I criticize, but uphold the conclusion of, an argument by McLarty to the effect that New Foundations style set theories don’t form a suitable foundation for category theory. McLarty’s argument is from the fact that Set and Cat are not Cartesian closed in NF-style set theories. I point out that these categories do still have a property approximating Cartesian closure, making McLarty’s argument not conclusive. After considering and attempting to address other problems with developing category theory (...) in NF-style set theories, I conclude that NF-style set theories are not a good foundation for category theory, because of numerous limitations introduced by their stratification restrictions. (shrink)

We introduce the notion of an alphabetic trace of a cut-free intuitionistic prepositional proof and show that it serves to characterize the equality of arrows in cartesian closed categories. We also show that alphabetic traces improve on the notion of the generality of proofs proposed in the literature. The main theorem of the paper yields a new and considerably simpler solution of the coherence problem for cartesian closed categories than those in [11, 14].

A construction is described of a cartesian closed category A with exactly two elements out of a C-monoid such that can be recovered from A without reference to the construction.

When Gödel developed his functional interpretation, also known as the Dialectica interpretation, his aim was to prove consistency of first order arithmetic by reducing it to a quantifier-free theory with finite types. Like other functional interpretations Gödel’s Dialectica interpretation gives rise to category theoretic constructions that serve both as new models for logic and semantics and as tools for analysing and understanding various aspects of the Dialectica interpretation itself. Gödel’s Dialectica interpretation gives rise to the Dialectica categories , in: (...) Contemp. Math., vol. 92, Amer. Math. Soc., Providence, RI, 1989, pp. 47–62] and J.M.E. Hyland in [J.M.E. Hyland, Proof theory in the abstract, Ann. Pure Appl. Logic 114 43–78, Commemorative Symposium Dedicated to Anne S. Troelstra ]). These categories are symmetric monoidal closed and have finite products and weak coproducts, but they are not Cartesian closed in general. We give an analysis of how to obtain weakly Cartesian closed and Cartesian closed Dialectica categories, and we also reflect on what the analysis might tell us about the Dialectica interpretation. (shrink)

Cartesian closed categories (CCCs) have played and continue to play an important role in the study of the semantics of programming languages. An axiomatization of the isomorphisms which hold in all Cartesian closed categories discovered independently by Soloviev and Bruce, Di Cosmo and Longo leads to seven equalities. We show that the unification problem for this theory is undecidable, thus settling an open question. We also show that an important subcase, namely unification modulo the (...) linear isomorphisms, is NP-complete. Furthermore, the problem of matching in CCCs is NP-complete when the subject term is irreducible. CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by Rittri. It also has potential applications to the problem of polymorphic type inference and polymorphic higher-order unification, which in turn is relevant to theorem proving and logic programming. (shrink)

Working in the fragment of Martin-Löfs extensional type theory [12] which has products (but not sums) of dependent types, we consider two additional assumptions: firstly, that there are (strong) equality types; and secondly, that there is a type which is universal in the sense that terms of that type name all types, up to isomorphism. For such a type theory, we give a version of Russell's paradox showing that each type possesses a closed term and (hence) that all terms (...) of each type are provably equal. We consider the kind of category theoretic structure which corresponds to this kind of type theory and obtain a categorical version of the paradox. A special case of this result is the degeneracy of a locally cartesian closed category with a morphism which is generic in the sense that every other morphism in the category can be obtained from it via pullback. (shrink)

