal of this article is to show that at least some important details in Kant’s theory of arithmetic can be picked up, improved by reconstruction and defended under a contemporary perspective: the theory of numbers as products of rule following construction presupposing successive synthesis in time and the theory of arithmetic equations, sentences or “formulas”—as Kant says—as synthetic a priori. In order to do so, two calculi in terms of modern mathematics are introduced which formalise Kant’s theory of addition as a form of synthetic operation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The significance of Kant’s theory is even more remarkable—as has often been mentioned—since he was no mathematician, despite having held lectures on mathematics. While he deals more often with geometry, arithmetic and even more algebra are mentioned less, but they are not the less important.
Cf. Hanna (2002, p. 337).
Benacerraf (1973).
Even an empiricist or a physicalist account of arithmetic faces this question as its theory of numbers does not ascribe the same ontological status to arithmetical entities as to genuine empirical objects; for instance, Bigelow has a relational account of numbers. He attributes to numbers empirical reality because they are abstracted from relations in empirical reality, which he takes to be internal properties of empirical objects (Bigelow 2001). The broadly debated indispensability argument of Quine and Putnam is just another symptom of the application-problem, which states the reality of numbers by an inference to the best explanation based on the fact of the indispensability of mathematics in physics as our best theory of the world (Quine 1960; Putnam 1971).
Since Kant’s examples in his remarks on geometry are all Euclidean this criticism is justified. But there still is much controversy about the question, if this also holds for his general theory of space in the “Transcendental Aesthetics” in the Critique of pure Reason. This depends on the answer to the question concerning which internal relation his theory of space has to geometry (apologetic are: Barker (1992), Melnick (1992), Hintikka (1992a, b), Allison (2004), critics are: Parsons (1992a, b) and Friedman (1992)).
By reconstruction we mean a method of amending missing parts in Kant’s theory of arithmetic by drawing conclusions out of his own theory that he did not explicitly draw, and if necessary, complementing it with modern means. This method is contrary to a historic reconstruction that complements missing parts of the theory from predecessors (cf. Sutherland 2004a, b, 2005, 2006). We think that both methods go beyond the written text relying on justifying arguments and that both methods can be complementary. Whereas a backward reconstruction presupposes that missing parts of a theory are missing because the author relies on contemporary shared assumptions or non-reflecting traditions aiming at uncovering them, a forward reconstruction abandons this precondition aiming at finding out whether and how far a theory is compatible with modern thought. This method at least aims at exposing the progressive potential of a theory. Similar to our method is Hanna’s (2002).
Hanna rejects the possibility of taking Kant’s theory in the vein of structuralism or Brouwer’s intuitionism because this amendment would expose Kant’s theory to the objection of psychologism (Hanna 2002, pp. 341, 344).
Frege’s criticism on Kant’s syntheticity-thesis in arithmetic was focussed on advocating for the contrary thesis, that arithmetic is analytic (Frege 1988, pp. 92–95). The background of this thesis is the idea of logicism, which is to deduce arithmetic from logic or to reduce arithmetic to logic. But shortly after, Russell’s paradox raised doubts about the truth of this assumption and finally Gödel’s incompleteness theorem showed that it is impossible. On the basis of the externalist semantics put forward by Quine and others, the critique of Kant is not directed against Kant’s syntheticity thesis, but against the apriority thesis. In the contemporary debate on the foundations of mathematics it is still an open question whether mathematic judgments are analytic or synthetic.
Along similar lines, but different in method and execution is Hanna (cf. Hanna 2002, p. 328 f.).
Letter to Johann Schultz from 25th of November 1788 (AA X, 555).
Cf. Sutherland (2004a).
