In his argument for the possibility of knowledge of spatial objects, in the Transcendental Deduction of the B-version of the Critique of Pure Reason, Kant makes a crucial distinction between space as “form of intuition” and space as “formalintuition.” The traditional interpretation regards the distinction between the two notions as reflecting a distinction between indeterminate space and determinations of space by the understanding, respectively. By contrast, a recent influential reading has argued that the two notions can (...) be fused into one and that space as such is first generated by the understanding through an act of synthesis of the imagination. Against this reading, this article argues that a key characteristic of space as a form of intuition is its nonconceptual unity, which defines the properties of space and is as such necessarily independent of determination by the understanding through the transcendental synthesis of the imagination. The conceptual unity that the understanding prescribes to the manifold in intuition, by means of the categories, defines the formalintuition. Furthermore, this article argues that it is the sui generis, nonconceptual unity of space, when taken as a unity for the understanding by means of conceptual determination, that first enables geometric knowledge and knowledge of spatially located particulars. (shrink)
In the first edition of his book on the completeness of Kant’s table of judgments, Klaus Reich shortly indicates that the B-version of the metaphysical exposition of space in the Critique of pure reason is structured following the inverse order of the table of categories. In this paper, I develop Reich’s claim and provide further evidence for it. My argumentation is as follows: Through analysis of our actually given representation of space as some kind of object (the formal (...) class='Hi'>intuition of space in general), the metaphysical exposition will show that this representation is secondary to space considered as an original, undetermined and as such unrepresentable intuitive manifold. Now, following Kant, the representation of any kind of object involves diversity, synthesis and unity. In the case of our representation of space as formalintuition, this involves, firstly, a manifold a priori, i.e. space as pure form, delivered by the transcendental Aesthetic, secondly, a figurative, productive synthesis of that manifold, and, thirdly, the unity provided by the categories. Analysing our given representation of space – the task of the metaphysical exposition – amounts to dismantling its unity and determine its characteristics with respect to the categories. (shrink)
Revised version of chapter in J. N. Mohanty and W. McKenna (eds.), Husserl’s Phenomenology: A Textbook, Lanham: University Press of America, 1989, 29–67. -/- Logic for Husserl is a science of science, a science of what all sciences have in common in their modes of validation. Thus logic deals with universal laws relating to truth, to deduction, to verification and falsification, and with laws relating to theory as such, and to what makes for theoretical unity, both on the side of (...) the propositions of a theory and on the side of the domain of objects to which these propositions refer. This essay presents a systematic overview of Husserl’s views on these matters as put forward in his Logical Investigations. It shows how Husserl’s theory of linguistic meanings as species of mental acts, his formal ontology of part, whole and dependence, his theory of meaning categories, and his theory of categorial intuition combine with his theory of science to form a single whole. Finally, it explores the ways in which Husserl’s ideas on these matters can be put to use in solving problems in the philosophy of language, logic and mathematics in a way which does justice to the role of mental activity in each of these domains while at the same time avoiding the pitfalls of psychologism. (shrink)
The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the (...) foundational analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition. (shrink)
Mit dem Terminus 'ursprünglicher Raum' wird der Raum bezeichnet, der Kant innerhalb der transzendentalen Ästhetik als reine subjektive Form der Anschauung des äußeren Sinnes bestimmt. Man könnte ihn auch den 'ästhetischen Raum' nennen. Auf jeden Fall muss er vom (proto-)geometrischen Raum unterschieden werden, da letzterer eine Einheit voraussetzt die auf einer Synthesis beruht, und dadurch – weil bei Kant alle Synthesis unter den Kategorien steht – weniger ursprünglich zum Anschauungsvermögen gehört. Es ist diese Unterscheidung zwischen dem ursprünglichen Raum, der „Form (...) der Anschauung“ ist, und dem (proto-)geometrischen Raum, der „formale Anschauung“ ist, auf die Kant in einer bekannten Fußnote im §26 der transzendentalen Deduktion der B-Auflage anspielt. -/- Die Bedeutung der Unterscheidung zwischen (proto-)geometrischem und ursprünglichem Raum liegt unter anderem darin, dass sie stipuliert, dass das ursprüngliche Wesen des Raumes vor und unabhängig von dem erreichbar ist, was durch jedwede Mathematik der Ausdehnung von ihm ausgesagt wird. Das bedeutet nun aber nicht, dass diese Unterscheidung uns zwingt, anzunehmen, dass das ursprüngliche Wesen des Raumes auch von uns erreichbar ist. Und nehmen wir mal an, dass wir tatsächlich über eine Art Zugang zu diesem Wesen verfügen, dann noch stellt sich überdies die Frage, ob ein solcher Zugang sich innerhalb der Sphäre der Erkenntnis befindet, mit anderen Worten: ob das ursprüngliche Wesen des Raumes vom Philosophen auch wirklich erkannt – das heißt: in Erkenntnisurteile gefasst und ausgedrückt – werden kann. (shrink)
