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)

When we ask ourselves about the meaning of life, two analyses are possible in principle: 1. that we are asking something about the purpose or the reason of being of life or of a life, or 2. that we are asking something the value of life or of a life. At the present article, I do not approach 1 neither the life as a whole, but I take the individual lives in the context of 2. I briefly explain what would (...) be to someone to have an objectivist conception of the meaning of life and what would be to someone to have a subjectivist one; and, then, I try to show how we can think in formulas that can capture the intuitions of those conceptions, without acquiring commitment with none of them. My way is to show that both objectivist and subjectivist conceptions think the meaning of life as related to projects or realizations that fulfill principles of value (the value can be objective or subjective according to the adopted conception), and ten I try to indicate that the projects or realizations from a person’s life come together through a simple sum to make the meaning of life of such a person’s life. The way I suggested for us to find the value of one of those realizations does not seem to present none theoretical problem. But the simple sum of realizations (and/or projects) to represent the value of the meaning of a person’s life simply did not work. That’s because it seems to lack some kind of index that is able to formalize the relations between realizations. Thus, although we couldn’t properly formalize the relation between the realizations that achieve the meaning of someone’s life at a certain moment, we get a good intuitive formalization of the value of a realization and we hope to have started an interesting investigation for future logicians, metaphysicians, mathematicians and theorists of value. (shrink)

The authors examined cultural preferences for formal versus intuitive reasoning among East Asian (Chinese and Korean), Asian American, and European American university students. We investigated categorization (Studies 1 and 2), conceptual structure (Study 3), and deductive reasoning (Studies 3 and 4). In each study a cognitive conflict was activated between formal and intuitive strategies of reasoning. European Americans, more than Chinese and Koreans, set aside intuition in favor of formal reasoning. Conversely, Chinese and Koreans relied on (...) intuitive strategies more than European Americans. Asian Americans' reasoning was either identical to that of European Americans, or intermediate. Differences emerged against a background of similar reasoning tendencies across cultures in the absence of conflict between formal and intuitive strategies. (shrink)

Linguists often sharply distinguish the different modules that support linguistics competence, e.g., syntax, semantics, pragmatics. However, recent work has identified phenomena in syntax (polarity sensitivity) and pragmatics (implicatures), which seem to rely on semantic properties (monotonicity). We propose to investigate these phenomena and their connections as a window into the modularity of our linguistic knowledge. We conducted a series of experiments to gather the relevant syntactic, semantic and pragmatic judgments within a single paradigm. The comparison between these quantitative data leads (...) us to four main results, (i) Our results support a departure from one element of the classical Gricean approach, thus helping to clarify and settle an empirical debate. This first outcome also confirms the soundness of the methodology, as the results align with standard contemporary accounts of scalar implicature (SI), (ii) We confirm that the formal semantic notion of monotonicity underlies negative polarity item (NPI) syntactic acceptability, but (iii) our results indicate that the notion needed is perceived monotonicity. We see results (ii) and (iii) as the main contribution of this study: (ii) provides an empirical interpretation and confirmation of one of the insights of the model-theoretic approach to semantics, while (iii) calls for an incremental, cognitive implementation of the current generalizations, (iv) Finally, our results do not indicate that the relationship between NPI acceptability and monotonicity is mediated by pragmatic features related to Sis: this tells against elegant attempts to unify polarity sensitivity and Sis (pioneered by Krifka and Chierchia). These results illustrate a new methodology for integrating theoretically rigorous work in formal semantics with an experimentally-grounded cognitively-oriented view of linguistic competence. (shrink)

In a recent article I detailed at length the methodology employed to explore the reflective and pre-reflective contents of singular intuitive experiences in contemporary mathematics in order to propose an essential structure of intuition arousal in mathematics. In this paper I present the phenomenological assessment of the essential structure according to the three formal structures as proposed by Sokolowski's scheme and show their relevance in the description of the intuitive experience in mathematics. I also show that this essential (...) structure acknowledges the perceptualist view of intuition, as proposed by Chudnoff, and discuss the analogy that is often proposed and argued between mathematical intuition and ordinary perceptive intuition. (shrink)