Kant’s use of the German term “Menge” poses a problem for contemporary translations. Since “set” is a technical term in modern mathematics and since it is obvious that Kant did not refer to this specific concept, most translations render it by “multitude”. But since Kant draws distinctions explicitly or implicitly between all related concepts, like “Mannigfaltigkeit” (“multiplicity”), “Vielheit” (“plurality”), “Größe” (“magnitude”), “Quantität” (“quantity”), quantum (“quantum”), in the end there is a specific term missing in English translations. Therefore, we use the term ‘set*’ to indicate that Kant’s concept of “Menge” has a specific meaning that is familiar but not identical with the modern term “set”. This difference is obvious in Kant’s missing differentiation between elements and units. Kant only speaks of “units”. In our interpretation we introduce the term ‘element’ for clarification. We will explicate Kant’s term of set* later.
Prolegomena § 57, AA IV, 352.
Letter to Schultz (AA X, 554). It is a problem whether Kant held that algebra is referring to proper objects, that is, infinitely large sets of numbers, and is therefore a concrete science, or whether he sees algebra just as a tool for solving problems in arithmetic and geometry. According to this latter view, commonly held in the eighteenth century, algebraic symbols denote lengths of geometric figures. This means algebra is just an abstract science without proper objects. Shabel argues for the latter (Shabel 1998, p. 617). Cf. also Sutherland (2006, p. 551). During the last years there has been much research on the problem concerning which relations between geometry, arithmetic and algebra are supposed by Kant. The text-basis in the end is too thin to give a detailed and reliable answer to this question. Sutherland draws the conclusion out of his historic reconstructions that Kant’s alleged theory of numbers was based on popular writings of his time on the theory of numbers by Wolff, Euler and Segner, which led to a conservative Greek concept of number not allowing him to think of them as continuous magnitudes (Sutherland 2006, p. 554). It is his thesis that Kant’s aim was to “explain the presuppositions underlying the Greek conception of magnitudes” (170). The role of irrational numbers in Kant stays vague (cf. Sutherland 2006, pp. 543, 554, 557).
Due to the German language, Kant is able to draw a distinction between several meanings of ‘principle’ (this remark is of some importance): In Kantian terms ‘Grundsatz’ is a principle in a broad sense and may also mean a sentence that serves as principle in a certain discipline, but may presuppose sentences of other disciplines. The German ‘Prinzip’ is a principle that is a first principle that is evident in itself. Contrary to modern terminology, Kant calls them “Axiome” if they are evident due to intuition, but “Akroame” if they are expressible only through concepts (Logic-Blomberg, AA XXIV, 278; Lecture on Logic (Jäsche) §§ 34 ff., AA IX, 110; also R 3132, AA XVI, 673; also CpR, B 760). Kant is well aware of the fact that geometry contains or even presupposes principles of logic. But due to his basic assumption about the dual structure of the capacity of knowledge, he nevertheless assumes that evidence of the central mathematical principles relies on intuition. Kant’s use of “Axiom” is therefore different to modern mathematics where axiom is an arbitrary mathematical assumption that functions as basis for mathematical reasoning in a formalism. Axioms in modern mathematics do not need any evidence, neither intuitive nor conceptual, to count as true, but they are true according their function in a formalism.
Hanna differentiates between two senses of ‘construct’: ‘construction’ in the sense of creating something new and ‘construal’ in the grammatical sense (Hanna 2002, pp. 342 ff.). Hanna thinks that Kant’s concept of construction should be taken in the sense of construal, because the creation-sense of construction in Kant’s theory of number leads to a fatal dependence of arithmetical objects on epistemic capacities, which in his eyes leads to psychologism, a view Kant clearly rejects.
It is necessary to note that ‘object’ is in Kant’s use understood as that to which knowledge refers, which may be individuals in space and time, but may also be laws of nature as general judgments on relations referring to objects in space and time. Kant’s central determination of ‘object’ says: “an object, however, is that in the concept of which the manifold of a given intuition is united.” (CpR, B 137). This applies to substances as well as to universal relations.
His examples for analytic judgments in mathematics are algebraic formulas like ‘a = a’ or ‘(a + b) > a’.
