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 “formal intuition.” 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 formal intuition. Furthermore, this article argues that it is the sui generis, nonconceptual unity of space, when takenas a unity for the understanding by means of conceptual determination, that first enables geometric knowledge and knowledge of spatially located particulars. (shrink)

I am interested in the use Kant makes of the pure intuition of space, and of properties and principles of space and spaces (i.e. figures, like spheres and lines), in the special metaphysical project of MAN. This is a large topic, so I will focus here on an aspect of it: the role of these things in his treatment of some of the laws of matter treated in the Dynamics and Mechanics Chapters. In MAN and other texts, (...) Kant speaks of space as the “ground,” “condition,” and “basis” of various laws, including the inverse-square and inverse-cube laws of attractive and repulsive force, and the Third Law of Mechanics. Moreover, in his proofs of all the laws just mentioned, the language of “construction” figures prominently, which suggests that Kant’s proofs (somehow) rest on or involve mathematical construction in his technical sense. Such claims give rise to a number of questions. How do properties and principles of space and spaces serve to ground this particular set of laws? Which spatial properties and principles is Kant appealing to? What, if anything, does the spatial grounding of the inverse-square and inverse-cube laws of diffusion (treated in the Dynamics Chapter) have in common with that of the Third treated in the Mechanics Chapter)? What role—if any—does mathematical construction play in Kant’s proofs of these laws? Finally, how if at all, are Kant’s grounding claims consistent with his other commitments—for example, how are they consistent with his notorious denial in Prolegomena §38 that there are any laws that “lie in space” (Prol 4:321)? I offer answers to these questions. (shrink)

Kant states that space and time are a priori conditions of experience, while apparently being committed to the euclidean nature of space and absolute simultaneity. His defense of the a priori character of spatio-temporal notions stems from taking them as pure intuitions, so its newtonian nature would derive from the configuration of what Kant names as intuition. Nevertheless, according to some recent discussions, it is not clear what intuition means for Kant and how space-time is (...) determined from it. In this paper I look into the debate about the origin of the synthesis of pure intuition that, according to Kant, would determine the spatio-temporal structure. I discuss to what extent taking into account the participation of categories in such a synthesis might have an effect on the commitment that, according to the kantian perspective, one should have with respect to the metric being determined a priori. My conclusion is that kantian analysis can incorporate the idea of a spatio-temporal metric that is not given, in the sense of universally and necessarily, a priori. (shrink)

One of the reasons for Carnap’s (1922) analysis of space was the confounding status of many arguments around the state of the art on that topic at that time. The unsatisfactory views supplied by mathematicians, physicists and philosophers led Carnap to propose a new conception of space. His proposal, which employs the notion of intuition as a fundamental tool, fared better, but clashed with his conventionalists intentions derived from an allegedly tolerant attitude. The notion of intuition (...) here examined allows us to claim that Carnap’s target was also an objectivist epistemological program dependent on certain realistic reminiscences embodied in the notion of evidence or matter of fact (Tatbestand). (shrink)

Les algèbres de dimensions supérieures libèrent les mathématiques de la restriction d'une notation purement linéaire. Elles aident ainsi à la modélisation de la géométrie et procurent une meilleure compréhension et plus de possibilités pour les calculs. Elles nous donnent de nouveaux outils pour l'étude de problèmes non-commutatifs, de dimension supérieure qui assurent le passage du local au global, en utilisant la notion d' «inverse algébrique de subdivision». Nous allons exposer comment ces idées sont venues aux auteurs en prolongeant initialement la (...) notion classique de groupe abstrait à celle de groupoïde abstrait, dont la composition n'est que partiellement définie, et qui ajoute une composante spatiale à la théorie habituelle des groupes. La théorie des noeuds nous fournit un exemple en indiquant comment une telle algèbre peut être utilisée pour décrire la structure d'un espace. Le prolongement à la dimension 2 utilise des compositions de carrés dans deux directions et la richesse de l'algèbre qui en résulte est montrée par certains calculs de dimension 2. La difficulté de la transition de la dimension 1 à la dimension 2 est également illustrée par la comparaison de la notion de carré commutatif à celle de cube commutatif - le traitement de cette dernière nécessitant de nouvelles notions. L'importance de la théorie des catégories est expliquée, de même que les possibilités de l'application d'algèbres de dimensions supérieures. (shrink)