Kant notably holds that arithmetic is synthetic a priori and has to do with the pure intuition of time. This seems to run against our conception of arithmetic as universal and topic neutral. Moreover, trained in the tradition constituting the aftermath of W.V. Quine's attack on the the a priori and on the analytic/synthetic distinction, the modern philosopher of arithmetic is likely to consider Kant's position a nonstarter, and leave settling the question of what Kant's philosophy of arithmetic is (...) exclusively to the Kant scholar and the historian of the philosophy of arithmetic. I argue that this conclusion is misguided because it rests on the unfounded supposition that the pure intuition of time is the basis for Kant's syntheticity and a priority theses. I recover Kant's grounds for holding those theses and their significance to contemporary philosophy of arithmetic. ;I consider and reject Friedman's eliminativist attempt at making Kant palatable to the contemporary philosopher. I argue that Kant's ideas about the mathematical method in 1763, before he explicitly draws the analytic/synthetic distinction, inform the appreciation of Kant's mature view. The idea of construction in intuition is a key to Kant's Critical position that explains the relation between the intellectual and the sensible aspects in Kant's thought. I show that Kant employs a distinct notion of pure formalintuition that is associated with arithmetical necessity construed as peculiarly mathematical; irreducible to logical or sensible modality. Kant's claim is not that the intuition of time serves to justify arithmetical judgments, I argue, but that we cannot represent time as we do unless we think of it arithmetically. According to Kant, arithmetic is not reducible to logic but it is nonetheless just as fundamental to thought in general. The singularity numerical judgments in relation to the category of quantity is shown to involve a notion of a form of an object that is primary with respect to the concept of an object in general. Finally, I reconstruct Kant's notion of symbolic construction and explicates Kant's conception of a constructive procedure. I argue that a Kantian view of the ontology of arithmetic takes the numbers to be nominalizations of construction procedures for intuitable symbolic types. (shrink)
Could a computer be programmed to make moral judgments about cases of intentional harm and unreasonable risk that match those judgments people already make intuitively? If the human moral sense is an unconscious computational mechanism of some sort, as many cognitive scientists have suggested, then the answer should be yes. So too if the search for reflective equilibrium is a sound enterprise, since achieving this state of affairs requires demarcating a set of considered judgments, stating them as explanandum sentences, and (...) formulating a set of algorithms from which they can be derived. The same is true for theories that emphasize the role of emotions or heuristics in moral cognition, since they ultimately depend on intuitive appraisals of the stimulus that accomplish essentially the same tasks. Drawing on deontic logic, action theory, moral philosophy, and the common law of tort, particularly Terry's five-variable calculus of risk, I outline a formal model of moral grammar and intuitive jurisprudence along the foregoing lines, which defines the abstract properties of the relevant mapping and demonstrates their descriptive adequacy with respect to a range of common moral intuitions, which experimental studies have suggested may be universal or nearly so. Framing effects, protected values, and implications for the neuroscience of moral intuition are also discussed. (shrink)
This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the property of knowledge in epistemic logic, I argue that rational intuition can be codified by a modal operator governed by the axioms of a dynamic provability logic, which augments GL with the modal $\mu$-calculus. Via correspondence results between modal logic and first-order logic, a precise translation can then be provided between the notion of 'intuition-of', i.e., the cognitive phenomenal properties of thoughts, (...) and the modal operators regimenting the notion of 'intuition-that'. I argue that intuition-that can further be shown to entrain conceptual elucidation, by way of figuring as a dynamic-interpretational modality which induces the reinterpretation of both domains of quantification and the intensions of mathematical concepts that are formalizable in monadic first- and second-order formal languages. (shrink)
In this article I deal with the notion of observation, from a phenomenologically motivated point of view, and its representation mainly by means of the formal language of quantum mechanics. In doing so, I have taken the notion of observation in two diverse contexts. In one context as a notion related with objects of a logical-mathematical theory taken as registered facts of phenomenological perception ( Wahrnehmung ) inasmuch as this phenomenological idea can also be linked with a process of (...) measurement on the quantum-mechanical level. In another context I have taken it as connected with a notion of temporal constitution basically as it is described in E. Husserl’s texts on the phenomenology of temporal consciousness. Given that mathematical objects as formal-ontological objects can be thought of as abstractions of perceptual objects by means of categorial intuition, the question is whether and under what theoretical assumptions we can, in principle, include quantum objects in abstraction in the class of formal-ontological objects and thus inquire on the limits of their description within a formal-axiomatical theory. On the one hand, I derive an irreducibility on the level of individuals taken in formal representation as syntactical atoms-substrates without any further content and on the other hand a transcendental subjectivity of consciousness objectified as a self-constituted temporal unity upon which it is ultimately grounded the possibility of generation of an abstract predicative universe of discourse. (shrink)