As only very few philosophers are contrary to this claim, like Mill, the evidence of it seems quite unquestionable.
Quine and Kripke are only two but prominent figures maintaining—according to their semantic accounts—that Kant’s distinction is in one or the other way misleading. Their criticism is twofold: first, they try to show that there are no a priori concepts but that all concepts are empirical; and secondly, they try to show that there are no analytic judgments concerning empirical concepts. As we do not deal with empirical concepts only the first criticism is of interest. Connected to this criticism is the problem of the necessity of mathematics.
It seems obvious that the law of non-contradiction is not sufficient for the truth of analytic judgements. So, Kant here must be taken as considering it a representative for the entire logic.
The principle of all synthetic judgments, which Kant formulates in CpR, B 197 is not parallel to the highest principle of the analytic judgments insofar as it does not serve as their sufficient truth-condition.
CpR, B 15, B 205; Prolegomena, AA IV, 268. In his letter to Schultz the example is 3 + 4 = 7 (AA X, 556).
In the introduction to the first Critique, Kant presupposes that the intuitive evidence in mathematics may rely on contemporary theories. Accordingly, he refers to Johannes Andreas Segner, who’s “Elementa Analyseos Finitorum” and “Deutliche und vollständige Vorlesungen über die Rechenkunst und Geometrie” Kant made use of in his lectures on mathematics (CpR, B 15; Prolegomena, AA IV, 269; etc.). He gives a simple explanation of the necessity of sensible intuition in arithmetic: to think the number ‘5’, it is necessary to count taking your fingers or coral-beads on an abacus as units in empirical sensation (CpR, B 299).
The term ‘representation’ is essential to Kant’s transcendental philosophy, but it does not refer to empirical cognitive phenomena. Consequently, Kant’s philosophy should not be taken as holding a kind of representationalism.
But this third feature is not necessarily connected with intuition: if like in human beings intuition is sensible, then it is given receptively, while concepts are spontaneously produced by the understanding (CpR, B 33, B 92 f.). But it is possible to conceive of intellectual intuition even if this idea might not make sense to us finite beings.
We mention here in particular the critique of the intuition-thesis by Frege (1988, pp. 16–17). For illustrating his doubts, Frege raised the question whether an arithmetic “formula” with large numbers, e.g. the equation 135664 +37863 = 173527, may be considered as immediately evident and he denies this. We think, however, that Kant’s intuition-thesis and our reconstruction of the theory of arithmetic in Sects. 3 and 4 are not exposed to this objection. Rather, it seems that Frege was not fully aware of the ambiguity of Kant’s terminology with respect to the word “intuition”. Indeed, Kant did not state the intuitiveness of any arbitrary number, but he emphasizes at several places that the successive construction of numbers according to a rule is performed in time, where time is considered as a pure form of intuition. (For more details we refer to Sect. 4).
The famous debate between Parsons and Hintikka discusses this problem (Parsons 1992a, b; Hintikka 1992a, b). Hintikka thinks that in Kant intuition is primarily distinguished from concepts through simplicity, whereas immediacy is just a resulting determination of intuition. Parsons insists that Kant holds both criteria as equally essential for the concept of intuition. The latter position exposes Kant to the famous criticism of empiricism by Quine and others against immediacy and givenness as fundamental features of the basis of knowledge.
Carson investigates the impact of Kant’s changing view on the nature of intuition on the difference between Kant’s pre-critical and his critical philosophy of mathematics (Carson 1999). She assumes that there is an inconsistency in Kant’s pre-critical views about indemonstrable mathematical truths on the basis of his theory of intuition at that time, which is solved by the critical theory of intuition.
Hintikka’s interpretation of the intuitivity of arithmetical equations indicates the immediacy criterion of intuition concerning arithmetic as follows: “The immediacy of arithmetical truth […] is due to the fact that the only thing we have to do in order to establish such equations is to carry out the computation” (Hintikka 1992a, p. 32).