For the most part, the papers collected in this volume stern from presentations given at a conference held in Tucson over the weekend of May 31 through June 2, 1985. We wish to record our gratitude to the participants in that conference, as well as to the National Science Foundation and the University of Arizona SBS Research Institute for their financial support. The advice we received from Susan Steele on organizational matters proved invaluable and had many felicitous consequences for the (...) success of the con­ ference. We also would like to thank the staff of the Departments of Linguistics of the University of Arizona and the University of Massachusetts at Amherst for their help, as weIl as a number of individuals, including Lin Hall, Kathy Todd, and Jiazhen Hu, Sandra Fulmer, Maria Sandoval, Natsuko Tsujimura, Stuart Davis, Mark Lewis, Robin Schafer, Shi Zhang, Olivia Oehrle-Steele, and Paul Saka. Finally, we would like to express our gratitude to Martin Scrivener, our editor, for his patience and his encouragement. Vll INTRODUCTION The term 'categorial grammar' was introduced by Bar-Rillel as a handy way of grouping together some of his own earlier work and the work of the Polish logicians and philosophers Lesniewski and Ajdukiewicz, in contrast to approaches to linguistic analysis based on phrase structure grammars. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Breakdown of C i M phy " artes an eta sacs RICHARD A. WATSON WITHIN CARTESIANISMthere arose many problems deriving from conflicts between Cartesian principles. Inadequate attempts to solve these problems were crucial reasons for the breakdown of Cartesian metaphysics in the late seventeenth and early eighteenth centuries. The major difficulties derived from the acceptance of a dualism of substances seated in a system which included (...) epistemological and causal likeness principles plus an ontological framework in which the categories of substance and modification were exhaustive. The major solutions involved either denying the likeness principles or altering the ontological framework. The first led to unintelligibility ; the second, culminating in Hume, opened the way to non-Cartesian metaphysics. The first section of this study contains a characterization of late seventeenth -century Cartesian metaphysics. The second is an exposition of Foucher 's four major criticisms of this system. In the third section, monistic solutions suggested in Descartes and by Spinoza, Foucher, Leibniz, and Locke are considered; in the fourth, the dualistic solutions of the orthodox Cartesians Rohault, R~gis, Desgabets, La Forge, Le Grand, and Arnauld; and in the fifth, the occasionalist solution of Malebranche. The sixth section is an analysis of these orthodox and occasionalist solutions, showing how each ultimately fails because of dependence upon the ontology of substance and modification. The seventh section is a consideration of Berkeley and Hume in the Cartesian context. A consideration of their systems as offering solutions to Cartesian problems illuminates in new light both the systems and the problems. In broad terms, this study moves from a consideration of problems arising from an ontology of two substances, to a consideration of problems arising from the ontological pattern of substance and modification. The Cartesians were concerned with relationships between two substances, but they saw no difficulties connected with the relationship between a substance and its modifications (or properties). Berkeley's treatment (deriving from Male- [177] 178 HISTORY OF PHILOSOPHY branche) of the relationships between mind and idea, and between mind and notion, is illuminated by a consideration of the extent to which he took these relationships for granted in the Cartesian way, and of the extent to which he saw difficulties in the relationship between substance and modification. The Berkeleian solution to Cartesian problems, like that of Malebranche before him, was not successful; Malebranche belittled, and Berkeley denied the dualism of substances, but neither philosopher could break entirely with the ontological pattern of substance and modification. The final breakdown of Cartesian metaphysics came with Hume, who, in providing the culmination of a most important trend in modern philosophy, opened the way to new metaphysics and became the father of contemporary philosophy. I A model late seventeenth-century Cartesian metaphysical system. The last grand expositor of the Cartesian philosophy was Antoine Le Grand who in 1694 published An Entire Body of Philosophy. The characterization of the late seventeenth-century Cartesian system given below follows Le Grand's exposition more closely than others, but it also incorporates elements from those of Rohault, R6gis, Desgabets, La Forge, Malebranche, and Arnauld.1 None of these philosophers professed a system of exactly the sort this characterization pictures, nor is it implied here that they do. The guide and rationale for drawing from them such a model Cartesianism is the polemical writing of Simon Foucher. Foucher's series of attacks upon Cartesianism (of which the second is his criticism of Malebranche of 1675, Critique de la Rdcherche de la veritd) give clearly the most important objections against Cartesian metaphysics.2The system outlined below, while more com1Le Grand, Antoine, An Entire Body of Philosophy According to the Principles of the Famous Renate des Cartes (London, 1694); Rohault, Jacques, Traitd de physique (Paris, 1671); R6gis, Pierre-Sylvain, Syst~me de philosophie, contenant la logique, la mdtaphysique, la physique et la morale (Lyon, 1690); Desgabets, Robert, Critique de la Critique de la Recherche de la veritd, oCt l'on decouvre le chemin qui conduit aux connoissances solides. Pour servir de rdsponse gtla Lettre d'un academicien (Paris, 1675); La Forge, Louis de, Traitd de l'dme humaine, de ses facultds et fonctions et de son union avec le corps, d'apr~s les principes de... (shrink)

We can regard operations that discard information, like specializing to a particular case or dropping the intermediate steps of a proof, as projections, and operations that reconstruct information as liftings. By working with several projections in parallel we can make sense of statements like “Set is the archetypal Cartesian Closed Category”, which means that proofs about CCCs can be done in the “archetypal language” and then lifted to proofs in the general setting. The method works even when our (...) archetypal language is diagrammatical, has potential ambiguities, is not completely formalized, and does not have semantics for all terms. We illustrate the method with an example from hyperdoctrines and another from synthetic differential geometry. (shrink)