According to Kant, the axioms of intuition, i.e. space and time, must provide an organization of the sensory experience. However, this first orderliness of empirical sensations seems to depend on a kind of faculty pertaining to subjectivity, rather than to the encounter of these same intuitions with the real properties of phenomena. Starting from an analysis of some very significant developments in mathematical and theoretical physics in the last decades, in which intuition played an important role, we argue (...) that nevertheless intuition comes into play in a fundamentally different way to that which Kant had foreseen: in the form of a formal or “categorical” yet not sensible intuition. We show further that the statement that our space is mathematically three-dimensional and locally Euclidean by no means follows from a supposed a priori nature of the sensible or subjective space as Kant claimed. In fact, the three-dimensional space can bear many different geometrical and topological structures, as particularly the mathematical results of Milnor, Smale, Thurston and Donaldson demonstrated. On the other hand, it has been stressed that even the phenomenological or perceptual space, and especially the visual system, carries a very rich geometrical organization whose structure is essentially non-Euclidean. Finally, we argue that in order to grasp the meaning of abstract geometric objects, as n-dimensional spaces, connections on a manifold, fiber spaces, module spaces, knotted spaces and so forth, where sensible intuition is essentially lacking and where therefore another type of mathematical idealization intervenes, we need to develop a new form of intuition. (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)

The failure of current bootstrapping accounts to explain the emergence of the concept of natural numbers does not entail that no link exists between intuitive and formal number concepts. The epidemiology of representations allows us to explain similarities between intuitive and formal number concepts without requiring that the latter are directly constructed from the former.

Kant on Intuition: Western and Asian Perspectives on Transcendental Idealism consists of 20 chapters, many of which feature engagements between Kant and various Asian philosophers. Key themes include the nature of human intuition, the status of Kant's idealism/realism, and Kant's notion of an object. Roughly half of the chapters take a stance on the recent conceptualism/non-conceptualism debate. The chapters are organized into four parts, each with five chapters. Part I explores themes relating primarily to the early sections of (...) Kant's first Critique: three chapters focus mainly on Kant's theory of the "forms of intuition" and/or "formal intuition", especially as illustrated by geometry, while two examine the broader role of intuition in transcendental idealism. Part II continues to examine themes from the Aesthetic but shifts the main focus to the Transcendental Analytic, where the key question challenging interpreters is to determine whether intuition is ever capable of operating independently from conception ; each contributor offers a defense of either the conceptualist or the non-conceptualist readings of Kant's text. Part III includes three chapters that explore the relevance of intuition to Kant's theory of the sublime, followed by two that examine challenges that Asian philosophers have raised against Kant's theory of intuition, particularly as it relates to our experience of the supersensible. Finally, Part IV concludes the book with five chapters that explore a range of resonances between Kant and various Asian philosophers and philosophical ideas. (shrink)

This anthology consists of 20 chapters, many of which feature engagements between Kant and various Asian philosophers. Key themes include the nature of human intuition (not only as theoretical—pure, sensible, and possibly intellectual—but also as relevant to Kant’s practical philosophy, aesthetics, the sublime, and even mysticism), the status of Kant’s idealism/realism, and Kant’s notion of an object. Roughly half of the chapters take a stance on the recent conceptualism/non-conceptualism debate. The chapters are organized into four parts, each with five (...) chapters. Part I explores themes relating primarily to the early sections of Kant’s first Critique: three chapters focus mainly on Kant’s theory of the "forms of intuition" and/or "formal intuition", especially as illustrated by geometry, while two examine the broader role of intuition in transcendental idealism. Part II continues to examine themes from the Aesthetic but shifts the main focus to the Transcendental Analytic, where the key question challenging interpreters is to determine whether intuition (via sensibility) is ever capable of operating independently from conception (via understanding); each contributor offers a defense of either the conceptualist or the non-conceptualist readings of Kant’s text. Part III includes three chapters that explore the relevance of intuition to Kant’s theory of the sublime, followed by two that examine challenges that Asian philosophers have raised against Kant’s theory of intuition, particularly as it relates to our experience of the supersensible. Finally, Part IV concludes the book with five chapters that explore a range of resonances between Kant and various Asian philosophers and philosophical ideas. (shrink)