Parsons (1992a, p. 74) Cf. Brittan (1992).
De mundi sensibilis § 12, AA I, 397.
Kant does not make use of the distinction between elements of the set and units of a quantity. This is one decisive difference between Kant’s concept of ‘set’ and our modern understanding. Terminologically, he uses the term ‘unit’ for any element. But in effect this distinction is present in Kant’s theory as he strives to distinguish between the objective sphere of the constructed object in intuition (the symbols), the original object the concept refers to (the set), and the conceptual sphere of number itself. Therefore, we feel justified in introducing this refined terminology here (cf. n. 13).
It has often been discussed whether Kant had a concept of ‘set’, which is in any way comparable to its contemporary meaning (Parsons 1992b, pp. 141 ff., n. 22; Longuenesse 1998, p. 276, n. 73) (cf. n. 13). Parsons notes that the Kantian use of “Menge” on many occasions resembles Hilbert’s definition of ‘set’ as: “jedes Viele, welches sich als Eines denken läßt, d.h. jeden Inbegriff bestimmter Elemente, welcher durch ein Gesetz zu einem Ganzen verbunden werden kann, […]” (“every multiplicity, which is thinkable as unity, that is: every essence of determinate elements that can be unified by a law” (our transl.); Cantor (1932, p. 204). More prominent is Cantor’s definition: “Unter einer “Menge” verstehe ich jede Zusammenfassung M von bestimmten wohlunterschiedenen Objekten m unserer Anschauung oder unseres Denkens (welche die Elemente von M genannt werden) zu einem Ganzen.“ (“As a ‘set’ I take every collection M of well defined objects m in our intuition or of our thinking (which are called the elements of M) into a whole” (Cantor 1932, p. 282). Even if many passages in Kant resemble these definitions Kant’s use of “Menge” is terminologically not restricted to a precise definition like in modern mathematics. However, Kant is well aware that numbers may be thought as elements of sets and that such a set of numbers may be an ordered series of units following a rule, or a “set of numbers” may be thought as disperse (AA XXIII, 199).
CpR, B 111; B 179; A 103; R 4322, AA XVII, 505.
Kant is not very proper in his use of these terms, as he states that 13 coins are not a quantum, but an “aggregate, that is a number of coins”, his argument is that a quantum is always continuous (CpR, B 212). Nowhere else he sticks to this differentiation. Longuenesse (1998, pp. 263–271) investigates Kant’s use of this family of terms more closely. Furthermore, she tries to point out the connection between quantifiers in judgments and the categories of quantity in mathematics. Cf. Sutherland (2004a, p. 427).
Parsons criticises Kant for taking number as a schema, because, as schema, number has “temporal content” (Parsons 1992b, p. 148). But the transcendental schema of number is, following Kant, just the representation of a certain method: “On the contrary, if I only think a number in general, which could be five or a hundred, this thinking is more the representation of a method for representing a multitude (e.g., a thousand) in accordance with a certain concept than the image itself which in this case I could survey and compare with the concept only with difficulty. Now this representation of a general procedure of the imagination for providing a concept with its image is what I call the schema for this concept.” (CpR, B 179 f.). It is the procedure of construction, which is the precondition of understanding the meaning of numbers that affords time; in number itself we abstract from this precondition.
The empirical schema of a dog is the general idea of an animal on four legs that barks. Since nearly all phenomenal properties of a dog vary between different dog-races, it is impossible to give an adequate image of the concept ‘dog’ in intuition. In opposition to that even if we may think ‘5’ as ‘. . . . .’ as well as ‘|||||’ both exhibit the concept ‘5’ adequately, as in all cases there are five elements.
Therefore mathematical objects cannot be underdetermined quasi-objects as Hanna thinks (Hanna 2002, p. 346).
