Cook and Reckhow [5] pointed out that $\mathcal {N}\mathcal {P} \neq co\mathcal {N}\mathcal {P}$ iff there is no propositional proof system that admits polynomial size proofs of all tautologies. The theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus on a conjecture from [16] in foundations of the theory that there is a proof complexity generator hard for all proof (...) systems. This can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows:•There exists a p-time function g stretching each input by one bit such that its range $rng(g)$ intersects all infinite $\mathcal {N}\mathcal {P}$ sets.We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results), to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from [18] is a good candidate for g. We define a new hardness property of generators, the $\bigvee $ -hardness, and show that one specific gadget generator is the $\bigvee $ -hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite $\mathcal {N}\mathcal {P}$ sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite $\mathcal {N}\mathcal {P}$ sets. (shrink)

Let g be a map defined as the Nisan–Wigderson generator but based on an NP ∩ coNP -function f. Any string b outside the range of g determines a propositional tautology τb expressing this fact. Razborov [27] has conjectured that if f is hard on average for P/poly then these tautologies have no polynomial size proofs in the Extended Frege system EF. We consider a more general Statement that the tautologies have no polynomial size proofs in any propositional proof (...) system. This is equivalent to the statement that the complement of the range of g contains no infinite NP set. We prove that Statement is consistent with Cook' s theory PV and, in fact, with the true universal theory T PV in the language of PV. If PV in this consistency statement could be extended to "a bit" stronger theory then Razborov's conjecture would follow, and if TPV could be added too then Statement would follow. We discuss this problem in some detail, pointing out a certain form of reflection principle for propositional logic, and we introduce a related feasible disjunction property of proof systems. (shrink)

We prove that the Nisan-Wigderson generators based on computationally hard functions and suitable matrices are hard for propositional proof systems that admit feasible interpolation. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

Complex frameworks for defining programming languages aim to generate various tools (e.g. interpreters, symbolic execution engines, deductive verifiers, etc.) using only the formal definition of a language. When used at an industrial scale, these tools are constantly updated, and at the same time, it is required to be trustworthy. Ensuring the correctness of such a framework is practically impossible. A solution is to generate proof objects as correctness artefacts that can be checked by an external trusted checker. A logic (...) suitable for developing such frameworks is matching logic. K framework is a canonical example having matching logic-based foundation. Since the (symbolic) configurations of the programs are represented by matching logic patterns, the algorithms computing the dynamics of these configurations can be seen as pattern transformers and a proof object should be generated for the relationship between these patterns. In this paper, we show that conjunctions and disjunctions of patterns, produced by semantics or analysis rules, can be safely normalized using unification and antiunification algorithms. We also provide a prototype implementation of our proof object generation technique and a checker for certifying the generated objects. (shrink)

Planning with incomplete knowledge becomes a very active research area since late 1990s. Many logical formalisms introduce sensing actions and conditional plans to address the problem. The action language $\mathcal{A}_{K}$ invented by Son and Baral is a well-known framework for this purpose. In this paper, we propose so-called cautious and weakly cautious semantics for $\mathcal{A}_{K}$ , in order to allow an agent to generate and execute reliable plans in safety-critical environments. Intuitively speaking, cautious and weakly cautious semantics enable the agent (...) to know exactly what happens after the execution of an action. Computational complexity analysis shows that cautious semantics reduces the reasoning complexity of $\mathcal{A}_{K}$ , it is also worth to point out that many useful domains could still be expressed with this setting. Another important contribution of our work is the development of Hoare style proof systems. These proof systems are served as inference mechanisms for the verification of conditional plans, and proved to be sound and complete. In addition, they could also be used for plan generation, in the sense that constructing a derivation is indeed a procedure to finding a plan. We point out that the proof systems posses a nice property for off-line planning, that is, the agent could generate and store short proofs in her spare time, and perform quick plan query by easily constructing a long proof from the stored shorter ones (under the assumption that sufficient proofs are stored). (shrink)

We consider tautologies formed form a pseudo-random number generator, defined in Krajicek [11] and in Alekhnovich et al. [2]. We explain a strategy of proving their hardness for Extended Frege systems via a conjecture about bounded arithmetic formulated in Krajicek [11]. Further we give a purely finitary statement, in the form of a hardness condition imposed on a function, equivalent to the conjecture. This is accompanied by a brief explanation, aimed at non-specialists, of the relation between prepositional proof (...) class='Hi'>complexity and bounded arithmetic. (shrink)

A formal sequent system dealing with Menelaus' configurations is introduced in this paper. The axiomatic sequents of the system stem from 2-cycles of Δ-complexes. The Euclidean and projective interpretations of the sequents are defined and a soundness result is proved. This system is decidable and its provable sequents deliver incidence results. A cyclic operad structure tied to this system is presented by generators and relations.