A total of 1918 loci, detected by the hybridization of 938 expressed sequence tag unigenes from 26 Triticeae cDNA libraries, were mapped to wheat homoeologous group 4 chromosomes using a set of deletion, ditelosomic, and nulli-tetrasomic lines. The 1918 EST loci were not distributed uniformly among the three group 4 chromosomes; 41, 28, and 31% mapped to chromosomes 4A, 4B, and 4D, respectively. This pattern is in contrast to the cumulative results of EST mapping in all homoeologous groups, as reported (...) elsewhere, that found the highest proportion of loci mapped to the B genome. Sixty-five percent of these 1918 loci mapped to the long arms of homoeologous group 4 chromosomes, while 35% mapped to the short arms. The distal regions of chromosome arms showed higher numbers of loci than the proximal regions, with the exception of 4DL. This study confirmed the complex structure of chromosome 4A that contains two reciprocal translocations and two inversions, previously identified. An additional inversion in the centromeric region of 4A was revealed. A consensus map for homoeologous group 4 was developed from 119 ESTs unique to group 4. Forty-nine percent of these ESTs were found to be homoologous to sequences on rice chromosome 3, 12% had matches with sequences on other rice chromosomes, and 39% had no matches with rice sequences at all. Limited homology was found between wheat ESTs on homoeologous group 4 and the Arabidopsis genome. Forty-two percent of the homoeologous group 4 ESTs could be classified into functional categories on the basis of blastX searches against all protein databases. (shrink)

Non-well-founded trees are used in mathematics and computer science, for modelling non-well-founded sets, as well as non-terminating processes or infinite data structures. Categorically, they arise as final coalgebras for polynomial endofunctors, which we call M-types. We derive existence results for M-types in locally cartesian closed pretoposes with a natural numbers object, using their internal logic. These are then used to prove stability of such categories with M-types under various topos-theoretic constructions; namely, slicing, formation of coalgebras , and (...) sheaves for an internal site. (shrink)

Image factorizations in regular categories are stable under pullbacks, so they model a natural modal operator in dependent type theory. This unary type constructor [A] has turned up previously in a syntactic form as a way of erasing computational content, and formalizing a notion of proof irrelevance. Indeed, semantically, the notion of a support is sometimes used as surrogate proposition asserting inhabitation of an indexed family. We give rules for bracket types in dependent type theory and provide complete semantics (...) using regular categories. We show that dependent type theory with the unit type, strong extensional equality types, strong dependent sums, and bracket types is the internal type theory of regular categories, in the same way that the usual dependent type theory with dependent sums and products is the internal type theory of locally Cartesian closed categories. We also show how to interpret first-order logic in type theory with brackets, and we make use of the translation to compare type theory with logic. Specifically, we show that the propositions-as-types interpretation is complete with respect to a certain fragment of intuitionistic first-order logic, in the sense that a formula from the fragment is derivable in intuitionistic first-order logic if, and only if, its interpretation in dependent type theory is inhabited. As a consequence, a modified double-negation translation into type theory (without bracket types) is complete, in the same sense, for all of classical first-order logic. (shrink)

The [lambda]-calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of [lambda]-conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a ‘minimal’ topological model in which every continuous function is [lambda]-definable. These results subsume earlier ones using cartesian closed (...) class='Hi'>categories, as well as those employing so-called Henkin and Kripke [lambda]-models. (shrink)

We show that proofs in the intuitionistic propositional logic factor through interpolants-in this way we prove a stronger interpolation property than the usual one which gives only the existence of interpolants.Translating that to categorical terms, we show that Pushouts (bipushouts) of bicartesian closed categories have the interpolation property (Theorem 3.2).

In this book the authors reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher order logic, and cartesian closed categories are essentially the same. In Part II, it is demonstrated that another formulation of higher order logic is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional logic and (...) the algebraic language of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained. Detailed historical references are provided throughout, and each section concludes with a set of exercises. Thus it is well-suited for graduate courses and research in mathematics and logic. Researchers in theoretical computer science, artificial intelligence and mathematical linguistics will also find this an accessible introduction to a subject of increasing application to these disciplines. (shrink)

We present a general notion of realizability encompassing both standard Kleene style realizability over partial combinatory algebras and Kleene style realizability over more general structures, including all partial cartesian closed categories. We shown how the general notion of realizability can be used to get models of dependent predicate logic, thus obtaining as a corollary (the known result) that the category Equ of equilogical spaces models dependent predicate logic. Moreover, we characterize when the general notion of realizability gives (...) rise to a topos, i.e., a model of impredicative intuitionistic higher-order logic. (shrink)

In-principle restrictions on the amount of information that can be gathered about a system have been proposed as a foundational principle in several recent reconstructions of the formalism of quantum mechanics. However, it seems unclear precisely why one should be thus restricted. We investigate the notion of paradoxical self-reference as a possible origin of such epistemic horizons by means of a fixed-point theorem in Cartesian closed categories due to Lawvere that illuminates and unifies the different perspectives on (...) self-reference. (shrink)