This paper provides a précis of Ernst Cassirer's concept of art as a symbolic form. It does so, though, in a specific respect. It points to the fact that Cassirer's concept of "symbolic form" is two-sided. On the one hand, the concept captures general cultural phenomena that are not only meaningful but also manifest the way man makes sense of the world; thus myth, religion, and art are considered general symbolic forms. On the other hand, it captures the formal (...) structures and semiotic tools thanks to which meaning is constructed within each general symbolic form (Cassirer called these structures "modes of objectivation"); thus, in art, perspective or the golden section are well-known examples of symbolic forms, now in a narrow sense, i.e. they are means to configure parts into an organized, meaningful whole. The paper will comment on art along both these two dimensions, but its main goal is to provide with concrete examples of aesthetic symbolic forms in the narrow sense in order to show how conceptual meaning can be inscribed in the space of aesthetic intuition. (shrink)
often insisted existence in mathematics means logical consistency, and formal logic is the sole guarantor of rigor. The paper joins this to his view of intuition and his own mathematics. It looks at predicativity and the infinite, Poincaré's early endorsement of the axiom of choice, and Cantor's set theory versus Zermelo's axioms. Poincaré discussed constructivism sympathetically only once, a few months before his death, and conspicuously avoided committing himself. We end with Poincaré on Couturat, Russell, and Hilbert.
In this thesis, I investigate the nature of geometric knowledge and its relationship to spatial intuition. My goal is to rehabilitate the Kantian view that Euclid's geometry is a mathematical practice, which is grounded in spatial intuition, yet, nevertheless, yields a type of a priori knowledge about the structure of visual space. I argue for this by showing that Euclid's geometry allows us to derive knowledge from idealized visual objects, i.e., idealized diagrams by means of non-formal logical (...) inferences. By developing such an account of Euclid's geometry, I complete the "standard view" that geometry is either a formal system or an empirical science, which was developed mainly by the logical positivists and which is currently accepted by many mathematicians and philosophers. My thesis is divided into three parts. I use Hans Reichenbach's arguments against Kant and Edmund Husserl's genetic approach to the concept of space as a means of arguing that the "standard view" has to be supplemented by a concept of a geometry whose propositions have genuine spatial content. I then develop a coherent interpretation of Euclid's method by investigating both the subject matter of Euclid's geometry and the nature of geometric inferences. In the final part of this thesis, I modify Husserl's phenomenological analysis of the constitution of visual space in order to define a concept of spatial intuition that allows me not only to explain how Euclid's practice is grounded in visual space, but also to account for the apriority of its results. (shrink)
In formal semantics intuition plays a key role, in two ways. Intuitions about semantic properties of expressions are the primary data, and intuitions of the semanticists are the main access to these data. The paper investigates how this dual role is related to the concept of competence and the role that this concept plays in semantics. And it inquires whether the self-reflexive role of intuitions has consequences for the methodology of semantics as an empirical discipline.
This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic . The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements (...) of the genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. This issues in an elaboration on a number of Gödel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics. (shrink)
In the paper, a detailed analysis of some new logical aspects of Cantor’s diagonal proof of the uncountability of continuum is presented. For the first time, strict formal, axiomatic, and algorithmic definitions of the notions of potential and actual infinities are presented. It is shown that the actualization of infinite sets and sequences used in Cantor’s proof is a necessary, but hidden, condition of the proof. The explication of the necessary condition and its factual usage within the framework of (...) Cantor’s proof makes , Cantor’s proof invalid. It’s shown that traditional Cantor’s proof has a second necessary, but hidden as well, condition which is teleological by its nature, i.e., is not mathematical. The explication of the second necessary condition makes Cantor’s statement on the uncountability of continuum unprovable from the point of view of classical logic. One of the most dramatic facts in history of science is connected just with the notion of actual infinity and consists in the following. On the one hand, Aristotle, Berkeley, Locke, Descartes, Spinoza, Gauss, Kant, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and a lot of other outstanding creators of classical logic and classical mathematics, during millenniums, stated categorically and insistently warned about that the actual infinity is a self-contradictory notion and its usage in mathematics is inadmissible as a disease from which one has recovered”). On the other hand, a lot of outstanding scientists of the XX c.c., such as Hilbert, Church, Turing, Gödel, etc., and modern axiomatic set theory as a whole ignored this opinion of genii and accepted Cantor’s “transfinite paradise”, based on the actual infinity. In the paper some logical, philosophical, and psychological aspects of this historical “favorable opposition” are analyzed, and the fact that the scientific intuition of genii has won the historical battle against the mytho-’logic’ of Cantor’s transfinite ‘paradise’ is explained and logically justified. (shrink)