This paper examines Berkeley's attitude toward our perception of spatial relations on the two- dimensional visual field. This is a topic on which there has been some controversy. Historians of visual theory have tended to suppose that Berkeley took "all" spatial relations to be derived in the way our knowledge of depth is: from association of more primitive sensations which are themselves in no way spatial. But many philosophers commenting on Berkeley have supposed that he takes our awareness of two- (...) dimensional visual layout to be directly perceived. I argue that Berkeley's position on this issue is deeply ambivalent and that neither alternative sits well with the rest of his philosophy. (shrink)

This paper engages with Kant‘s account of space as a continuum. The stage is set by looking at how the question of spatial continuity comes up in a debate from the 1920s between Ernst Cassirer and logical empiricist thinkers about Kant‘s conception of spatial representation as a pure intuition. While granting that concrete features of space can only be known empirically, Cassirer attempted to save Kant‘s conception by restricting it to the core commitment of space as (...) a continuous coexistent manifold. Cassirer did not however come up with a transcendental argument for spatial continuity. The paper presents such an argument by providing a reading of Kant‘s from which it transpires that Kant does not simply rely on supposed into the continuity of space. It is by way ofinstead that we can know space to be continuous and Kant’s distinction between intuitions and concepts does hinge on such knowledge. (shrink)

Defences of Kant's foundations of geometry fall short if they are unable to equally ground Euclidean and non-Euclidean geometries. Thus, Kant's account must be separated from geometrical postulates. I argue that characterizing space as the form of outer intuition must be independent of postulates. Geometrical postulates are then expressions of particular spatializing activities made possible by the a priori intuition of space. While Amit Hagar contends that this is to speak of noumena, I argue that a (...) Kantian account of space as the form of outer attention-directing remains seated in the subject. (shrink)

In the Transcendental Aesthetic of the Critique of Pure Reason, Immanuel Kant famously claims that we have an a priori intuition of space as an ‘infinite given magnitude’. Later on, in the Transcendental Analytic, he seems to add that the intuition of space presupposes a synthetic activity of the transcendental imagination. Several authors have recently pointed out that these two claims taken together give rise to two problems. First, it is unclear how the transcendental imagination of (...) a finite mind could ever result in the intuition of an entity that is infinitely large. Second, Kant claims that our intuition of space has a ‘whole-prior-to-its-parts’ structure, such that its parts are given only as limitations of the whole, while synthesis is compositional and has a ‘parts-prior-to-their-whole’ structure, because it consists in first running through and then taking together the parts of a sensible manifold. I will solve these two problems by showing that Kant thought that synthesis does not always have a compositional structure but that there is also a form of ‘decomposing’ synthesis, which has a whole-prior-to-its-parts structure. Building on similarities between Kant, Edmund Husserl and G. W. F. Hegel, I will argue that infinite space is given to us in intuition by precisely such an activity of decomposition, one that allows us to differentiate between finite spatial objects and the unlimited phenomenal horizon in which they appear. (shrink)

Des branches entières des mathématiques sont fondées sur des liens posés entre les nombres et l’espace : mesure de longueurs, définition de repères et de coordonnées, projection des nombres complexes sur le plan… Si les nombres complexes, comme l’utilisation de repères, sont apparus relativement récemment (vers le XVIIe siècle), la mesure des longueurs est en revanche un procédé très ancien, qui remonte au moins au 3e ou 4e millénaire av. J-C. Loin d’être fortuits, ces liens entre les nombres et l’espace (...) reflèteraient une intuition fondamentale, universelle, façonnée au cours des millénaires par la sélection naturelle, et qui aurait servi de guide et d’inspiration aux mathématiciens au fil des siècles. (shrink)