Building a meaningful model of biological regulatory network is usually done by specifying the components and their interactions, by guessing the values of parameters, by comparing the predicted behaviors to the observed ones, and by modifying in a trial-error process both architecture and parameters in order to reach an optimal fitness. We propose here a different approach to construct and analyze biological models avoiding the trial-error part, where structure and dynamics are represented as formal constraints. We apply the method (...) to Hopfield-like networks, a formalism often used in both neural and regulatory networks modeling. The aim is to characterize automatically the set of all models consistent with all the available knowledge. The available knowledge is formalized into formal constraints. The latter are compiled into Boolean formula in conjunctive normal form and then submitted to a Boolean satisfiability solver. This approach allows to formulate a wide range of queries, expressed in a high level language, and possibly integrating formalized intuitions. In order to explore its potential, we use it to find cycles for 3-nodes networks and to determine the flower morphogenesis regulatory network of Arabidopsis thaliana. Applications of this technique are numerous and concern the building of models from data as well as the design of biological networks possessing specified behaviors. (shrink)

We contrast Boyer's intuitive account of ownership with formal legal accounts based on institutions of ownership. Boyer's emphasis on social aspects of ownership intuitions may have a bearing on recent arguments that property institutions are justified by their capacity to promote human flourishing. Moreover, Boyer's account of property intuitions facilitates the study of acquisition and mental representation of formal ownership concepts.

We discuss various ways, which have been plainly justified in the secondhalf of the twentieth century, to introduce infinitesimals, and we considerthe new style of reasoning in mathematical analysis that they allow.

Humans have a tendency to reason teleologically. This tendency is more pronounced under time pressure, in people with little formal schooling and in patients with Alzheimer’s. This has led some cognitive scientists of religion, notably Kelemen, to call intuitive teleological reasoning promiscuous, by which they mean teleology is applied to domains where it is unwarranted. We examine these claims using Kant’s idea of the transcendental illusion in the first Critique and his views on the regulative function of teleological reasoning (...) in the third Critique. We examine whether a Kantian framework can help resolve the tension between the apparent promiscuity of intuitive teleology and its role in human reasoning about biological organisms and natural kinds. (shrink)

This book examines the evidential status and use of linguistic intuitions, a topic that has seen increased interest in recent years. Linguists use native speakers' intuitions - such as whether or not an utterance sounds acceptable - as evidence for theories about language, but this approach is not uncontroversial. The two parts of this volume draw on the most recent work in both philosophy and linguistics to explore the two major issues at the heart of the debate. Chapters in the (...) first part address the 'justification question', critically analysing and evaluating the theoretical rationale for the evidential use of linguistic intuitions. The second part discusses recent developments in the domain of experimental syntax, focusing on the question of whether formal and systematic models of gathering intuitions are epistemically and methodologically superior to the informal methods that have traditionally been used. -/- The volume provides valuable insights into whether and how linguistic intuitions can be used in theorizing about language, and will be of interest to graduate students and researchers in linguistics, philosophy, and cognitive science. (shrink)

Despite his unreserved appreciation of Turing’s analysis for being a “precise and unquestionably adequate definition” of formal system or mechanical computability, Gödel nevertheless published a short note in 1972 claiming to have found a “philosophical error” in Turing’s argument with regard to the finite nature of mental states and memory. A natural question arises: how could Gödel enjoy the generality conferred on his results by Turing’s work, despite the error of its ways? Previous interpretative strategies by Feferman, Shagrir and (...) others have mainly tried to resolve the disparity by distinguishing different types of arguments in Turing and taking Gödel to approve only some of them. By a more integral examination of their ideas, especially Turing’s response to the “mathematical objection” based on Gödel’s incompleteness theorem and Gödel’s own conception of finite yet non-mechanical procedures, and taking some of the main ideas of current developments in machine learning into consideration, I will try to present a new explanation for the apparent disparity, arguing that there is no “error” on Turing’s side and the seemingly conflicting views held by Turing and Gödel should best be seen as complementary, keeping intuition and ingenuity together. (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 (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) 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 (intuitive though) 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)

