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.

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)

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)

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)

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)

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)

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)

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)

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 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)

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)

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)

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)

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.

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)

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)

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)

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).

A close analysis of the specifically cinematographic procedure in Akira Kurosawa’s ‘Dream’ Crows reveals it as an articulated and insightful philosophical statement, endowed with general relevance conceming ‘natural’ perception, phenomenological Erlebnis, mechanical image and aesthetic rapture. The antagonism between the Benjarninian lineage of a mechanical irreducibility of the cinematic image to anthropocentric categories, and the Cartesian tradition of a film-philosophy still relying on the equally irreducible structure of the intentional act, be it the one of a deeply embodied (...) and enworlded counsciousness, in accounting for the essential structure of film and spectator (and their relation), i.e., the antagonism between the decentering primacy of the image and the self-centered primacy of perception, cannot be settled through a simple Phenomenological shift from occularcentric, intentional counsciousness to its embodyment ‘ in-the-world’ as yet another carrier of intentionality. Still it remains to be explained what is it in the mechanical image that is able to so deeply affect the human flesh, and conversely, to what features in the human bodily experience is its mechanical other, the fascinating image, so successfuly adressing? It should be expected from the anti-Cartesianism of both the early and the late Merleau- Ponty the textual support for an approach to the essential condition of passivity in movie watching, that would be convergent with Benjamin. The Chapter ‘Le sentir’, in Phénoménologie de la perception, will offer us the proper guide to elucidate what we are already perceiving and conceiving in Kurosawa’s film, where the ex-static phenomenological body of the aesthetical contemplator ‘ enters the frame’ like the Benjaminian surgeon enters the body and like the painter - and always already like our deepest levei of ‘sensing’, previously to any act of cousciousness - ‘just looses himself in the scene before him’. The Polichinello secret of cinema watching is nonetheless too evident to be seen, and that is where Phenomenological description and reduction are still required. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Anthony Paul Kerby THE ADEQUACY OF SELF-NARRATION: A HERMENEUTICAL APPROACH An important question that arises from the increasing contemporary emphasis on the self as a narrative construct concerns the adequacy or truthfulness of the narrative accounts we give ofourselves. What, for example, stops our self-narrations and self-characterizations from becoming, in many cases, mere flights of fancy or fictions? If, on a fairly radical view, the self is taken to (...) be a product of narrative emplotment, then clearly the above question cannot be answered by a simple appeal to what the narrative is supposed to represent or copyas though the selfwere first given to us external or prior to the narrative. It is not immediately clear, however, just what a narrative account of the human subject could be adequate to, and, if we were to grant some prenarrative status to the human subject, it is also not clear whether the imposition of narrative structures falsifies the "truth" of this latter subject. What we shall examine, then, is precisely this important question of adequacy and various problems that surround it. To bring these problems into focus and offer some solutions, we shall consider both self-narration and the writing of history. These solutions will have a decidedly hermeneutical flavor, and for one principal reason: hermeneutics, especially as represented by Gadamer and Ricoeur, is particularly suited to addressing questions of self-expression while avoiding traditional metaphysical presuppositions about the human subject. Let us begin with a few words on hermeneutics. To offer a hermeneutics of something will presumably commit one to certain philosophical and methodological presuppositions. To see what these presuppositions are we might briefly compare hermeneutics to two of its close neighbors: phenomenology and semiotics. I believe 232 Anthony Paul Kerby233 diät these three positions are all of value and that their central insights must be integrated. Contemporary hermeneutics can be viewed as usefully fulfilling this latter task. On the one hand, a hermeneutic philosophy can accept the phenomenological starting point of immediacy to phenomena. However, it does not delude itself into thinking that there is a privileged mode of access to phenomena that would disconnect the categories of our particular historical and linguistic heritage. On the other hand, while recognizing that our understanding is mediated by language and semiotic systems generally—the signifying networks of exchange and communication—hermeneutics need notlose contactwith the life-experiences of the subjects within this semiotic realm. That is, we not only come to understand the sign systems operative in our world, but also, to some degree, ourselves as we act in and through them. As Ricoeur often stresses, semiotics (and structuralism) have still to take the final step of accounting for the experiencing subjects presupposed by any structural social system. Semiotics would, at its extreme point, remain arbitrary with respect to phenomenological experience, just as phenomenology would remain silent if it were consistendy to avoid the conceptual and cultural biases of our languages. Thus, while presupposing individual human experience (and worldliness ), hermeneutics can still accept that such experience only achieves expression and explicit understandingin the semiotic realm, and further that this expression is not the duplication of a fully formed but preexpressed understanding, for understanding is only first grasped at the level of expression. As Gadamer has written: "Being that can be understood is language," but this is not to deny the existence of some matter at hand, a hermeneutic Sache selbst, that thereby comes to language ; in our case the human subject itself. Hermeneutics is essentially a reflexive philosophy, but one which rejects the absolute starting point of the Cartesian subject. The subject's self-understanding is an interpretive process, where we must recognize interpretation as an integral part of the subject's very being. Charles Taylor sums up this hermeneutic situation by saying that man "cannot be understood simply as an object among objects, for his life incorporates an interpretation, an expression of which cannot exist unexpressed, because the self that is to be interpreted is essentially that of a being who self-interprets." ' This position, which implies a rejection of both subjectivism and die metaphysics of presence, problematizes self-understanding by viewing the self as mediated by its... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:From Metaphysical Representations to Aesthetic Life: Toward the Encounter with the Other in the Perspective of Daoism by Massimiliano LacertosaRenjie Li (bio)From Metaphysical Representations to Aesthetic Life: Toward the Encounter with the Other in the Perspective of Daoism. By Massimiliano Lacertosa. Albany: SUNY Press, 2023. Pp. 220, Paperback $34.95, isbn 978-1-4384-9364-0.The title of Massimiliano Lacertosa's From Metaphysical Representations to Aesthetic Life: Toward the Encounter with the Other in (...) the Perspective of Daoism clearly points to a comparison between "Metaphysical Representations" and the "Aesthetic Life," or even to a juxtaposition between two different philosophical traditions, namely, the Western and the Chinese or, more specifically, the Platonic-Cartesian and the Daoist. Attempting to reflect on philosophy per se, the author takes Daoism to be the Other that transcends Western philosophical categories. Reiterating an oft-expressed concern, Lacertosa argues that interpretations of Chinese thinking confined to Western frameworks are detrimental for a broader understanding of what philosophy can be, and are ultimately, on the part of the West, merely self-repetition. Drawing on the philosophies of Nietzsche and Heidegger, and key passages from the Daodejing 道德經 (or Laozi 老子) and the Zhuangzi 莊子, Lacertosa unpacks Western philosophy's self-reflection, and reevaluates the conceptual framework of thinking between East and West as more than a mere juxtaposition. The book consists of an introduction, five chapters, and a concise conclusion.The Introduction first clarifies the use of some key terms. For any discussion of "philosophy," the author argues, addressing what is "ethics" and "aesthetics" is a good starting point. Ethics, in Lacertosa's work, is not merely restricted to a domain of moral values. Drawing on Aristotle's understanding in the Nicomachean Ethics, the author emphasizes that ethics derives from the word ēthos (ἦθος), meaning "habit" or "custom." Resorting to Michel Foucault, Lacertosa considers the meaning of "ethics" (i.e., ēthos) as a "way of being," a certain way of acting that corresponds to one's environment (pp. 1-2). Moreover, as the author explains, "aesthetics" in his philosophical analysis distances itself from Kant's analysis of the subjective power of judgment and historical debates on theories of beauty. For Lacertosa, aesthetics means sense perception or, more specifically, an aesthetic gesture towards nature. The main concern of the book is thus to identify the relationship between ethical implications and aesthetic awareness beyond Western traditions (pp. 3-4). The Introduction closes with critical remarks on current comparative practices. Existing comparative analyses, [End Page 1] the author argues, between Western onto-theological thinking and Lao-Zhuang (i.e., Daoist) aesthetic approaches, remain highly limited and questionable due to their reliance on anthropocentric and teleological views imported from Western traditions (pp. 7-8).Chapter 1 begins by pointing out a crucial problem in comparative studies, namely, the language barrier. Lacertosa argues that translation inevitably involves misunderstandings since the meanings of concepts are deeply rooted in their native cultures. The author then goes on to show the difficulty of seeking a symmetrical translation between Indo-European languages and the Chinese language by looking at three key Daoist terms: you 有, wu 無, and dao 道. Translating them as "Being", "Non-Being", and "the Dao" respectively is, according to the author, merely self-repetition that subordinates Chinese thinking to Western metaphysics. Lacertosa suggests that dao in particular is better appreciated if understood as a process of becoming, which is also a constant process of forming and un-forming (p. 27). The implication is that any new translation or interpretation requires adequate affirmation of the otherness of other languages.Chapter 2 then turns to the more general issue of how interpretation through language shapes the way we perceive the world, as language is the most significant way to represent reality, consequently forming what Lacertosa emphasizes: one's "worldview" (p. 31). With reference to Heidegger's ontological difference, the author asserts that the worldview, or the psychology of worldviews, is ontic, since it presents itself as a specific or certain interpretation of the world, which can never be conceived as a universal ontology that considers all possibilities. This objectification of the world implies the separation of the subject and the object, because this objectification is always the... (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)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Descartes’s Concept of MindJoanna ForstromLilli Alanen. Descartes’s Concept of Mind. Cambridge, MA: Harvard University Press, 2003. pp. xv + 355. Cloth, $65.00.Descartes's Concept of Mind takes as its task that of redressing "the distortions of Descartes's concept of human mind and thinking caused by the Cartesian myth that Ryle justly sought to correct, but that his gripping caricature has helped keep alive" (x). Offering a close reading (...) of the text in its philosophical and historical context and examining its place in contemporary literature, Alanen develops Descartes's philosophy of mind, broadly construed, with a focus on the embodiment of the human mind.Alanen sees Descartes's philosophy of mind as "the result of an ongoing struggle to adjust some basic views about human nature, thought and agency rooted in experience and tradition to the new picture of the universe" (8). Taking this perspective reveals a greater continuity in his fundamental views. Changes, she suggests, are a matter of emphasis, subject matter, and perspective. She grants Descartes's dualism and his view of the mind-body union and chooses to focus instead on the mind as embodied.In developing the analysis Alanen takes a somewhat chronological approach in her seven chapters. Chapter one is a prehistory of Descartes's philosophy of mind focusing on his unpublished, early writings. She looks at his early intellectual ambitions and interests beginning with mathematics and a science of nature and then expanding into metaphysics. The distinctive views of his later work are formed, according to Alanen, during this time. In