The paper focuses on some puzzles about Carnap's intended epistemological point in the "completion" and "generalization" of the Anschauungsraum in sec. II of Der Raum (leaving aside the technical problems which also arise). Since any global structure at all requires that eidetic intuition be supplemented with freely-chosen postulates and/or intuitively unmotivated generalizations, it is unclear, as several authors have pointed out, how and in what sense "intuitive space" as a whole represents a distinctive, a priori contribution to our (...) knowledge. I suggest a way of approaching this issue based on Carnap's sources -- in particular, Husserl and Driesch, both of whom he repeatedly claims to be following. The idea of a severely finite realm of possible intuition, which both requires and allows supplementation with an infinite conceptual structure, is central to Husserl's thought, and, I argue, it would be natural for Carnap to rely on it in attempting to reconcile Husserlian eidetic intuition with the general theory of relativity. That this larger conceptual structure owes its details to free postulation is, on the other hand, decidedly un-Husserlian. But here, I claim, Carnap takes his cue from Driesch's view, in the first edition of the Ordnungslehre, that natural actuality is the result of a certain demand for order: we demand such order in natural things and reject as non-actual (hallucinatory, dreamed, etc.) whatever nature-like elements of experience fail to fit into it. In conclusion I suggest that more radical versions of this particular constellation of ideas provide the key to understanding much of Carnap's later thought. (shrink)

Recently, there has been much research into conceptual engineering in connection with feminist inquiry and activism, most notably involving gender issues, but also sexism and misogyny. Our paper contributes to this research by explicating, in a principled manner, a series of other concepts important for feminist research and activism, to wit, feminist political identity terms. More specifically, we show how the popular Conceptual Spaces Framework (CSF) can be used to identify and regiment concepts that are central to feminist research, focusing (...) especially on feminism in France. According to the CSF, concepts can be represented geometrically, as regions in similarity spaces. A particular strength of the CSF framework is its empirically-focused methodology, which allows researchers to infer the boundaries of concepts from empirical data, thus eliminating the need to strongly rely on intuitions about meanings. This is shown to be especially valuable for the explication of concepts relating to feminist political identity, given that the intuitions of feminist scholars and activists about what would appear to be core concepts in the area tend to be poorly aligned or even conflicting. We report the results from an empirical categorization study conducted among French feminists and show how they support the view that the CSF can contribute to both the conceptual engineering project and our understanding of the structure of social reality. (shrink)

Elsewhere I have argued that standard theories of linguistic competence are committed to taking seriously talk of “representations of” standard linguistic entities (“SLEs”), such as NPs, VPs, morphemes, phonemes, syntactic and phonetic features. However, it is very doubtful there are tokens of these “things” in space and time. Moreover, even if were, their existence would be completely inessential to the needs of either communication or serious linguistic theory. Their existence is an illusion: an extremely stable perceptual state we regularly (...) enter as a result of being stimulated by the wave forms we regularly produce when we execute our intentions to utter such tokens (a view I call “Folieism”). In his Ignorance of Language, Michael Devitt objects to this view, arguing that, “On Rey’s view, communication seems to rest on miraculous guesses.” I argue here that my view is not prey to his objections, and actually affords a scientifically more plausible view than his “empiricist” alternative. Specifically, I reply to his objections that my view couldn’t explain the conventionality of language and success of communication (§2.1), that I am faced with intractable difficulties surrounding the identity of intentional inexistents (§2.2), and that, contrary to my view, SLEs can be relationally defined (§2.3). Not only can Folieism survive Devitt’s objections, but (§3) it also provides a more satisfactory account of the role of linguistic intuitions than the “empirical” account on which he insists. (shrink)

This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. (...) However, such later scientific developments as non-Euclidean geometries and Einstein’s general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz’s epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen’s account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer’s reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism. (shrink)

Our aim in this paper is to take quite seriously Heinz Post’s claim that the non-individuality and the indiscernibility of quantum objects should be introduced right at the start, and not made a posteriori by introducing symmetry conditions. Using a different mathematical framework, namely, quasi-set theory, we avoid working within a label-tensor-product-vector-space-formalism, to use Redhead and Teller’s words, and get a more intuitive way of dealing with the formalism of quantum mechanics, although the underlying logic should be modified. We (...) build a vector space with inner product, the Q-space, using the non-classical part of quasi-set theory, to deal with indistinguishable elements. Vectors in Q-space refer only to occupation numbers and permutation operators act as the identity operator on them, reflecting in the formalism the fact of unobservability of permutations. Thus, this paper can be regarded as a tentative to follow and enlarge Heinsenberg’s suggestion that new phenomena require the formation of a new “closed” (that is, axiomatic) theory, coping also with the physical theory’s underlying logic and mathematics. (shrink)