Various formally valid counterexamples have been adduced against the Humean dictum that one cannot derive an “ought” from an “is.” There are formal rebuttals—some very sophisticated now (e.g., Charles R. Pigden’s and Gerhard Schurz’s)—to such counterexamples. But what follows is an intuitive and informal argument against them. I maintain that it is better than these sophisticated formal defenses of the Humean dictum and that it also helps us see why it implausible to think that we can be as (...) decisive about the truth or falsity of the dictum as both the formal counterexamples or formal barriers to them purport to be. (shrink)

In general, intuition and demonstration are explained in terms of the class of relations of ideas that remain the same as long as the ideas remain the same. The main negative point is that our concept of a deductively valid argument, even one with necessarily true premises, has little to do with Hume's conception of demonstration. Following Descartes and Locke, the emphasis is on content and certainty, not necessity and formal validity. Two ideas are intuitively related if the (...) relation between them is immediately conceived. Two ideas are demonstratively related if the relation between them is conceived, not immediately, but via other intermediate ideas. The link between each pair of adjacent ideas in the resulting chain must be intuitive. With this background, some of Hume's more infamous claims are explained: we know in advance that two ideas cannot be demonstratively related if we can conceive one without the other; the causal maxim is neither intuitively nor demonstratively known; the only relation capable of sustaining demonstrations is that of quantity and number. (shrink)

This paper introduces new logical systems which axiomatize a formal representation of inconsistency (here taken to be equivalent to contradictoriness) in classical logic. We start from an intuitive semantical account of inconsistent data, fixing some basic requirements, and provide two distinct sound and complete axiomatics for such semantics, LFI1 and LFI2, as well as their first-order extensions, LFI1* and LFI2*, depending on which additional requirements are considered. These formal systems are examples of what we dub Logics of (...) class='Hi'>Formal Inconsistency (LFI) and form part of a much larger family of similar logics. We also show that there are translations from classical and paraconsistent first-order logics into LFI1* and LFI2*, and back. Hence, despite their status as subsystems of classical logic, LFI1* and LFI2* can codify any classical or paraconsistent reasoning. (shrink)

We discuss various ways, which have been plainly justified in the secondhalf of the twentieth century, to introduce infinitesimals, and we considerthe new style of reasoning in mathematical analysis that they allow.

As one commentator notes, Spinoza’s conception of “the third kind of knowledge”—intuition, has been “regarded as exceptionally obscure. Some writers regard it as a kind of mystic vision; others regard it as simply unintelligible.” For Spinoza, the first kind of knowledge, which he calls “imagination,” is a kind of sense-experience of particulars; the second kind, which he calls “understanding,” involves the rational grasp of universals, and the third, in his words, “proceeds from an adequate idea of the formal (...) essence of some of the attributes of God to an adequate knowledge of the essence of things.” In this essay I will attempt to show, through an explication of Spinoza’s concept of intuition, how a prime example of intuition can be found in the art of poetry. More specifically, I will examine resonances between the work of the poet Gerard Manley Hopkins (1844–1889) and Baruch de Spinoza (1632–1677) by exploring the ways in which Hopkins’ poetry can work as both (1) an exemplar of poetry qua Spinozistic intuition, and (2) an intuition-based access to Spinoza’s thought. The upshot of this essay, then, is that there is a kind of knowledge, and by implication a kind of education (through which to acquire that knowledge) which—even for a philosopher as mathematically rigorous as Spinoza—may require recourse to the art of poetry. (shrink)

