The role and place of transcendental psychology in Kant's Critique of Pure Reason has been a source of some contention. This work presents a detailed argument for restoring transcendental psychology to a central place in the interpretation of Kant's Analytic, in the process providing a detailed response to more "austere" analytic readings.

The paper provides a reconstruction of proof by contradiction in Kant’s pure general logic. A seemingly less-explored point of view on this topic is how apagogical proof can account for the formal truth of a judgement. Integrating the argument held by Kjosavik (2019), I intend to highlight how one can use proof by contradiction, conceived as a modus tollens, to establish the logical actuality (logical or formal truth) of a cognition. Although one might agree on the capacity of the proof (...) to prove formal falsity, the logical actuality of a judgement is assessable based on a logically grounded judgement and, as for transcendental logic, this cognitive operation has to presuppose the real possibility of an object. (shrink)

Este artigo defende que, na Investigação sobre os princípios da teologia natural e da Moral, o conceito de “sentido interno” é central à reflexão kantiana sobre a matemática. Ele não deve ser entendido em sua acepção crítica, mas corresponde antes àquela derivada de Locke, pelo intermédio de Crusius, e se resume a uma reflexão mental sobre figuras e conceitos matemáticos. Ao incluir no sentido interno a dimensão simbólica do conhecimento matemático, Kant podia ainda ignorar um dos problemas centrais de sua (...) reflexão madura sobre a matemática, a saber, como a inspeção de objetos matemáticos particulares permite formular provas e leis de caráter universal. Isto acentua diferenças, muitas vezes ignoradas, entre a filosofia da matemática crítica e pré-crítica. Sign and Internal Sense in the pre-critical philosophy of mathematics Abstract In this article it is argued that, in the Inquiry concerning the Principles of natural theology and of morals, the concept of “inner sense” is central to Kant´s theses about mathematics. It must not, however, be taken in its critical meaning, corresponding rather to a loosely lockean influx, trough Crusius, indicating only that mental reflexion about mathematical objects or concepts is involved in mathematical knowledge. The concept of inner sense incorporates, in particular, the symbolic aspect of mathematical knowledge, and, in so doing, allows Kant to ignore what would become one of the central problems of his mature tought, namely how, in mathematics, the inspection of particulars provides us with proofs and laws which are general in character. This indicates to some relevant differences between the critical and pre-critical theory of mathematics which have been regularly overlooked. (shrink)

Officially, for Kant, judgments are analytic iff the predicate is "contained in" the subject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierarchy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of containment: (...) genera are contained in the species formed from them. Kant's thesis then amounts to the claim that no concept hierarchy conforming to division rules can express truths like '7+5=12.' Kant is correct. Operation concepts ( ) bear two relations to number concepts: and are inputs, is output. To capture both relations, hierarchies must posit overlaps between concepts that violate the exclusion rule. Thus, such truths are synthetic. (shrink)