Hanna points out that the intuition-thesis implies that mathematical truths are not logically necessary in the strict sense, but that it must be possible to think of a world in which they do not hold (Hanna 2002, pp. 331 ff.). This way of putting the conditionality of mathematical truths presupposes a certain kind of theory of truth which is questionable in this context.
This is Kant’s discovery of his dissertation “De mundi sensibilis” of 1770 (§ 12, AA II, 398). Here Kant stresses, that not only time but also space is inevitable to “realize” numbers that are proper concepts of the understanding. Space and time are necessary for the realisation of numbers, because “multiples have to be posited after one another and simultaneously”, meaning in order to be represented as multiples.
Sutherland points out the relevance of homogeneity for Kant’s theory of number (Sutherland 2004b, pp. 164 ff.).
In translations of Kant’s writings it is common to translate “Mannigfaltigkeit” literally by ‘manifold’; but in mathematical contexts it seems more fitting to translate it by ‘multiplicity’. Therefore, we would like to speak of ‘multiplicity’ concerning mathematical units, and of ‘manifold’ concerning parts of space and time. Parsons points out that the translation of Kant’s use of the whole family of terms is problematic. In other cases he himself does not seem to be consistent in his use of terms (Parsons 1992b, p. 141).
Parsons thinks this thesis forces Kant to the claim that like in geometry construction in arithmetic is ostensive construction, which is contrary to what Kant says (Parsons 1992b, p. 138).
The categories are concepts based on the same function of thinking as the logical forms (this is Kant’s rather abbreviated argument for the deduction of the categories in the “Leitfaden-chapter” in the Critique of Pure Reason). They are logical functions understood as determinations of the manifold of intuition. Therefore, they are determinations a priori of objects or of the object in general (“Gegenstand überhaupt”). Their function is to give the synthesis of the multiplicity of representations a rule in order to create one intuition (CpR, B 104).
This much is Kant’s theory in his pre-critical thinking on the nature of numbers, like it is to be found in his essay “Investigation on the Clearness of Principles” from 1764 (AA II, 290). Even if he calls the function of this operation “synthesis”, it is not thought of as an operation of the faculty of knowledge and that the rules for these operations are the categories. This is Kant’s discovery of critical philosophy. Without this thought the theory of construction is yet unclear and only this thought makes the solution of the application problem possible.
Letter to Schultz, AA X, 556.
Kant thinks of ciphers as a kind of “arbitrary artificial signs” that are arranged by “agreement between individuals just for sight” (“zwischen Einzelnen verabredete Zeichen blos fürs Gesicht (Ziffern)” Anthropology from a Pragmatic Point of View. AA VII, 192). As numeral-systems he takes into account what he calls decadic and tetractic (Critique of Judgment, AA V, 254). Decadic is Kant’s example for a concept that functions as a rule producing unity through synthesis (CpR, B 104).
He determines geometrical construction as ostensive and algebraic construction as symbolic (CpR, B 745).
In § 59 of the Critique of Judgment, where he treats the concept of symbol in more detail, he distinguishes between schemata as illustrations of concepts of the understanding and symbols as illustrations of concepts of reason (Critique of Judgment, AA V, 351). ‘Symbolic construction’ means that arithmetic is construction with symbols. Symbols are used because there has to be a representative of the qualitatively undetermined elements of a set*, which functions as unit in counting. Kant calls symbols in algebra also “characters”.
This is not just an accidental idea as the Critique of Practical Reason shows (AA V, 8 n), where he explains the concept of formula, which “determines that, what has to be done, to stay exactly at a problem” (our emphasis; also: AA V, 11 n).
It is somehow unsatisfying that Kant only deals in detail with addition as a form of synthesis, but does not investigate its relation to the other arithmetical operations: subtraction, multiplication and division. In his letter to Schultz, he mentions that addition as well as subtraction is synthesis (AA X, 555). In two reflections, Kant strikingly assimilates the four arithmetical operations with the four groups of categories: addition resembles quantity, subtraction resembles quality, multiplication resembles relation and division resembles modality (R 5651 f., AA XVIII, 302).