Paradox is a logical phenomenon. Usually, it is produced in type theory, on a type Ω of “truth values”. A formula Ψ (i.e., a term of type Ω) is presented, such that Ψ↔¬Ψ (with negation as a term¬∶Ω→Ω)-whereupon everything can be proved: In Sect. 1 we describe a general pattern which many constructions of the formula Ψ follow: for example, the well known arguments of Cantor, Russell, and Gödel. The structure uncovered behind these paradoxes is generalized in Sect. 2. This (...) allows us to show that Reynolds' [R] construction of a typeA ≃℘℘A in polymorphic λ-calculus cannot be extended, as conjectured, to give a fixed point ofevery variable type derived from the exponentiation: for some (contravariant) types, such a fixed point causes a paradox.Pursueing the idea that $$\frac{{{\text{type theory}}}}{{{\text{categorical interpretation}}}} = \frac{{{\text{(propositional) logic}}}}{{{\text{Lindebaum algebra}}}}$$ the language of categories appears here as a natural medium for logical structures. It allows us to abstract from the specific predicates that appear in particular paradoxes, and to display the underlying constructions in “pure state”. The essential role of cartesian closed categories in this context has been pointed out in [L]. The paradoxes studied here remain within the limits of the cartesian closed structure of types, as sketched in this Lawvere's seminal paper — and do not depend on any logical operations on the type Ω. Our results can be translated in simply typed λ-calculus in a straightforward way (although some of them do become a bit messy). (shrink)

Mitchell, J.C. and E. Moggi, Kripke-style models for typed lambda calculus, Annals of Pure and Applied Logic 51 99–124. The semantics of typed lambda calculus is usually described using Henkin models, consisting of functions over some collection of sets, or concrete cartesian closed categories, which are essentially equivalent. We describe a more general class of Kripke-style models. In categorical terms, our Kripke lambda models are cartesian closed subcategories of the presheaves over a poset. To those (...) familiar with Kripke models of modal or intuitionistic logics, Kripke lambda models are likely to seem adequately ‘semantic’. However, when viewed as cartesian closed categories, they do not have the property variously referred to as concreteness, well- pointedness or having enough points. While the traditional lambda calculus proof system is not complete for Henkin models that may have empty types, we prove strong completeness for Kripke models. In fact, every set of equations that is closed under implication is the theory of a single Kripke model. We also develop some properties of logical relations over Kripke structures, showing that every theory is the theory of a model determined by a Kripke equivalence relation over a Henkin model. We discuss cartesian closed categories but present the main definitions and results without the use of category theory. (shrink)

We show that a standard axiomatization of mereology is equivalent to the condition that a topological space is discrete, and consequently, any model of general extensional mereology is indistinguishable from a model of set theory. We generalize these results to the Cartesian closed category of convergence spaces.

We define a new class of Kripke structures for the second-order λ-calculus, and investigate the soundness and completeness of some proof systems for proving inequalities as well as equations. The Kripke structures under consideration are equipped with preorders that correspond to an abstract form of reduction, and they are not necessarily extensional. A novelty of our approach is that we define these structures directly as functors A: → Preor equipped with certain natural transformations corresponding to application and abstraction . We (...) make use of an explicit construction of the exponential of functors in the Cartesian-closed category Preor, and we also define a kind of exponential ΠΦs ε T to take care of type abstraction. However, we strive for simplicity, and we only use very elementary categorical concepts. Consequently, we believe that the models described in this paper are more palatable than abstract categorical models which require much more sophisticated machinery . We obtain soundness and completeness theorems that generalize some results of Mitchell and Moggi to the second-order λ-calculus, and to sets of inequalities. (shrink)

A natural extension of Natural Deduction was defined by Schroder-Heister where not only formulas but also rules could be used as hypotheses and hence discharged. It was shown that this extension allows the definition of higher-level introduction and elimination schemes and that the set $\{ \vee, \wedge, \rightarrow, \bot \}$ of intuitionist sentential operators forms a {\it complete} set of operators modulo the higher level introduction and elimination schemes, i.e., that any operator whose introduction and elimination rules are instances of (...) the higher-level schemes can be defined in terms of $\{ \vee, \wedge, \rightarrow, \bot \}$.The aim of the present work is to extend the well-known connections between Proof Theory and Category Theory to higher-level Natural Deduction. To be precise, we will show how an adjointness between cartesian closed categories with finite coproducts can be associated, in a systematic way, with any operator $\phi$ defined by the higher-level schemes. The objects in the categories will be rules instead of formulas. (shrink)

An intuitionistic version of Cantor's theorem, which shows that there is no surjective function from the type of the natural numbers N into the type N → N of the functions from N into N, is proved within Martin-Löf's Intuitionistic Type Theory with the universe of the small types.