The main purpose of this article is to offer an interpretation of the Kantian notion of formal intuition, which admits an analytical distinction from the concept of form of intuition. I hold that formal intuition is the result of the intersection between sensibility and understanding as mutually irreducible faculties. In this way, the necessary cooperation between faculties is emphasized. The radical heterogeneity between the sensible contribution and the intellectual contribution is also indicated. This interpretation allows (...) to take distance from strongly conceptualist interpretations and from non-conceptualist readings. (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 (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) 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 (intuitive though) 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)

The two volumes in this advanced textbook present results, proof methods, and translations of motivational and philosophical considerations to formal constructions. In this Vol. I the author explains preferential structures and abstract size. In the associated Vol. II he presents chapters on theory revision and sums, defeasible inheritance theory, interpolation, neighbourhood semantics and deontic logic, abstract independence, and various aspects of nonmonotonic and other logics. In both volumes the text contains many exercises and some solutions, and the author limits (...) the discussion of motivation and general context throughout, offering this only when it aids understanding of the formal material, in particular to illustrate the path from intuition to formalisation. Together these books are a suitable compendium for graduate students and researchers in the area of computer science and mathematical logic. (shrink)

To ascertain that a formalization of the intuitive notion of a ‘concept’ is linguistically interesting, one has to check whether it allows to get a grip on distinctions and notions from lexical semantics. Prime candidates are notions like ‘prototype’, ‘stereotypical attribute’, ‘essential attribute versus accidental attribute’, ‘intension versus extension’. We will argue that although the current paradigm of formal concept analysis as an application of lattice theory is not rich enough for an analysis of these notions, a lattice theoretical (...) approach to concepts is a suitable starting point for formalizing them. (shrink)

This volume is a collection of original contributions about the core knowledge in fundamental domains. It includes work on naive physics, such as formal specifications of intuitive theories of spatial relations, time causality, substance and physical objects, and on naive psychology.

Computation and formalization are not modalities of pure abstractive operations. The essay tries to revise the assumption of the constitutive nonsensuality of the formal. The argument is that formalization is a kind of linear spatialization, which has significant visual dimensions. Thus, a connection can be discovered between visualization by figurative graphism and formalization by symbolic calculations: Both use spatial relations not only to represent but also to operate on epistemic, nonspatial, nonvisual entities. Descartes was one of the pioneers of (...) using this kind of two-dimensional spatiality as a cognitive instrument. (shrink)

McDowell’s “Avoiding the Myth of the Given” (2008, 2009) attempts to reconcile two claims: 1) what we most fundamentally experience is a fundamental level of invariable simple objects and their sensible properties; experience of these objects and properties is the ultimate ground of our knowledge of the world; 2) experience is through-and-through conceptually structured. This leads McDowell to endorsing the incoherent notion of intuitional content – necessary and thus irrevisable basic empirical conceptually structured contents or empirical categories. The notion requires (...) the necessity and irrevisability of purely formal concepts and the on-going responsiveness to experience of empirical concepts. This reveals a dilemma: If the fundamental objects of experience are invariable then they cannot have empirically conceptual form; for revisability is a mark of the empirically conceptual. But purely formal concepts cannot give us the fundamental empirical structure of the world. The dilemma forces us to choose between claiming that the fundamental objects of experience are invariable and claiming that they are conceptually formed. I conclude by briefly describing how Kant successfully faces the challenge of accounting for the most fundamental invariable level of objects of experience without yielding to the myth of the given. (shrink)

We discuss various ways, which have been plainly justified in the second half of the twentieth century, to introduce infinitesimals, and we consider the new style of reasoning in mathematical analysis that they allow.