chapter two, three different perspectives about the nature of man are distinguished: man as an object of mechanistic physics, man as mind or intellect and so an epistemic subject, and man as a real union between the mind and body. She focuses on the third, as it is the real union that has been overlooked or misconstrued by most. How this union is to [End Page 115] be conceived, and the epistemological implications of it are explored. By highlighting the consequences of Descartes's distinction between the mind and the body in terms of knowledge, Alanen directly confronts Ryle and his legacy. In short, because we cannot expect to have knowledge of the mind—body union, we cannot conceive it—there is no category mistake to be made. She elaborates much more in the chapter.Chapters three, four and five, are centered on the writings from 1637-44. Here Alanen focuses on Descartes's notion of thinking and modes of thoughts. The way that thinking is defined, its intentional and representative nature, and the status of sensations as mental phenomena are carefully explored in light of discussions of these issues in contemporary literature and in the literature of Descartes's time. The issues are complex, but the writing is clear and easy to follow. Her discussions of representation, sense perception, and material falsity are particularly good in these chapters.Chapters six and seven reflect the issues that Princess Elisabeth raised: emotions, will, and related moral issues. These two chapters are perhaps the most significant in the book. Chapter six centers on The Passions. The passions depend on the union of the mind and body and do not consist in thought alone. Alanen sees them as a subclass of thoughts. She examines carefully the intentionality of emotions, the distinction between action and passions, and the functions attributed to the body and the soul. She raises a number of questions, some of which chapter seven begins to answer, and others which are left to challenge the reader. Chapter seven is about free will and virtue. Alanen places Descartes's conception of the will and agency in the context of the debate between the voluntarists and the intellectualists. She considers the ancient and medieval antecedents, with a focus on the Stoics as she develops her analysis. She also includes a good discussion of Descartes's contemporaries, and their reactions to his work. Her inclusion of secondary literature is well done. One is left with a better understanding of the complexities of Descartes's understanding of human nature and free will.Overall, Alanen's book is an excellent contribution to... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Pretexts of Authority: The Rhetoric of Authorship in the Renaissance PrefaceSteven RendallPretexts of Authority: The Rhetoric of Authorship in the Renaissance Preface, by Kevin Dunn; xii & 198 pp. Stanford: Stanford University Press, 1994, $32.50.This study is of broader interest than its title might suggest; it engages many of the current issues in literary and cultural studies, and does so with exceptional intelligence. Drawing on Jürgen Habermas’s notion (...) of “the public sphere” and recent research on the history of private life, Dunn focuses on prefatory rhetoric in order to study the ways in which writers of the sixteenth and seventeenth centuries struggled to establish their authority, to define themselves as authors and to regulate their relations with their readers. As they challenged collapsing medieval models of authority, Dunn argues, Renaissance writers had to invent different strategies of self-presentation and discursive legitimation, developing in the process the Renaissance conception of authorship, and eventually a new model of intellectual activity in the service of the public good. [End Page 181]The first part of the book discusses Luther’s effort to ground an oppositional discourse in a personal narrative of suffering and redemption, and Milton’s later recognition of himself as a “member incorporate” of a (Puritan) body for which he can speak as a representative; these represent two termini of the development of Protestant prefatory rhetoric. The second part traces a parallel development in “scientific” writers (Descartes, Bacon, Hobbes, Locke, Sprat), moving from the Cartesian effort to ground science in a personal narrative of discovery to Bacon’s institutionalization of science and Hobbes’s theory of representation based on “common sense.”These chapters offer many insights that deserve close attention. However, since I lack space here to discuss each of them in detail, I will mention only a few points that suggest the wider implications of Dunn’s study. First: he is alive to the role of the body in prefatory rhetoric, as a “private” proof of spiritual authority (in Paul), as a flawed textual corpus (in Luther), and as a model of the body politic (in Hobbes and others). Second: he challenges—correctly, in my view—Foucault’s famous claim about the role of “penal appropriation” in the development of the modern conception of the author, arguing that “texts, books, and discourses begin to need authors at the moment when their transgression itself seeks institutionalization. In other words, it is when the subversive discourse finds itself on the brink of empowerment, of articulating something more than a negative critique of reigning orthodoxies, that the authority of the author becomes necessary” (p. 10). Third: he describes the conflict in Renaissance authors between self-authorization and the desire to prevent readers from following their example—that is, they strive both to open up and to shut off a discursive potential—and offers a subtle account of the persistence of this desire, in the form of the claim to authorial ownership, in writers fully involved in the literary marketplace. Fourth: most of the authors examined here are not traditionally regarded as “literary”; Dunn is casting a broader net, including theological, philosophical, political, and scientific writing, and this is, of course, very much in the spirit of the times—both of our own, and that of the writers discussed. For as Dunn observes, the modern notion of “literature” arises sometime around the end of the Period with which he is concerned in this book—and, perhaps, as a result of the development he describes in it. He argues that “with science having secured its discursive boundaries through the complementary myths of objectivity and public utility, literature became the new site for working out the completing claims of public and private. It is at such a site that literature begins to emerge as an independent category, a place crucial for cultural and political negotiations” (p. 16). As Dunn shows, this is reflected in the way late seventeenth-century French and English prefaces abandon advocacy for critical judgment, handing down sentences based on publicly accepted, secular norms.This fine, illuminating, readable book makes a stimulating contribution to our understanding of the history of cultural authority.Steven RendallUniversity... (shrink)