The essay addresses the well‐known idea that there has to be a place for intuition, thought of as a kind of non‐inferential rational insight, in the epistemology of basic logic if our knowledge of its principles is non‐empirical and is to allow of any finite, non‐circular reconstruction. It is argued that the error in this idea consists in its overlooking the possibility that there is, properly speaking, no knowledge of the validity of principles of basic logic. When certain important (...) distinctions are observed, for instance that between recognising that Modus Ponenes is sound and recognising that it is proof against the competent discovery of basic counterexamples, the case for thinking that there is indeed no space for genuine recognition of the validity of Modus Ponens becomes increasingly impressive. It is argued however that, the impossibility of knowledge notwithstanding, we are, in an important sense, entitled to take it that Modus Ponens is sound and that this notion of entitlement can help break the trichotomy ‐ intuition, inference, experience ‐ which imprisons our ordinary thinking about logical knowledge. (shrink)

This chapter will consider three themes relating to the significance of intuitions in contemporary philosophy. In §1, I’ll review and explore the relationship between philosophical use of words like ‘intuitively’ and any kinds of mental states that might be called ‘intuitions’. In §2, I’ll consider the widely-discussed analogy between intuitive experience and perceptual experience, drawing out some interesting similarities and differences. Finally, in §3, I’ll introduce the recent movement of ‘experimental philosophy’, and consider to what extent its projects are tied (...) up with questions about the role of intuitions in philosophy. My primary aim will be to survey and explain; I will make no effort, however, to hide the fact that I’m a philosopher and a partisan to some of these debates. So there will be some arguments for controversial points of view, too—hopefully these will be easily perceptible, and my opponents fairly represented. Limitations of space demand a rather superficial treatment of most of our topics; endnotes will direct the reader to more comprehensive discussions. (shrink)

In this paper, I intend to show that it’s a serious mistake to construe the role of sensible representation in Kant’s work as a nonconceptual content (in the contemporary and technical sense of “content”), which, like a mental indexical would refer to what appears in space and time in the so-called de re form. The interpretation I advance and further support is this: without possessing a representational content, sensible representation must be understood as the basic epistemic relation between the (...) subject and the physical entities in space and time she is acquainted with. Immediate reference of sensible intuition to their respective objects is to be understood as a form of knowledge by acquaintance. I call my own interpretation relationalism. (shrink)

We define qualia space Q to be the space of all possible conscious experience. For simplicity we restrict ourselves to perceptual experience only, though other kinds of experience could also be considered. Qualia space is a highly idealized concept that unifies the perceptual experience of all possible brains. We argue that Q is a closed pointed cone in an infinite-dimensional separable real topological vector space. This quite technical structure can be explained for the most part in (...) a simple, intuitive way. The structure of qualia space allows us to consider and even answer in a precise way such questions as: Is there a continuous path from the sensation of blue to the sensation of pain? Once we fix a desired accuracy of approximation, do there exist finitely many perceptual experiences such that any possible perceptual experience is approximately equal to one of them? What should be meant by ‘fundamentally different’ perceptual experiences? There is the possibility of additional structure, such as a Hilbert space structure on the vector space in which Q is embedded. (shrink)

Gaining information can be modelled as a narrowing of epistemic space . Intuitively, becoming informed that such-and-such is the case rules out certain scenarios or would-be possibilities. Chalmers’s account of epistemic space treats it as a space of a priori possibility and so has trouble in dealing with the information which we intuitively feel can be gained from logical inference. I propose a more inclusive notion of epistemic space, based on Priest’s notion of open worlds yet (...) which contains only those epistemic scenarios which are not obviously impossible. Whether something is obvious is not always a determinate matter and so the resulting picture is of an epistemic space with fuzzy boundaries. (shrink)

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of (...) space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic (...) numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education. (shrink)