In this paper I compare two well studied approaches to topological semantics – the domain-theoretic approach, exemplified by the category of countably based equilogical spaces, Equ and Typ Two Effectivity, exemplified by the category of Baire space representations, Rep . These two categories are both locally cartesian closed extensions of countably based T0-spaces. A natural question to ask is how they are related.First, we show that Rep is equivalent to a full coreflective subcategory of Equ, consisting of (...) the so-called 0-equilogical spaces. This establishes a pair of adjoint functors between Rep and Equ. The inclusion Rep → Equ and its coreflection have many desirable properties, but they do not preserve exponentials in general. This means that the cartesian closed structures of Rep and Equ are essentially different. How ever, in a second comparison we show that Rep and Equ do share a common cartesian closed subcategory that contains all countably based-T0 spaces. Therefore, the domain-theoretic approach and TTE yield equivalent topological semantics of computation for all higher-order types over countably based T0-spaces. We consider several examples involving the natural numbers and the real numbers to demonstrate how these comparisons make it possible to transfer results from one setting to another. (shrink)

The standard presentation of topological spaces relies heavily on (naïve) set theory: a topology consists of a set of subsets of a set (of points). And many of the high-level tools of set theory are required to achieve just the basic results about topological spaces. Concentrating on the mathematical structures, category theory offers the possibility to look synthetically at the structure of continuous transformations between topological spaces addressing specifically how the fundamental notions of point and open come about. As a (...) byproduct of this, one may look at the different approaches to topology from an external perspective and compare them in a unified way. Technically, the category of sober topological spaces can be seen as consisting of (co)algebraic structures in the exact completion of the elementary category of sets and relations. Moreover, the same abstract construction of taking the exact completion, when applied to the category of topological spaces and continuous functions produces an extension of it which is cartesian closed. In other words, there is one general mathematical construction that, when applied to a very elementary category, generates the category of topological spaces and continuous functions, and when applied to that category produces a very suitable category where to deal with all sorts functions spaces. Yet, via such free constructions it is possible to give a new meaning to Marshall Stone’s dictum: “always topologize” as the category of sets and relations is the most natural way to give structure to logic and the category of topological spaces and continuous functions is obtained from it by a good mix of free – i.e. syntactic – constructions. (shrink)

It is proved that equations between arrows assumed for cartesian categories are maximal in the sense that extending them with any new equation in the language of free cartesian categories collapses a cartesian category into a preorder. An analogous result holds for categories with binary products, which may lack a terminal object. The proof is based on a coherence result for cartesian categories, which is related to model-theoretic methods of normalization. This maximality (...) of cartesian categories, which is analogous to Post completeness, shows that the usual equivalence between deductions in conjunctive logic induced by βη normalization in natural deduction is chosen optimally. (shrink)

Here, we present a category ${\mathbf {pEff}}$ which can be considered a predicative variant of Hyland's Effective Topos ${{\mathbf {Eff} }}$ for the following reasons. First, its construction is carried in Feferman’s predicative theory of non-iterative fixpoints ${{\widehat {ID_1}}}$. Second, ${\mathbf {pEff}}$ is a list-arithmetic locally cartesian closed pretopos with a full subcategory ${{\mathbf {pEff}_{set}}}$ of small objects having the same categorical structure which is preserved by the embedding in ${\mathbf {pEff}}$ ; furthermore subobjects in ${{\mathbf {pEff}_{set}}}$ are (...) classified by a non-small object in ${\mathbf {pEff}}$. Third ${\mathbf {pEff}}$ happens to coincide with the exact completion of the lex category defined as a predicative rendering in ${{\widehat {ID_1}}}$ of the subcategory of ${{\mathbf {Eff} }}$ of recursive functions and it validates the Formal Church’s thesis. Hence pEff turns out to be itself a predicative rendering of a full subcategory of ${{\mathbf {Eff} }}$. (shrink)

The "Cartesian Meditations" translation is based primarily on the printed text, edited by Professor S. Strasser and published in the first volume of Husserliana: Cartesianische Meditationen und Pariser Vorträge, ISBN 90-247-0214-3. Most of Husserl's emendations, as given in the Appendix to that volume, have been treated as if they were part of the text. The others have been translated in footnotes. Secondary consideration has been given to a typescript (cited as "Typescript C") on which Husserl wrote in 1933: "Cartes. (...) Meditationen / Originaltext 1929 / E. Husserl / für Dorion Cairns". Its use of emphasis and quotation marks conforms more closely to Husserl’s practice, as exemplified in works published during his lifetime. In this respect the translation usually follows Typescript C. Moreover, some of the variant readings n this typescript are preferable and have been used as the basis for the translation. Where that is the case, the published text is given or translated in a foornote. The published text and Typescript C have been compared with the French translation by Gabrielle Pfeiffer and Emmanuel Levinas (Paris, Armand Collin, 1931). The use of emphasis and quotation marks in the French translation corresponds more closely to that in Typescript C than to that in the published text. Often, where the wording of the published text and that of Typescript C differ, the French translation indicates that it was based on a text that corresponded more closely to one or the other – usually to Typescript C. In such cases the French translation has been quoted or cited in a foornote. (shrink)