Great intuitions are fundamental to conjecture and discovery in mathematics. In this paper, we investigate the role that intuition plays in mathematical thinking. We review key events in the history of mathematics where paradoxes have emerged from mathematicians' most intuitive concepts and convictions, and where the resulting difficulties led to heated controversies and debates. Examples are drawn from Riemannian geometry, set theory and the analytic theory of the continuum, and include the Continuum Hypothesis, the Tarski-Banach Paradox, and several works (...) by GÃ¶del, Cantor, Wittgenstein and Weierstrass. We examine several fallacies of intuition and determine how far our intuitive conjectures are limited by the nature of our sense-experience, and by our capacities for conceptualization. Finally, I suggest how we can use visual and formal heuristics to cultivate our mathematical intuitions and how the breadth of this new epistemic perspective can be useful in cases where intuition has traditionally been regarded as out of its depth. (shrink)
The progress in computer programming leads to the shift in traditional correlation between intuitive and formal components of mathematical knowledge. From epistemological point of view the role of intuition decreases in compare with formal representation of mathematical structures. The relevant explanation is to be found in D. Hilbert’s formalism and corresponding Kantian’s motives in it. The notion of sign belongs to both areas under consideration: on the one hand it is object of intuition in Kantian de (...) re sense, on the other hand, it is part of formal structure. Intuitive mathematical knowledge is expressed by primitive recursive reasoning. The W. Tait’s thesis, namely, that finitism as methodology of mathematics is equivalent to primitive recursive reasoning is discussed in connection with some explications of Kantian notion of intuition. The requirements of finitism are compared with normative role of logic. (shrink)
L’article s’interroge sur l’unité intrinsèque des concepts d’intuition, d’évidence et de remplissement dans la pensée de Husserl : existe-t-il un concept formel d’intuition qui soit valable pour toutes les sphères d’objets possibles ? Peut-on transposer aux différents types d’essences ou de catégories d’objets le paradigme de l’intuition élaboré dans la sphère de la perception sensible ? Cette question nous conduit à analyser, chez Husserl, la structure et les modalités du remplissement et de l’intuition pour les singularités (...) sensibles, les essences matériales et les essences mêlant sensibilité et forme catégoriale.In our paper, we ask whether or not the concepts of intuition, insight and fulfillment have an intrinsic unity in Husserl’s thought. Does a formal concept of intuition exist, which would have a validity in all possible spheres of objects ? Is it possible to transfer to the different kinds of essences the model of intuition, which has been elaborated in the sphere of sensory perception ? That problem leads us to analyze in Husserl’s thought the structure and modalities of fulfillment and intuition for sensory singularities, for material essences and for essences that are a mix of sensibility and categorical form. (shrink)
This article is composed of three sections that investigate the epistemological foundations of Husserl’s idea of logic from the Logical Investigations . First, it shows the general structure of this logic. Husserl conceives of logic as a comprehensive, multi-layered theory of possible theories that has its most fundamental level in a doctrine of meaning. This doctrine aims to determine the elementary categories that constitute every possible meaning (meaning-categories). The second section presents the main idea of Husserl’s search for an epistemological (...) foundation for knowledge, science and logic. Their epistemological clarification can only be reached through a detailed analysis of the structure of those intentions that give us what is meant in our intentions. To reveal the intuitive giveness of logical forms is the ultimate aim of Husserl’s epistemology of logic. Logical forms and meaning-categories can only be given in a certain higher-order intuition that Husserl calls categorical intuition. The third section of this article distinguishes different kinds of categorical intuition and shows how the most basic logical categories and concepts are given to us in a categorical abstraction. (shrink)
This paper addresses a complaint, by Prichard, against Plato and other ancients. The charge is that they commit a mistake is in thinking that we are capable of giving reasons for the requirements of duty, rather than directly and immediately apprehending those requirements. I respond in two ways. First, Plato does not make the egregious mistake of substituting interest for duty, and thus giving the wrong kind of reason for duty’s requirements, as Prichard alleges. Second, we should see that the (...) ancient ethical enterprise as being comprehensive in a sense Prichard simply ignores. The ancients sought what we now call wide reflective equilibrium in judgments about both duty and interest, and to see this I focus on a puzzle in how to understand much of ancient moral philosophizing. This puzzle is to make sense of the work that formal constraints on happiness do to support their preferred views of happiness (or interest). This way of engaging not only our thoughts about duty and interest, but about what it is to be human and to lead a human life, make the ancient model far more satisfying than Prichard’s recommendation that we give it all up as a mistake. (shrink)
En este artículo, analizo las nociones de forma de la intuición e intuición formal en el § 26 de la Crítica de la razón pura. Según Kant, la intuición formal contiene unidad de diversas representaciones, la cual no implica la forma de la intuición. Aclararé que la intuición formal es un híbrido de l..