In this paper, we will examine the role that intuitions and responses to thought experiments play in confirming or disconfirming theories of reference, using insights from both debates as our starting point. Our view is that experimental evidence of the type elicited by MMNS does play a central role in the construction of theories of reference. This, however, is not because such theory construction is accurately characterized by "the method of cases." First, experimental philosophy does not directly collect data about (...) intuitions, but rather about peoples responses to thought experiments, which may reflect their intuitions but may well not. Second, unusually in the case of the theory of reference, experimental prompts involve elicitation of the phenomenon under investigation—that is, referring—and it is the reference facts rather than the intuitions that a theory of reference should capture. Finally, "best fit" models like the method of cases are inconsistent with the actual practice of semantic theorists, who appeal to general principles of theory construction as well as considerations native to the theory of reference, often in the service of rejecting intuitions. Indeed, Kripke himself viewed several claims of his theory as unintuitive, but felt no pressure to alter them on that account. These facts, we'll argue, suggest that the relevance of experimental methods in the theory of reference cannot be straightforwardly extended to other fields: an experimental prompt containing Newtons two rocks tied together in empty space may elicit referring, and may also elicit intuitions, but it doesn't also elicit the phenomenon under investigation. Furthermore, the considerations native to the theory of reference are not part of physics, which has its own distinct set of principles and considerations. This suggests that an evaluation of the centrality of intuitions/ responses to thought experiments in philosophical methodology must proceed in a piecemeal fashion. We conclude that the potential contributions of experimental philosophy will take different forms in different sub-fields of philosophical inquiry. (shrink)

This article is devoted to the problem of ontological foundations of three-dimensional Euclidean geometry. Starting from Bertrand Russell’s intuitions concerning the sensual world we try to show that it is possible to build a foundation for pure geometry by means of the so called regions of space. It is not our intention to present mathematically developed theory, but rather demonstrate basic assumptions, tools and techniques that are used in construction of systems of point-free geometry and topology by means of (...) mereology and Whitehead-like connection structures. We list and briefly analyze axioms for mereological structures, as well as those for connection structures. We argue that mereology is a good tool to model so called spatial relations. We also try to justify our choice of axioms for connection relation. Finally, we briefly discuss two theories: Grzegorczyk’s point-free topology and Tarski’s geometry of solids. (shrink)

Thomas C. Vinci argues that Kant's Deductions demonstrate Kant's idealist doctrines and have the structure of an inference to the best explanation for correlated domains. With the Deduction of the Categories the correlated domains are intellectual conditions and non-geometrical laws of the empirical world. With the Deduction of the Concepts of Space, the correlated domains are the geometry of pure objects of intuition and the geometry of empirical objects.

Kant says patently conflicting things about infinity and our grasp of it. Infinite space is a good case in point. In his solution to the First Antinomy, he denies that we can grasp the spatial universe as infinite, and therefore that this universe can be infinite; while in the Aesthetic he says just the opposite: ‘Space is represented as a given infinite magnitude’. And he rests these upon consistently opposite grounds. In the Antinomy we are told that we (...) can have no intuitive grasp of an infinite space, and in the Aesthetic he says that our grasp of infinite space is precisely intuitive. (shrink)

Intuition. Originally an alleged direct relation, analogous to visual seeing, between the mind and something abstract and so not accessible to the senses. What are intuited (which can be derivatively called 'intuitions') may be abstract objects, like numbers or properties, or certain truths regarded as not accessible to investigation through the senses or calculation; the mere short circuiting of such processes in 'bank managers intuition' would not count as intuition for philosophy. Kant talks of our intuiting (...) class='Hi'>space and time, in a way which is direct and entirely free from any mediation by the intellect - but this must be distinguished from an alleged pure reception of 'raw data' from the senses; the intuiting is presupposed by, and so cannot depend upon, sensory experience. (shrink)

I examine how Kant argues for the transcendental ideality of space. I defend a reading on which Kant accepts the ideality of space because it explains our (actual) knowledge that mathematical judgments are necessarily true. I argue that this reading is preferable over the alternative suggestion that Kant can infer the ideality of space directly from the fact that we have an a priori intuition of space. Moreover, I argue that the reading I propose does (...) not commit Kant to incoherent modal views. If we carefully distinguish between different senses of modality, the fact that our spatial form of intuition is (in some sense) contingent does not undermine the claim that this form can explain how our mathematical judgments are (in some sense) necessary. (shrink)