This essay is an attempt to construct an artificial dialog loosely modeled after that sought by Robert Maynard Hutchins who was a significant influence on many of us including and especially Robert Rosen. The dialog is needed to counter the deep and devastating effects of Cartesian reductionism on today’s world. The success of such a dialog is made more probable thanks to the recent book by A. Louie. This book makes a rigorous basis for a new paradigm, the one (...) pioneered by the late Robert Rosen. If we are to make such a paradigm shift happen, it has to be in the spirit of the dialog. The relationship between science, economics, technology and politics has to be openly recognized and dealt with. The message that Rosen sent to us has to be told outside small select circles of devotees. The situation has even been described by some as resembling a cult. This is no way for universal truths like these to be seen. The essay examines why this present situation has happened and identifies the systemic nature of the problem in terms of Rosen’s concepts about systems. The dialog involves works by George Lakoff, W. Brian Arthur, N. Katherine Hayles, Robert Reich and Dorion Sagan. These scholars each have their own approach to identifying the nature of the interacting systems that involve human activity and the importance of identifying levels of abstraction in analyzing systems. Pooling their insights into different facets of a complex system is very useful in constructing a model of the self referential system that humans and their technology have shaped. The role of the human component in the whole earth system is the goal of the analysis. The impact of the Cartesian reductionist paradigm on science and the related aspects of human activity are examined to establish an explanation for the isolation of Rosen’s paradigm. The possible way to proceed is examined in the conclusion. (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)

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)

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)

: 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)

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.

It is proved that all the isomorphisms in the cartesian category freely generated by a set of objects (i.e., a graph without arrows) can be written in terms of arrows from the symmetric monoidal category freely generated by the same set of objects. This proof yields an algorithm for deciding whether an arrow in this free cartesian category is an isomorphism.

The gradual appearance and relative stabilisation of the names of different kinds of martial activities in different cultures and contexts has led to confusion and to an unhelpful and unjustifiable elision of meanings, which merges different modes of combat and other martial activities. To gain a clearer perspective on this area, we must enquire into the criteria according to which the various kinds of martial activities are classified. Our assessment of the literature suggests that there is no satisfactory and well-justified (...) overall cross-cultural account of the classification of martial activities. This paper provides a revisionary classification and offers an explanation and a justification of the five main categories identified: close combat, warrior arts, martial paths, martial arts and martial sports; as well as some minor ones, such as martial training, martial therapy, martial display, martial games and martial dance. (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)

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)