This paper argues that a Cartesian belief in the self-transparency of minds might actually be an innate aspect of our mind-reading faculty. But it acknowledges that some crucial evidence needed to establish this claim hasn’t been looked for or collected. What we require is evidence that a belief in the self-transparency of mind is universal to the human species. The paper closes with a call to anthropologists (and perhaps also developmental psychologists), who are in a position to collect such (...) evidence, encouraging them to do so. (shrink)

Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe­ maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an (...) adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general­ ized monoid. Chapters VI and VII explore this notion and its generaliza­ tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure lead inter alia to the study of more convenient categories of topological spaces. (shrink)

HOME . ABOUT US . CONTACT US HELP . PUBLISH WITH US . LIBRARIANS Search in or Explore Browse Publications A-Z Browse Subjects A-Z Advanced Search University of Cambridge SIGN IN Register | Why Register? | Sign Out | Got a Voucher? prev abstract next Two Approaches to Reading the Historical Descartes A Devout Catholic? Knowledge of The Mental Thought and Language Descartes as A Natural Philosopher Substance Dualism Notes Two Approaches to Reading the Historical Descartes Author: Desmond M. Clarke (...) DOI: 10.1080/09608780902986680 Publication Frequency: 5 issues per year Published in: British Journal for the History of Philosophy, Volume 17, Issue 3 June 2009 , pages 601 - 616 John Cottingham: Cartesian Reflections: Essays on Descartes's Philosophy (Oxford: Oxford University Press, 2008) 40.00 (hb.). ISBN 978-0-19-922697-9 John Cottingham, in a new collection of essays, asks the question: 'what exactly did Descartes himself chiefly take himself to be doing?' (254). 1 While the question is relatively clear, and while it acknowledges implicitly that Descartes was probably doing a range of different things, the answer that is apparently proposed here emerges only on reading the whole collection. Cottingham distinguishes in Chapter 1 - which is a new, synoptic overview of what is discussed in the other chapters, all of which were previously published - between two approaches to reading Descartes. One is to see him as 'a dummy on which to drape various suspect doctrines (such as “Cartesian dualism”)' (3), which contemporary analytic philosophers have shown to be radically mistaken. Another approach is adopted by historians of ideas 'who make it their life's work to pay meticulous scholarly attention to the philosophical works of past ages' (3). Cottingham does not explicitly criticize either of these approaches, but he hints at situating his own as some kind of Aristotelian middle course between the two. Since the two reference points are dangerously close to straw men or what Cottingham calls 'extreme positions', the proposed middle way may simply combine elements of two approaches, each of which is entirely legitimate. I return to this question at the conclusion. In fact, many of these essays were intended to show (rightly!) that Descartes never held the philosophical positions that are often attributed to him. The interpretation of Karol Wojtyla, later Pope John Paul II, provides a good example of how mistaken one can be: The Cogito ergo sum radically changed the way of doing philosophy … After Descartes, philosophy became a science of pure thought: all that is being- the created world, and even the Creator, is situated within the ambit of the Cogito, as contents of human consciousness. Philosophy is concerned with beings as contained in consciousness, and not as existing independently of it. (257) The only way to address such a caricature is to refer back to what Descartes actually wrote. Cottingham does precisely that, often quoting the original Latin or French texts. However, having shown successfully, by a close reading of the texts, that Descartes did not hold many of the views that are attributed to him, it still remains to say what Descartes did hold or teach about various philosophical problems that retain their perennial interest for us. This is how the question arises, intermittently, about what Descartes thought he was 'chiefly' doing, or what was his primary objective, in the course of an intellectual career that spanned three decades. However, the apparently legitimate desire to bring Descartes' intellectual endeavours into sharp focus may be frustrated by the evidence. His life and work manifestly lack the coherence or unity of purpose that one finds, for example, among many of his French or Dutch contemporaries. It is comparatively easy to 'read' the life of Gisbertus Voetius as that of an unwavering Calvinist theologian, to see Antoine Arnauld as a staunch and consistent theological defender of Port Royal, and even to interpret the obviously fragmentary contents of Pascal's unpublished notebooks (subsequently, the Penses) as an extended search for authentic religious faith, in opposition to what he perceived as the corruption of ecclesial structures. In contrast, Descartes' life reveals features that are difficult to integrate into a coherent pattern. He lived and published during a critical juncture in the Scientific Revolution of the seventeenth century. Although baptized into the Catholic Church soon after his birth in the Loire district of France, he chose to live most of his life in the aggressively Calvinist United Provinces, in which other religious practices were officially (though often ineffectively) banned. Descartes