Criteria of acceptability for mathematical proofs are field-dependent. In topology, though not in most other domains, it is sometimes acceptable to appeal to visual intuition to support inferential steps. In previous work :829–842, 2014; Lolli, Panza, Venturi From logic to practice, Springer, Berlin, 2015; Larvor Mathematical cultures, Springer, Berlin, 2016) my co-author and I aimed at spelling out how topological proofs work on their own terms, without appealing to formal proofs which might be associated with them. In this (...) article, I address two criticisms that have been raised in Tatton-Brown against our approach: that it leads to a form of relativism according to which validity is equated with social agreement and that it implies an antiformalizability thesis according to which it is not the case that all rigorous mathematical proofs can be formalized. I reject both criticisms and suggest that our previous case studies provide insight into the plausibility of two related but quite different theses. (shrink)

This paper considers ways that experimental design can affect judgments about informally presented context shifting experiments. Reasons are given to think that judgments about informal context shifting experiments are affected by an exclusive reliance on binary truth value judgments and by experimenter bias. Exclusive reliance on binary truth value judgments may produce experimental artifacts by obscuring important differences of degree between the phenomena being investigated. Experimenter bias is an effect generated when, for example, experimenters disclose (even unconsciously) their own beliefs (...) about the outcome of an experiment. Eliminating experimenter bias from context shifting experiments makes it far less obvious what the “intuitive” responses to those experiments are. After it is shown how those different kinds of bias can affect judgments about informal context shifting experiments, those experiments are revised to control for those forms of bias. The upshot of these investigations is that participants in the contextualist debate who employ informal experiments should pay just as much attention to the design of their experiments as those who employ more formal experimental techniques if they want to avoid obscuring the phenomena they aim to uncover. (shrink)

The two volumes in this advanced textbook present results, proof methods, and translations of motivational and philosophical considerations to formal constructions. In the associated Vol. I the author explains preferential structures and abstract size. In this Vol. II he presents chapters on theory revision and sums, defeasible inheritance theory, interpolation, neighbourhood semantics and deontic logic, abstract independence, and various aspects of nonmonotonic and other logics. In both volumes the text contains many exercises and some solutions, and the author limits (...) the discussion of motivation and general context throughout, offering this only when it aids understanding of the formal material, in particular to illustrate the path from intuition to formalisation. Together these books are a suitable compendium for graduate students and researchers in the area of computer science and mathematical logic. (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)

Objectives: Medical decision support and other intelligent applications in the life sciences depend on increasing amounts of digital information. Knowledge bases as well as formal ontologies are being used to organize biomedical knowledge and data. However, these two kinds of artefacts are not always clearly distinguished. Whereas the popular RDF(S) standard provides an intuitive triple-based representation, it is semantically weak. Description logics based ontology languages like OWL-DL carry a clear-cut semantics, but they are computationally expensive, and they are often (...) misinterpreted to encode all kinds of statements, including those which are not ontological. Method: We distinguish four kinds of statements needed to comprehensively represent domain knowledge: universal statements, terminological statements, statements about particulars and contingent statements. We argue that the task of formal ontologies is solely to represent universal statements, while the non-ontological kinds of statements can nevertheless be connected with ontological representations. To illustrate these four types of representations, we use a running example from parasitology. Results: We finally formulate recommendations for semantically adequate ontologies that can efficiently be used as a stable framework for more context-dependent biomedical knowledge representation and reasoning applications like clinical decision support systems. (shrink)

We use formal semantic analysis based on new constructions to study abstract realizability, introduced by Läuchli in 1970, and expose its algebraic content. We claim realizability so conceived generates semantics-based intuitive confidence that the Heyting Calculus is an appropriate system of deduction for constructive reasoning.Well-known semantic formalisms have been defined by Kripke and Beth, but these have no formal concepts corresponding to constructions, and shed little intuitive light on the meanings of formulae. In particular, the completeness proofs for (...) these semantics do not generate confidence in the sufficiency of the Heyting Calculus, since we have no reason to believe that every intuitively constructive truth is valid in the formal semantics.Läuchli has proved completeness for a realizability semantics with formal concepts analogous to constructions. We argue in some detail that, in spite of a certain inherent inexactness of the analogy, every intuitively constructive truth is valid in Läuchli semantics, and therefore the Heyting Calculus is powerful enough to prove all constructive truths. Our argument is based on the postulate that a uniformly constructible object must be communicable in spite of imprecision in our language, and that the permutations in Läuchli's semantics represent conceivable imprecision in a language, while allowing a certain amount of freedom in choosing the particular structure of the language.We give a detailed generalization of Läuchli's proof of completeness for the propositional part of the Heyting Calculus, in order to make explicit constructive and algebraic content.In our treatment, we establish several new results about Läuchli models. We show how to extend the sconing and gluing constructions familiar from Kripke and Frame semantics and Topos theory, to Läuchli models, and use them to give an algebraic approach to countermodel construction. In particular, the Läuchli arguments are given without the restriction to the integers, Z, as a group of permutations, which makes much of the coding scheme used in Läuchli's original paper transparent.We also make use of a new propositions-as-types syntax for the Heyting calculus, with limited nondeterminism, in which validity of formulae can be decided without loop-detection. (shrink)