El artículo presenta y analiza un conjunto de notas manuscritas de clases para cursos sobre geometría, dictados por David Hilbert entre 1891 y 1905. Se argumenta que en estos cursos el autor elabora la concepción de la geometría que subyace a sus investigaciones axiomáticas en Fundamentos de la geometría . Por un lado, afirmo que lo que caracteriza esta concepción de la geometría es: i) una posición axiomática abstracta o formal; ii) una posición empirista respecto del origen de la (...) geometría y de su lugar dentro de las distintas teorías matemáticas. Por otro lado, sostengo que el papel que Hilbert le confiere a la intuición geométrica en el proceso de axiomatización de esta teoría, permite apreciar claramente su oposición respecto de las posiciones formalistas con las que habitualmente es identificado. (shrink)
This paper contains a critical analysis of the interpretation of Kant's second edition version of the Transcendental Deduction offered by Béatrice Longuenesse in her recent book: Kant and the Capacity to Judge. Though agreeing with much of Longuenesse's analysis of the logical function of judgment, I question the way in which she tends to assign them the objectifying role traditionally given to the categories. More particularly, by way of defending my own interpretation of the Deduction against some of her criticisms, (...) I argue that Longuenesse fails to show how either part of the two-part proof may be plausibly thought to have established the necessity of the categories (as opposed to the logical functions). Finally, I question certain aspects of her 'radical' interpretation of the famous footnote at B160-1, where Kant distinguishes between 'form of intuition' and 'formalintuition'. (shrink)
In this paper, I reconstruct Edmund Husserl's view on the relationship between formal inquiry and the life-world, using the example of formal geometry. I first outline Husserl's account of geometry and then argue that he believed that the applicability of formal geometry to intuitive space (the space of everyday-experience) guarantees the conceptual continuity between different notions of space.
Relevance of premises to conclusion can be explicated through Toulmin’s notion of warrant, understood as an inference rule, albeit not necessarily formal. A normative notion of relevance requires the warrant to be reliable. To determine reliability, we propose a fourfold classification of warrants into a priori, empirical, institutional, and evaluative, with further subdivisions possible. This classification has its ancestry in classical rhetoric and recent epistemology. Distinctive to each type of warrant is the mode by which such connections are intuitively (...) discovered and the grounds on which we ultimately justify them. The classification of warrants is thus epistemic. We illustrate the difference by contrasting empirical physical with institutional intuition, and argue for the advantages of this approach over Toulmin’s conception of field dependence. (shrink)
There are some puzzles about G¨ odel’s published and unpublished remarks concerning finitism that have led some commentators to believe that his conception of it was unstable, that he oscillated back and forth between different accounts of it. I want to discuss these puzzles and argue that, on the contrary, G¨ odel’s writings represent a smooth evolution, with just one rather small double-reversal, of his view of finitism. He used the term “finit” (in German) or “finitary” or “finitistic” primarily to (...) refer to Hilbert’s conception of finitary mathematics. On two occasions (only, as far as I know), the lecture notes for his lecture at Zilsel’s [G¨ odel, 1938a] and the lecture notes for a lecture at Yale [G¨ odel, *1941], he used it in a way that he knew—in the second case, explicitly—went beyond what Hilbert meant. Early in his career, he believed that finitism (in Hilbert’s sense) is openended, in the sense that no correct formal system can be known to formalize all finitist proofs and, in particular, all possible finitist proofs of consistency of first-order number theory, P A; but starting in the Dialectica paper.. (shrink)
In this thesis, I examine the perceptibility of the Platonic Ideas in the thought of Arthur Schopenhauer. The work is divided into four chapters, each focusing and building upon a specific aspect related to this question. The first chapter ("Plato and the Primacy of Intellect") deals with Schopenhauers interpretation specific to Platonic thought. I there address the question of why it is that Schopenhauer should consider Plato to have interpreted the Ideas as 'perceptible', particularly in view of evidence which seems (...) to testify to the contrary. Does Schopenhauer misinterpret Plato, or are there sufficient grounds for considering his interpretation consistent? This is important in light of Schopenhauers reinterpretation of the Platonic Idea on the basis of the primacy of the will in contradistinction to Plato's own emphasis upon the intelligible. In effect, for Schopenhauer, Plato confuses the nature of imperceptible concepts with the more perceptually grounded Ideas. In the second chapter ("On the Direct Path to Knowledge"), I explore Schopenhauer's methodological approach on the basis of the will. I there discuss the difference between Schopenhauer and Kant with respect to Transcendental Idealism, particularly in terms of their separate analysis and derivation of the thing-in-itself. From there, I turn to a consideration of Schopenhauer's discussion of the nature of formal and empirical intuitions, of abstraction and the nature of universal concepts, and finally of the