In this paper we define Martin-L¨of complexes to be algebras for monads on the category of (reflexive) globular sets which freely add cells in accordance with the rules of intensional Martin-L¨of type theory. We then study the resulting categories of algebras for several theories. Our principal result is that there exists a cofibrantly generated Quillen model structure on the category of 1-truncated Martin-L¨of complexes and that this category is Quillen equivalent to the category of groupoids. In particular, 1-truncated Martin-L¨of complexes (...) are a model of homotopy 1-types. In order to establish these facts we give a proof-theoretic analysis, using a modified version of Tait’s logical predicates argument, of the propositional equality classes of terms of identity type in the 1-truncated theory. (shrink)

Proof complexity is a rich subject drawing on methods from logic, combinatorics, algebra and computer science. This self-contained book presents the basic concepts, classical results, current state of the art and possible future directions in the field. It stresses a view of proof complexity as a whole entity rather than a collection of various topics held together loosely by a few notions, and it favors more generalizable statements. Lower bounds for lengths of proofs, often regarded as (...) the key issue in proof complexity, are of course covered in detail. However, upper bounds are not neglected: this book also explores the relations between bounded arithmetic theories and proof systems and how they can be used to prove upper bounds on lengths of proofs and simulations among proof systems. It goes on to discuss topics that transcend specific proof systems, allowing for deeper understanding of the fundamental problems of the subject. (shrink)

Default logic is one of the most popular and successful formalisms for non-monotonic reasoning. In 2002, Bonatti and Olivetti introduced several sequent calculi for credulous and skeptical reasoning in propositional default logic. In this paper we examine these calculi from a proof-complexity perspective. In particular, we show that the calculus for credulous reasoning obeys almost the same bounds on the proof size as Gentzen’s system LK. Hence proving lower bounds for credulous reasoning will be as hard as (...) proving lower bounds for LK. On the other hand, we show an exponential lower bound to the proof size in Bonatti and Olivetti’s enhanced calculus for skeptical default reasoning. (shrink)

In the first section of this paper I define a set of measures for proof complexity, which combine measures in terms of length and space. In the second section these measures are generalized to the broader category of formal texts. In the third section of the paper I outline several applications of the proposed theory.

We introduce three formal theories of increasing strength for linear algebra in order to study the complexity of the concepts needed to prove the basic theorems of the subject. We give what is apparently the first feasible proofs of the Cayley–Hamilton theorem and other properties of the determinant, and study the propositional proof complexity of matrix identities such as AB=I→BA=I.

A Ramsey statement denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \longrightarrow (k)^2_2}$$\end{document} says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formula RAM(n, k) of size O(nk) and with terms of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}$$\end{document}. Let rk be the minimal n for which the statement holds. We prove that (...) RAM(rk, k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlák’s work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4k, k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R. (shrink)

Statman and Orevkov independently proved that cut-elimination is of nonelementary complexity. Although their worst-case sequences are mathematically different the syntax of the corresponding cut formulas is of striking similarity. This leads to the main question of this paper: to what extent is it possible to restrict the syntax of formulas and — at the same time—keep their power as cut formulas in a proof? We give a detailed analysis of this problem for negation normal form , prenex normal (...) form and monotone formulas. We show that proof reduction to NNF is quadratic, while PNF behaves differently: although there exists a quadratic transformation into a proof in PNF too, additional cuts are needed; the elimination of these cuts requires nonelementary expense in general. By restricting the logical operators to {,,,}, we obtain the type of monotone formulas. We prove that the elimination of monotone cuts can be of nonelementary complexity . On the other hand, we define a large class of problems where cut-elimination of monotone cuts is only exponential and show that this bound is tight. This implies that the elimination of monotone cuts in equational theories is at most exponential. Particularly, there are no short proofs of Statman's sequence with monotone cuts. (shrink)