may have adopted the motto from Ovid, at least early in his career: 'bene vixit qui bene latuit' (he lives well who conceals himself well), and he seems genuinely to have wished to avoid theological controversies. However, he engaged in very public controversies with so many of his contemporaries - including Hobbes, Gassendi, the French Jesuits (collectively) and Father Dinet (in particular), Voetius and the University of Utrecht, Regius, Fermat, Roberval, Revius and other theologians at Leiden - that one might conclude that his claimed preference for a quiet life was a disingenuous mask. 2 Descartes published four books during his lifetime, and wrote at least one other that he had intended to publish. The latter was his first major composition, Le Monde, which he suppressed when he heard about Galileo's condemnation by Rome in 1633. This was followed, in 1637, by Descartes' first book (also written in French), which he tried to publish anonymously by withholding his name from the title page: the Discours de la methode pour bien conduire sa raison, & chercher la verit dans les sciences. Plus la dioptrique, les meteors, et la geometrie, qui sont des essais de cete methode. Four years later Descartes published the first edition (in Latin) of Meditationes de prima philosophia, in quibus Dei existentia et animae immortalitas demonstrator, which also included six sets of objections and replies. The Principia philosophiae appeared (in Latin) in Amsterdam in 1644, and was followed five years later by Les Passions de l'Ame (in French). 3 In parallel with these publications, Descartes carried on a very extensive correspondence over a thirty-year period (in both Latin and French), and preserved copies or drafts of his letters with a view to future publication. Given the range and variety of his interests, and the sheer volume of writings, published or otherwise, that have survived from his pen, one may be tempted to engage in a comparative evaluation, as Cottingham does, by selecting one of Descartes' books as his primary contribution to philosophy. Cottingham claims that the Meditations was Descartes''masterpiece' (44), 'the definitive statement of Descartes's philosophy' (45) and the 'definitive statement of his metaphysics' (68). He also describes the Meditations more narrowly as 'his metaphysical masterpiece' (259, 303) and, more broadly, as 'his philosophical masterpiece' (289), and he lists it with the Discours as one of Descartes' two 'masterworks' (280). The Principia offers some competition in this comparative judgement when it is described as 'the canonical presentation of his metaphysical views' (114), while 'the construction of a moral system … was the crowning aim of his philosophy' (231). Having pitched repeatedly for the Meditations as Descartes' primary text, Cottingham claims that its author was 'a devout Catholic' (215), that he was 'a devoutly religious philosopher' (256), and that he could not free himself 'from the influence of the long years of theological study he had dutifully completed at La Flche' (62). Accordingly, the Meditations should be read as 'in essence a work of theodicy' (220); 'what has pride of place in the construction of his philosophical system is … an appeal to God … the nature and existence of the Deity is something that lies at the very heart of his entire philosophical system' (255). Without quite saying so, there are hints here that Descartes was a devout Christian whose primary intellectual contribution was to write a work of metaphysics, in which God is central and in the course of which the author alternates between proving God's existence and contemplating God - the latter a seventeenth-century version of Bonaventure's Journey of the Soul to God. 'Descartes's attraction to a contemplative mode of philosophizing' (305) is reflected, in the Meditations, in 'the language of the soul's coming to rest in adoring contemplation of the light' (306). Cottingham also accepts the overwhelming evidence from Descartes' correspondence that he 'devoted most of his career not to metaphysics but to science' (108), although he quibbles elsewhere with those who adopt the shorthand term 'science' to describe part of what was called 'natural philosophy' in the seventeenth century (282). He also refers to the '(notoriously lame) argument for the essential incorporeality of the thinking self' in one of Descartes' masterworks (60), and he describes the 'strange, seemingly isolated world of his metaphysical meditations' (139), with its 'creaking ontology' (147), when read in isolation from the rest of his work as a natural philosopher. How should we read him, then, in the twenty-first century? A Devout Catholic? That Descartes was a devout Catholic is possible, unlikely and undecidable. He seems not to have studied theology at all while at school at La Flche, although he completed the pre-theology college cycle in the company of Jesuit students who then continued their studies in theology. Descartes consistently attempted to avoid public entanglement with religious and theological controversies, and said so frequently. 4 Given the alignments that prevailed at the time, both in France. (shrink)

In this paper I examine contemporary accounts of noninferential justification in light of what I take to be the Cartesian project of building epistemology on foundations made secure by the impossibility of error. I argue that familiar abstract arguments for foundationalism, by themselves, don’t seem to motivate Cartesianism. But I further argue that there is one version of foundationalism that is more closely linked to the way in which Descartes sought ideal knowledge.