Der Kommentar zur Kritik der reinen Vernunft bietet eine textnahe Erschließung der zentralen Begriffe, Thesen und Argumentationsgänge von Kants Hauptwerk auf aktuellem Forschungsstand. Es ist der erste Kommentar zur KrV, der den gesamten Text in der Fassung der ersten und zweiten Auflage gleichmäßig und lückenlos berücksichtigt. Davon profitieren vor allem die „Transzendentale Dialektik“ und die „Methodenlehre“, die in früheren Gesamtkommentaren meist nicht hinreichend berücksichtigt worden sind. Die Beiträge wurden nach einheitlichen Richtlinien verfasst, wobei unterschiedliche Herangehensweisen und Interpretationsansätze zur Geltung kommen. (...) Um dem Leser dieses Kommentars die Orientierung zu erleichtern, ist jeder Beitrag in drei bzw. vier Teile untergliedert: Der erste Teil behandelt die Stellung und Funktion des kommentierten Textabschnitts in der KrV, der zweite Teil gibt einen Überblick über Inhalt und Aufbau des Abschnitts, der dritte Teil enthält den eigentlichen Textkommentar. Wo dies sinnvoll erschien, wurden in einem vierten Teil wichtige Interpretationsfragen angesprochen. Hinweise auf weiterführende Spezialliteratur am Ende eines jeden Beitrags, eine Auswahlbibliographie zu Kant und zur Kritik der reinen Vernunft sowie ein Namen- und ein Sachregister machen den Band zu einem komfortablen Arbeitsmittel für den Benutzer. Die vorliegende zweite Auflage trägt der Tatsache Rechnung, dass der Band mittlerweile als Standardwerk der Kant-Forschung gilt. Mit Beiträgen von Henry Allison, Karl Ameriksm Reinhard Brandt, Wolfgang Carl, Konrad Cramer, Jean Ferrari Eckart Förster, Volker Gerhardt, Paul Guyer, Otfried Höffe, Hansgeorg Hoppe, Rolf-Peter Horstmann, Heiner F. Klemme, Lothar Kreimendahl, Béatrice Longuenesse, Georg Mohr, Birgit Recki, Alain Renaut, Peter Rohs, Gerhard Seel, Dieter Sturma, Bernhard Thöle, Eric Watkins, Marcus Willaschek. (shrink)

En este artículo se examina la noción kantiana de las demostraciones matemáticas. Esta noción se encuentra desarrollada en el apartado titulado “Disciplina de la razón pura en su uso dogmático” de la _Crítica de la razón pura. _En este texto, Kant explica por qué los procedimientos exitosos en el conocimiento matemático resultan impracticables en metafísica. En primer lugar se estudian dos pasajes en los que el filósofo describe dos demostraciones: la demostración de la congruencia de los ángulos de la base (...) de un triángulo isósceles, y la demostración de que la suma de los ángulos internos de cualquier triángulo es igual a dos rectos. A continuación, se examina la descripción de la demostración matemática como una prueba apodíctica en la intuición. Este análisis muestra que la demostración matemática no constituye la misma clase de procedimiento presentado como demostración en los §§57 y 59 de la _Crítica del Juicio._. (shrink)

The Univalent Foundations of Mathematics provide not only an entirely non-Cantorian conception of the basic objects of mathematics but also a novel account of how foundations ought to relate to mathematical practice. In this paper, I intend to answer the question: In what way is UF a new foundation of mathematics? I will begin by connecting UF to a pragmatist reading of the structuralist thesis in the philosophy of mathematics, which I will use to define a criterion that a formal (...) system must satisfy if it is to be regarded as a “structuralist foundation.” I will then explain why both set-theoretic foundations like ZFC and category-theoretic foundations like ETCS satisfy this criterion only to a very limited extent. Then I will argue that UF is better-able to live up to the proposed criterion for a structuralist foundation than any currently available foundational proposal. First, by showing that most criteria of identity in the practice of mathematics can be formalized in terms of the preferred criterion of identity between the basic objects of UF. Second, by countering several objections that have been raised against UF’s capacity to serve as a foundation for the whole of mathematics. (shrink)

The aim of this paper is to show that attention to Kant's philosophy of mathematics sheds light on the doctrine that there are two stems of the cognitive capacity, which are distinct, but equally necessary for cognition. Specifically, I argue for the following four claims: The distinctive structure of outer sensible intuitions must be understood in terms of the concept of magnitude. The act of sensibly representing a magnitude involves a special act of spontaneity Kant ascribes to a capacity he (...) calls the productive imagination. Contrary to what is assumed by many commentators, it is not the case that the Two Stems Doctrine implies that a representation is either sensible or spontaneity-dependent, but not both. Outer sensible intuitions are both sensible and spontaneity-dependent – they are sensible because they exhibit the kind of structure Kant takes to be distinctive of outer sensible intuitions, and they depend on spontaneity because they are cases of sensibly representing a magnitude. (shrink)