Kant says patently conflicting things about infinity and our grasp of it. Infinite space is a good case in point. In his solution to the First Antinomy, he denies that we can grasp the spatial universe as infinite, and therefore that this universe can be infinite; while in the Aesthetic he says just the opposite: ‘Space is represented as a given infinite magnitude’ (A25/B39). And he rests these upon consistently opposite grounds. In the Antinomy we are told that (...) we can have no intuitive grasp of an infinite space, and in the Aesthetic he says that our grasp of infinite space is precisely intuitive. (shrink)

In the Critique of Pure Reason Kant appears to make incompatible claims regarding the unitary natures of what he takes to be our a priori representations of space and time. I argue that these representations are unitary independently of all synthesis and explain how this avoids problems encountered by other positions regarding the Transcendental Deduction and its relation to the Transcendental Aesthetic in that work. Central is the claim that these representations (1) contain, when characterized as intuitions and considered (...) as prior to any affections of sensibility, only an infinitude of merely possible finite spatial and temporal representations, and (2) are representations that are merely transcendental grounds for the possibilities for receiving or generating finite representations in sensibility that are determined (immediately, in the case of reception) by means of syntheses that accord with the categories. (shrink)

A natural way to think about epistemic possibility is as follows. When it is epistemically possible (for a subject) that p, there is an epistemically possible scenario (for that subject) in which p. The epistemic scenarios together constitute epistemic space. It is surprisingly difficult to make the intuitive picture precise. What sort of possibilities are we dealing with here? In particular, what is a scenario? And what is the relationship between scenarios and items of knowledge and belief? This chapter (...) tries to make sense of epistemic space. It explores different ways of making sense of scenarios and of their relationship to thought and language. It discusses some issues that arise and outlines some applications to the analysis of the content of thought and the meaning of language. (shrink)

Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea (...) of mathematics as the science of structure. The article closes with some remarks on structuralist and nonstructuralist themes in Frege's own development of arithmetic. (shrink)

The Archimedean Axiom is often held to be an intuitively obvious truth about the geometry of physical space. After a general discussion of the varieties of geometrical intuition that have been proposed, I single out one variety which we can plausibly be held to have and then argue that it does not underwrite the intuitive obviousness of the Archimedean Axiom. Generalizing that result, I conclude that the Axiom is not intuitively obvious.

Se pretende reconstruir los argumentos de la exposición metafísica de la Crítica de la razón pura, según los cuales nuestras representaciones de espacio y tiempo deben ser consideradas como intuiciones y no como conceptos. Para cumplir con este objetivo, en primer lugar, se determina un criterio de la intuición y, en segundo lugar, se demuestra que las representaciones de espacio y de tiempo cumplen con ese criterio. The paper reconstructs the arguments of the metaphysical presentation of the Critique of Pure (...) Reason, according to which our representations of space and time should be considered as intuitions and not as concepts. To this effect, a criterion for intuition is established in the first place, and then it is demonstrated that the representations of space and time do in fact fulfill that criterion. (shrink)

The ArgumentThis article compares the corresponding effects in science and art of a change in the intuition of time at the beginning of this century. McTaggart's distinction between linear time and tense time is applied to the question of whether linear perspective requires a notion of time as succession. It is argued that the problem of self-representation is a basic problem for this kind of uniform space-time because of the contradiction between this model's need for a privileged point (...) of view and its simultaneous denial of such a possibility. (shrink)

ABSTRACT Because he claims that intuition can give us insight into “life” apart from space, if not time, Bergson seems to transgress Kant's limits on possible knowledge. This article argues, however, that there is a “critical” way to read Bergson that allows him to evade accusations of epistemologically and politically problematic vitalism. When compared with the archaeological and genealogical treatment of “life” found in Michel Foucault's work, Bergson's effort to save humans from the destructive effects of intellect converges (...) in interesting ways with Foucault's displacement of life as a foundational philosophical or scientific term. (shrink)