Shabel (2006).
In order to avoid misunderstandings, Weyl (1966, pp. 54–55) pointed out that by using spatial figures like strokes the concept of number is not essentially connected with space as a form of intuition, since we could have used equally sounds for counting.
Letter to Schultz (AA X, 555).
CpR, B 17; B 745.
Investigation on the Clearness of Principles, AA II, 278.
Leibniz, G. W., Nouveaux Essays II, chap. IV, § 10.
This is an important difference between Kant’s semi-critical position of his dissertations-thesis “De Mundi Sensibilis” from 1770 to his critical philosophy: in 1770 he explicitly rejects the inference from epistemical inaccessibility to ontological inexistence of the infinite, despite already holding that mathematical entities are to be constructed (AA II, 387 ff.). So his construction-thesis up to 1770 is conceived of as a purely subjective condition that has no impact on the ontological sphere.
CpR, B 555; Metaphysical Foundations of Natural Science, AA IV, 506. Falkenburg (2000, pp. 164–172) explains the Kantian conception of the infinite by introducing the difference between ordinality and cardinality and shows why Kant develops the theory of the potential infinite in his critical period.
Transcendental contradiction means that the concept of something that entails being a possible object of knowledge, does not meet the transcendental conditions of knowledge.
Kant expresses quite interesting thoughts in his lectures on the philosophy of religion, where he considers it as obvious that the mathematical infinite is not an absolute quantity, a given set* or object, but that it is just a relative quantity. The quantity of units depends on the scale that we choose to measure the size of something, but infinite sizes cannot be measured by a scale. Nevertheless, he thinks that infinite size expressed in the scale of miles, as ‘infinitely many miles’ or in the scale of solar distances (he seems to think of the distance between earth and sun) as ‘infinitely many solar-distances’, are different in size. So he thinks that we imagine a plurality of infinities that may be smaller or larger depending on scale. It is tempting to see in this statement, erroneously, rough outlines of Hilbert’s conception of a plurality of infinities with different cardinalities.
Leibniz thinks that the knowledge of the ground of a rule that entails the infinite is knowledge of the infinite itself (Nouveaux Essays. GP V, 143–146).
Perhaps, except the short and not very convincing essay of 1763 “Essay to introduce the Concept of negative Quantities into Philosophy”, AA II, 165–204.
Letter by August Wilhelm Rehberg before September 25, 1790 (AA XI, 207).
For example Thiel (1995, pp. 141 f.).
This point was emphasized in particular by Tetens (1994).
There are strong indications that in the realm of quantum physics observable objects cannot be individualised and that in a set of several objects of the same kind, quantum objects are completely indistinguishable. Hence, the justification of the applicability of arithmetic discussed here seems to be restricted to objects of classical physics.
We are indebted to Gabriel Gottlieb for checking this article linguistically.
References
Allison, H. E. (2004). Kant’s transcendental idealism. An interpretation and defense (revised and enlarged edition. New Haven, London: Yale University Press.
Barker, S. (1992). Kant's view of geometry: a partial defense. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 221–243). Dordrecht: Kluwer.
Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, LXX(19), 661–679.
Bigelow, J. (2001). The reality of numbers. A physicalist’s philosophy of mathematics. Oxford: Oxford University Press (reprint of 1988).
Brittan, G. (1992). Algebra and intuition. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 315–340). Dordrecht: Kluwer.
Cantor, G. (1932). Gesammelte Abhandlungen mathematischen und philosophischen Inhalts. Ed. E. Zermelo. Berlin: Springer.
Carson, E. (1999). Kant on the method of mathematics. Journal of the History of Philosophy, XXXVII(4), 629–652.
Falkenburg, B. (2000). Kants Kosmologie. Frankfurt: Klostermann.