This paper argues that a Cartesian belief in the self-transparency of minds might actually be an innate aspect of our mind-reading faculty. But it acknowledges that some crucial evidence needed to establish this claim hasn’t been looked for or collected. What we require is evidence that a belief in the self-transparency of mind is universal to the human species. The paper closes with a call to anthropologists (and perhaps also developmental psychologists), who are in a position to collect such (...) evidence, encouraging them to do so. (shrink)

WHATEVER one may think of the merits and demerits of the Cartesian system one must acknowledge the great vitality of the Cartesian principles. They were launched with a passion, a sincerity, an engagement rarely equalled. The principles in some way met a deeply-felt need stirring in many breasts in the 17th century; a half-unconscious aspiration which many struggled to articulate and expressed in a variety of ways. Bacon, Galileo, Descartes, each in his own way helped to formulate and (...) create the new order, and in doing so stimulated the general appetite for it so that it grew apace. Descartes prepared the ground, Galileo planted the tree, Bacon looked after the ‘public relations’. The scientific age is the outcome of their work. These men were themselves too close to things to see clearly the consequences of what they were doing; some of these consequences would have greatly surprised them. Could Descartes, for instance, personally the most ‘existentialist’ of philosophers, have foreseen that his principles would give rise to a diametrically opposite regime: that forcing the radical separation and disengagement of man and the material world would lead to a state of brittle ‘facts’ and shallow ‘emotions’, and would provoke the contemporary ‘existentialist’ reaction? (shrink)

: Descartes provides an original and puzzling argument for the traditional theological doctrine that the world is continuously created by God. His key premise is that the parts of the duration of anything are "completely independent" of one another. I argue that Descartes derives this temporal independence thesis simply from the principle that causes are necessarily simultaneous with their effects. I argue further that it follows from Descartes's version of the continuous creation doctrine that God is the instantaneous and total (...) cause of everything that happens, and that this is just what his physics demands. But although God is the total cause of everything, he is not the only cause, since Descartes considers it obvious that finite minds have the power to move bodies. In the face of this apparent paradox, several recent commentators have suggested that Descartes accepted the late scholastic view that God and finite causes somehow collaborate or concur in the production of numerically identical effects. But close examination reveals that the case for Cartesian concurrentism is weak. Fortunately, there is a simpler and more fruitful solution to the problem of reconciling divine and human action. I argue that that in Descartes's world certain motions, such as voluntary movements of our bodies, are causally overdetermined This account allows Descartes to avoid occasionalism without embracing an elaborate metaphysics of secondary causality. Finally, I argue that the overdeterminist interpretation sheds new light on two longstanding problems of Cartesian metaphysics. First, it resolves a serious difficulty with a standard reading of Descartes's conception of human freedom. Second, it offers a novel approach to the old problem of reconciling genuine human action with the principle of the conservation of total quantity of motion. (shrink)

This book argues that science and metaphysics are closely and inseparably interwoven in the work of Descartes, such that the metaphysics cannot be understood without the science and vice versa. In order to make his case, Thomas Vinci offers a careful philosophical reconstruction of central parts of Descartes' metaphysics and of his theory of perception, each considered in relation to Descartes' epistemology. Many authors of late have written on the relation between Descartes' metaphysics and his physics, especially insofar as the (...) former was intended to justify the latter. Vinci's work does not focus on this relation. It takes as a broad interpretive principle that Descartes wanted to justify a certain picture of matter with his metaphysics, but it focuses its own efforts on the way in which metaphysics and science meet in Descartes' theory of sense-perception. Vinci aims to show that Descartes gave an important positive role to sense-perception in his epistemology, and also that he used his reflections on sense-perception to frame his criticism of previous theories of the sensory qualities of objects. (shrink)

Category mistakes are sentences such as ”The number two is blue’ or ”Green ideas sleep furiously’. Such sentences are highly infelicitous and thus a prominent view claims that they are meaningless. Category mistakes are also highly prevalent in figurative language. That is to say, it is very common for sentences which are used figuratively to be such that, if taken literally, they would constitute category mistakes. In this paper I argue that the view that category mistakes are meaningless is inconsistent (...) with many central and otherwise plausible theories of figurative language. Thus if the meaninglessness view is correct, the theories in question must each be rejected, and conversely, if any of the theories in question is correct, the meaninglessness view must be wrong. The debates concerning the semantics of figurative language and concerning the semantic status of category mistakes are closely connected. (shrink)