Are the pure intuitions of space and time, for Kant, dependent upon the understanding's activity? This paper defends the recently popular Self-Affection Thesis : namely, that the pure intuitions require an activity of self-affection—an influence of the understanding on the inner sense. Two systematic objections to this thesis have been raised: The Independence objection claims that SAT undermines the independence of sensibility; the Compatibility objection claims that certain features of space and time are incompatible with being the products (...) of the understanding's activity. I show that the resources to rebut these objections can be found in Kant's account of causal influence. (shrink)

In a possible world framework, an agent can be said to know a proposition just in case the proposition is true at all worlds that are epistemically possible for the agent. Roughly, a world is epistemically possible for an agent just in case the world is not ruled out by anything the agent knows. If a proposition is true at some epistemically possible world for an agent, the proposition is epistemically possible for the agent. If a proposition is true at (...) all epistemically possible worlds for an agent, the proposition is epistemically necessary for the agent, and as such, the agent knows the proposition. -/- This framework presupposes an underlying space of worlds that we can call epistemic space. Traditionally, worlds in epistemic space are identified with possible worlds, where possible worlds are the kinds of entities that at least verify all logical truths. If so, given that epistemic space consists solely of possible worlds, it follows that any world that may remain epistemically possible for an agent verifies all logical truths. As a result, all logical truths are epistemically necessary for any agent, and the corresponding framework only allows us to model logically omniscient agents. This is a well-known consequence of the standard possible world framework, and it is generally taken to imply that the framework cannot be used to model non-ideal agents that fall short of logical omniscience. -/- A familiar attempt to model non-ideal agents within a broadly world involving framework centers around the use of impossible worlds where the truths of logic can be false. As we shall see, if we admit impossible worlds where “anything goes” in epistemic space, it is easy to avoid logical omniscience. If any logical falsehood is true at some impossible world, then any logical falsehood may remain epistemically possible for some agent. As a result, we can use an impossible world involving framework to model extremely non-ideal agents that do not know any logical truths. -/- A much harder, and considerably less investigated challenge is to ensure that the resulting epistemic space can also be used to model moderately ideal agents that are not logically omniscient but nevertheless logically competent. Intuitively, while such agents may fail to rule out impossible worlds that verify complex logical falsehoods, they are nevertheless able to rule out impossible worlds that verify obvious logical falsehoods. To model such agents, we need a construction of a non-trivial epistemic space that partly consists of impossible worlds where not "anything goes". This involves imposing substantive constraints on impossible worlds to eliminate from epistemic space, say, trivially impossible worlds that verify obvious logical falsehoods. -/- The central aim of this dissertation is to investigate the nature of such non-trivially impossible worlds and the corresponding epistemic spaces. To flag my conclusions, I argue that successful constructions of epistemic spaces that can safely navigate between the Charybdis of logical omniscience and the Scylla of of “anything goes” are hard, if not impossible to find. (shrink)

Conceptualism, like any other philosophical doctrine of comparable scope, has both ontological and epistemological aspects. Ontologically, however, conceptualism does not differ significantly from certain forms of nominalism. 1 At its root lies an epistemological thesis: All objects of sensory intuition are localized in space and time. 2 In this paper, I wish to explore some of the consequences of this thesis.

It's well-known that Kant believed that intuition was central to an account of mathematical knowledge. What that role is and how Kant argues for it are, however, still open to debate. There are, broadly speaking, two tendencies in interpreting Kant's account of intuition in mathematics, each emphasizing different aspects of Kant's general doctrine of intuition. On one view, most recently put forward by Michael Friedman, this central role for intuition is a direct result of the limitations (...) of the syllogistic logic available to Kant. On this view, Kant's reasons for introducing intuition are taken to be logical or mathematical, rather than philosophical. The other tendency, which I shall try to develop here, emphasizes an epistemological or phenomenological role for intuition in mathematics arising out of what may loosely be called Kant's ‘antiformalism.’This paper, which focuses specifically on the case of geometry, falls into two parts. First, I consider Kant's discussion of intuition in the Metaphysical Exposition of the concept of space. (shrink)