Frege, G. (1988). Grundlagen der Arithmetik. Ed. Ch. Thiel. Hamburg: Meiner.
Friedman, M. (1992). Kant’s theory of geometry. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 177–220). Dordrecht: Kluwer.
Hanna, R. (2002). Mathematics for humans. European Journal of Philosophy, 10(3), 328–352.
Hintikka, J. (1992a). Kant on the mathematical method. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 21–42). Dordrecht: Kluwer.
Hintikka, J. (1992b). Kant’s transcendental method and his theory of mathematics. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 341–360). Dordrecht: Kluwer.
Kamlah, W., & Lorenzen, P. (1967). Logische Propädeutik. Mannheim: Bibliographisches Institut.
Kant, I. (1998). Critique of pure reason (P. Guyer & A. W. Wood, Trans.). Cambridge: Cambridge University Press. Cited as CpR.
Kant, I. (1900 ff.). Gesammelte Schriften. Ed. by Deutsche Akademie der Wissenschaften (Cited as AA with roman numerals for volume and arabic numerals for pages, and n for note).
Leibniz, G. W. (1882). Nouveaux essays. Ed. C. J. Gerhardt, Philosophische Schriften (Vol. V). Berlin (Reprint Hildesheim 1960).
Longuenesse, B. (1998). Kant and the capacity to judge. Sensibility and discursivity in the transcendental analytic of the critique of pure reason. Princeton: Princeton University Press.
Lorenzen, P. (1955). Einführung in die operative Logik und Mathematik. Heidelberg: Springer.
Melnick, A. (1992). The geometry of a form of intuition. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 245–256). Dordrecht: Kluwer.
Parsons, C. (1992a). Kant’s philosophy of arithmetic (and postscript). In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 43–80). Dordrecht: Kluwer.
Parsons, C. (1992b). Arithmetic and the categories. In C. J. Posy (Ed.), Kant’s philosophy of mathematics. Modern essays (pp. 135–158). Dordrecht: Kluwer.
Putnam, H. (1971). Philosophy of logic. New York: Harper.
Quine, W. V. O. (1960). Word and object. Cambridge, MA: MIT Press.
Shabel, L. (1998). Kant on the ‘symbolic construction’ of mathematical concepts. Studies in the History and Philosophy of Science, 29, 589–621.
Shabel, L. (2006). Kant’s philosophy of mathematics. In P. Guyer (Ed.), The Cambridge companion to Kant (2nd ed.). Cambridge: Cambridge University Press.
Sutherland, D. (2004a). The role of magnitude in Kant’s critical philosophy. Canadian Journal of Philosophy, 34(4), 411–442.
Sutherland, D. (2004b). Kant’s philosophy of mathematics and the Greek mathematical tradition. Philosophical Review, 113(2), 157–201.
Sutherland, D. (2005). The point of Kant’s axioms of intuition. Pacific Philosophical Quarterly, 86, 135–159.
Sutherland, D. (2006). Kant on arithmetic, algebra, and the theory of proportions. Journal of the History of Philosophy, 44(4), 533–558.
Tetens, H. (1994). Arithmetik: ein Apriori der Erfahrung? Dialektik, 3, 125–146.
Thiel, C. (1995). Philosophie und Mathematik. Darmstadt: Wissenschaftliche Buchgesellschaft.
Weyl, H. (1927). Philosophie der Mathematik und Naturwissenschaft. In A. Baeumler & M. Schröter (Eds.), Handbuch der Philosophie, Abteilung II, Natur, Geist, Gott (pp. 1–162). München: Oldenburg.
Weyl, H. (1966). Philosophie der Mathematik und Naturwissenschaft. (3rd edition). München: Oldenburg.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Engelhard, K., Mittelstaedt, P. Kant’s Theory of Arithmetic: A Constructive Approach?. J Gen Philos Sci 39, 245–271 (2008). https://doi.org/10.1007/s10838-008-9072-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10838-008-9072-y