The paper attempts to summarize the debate on Kant’s philosophy of geometry and to offer a restricted area of mathematical practice for which Kant’s philosophy would be a reasonable account. Geometrical theories can be characterized using Wittgenstein’s notion of pictorial form . Kant’s philosophy of geometry can be interpreted as a reconstruction of geometry based on one of these forms — the projective form . If this is correct, Kant’s philosophy is a reasonable reconstruction of such theories as projective geometry; (...) and not only as they were practiced in Kant’s time, but also as architects use them today. (shrink)

According to Kant, we gain mathematical knowledge by constructing objects in pure intuition. This is true not only of geometry but arithmetic and algebra as well. Construction has prominent place in scholarly accounts of Kant’s views of mathematics. But did Kant have a clear vision of what construction is? The paper argues that Kant employed two different, even conflicting models of construction, depending on the philosophical issue he was dealing with. In the equivalence model, Kant claims that the object constructed (...) in intuition is equivalent to the properties included in the conceptual rule. In the overstepping model of construction, Kant argues that construction goes beyond the concept which is a “mere definition”. What is more, both models of construction can be found in the Doctrine of Method in the first Critique. The paper examines reasons that have led Kant to adopt the two models of construction, and proposes a reading that alleviates the apparent contradiction between the two models. (shrink)

Why does Frege claim that logical axioms are ‘self‐evident,’ to be recognized as true ‘independently of other truths,’ and then offer arguments for those axioms? I argue that he thinks the arguments provide us with the justification that we need for accepting the axioms and that this is compatible with his remarks about self‐evidence. This compatibility depends on philosophical considerations connected with the ‘critical method’: an interesting approach to the justification of axioms endorsed by leading philosophers at the time.

Famously, in the essay Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality (Prize Essay), Kant attempts to distance himself from the Wolffian model of philosophical inquiry. In this respect, Kant scholars have pointed out Kant’s claim that philosophy should not imitate the method of mathematics and his appeal to Newton’s “analytic method.” In this article, I argue that there is an aspect of Kant’s critique of the Wolffian model that has been neglected. Kant presents a powerful (...) attack of the idea that philosophy should proceed through the analysis of concepts and argues that we should give up the aim of arriving at “complete” concepts as one of its fundamental desiderata. Importantly, this attack of philosophy conceived as conceptual analysis is distinctive and original if one compares it with Kant’s critique of analysis in the Critique of Pure Reason, which rests on the idea that analysis is insufficient to ground substantial metaphysical truths. I also make an additional point: Scholars have debated whether the appeal to Newton’s analytic method involves a form of empiricism. I submit that this debate is ill-conceived because Kant uses a notion of “given concepts” that is indeterminate with respect to their a priori or a posteriori status. (shrink)

En este trabajo me propongo mostrar, en primer lugar, una interpretación de la distinción kantiana entre juicio de percepción y juicio de experiencia. Posteriormente reconstruiré el proceso de síntesis categorial e intentaré explicar qué papel juegan en dicha acción los conceptos empíricos. En definitiva, la tesis que se va a defender es que en el nivel de constitución de la experiencia tanto los conceptos empíricos como los juicios de percepción hacen posible la síntesis objetiva y, al mismo tiempo, ésta provee (...) a los conceptos involucrados en dicha síntesis, tanto empíricos como puros, sentido y referencia. (shrink)

This review of contemporary discussions of Kantian philosophy of mathematics is timed for the publication of the essay Kant’s Philosophy of Mathematics. Volume 1: The Critical Philosophy and Its Roots (2020) edited by Carl Posy and Ofra Rechter. The main discussions and comments are based on the texts contained in this collection. I first examine the more general questions which have to do not only with the philosophy of mathematics, but also with related areas of Kant’s philosophy, e. g. the (...) question: What is intuition and singular term? Then I look at more specific questions, e. g.: What is the subject of arithmetic and what is the significance of diagrams in mathematical reasoning? As a result, the reader is presented with a fairly complete overview of modern discussions which can be used as an introduction to the problem field of Kant’s philosophy of mathematics. (shrink)

This paper considers Kant's understanding of conceptual representation in light of his view of geometry.