intuitively grounded nature of mathematical demonstration which serves as an analogy for the manner in which Schopenhauer deals with knowledge of the Ideas. In the third chapter ("The Perceptibility of the Ideas"), I discuss the manner in which the Ideas become accessible as the manifestation of the Will, arising through the representation of perception. There a number of distinctions are made between Ideas and concepts, as well as the nature of contemplation through Genius. The Ideas are found to be perceptible as aesthetic intuitions which relate not only to the in-itself nature inherent to phenomena, but serve also as the essential basis and foundation for all genuine forms of art. In the fourth and final chapter ("Critical Discussion of Schopenhauers Ideas"), I consider and outline a number of potential problems with respect to Schopenhauers analysis. In particular, Schopenhauer encounters difficulties on the basis of the simultaneous immanence and transcendence of the Ideas; of the knowledge of the Ideas characterized as intuitive while yet having a strangely abstract basis; of the unity of the will in face of the plurality of Ideas and phenomena; and finally of the annihilation of the will which creates essential problems in terms of the desideratum of knowledge itself. I here also attempt to offer various solutions to these problems where relevant. The conclusion which I arrive at in this work is that any radical reorientation of ontology must bear an essential effect upon the character and methodology of knowledge itself. In other words, Schopenhauers interpretation of the perceptibility of the Ideas is a direct consequence of hi s analysis of the will as the metaphysical thing-in-itself. In fact, this essential change colors his entire epistemological approach, from science and mathematics, to logic and the Ideasfrom the ground the up. (shrink)
The project of this paper is to address a complaint, by Prichard, against Plato and other ancients, as committing a basic “mistake” in moral philosophy. The basic mistake is in thinking that we are capable of giving reasons for the requirements of duty, rather than directly and immediately apprehending those requirements. Prichard’s argument that this is a mistake consists in an argument that attempts to give reasons for such requirements always fail. He classes those attempts into two kinds, and one (...) of those kinds is exemplified by Plato. (I leave aside here the second kind.) My response is to show two things. First, Plato does not make the egregious mistake of substituting interest for duty, and thus giving the wrong kind of reason for duty’s requirements, as Prichard alleges. This allegation assumes, first, that they duty and interest are entirely distinct notions, and, second, that we have a clear and accurate sense of the contents and bounds of each. Neither of these assumptions is accepted by Plato, and appreciating what their denial involves is essential to grasping the enterprise of moral philosophy as the ancients practiced it. Second, we should see that enterprise as being comprehensive in a sense Prichard simply ignores. They are seeking what we now call wide reflective equilibrium in judgments about both duty and interest, and to see this I focus on a puzzle in how to understand much of ancient moral philosophizing. This puzzle is how to make sense of the work the ancients see formal constraints on happiness as supporting their preferred views of happiness (or interest). I claim that they way they do so is a mess on any picture other than one of a search for wide reflective equilibrium, and this way of engaging not only our thoughts about duty and interest, but what it is to be human, and to lead a human life, make the ancient model far more satisfying than Prichard’s recommendation that we give it all up as a mistake. (shrink)
We aim to show that Kant’s theory of time is consistent by providing axioms whose models validate all synthetic a priori principles for time proposed in the Critique of Pure Reason. In this paper we focus on the distinction between time as form of intuition and time as formalintuition, for which Kant’s own explanations are all too brief. We provide axioms that allow us to construct ‘time as formalintuition’ as a pair of continua, (...) corresponding to time as ‘inner sense’ and the external representation of time as a line Both continua are replete with infinitesimals, which we use to elucidate an enigmatic discussion of ‘rest’ in the Metaphysical foundations of natural science. Our main formal tools are Alexandroff topologies, inverse systems and the ring of dual numbers. (shrink)
In the Transcendental Aesthetic, Kant famously characterizes space as a unity, understood as an essentially singular whole. He further develops his account of the unity of space in the B-Deduction, where he relates the unity of space to the original synthetic unity of apperception, and draws an infamous distinction between form of intuition and formalintuition. Kant ’s cryptic remarks in this part of the Critique have given rise to two widespread and diametrically opposed readings, which I (...) call the Synthesis and Brute Given Readings. I argue for an entirely new reading, which I call the Part-Whole Reading, in part by considering the development of Kant ’s views on the unity of space from his earliest works up through crucial reflections written during the silent decade. (shrink)
Here I criticise Audi's account of self-evidece. I deny that understanding of a proposition can justify belief in it and offfer an account of intuition that can take the place of understanding in an account of self-evidence.