This paper seeks to determine the intuitive meaning of the concept of information by indicating its essential (definitional) features and relations with other concepts, such as that of knowledge. The term “information” – as with many other concepts, such as “process”, “force”, “energy” and “matter” – has a certain established meaning in natural languages, which allows it to be used, in science as well as in everyday life, without our possessing any somewhat stricter definition of it. The basic aim here (...) is thus to explicate what it amounts to in the context of its intuitive meaning as encountered in natural languages, what the subject of cognition implicitly presumes when using the term, and to which ontological situations it can be applied. I demonstrate that the essential features of the notion of information include the presence of a material medium, its transformation, the recording and reading of information encoded in the medium, and the grasp of what is recorded, coded and transmitted as an intentional object, where the latter is construed in terms broadly in line with the ontologies of Husserl and Ingarden. Along the way, a number of issues relating to the notion of information are also pointed out: the problem of informational identity, of the existence of virtual objects, and of the choice of an adequate information carrier, as well as formal-ontological problems, including those which concern relations between information carriers and intentional objects. (shrink)

Temporaryism—the view that not always everything always exists—comes in two main versions: presentism and expansionism (aka the growing block theory of time). Both versions of the view are commonly formulated using the notion of being present, which we, among others, find problematic. Expansionism is also sometimes accused of requiring extraordinary conceptual tools for its formulation. In this paper, we put forward systematic characterizations of presentism and expansionism which involve neither the notion of being present nor unfamiliar conceptual tools. These characterizations (...) are full-blown logics, each logic comprising an axiomatic proof system and an intuitive semantics with respect to which the system is both sound and complete. (shrink)

Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to (...) perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic or symbolic dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children and adults were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction. (shrink)

This dissertation constitutes an inquiry into the formal aspects of a particular form of conceptual intentional realism: Holistic Realist Ramified Conceptualism. Several axiomatic systems, which have this theory as their philosophical background are developed and/or studied from a syntactical and semantical point of view. ;Among systems studied are Cocchiarella's RRC$\sbsp{\lambda}{\*}$ and HRRC$\sbsp{\lambda}{\*}$. A set theoretical semantics for these systems is developed. Also, completeness theorems with respect to certain extensions of RRC$\sbsp{\lambda}{\*}$ and HRRC$\sbsp{\lambda}{\*}$ and certain notions of validity related to (...) that semantics are proved. A relative consistency proof of RRC$\sbsp{\lambda}{\*}$, HRRC$\sbsp{\lambda}{\*}$ and other systems is given. ;An axiomatic system for identity free realist ramified conceptualism involving indiscernibility relative to constructive concepts is developed. A set theoretical semantics for this system is constructed. A completeness theorem with respect to certain extensions of the identity-free system and a notion of validity related to that semantics is proved. ;Epistemic considerations are brought into this inquiry by introducing an epistemic operator into the logical syntax. Such an operator corresponds intuitively to the idea of constructive knowability. A philosophical discussion of this idea is presented and on the basis of this discussion an epistemic logical theory is formulated. It is shown that this system has HRRC$\sbsp{\lambda}{\*}$ as a proper subsystem. On the basis of the epistemic logical theory, an axiomatic system is introduced from which Peano's arithmetic is deduced. In this way, it is shown that arithmetic can be developed axiomatically on conceptualist foundations. (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)