In his 1763 Prize Essay, Kant is thought to endorse a version of formalism on which mathematical concepts need not apply to extramental objects. Against this reading, I argue that the Prize Essay has sufficient resources to explain how the objective reference of mathematical concepts is secured. This account of mathematical concepts’ objective reference employs material from Wolffian philosophy. On my reading, Kant's 1763 view still falls short of his Critical view in that it does not explain the universal, unconditional (...) applicability of mathematical concepts. (shrink)

The article examines how Salomon Maimon’s concept of number as ratio can be used to demonstrate that arithmetical judgments are analytical. Based on his critique of Kant’s synthetic a priori judgments, I show how this notion of number fulfills Maimon’s requirements for apodictic knowledge. Moreover, I suggest that Maimon was influenced by mathematicians who previously defined number as a ratio, such as Wallis and Newton. Following an analysis of the real definition of this concept, I conclude that within the framework (...) of Maimon’s philosophy, arithmetical judgments cannot be analytical, nor is arithmetic an objectively necessary science, but rather only subjectively necessary. We should also cast doubt on his claim that we can create real objects from pure concepts of the understanding. (shrink)

It is a common thought that mathematics can be not only true but also beautiful, and many of the greatest mathematicians have attached central importance to the aesthetic merit of their theorems, proofs and theories. But how, exactly, should we conceive of the character of beauty in mathematics? In this paper I suggest that Kant's philosophy provides the resources for a compelling answer to this question. Focusing on §62 of the ‘Critique of Aesthetic Judgment’, I argue against the common view (...) that Kant's aesthetics leaves no room for beauty in mathematics. More specifically, I show that on the Kantian account beauty in mathematics is a non-conceptual response felt in light of our own creative activities involved in the process of mathematical reasoning. The Kantian proposal I thus develop provides a promising alternative to Platonist accounts of beauty widespread among mathematicians. While on the Platonist conception the experience of mathematical beauty consists in an intellectual insight into the fundamental structures of the universe, according to the Kantian proposal the experience of beauty in mathematics is grounded in our felt awareness of the imaginative processes that lead to mathematical knowledge. The Kantian account I develop thus offers to elucidate the connection between aesthetic reflection, creative imagination and mathematical cognition. (shrink)

Kant’s investigations into so‐called a priori judgments of pure mathematics in the Critique of Pure Reason (KrV) are mainly confined to geometry and arithmetic both of which are grounded on our pure forms of intuition, space, and time. Nevertheless, as regards notions such as irrational numbers and continuous magnitudes, such a restricted account is crucially problematic. I argue that algebra can play a transcendental role with respect to the two pure intuitive sciences, arithmetic and geometry, as the condition of their (...) possibility. It follows that Kant’s schematism of the concept of magnitude ought to be quantitatively represented in algebraic formulas in general, and also undergo several modifications in order to suit continuities. (shrink)

What is central to the progression of a sequence is the idea of succession, which is fundamentally a temporal notion. In Kant's ontology numbers are not objects but rules (schemata) for representing the magnitude of a quantum. The magnitude of a discrete quantum 11...11 is determined by a counting procedure, an operation which can be understood as a mapping from the ordinals to the cardinals. All empirical models for numbers isomorphic to 11...11 must conform to the transcendental determination of time-order. (...) Kant's transcendental model for number entails a procedural semantics in which the semantic value of the number-concept is defined in terms of temporal procedures. A number is constructible if and only if it can be schematized in a procedural form. This representability condition explains how an arbitrarily large number is representable and why Kant thinks that arithmetical statements are synthetic and not analytic. (shrink)

This paper argues that kant's general epistemology incorporates a theory of algebra which entails a less constricted view of kant's philosophy of mathematics than is sometimes given. To extract a plausible theory of algebra from the "critique of pure reason", It is necessary to link kant's doctrine of mathematical construction to the idea of the "schematism". Mathematical construction can be understood to accommodate algebraic symbolism as well as the more familiar spatial configurations of geometry.