I suppose the natural way to interpret this question is something like “why do formal methods rather than anything else in philosophy” but in my case I’d rather answer the related question “why, given that you’re interested in formal methods, apply them in philosophy rather than elsewhere?” I started off my academic life as an undergraduate student in mathematics, because I was good at mathematics and studying it more seemed like a good idea at the time. I enjoyed (...) mathematics a great deal. At the University of Queensland, where I was studying, there was a special cohort of “Honours” students right from the first year. You were taught more research-oriented and rigourous subjects than were provided for the “Pass” students. This meant that we had a small cohort of students, who knew each other pretty well, studied together and learned a lot. I could see myself making an academic career in mathematics. (I surely couldn’t see myself doing anything other than an academic career. Being around the university was too much fun.) However, there was a fly in the ointment. I was doing well in my studies, but I was losing the feel for a great deal of the mathematics I was doing. Applied mathematics went first, and analysis soon after. I could do the work, but I didn’t understand it. I wrote assignments by matching patterns from what I had written in my lecture notes, or what was in the text with what we were asked. In exams, I just bashed away at the problem, sometimes when asked in an exam to prove that A = B, I’d work at A from the top of a page and keep manipulating it until I’d got stuck. Then I’d work backwards from B, hoping to meet at somewhere rather like where I’d got stuck. If I was honest, I’d write “I don’t know how to get from here to there”. If I was dishonest, I’d just leave the transition unexplained. Knowing what I know now about marking assignments, it doesn’t suprise me that I did very well. The areas where intuition and understanding lasted the longest (and which were most fun) were topology, probability theory, combinatorics, set theory and logic.. (shrink)
: In the first edition of his book on the completeness of Kant’s table of judgments, Klaus Reich shortly indicates that the B-version of the metaphysical exposition of space in the Critique of pure reason is structured following the inverse order of the table of categories. In this paper, I develop Reich’s claim and provide further evidence for it. My argumentation is as follows: Through analysis of our actually given representation of space as some kind of object, the metaphysical exposition (...) will show that this representation is secondary to space considered as an original, undetermined and as such unrepresentable intuitive manifold. Now, following Kant, the representation of any kind of object involves diversity, synthesis and unity. In the case of our representation of space as formalintuition, this involves, firstly, a manifold a priori, i.e. space as pure form, delivered by the transcendental Aesthetic, secondly, a figurative, productive synthesis of that manifold, and, thirdly, the unity provided by the categories. Analysing our given representation of space – the task of the metaphysical exposition – amounts to dismantling its unity and determine its characteristics with respect to the categories. (shrink)
The phenomenology of a priori intuition is explored at length (where a priori intuition is taken to be not a form of belief but rather a form of seeming, specifically intellectual as opposed to sensory seeming). Various reductive accounts of intuition are criticized, and Humean empiricism (which, unlike radical empiricism, does admit analyticity intuitions as evidence) is shown to be epistemically self-defeating. This paper also recapitulates the defense of the thesis of the Autonomy and Authority of Philosophy (...) given in the author’s “A Priori Knowledge and the Scope of Philosophy” (Philosophical Studies, 1996). (shrink)
Changes in an upper level ontology have obvious conse-quences for the domain ontologies that use it at lower levels. It is therefore crucial to document the changes made between successive versions of ontologies of this kind. We describe and apply a method for tracking, explaining and measuring changes between successive versions of upper level ontologies such as the Basic Formal Ontology (BFO). The proposed change-tracking method extends earlier work on Realism-Based Ontology Versioning (RBOV) and Evolutionary Terminology Auditing (ETA). We (...) describe here the application of this evaluation method to changes between BFO 1.0, BFO 1.1, and BFO 2.0. We discuss the issues raised by this application and describe the extensions which we added to the original evaluation schema in order to account for changes in an ontology of this type. Our results show that BFO has undergone eight types of changes that can be systematically explained by the extended evaluation schema. Finally, we discuss problematic cases, possible pitfalls and certain limits of our study that we propose to address in future work. (shrink)
The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main (...) problem facing Platonists—the problem of explaining how our experiences make contact with mathematical reality. (shrink)
What one finds intuitive changes--propositions initially found intuitive, counterintuitive, or neither intuitive nor counterintuitive can shift their status. In this paper I develop a puzzle about changes in what one finds intuitive: (1) Changes in what one finds intuitive partly consist in learning new facts; (2) If changes in what one finds intuitive partly consist in learning new facts, then these changes are changes in inferences not intuitions; (3) But changes in what one finds intuitive are changes in intuitions. I (...) argue that changes in what one finds intuitive are changes in the contents of one’s intuition experiences due to a form of restructuring familiar from the literature on problem solving, and that this provides grounds for denying step (2) in the puzzle. I consider and reject alternatives that target steps (1) or (3). And I explore the significance my view of changes in what one finds intuitive has for recent controversies about philosophical methodology. (shrink)
The “received wisdom” in contemporary analytic philosophy is that intuition talk is a fairly recent phenomenon, dating back to the 1960s. In this paper, we set out to test two interpretations of this “received wisdom.” The first is that intuition talk is just talk, without any methodological significance. The second is that intuition talk is methodologically significant; it shows that analytic philosophers appeal to intuition. We present empirical and contextual evidence, systematically mined from the JSTOR corpus (...) and HathiTrust’s Digital Library, which provide some empirical support for the second rather than the first hypothesis. Our data also suggest that appealing to intuition is a much older philosophical methodology than the “received wisdom” alleges. We then discuss the implications of our findings for the contemporary debate over philosophical methodology. (shrink)
We propose a formal representation of objects , those being mathematical or empirical objects. The powerful framework inside which we represent them in a unique and coherent way is grounded, on the formal side, in a logical approach with a direct mathematical semantics in the well-established field of constructive topology, and, on the philosophical side, in a neo-Kantian perspective emphasizing the knowing subject’s role, which is constructive for the mathematical objects and constitutive for the empirical ones.
It is often claimed that emotions are linked to formal objects. But what are formal objects? What roles do they play? According to some philosophers, formal objects are axiological properties which individuate emotions, make them intelligible and give their correctness conditions. In this paper, I evaluate these claims in order to answer the above questions. I first give reasons to doubt the thesis that formal objects individuate emotions. Second, I distinguish different ways in which emotions are (...) intelligible and argue that philosophers are wrong in claiming that emotions only make sense when they are based on prior sources of axiological information. Third, I investigate how issues of intelligibility connect with the correctness conditions of emotions. I defend a theory according to which emotions do not respond to axiological information, but to non-axiological reasons. According to this theory, we can allocate fundamental roles to the formal objects of emotions while dispensing with the problematic features of other theories. (shrink)
We consider the question: under what circumstances can the concept of adaptation be applied to groups, rather than individuals? Gardner and Grafen (2009, J. Evol. Biol.22: 659–671) develop a novel approach to this question, building on Grafen's ‘formal Darwinism’ project, which defines adaptation in terms of links between evolutionary dynamics and optimization. They conclude that only clonal groups, and to a lesser extent groups in which reproductive competition is repressed, can be considered as adaptive units. We re-examine the conditions (...) under which the selection–optimization links hold at the group level. We focus on an important distinction between two ways of understanding the links, which have different implications regarding group adaptationism. We show how the formal Darwinism approach can be reconciled with G.C. Williams’ famous analysis of group adaptation, and we consider the relationships between group adaptation, the Price equation approach to multi-level selection, and the alternative approach based on contextual analysis. (shrink)
The problem of concept representation is relevant for many sub-fields of cognitive research, including psychology and philosophy, as well as artificial intelligence. In particular, in recent years it has received a great deal of attention within the field of knowledge representation, due to its relevance for both knowledge engineering as well as ontology-based technologies. However, the notion of a concept itself turns out to be highly disputed and problematic. In our opinion, one of the causes of this state of affairs (...) is that the notion of a concept is, to some extent, heterogeneous, and encompasses different cognitive phenomena. This results in a strain between conflicting requirements, such as compositionality, on the one hand and the need to represent prototypical information on the other. In some ways artificial intelligence research shows traces of this situation. In this paper, we propose an analysis of this current state of affairs. Since it is our opinion that a mature methodology with which to approach knowledge representation and knowledge engineering should also take advantage of the empirical results of cognitive psychology concerning human abilities, we outline some proposals for concept representation in formal ontologies, which take into account suggestions from psychological research. Our basic assumption is that knowledge representation systems whose design takes into account evidence from experimental psychology may therefore give better results in many applications. (shrink)
An argument is epistemically self-defeating when either the truth of an argument’s conclusion or belief in an argument’s conclusion defeats one’s justification to believe at least one of that argument’s premises. Some extant defenses of the evidentiary value of intuition have invoked considerations of epistemic self-defeat in their defense. I argue that there is one kind of argument against intuition, an unreliability argument, which, even if epistemically self-defeating, can still imply that we are not justified in thinking (...) class='Hi'>intuition has evidentiary value. (shrink)
Formal epistemology is just what it sounds like: epistemology done with formal tools. Coinciding with the general rise in popularity of experimental philosophy, formal epistemologists have begun to apply experimental methods in their own work. In this entry, I survey some of the work at the intersection of formal and experimental epistemology. I show that experimental methods have unique roles to play when epistemology is done formally, and I highlight some ways in which results from (...) class='Hi'>formal epistemology have been used fruitfully to advance epistemically-relevant experimental work. The upshot of this brief, incomplete survey is that formal and experimental methods often constitute mutually informative means to epistemological ends. (shrink)