Kant’s philosophy of mathematics presents two fundamental problems of interpretation: (1) Kant claims that mathematical truths or “judgments” are synthetic a priori; and (2) Kant maintains that intuition is required for generating and/or understanding mathematical statements. Both of these problems arise for us because of developments in mathematics since Kant. In particular, the axiomatization of geometry--Kant’s paradigm of mathematical thinking--has made it seem to some commentators as, for example, Russell, that both (1) and (2) are false (Russell 1919, p. 145).2 (...) If virtually all of mathematics, including geometry, is axiomatizable, it would seem that mathematics results in analytic judgments that are totally independent of sensibility, the source, according to Kant, of intuition. In this paper I will address both of these difficulties. I shall argue that Kant’s understanding of both “synthetic” and “intuition” make his position immune to these criticisms. (shrink)

The Aristotelian origins of formal systems are outlined, together with Aristotle's use of causal terms in describing syllogisms. The precision and exactness of a formalism, based on the projection of logical forms into perceptive signs, is contrasted with foundational, abstract concepts, independent of any formalism, which are presupposed for the understanding of a formal language. The definition of a formal system by means of a Turing machine is put in the context of Wittgenstein's general considerations of a machine understood as (...) a sign. A modification of Łukasiewicz's logic Ł3 with the inclusion of justification terms is proposed in order to formally analyse some features of formalistic reasoning as a mechanical, causal affair within a wider context of "indeterminacy". (shrink)

In recent years non-conceptual content theorists have taken Kant as a reference point on account of his notion of intuition (§§ 1-2). The present work aims at exploring several complementary issues intertwined with the notion of non-conceptual content: of these, the first concerns the role of the intuition as an indexical representation (§ 3), whereas the second applies to the presence of a few epistemic features articulated according to the distinction between knowledge by acquaintance and knowledge by description (§ 4). (...) This work intends to dismiss the possibility that intuition may have an autonomous function of de re knowledge in support of an interpretative reading which can be labelled as weak conceptualism. To this end, the exploration will be conducted from a strictly transcendental perspective – i.e., by referring to the so called theory of the “concept of a transcendental object”. (shrink)

Kant advertises his Critique of Pure Reason as fulfilling reason's "most difficult" task: self-knowledge. As it is carried out in the Critique, this investigation is meant to be "scientific and fully illuminating"; for Kant, this means that it must follow a proper method. Commentators writing in English have tended to dismiss Kant's claim that the Critique is the scientific expression of reason's self-knowledge---either taking it to be sheer rhetoric, or worrying that it pollutes the Critique with an unfortunate residue of (...) rationalism. As a result, there is little sustained treatment of the method of the Critique in the secondary literature. Since Kant holds that the substantive insights of critical philosophy are not separable from the methodological context in which they come to light, this is a serious mistake. My dissertation corrects for this, by approaching the Critique through an examination of its method. In doing so, it yields a reading of the Transcendental Deduction that not only promises to resolve current debates about its "proof structure", but also fully accounts for the Deduction's pivotal role in the work as a whole. (shrink)

En este texto se busca poner una base para una interpretación de la intuición y la construcción en Kant, para lo cual se analiza la célebre interpretación desarrollada por Hintikka. Este análisis muestra que esta interpretación presenta algunas debilidades, sin embargo, de ella se rescata que se puede alcanzar una comprensión de la intuición y la construcción vinculándolas con algunos planteamientos de los antiguos filósofos y matemáticos griegos, especialmente Proclo y Euclides. Más específicamente, se muestra que un punto de partida (...) razonable para desarrollar una interpretación de la intuición y la construcción consiste en vincularlas con las dos conclusiones que Proclo plantea en su Comentario al Libro I de los Elementos de Euclides. Sin embargo, se debe tener en cuenta que tales vínculos son sólo razonables dentro del ámbito de las demostraciones, pues Kant afirma que la construcción también está presente en las definiciones y en los teoremas. (shrink)