Proof-theoretic models of grammar are based on the view that an explicit characterization of a language comes in the form of the recursive enumeration of strings in that language. That recursive enumeration is carried out by a procedure which strongly generates a set of structural descriptions Σ and weakly generates a set of strings S; a grammar is thus a function that pairs an element of Σ with elements of S. Structural descriptions are obtained by means of Context-Free phrase (...) structure rules or via recursive combinatorics and structure is assumed to beuniform: binary branching trees all the way down. In this work we will analyse natural language constructions for which such a rigid conception of phrase structure is descriptively inadequate and propose a solution for the problem of phrase structure grammars assigning too much or too little structure to natural language strings: we propose that the grammar can oscillate between levels of computational complexity in local domains, which correspond to elementary trees in a lexicalised Tree Adjoining Grammar. (shrink)

We give a proof-theoretic and algorithmic complexity analysis for systems introduced by Morrill to serve as the core of the CatLog categorial grammar parser. We consider two recent versions of Morrill’s calculi, and focus on their fragments including multiplicative connectives, additive conjunction and disjunction, brackets and bracket modalities, and the! subexponential modality. For both systems, we resolve issues connected with the cut rule and provide necessary modifications, after which we prove admissibility of cut. We also prove algorithmic undecidability (...) for both calculi, and show that categorial grammars based on them can generate arbitrary recursively enumerable languages. (shrink)

Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

In the human cognitive economy there are four grades of epistemic involvement. Knowledge partitions into distinct sorts, each in turn subject to gradations. This gives a fourwise partition on ignorance, which exhibits somewhat different coinstantiation possibilities. The elements of these partitions interact with one another in complex and sometimes cognitively fruitful ways. The first grade of knowledge I call “anselmian” to echo the famous declaration credo ut intelligam, that is, “I believe in order that I may come to know”. As (...) construed here, one knows in this anselmian way that E = mc2 just in case one knows that sentence expresses a true statement, but without having to understand the proposition it expresses. Most epistemologists ignore the significance of this grade of epistemic involvement. In a second grade of epistemic involvement, knowing that E = mc2 is knowing what that sentence means and understanding the proposition it express. This is knowledge in the propositional or semantic sense, and is the dominant target of epistemological investigation. Tacit and implicit knowledge occupies another tier. A typical example would be something that someone has “known all along” but, until now, hasn’t had occasion to put her mind to it or formulate in words. TI-knowledge remains a minority interest in today’s epistemology. Operating at a fourth grade of epistemic involvement is what I call “impact”-knowledge, which is the knowledge of a matter at its deepest and most widespread. An example, to be discussed below, is the knowledge that was generated by the Wiles proof of Fermat’s last theorem. Its true importance lies not only, or even mainly, in its verification of a commonly accepted fact about numbers, but rather in its enrichment of the mathematics of elliptical curves and the promise it holds for greater advancement into the mathematical unknown. Knowledge of this fourth grade has yet to find a seat in the parliaments of epistemology. Knowledge of the anselmian sort is independent of the other three. Tacit and implicit knowledge is incompatible with anselmian and semantic knowledge but coinstantiable with impact-knowledge. Semantic knowledge is incompatible with tacit and implicit knowledge but coinstantiable with the others. Impact-knowledge is pairwise coinstantiable with the others. Below I will bring the ignorance partitions into such alignment as they have with these ones. In doing so, I’ll propose a naturalized causal response epistemology designed to give these interactive distinctions the theoretical air they need to breathe. (shrink)

The unreduced many-body interaction problem solution, absent in usual science framework, reveals a new quality of emerging multiple, equally real but mutually incompatible system configurations, or “realisations”, giving rise to the universal concept of dynamic complexity and chaoticity. Their imitation by a single, “average” realisation or trajectory in usual theory (corresponding to postulated “exact” or perturbative problem solutions) is a rough simplification of reality underlying all stagnating and emerging problems of conventional (unitary) science, often in the form of missing, (...) or “dark”, entities and “hidden variables”. We show how the application of unreduced interaction problem solution and the ensuing concept of dynamic complexity provides the desired causally complete and intrinsically unified solutions to respective unitary science problems in the entire range of complexity levels, from elementary particles and cosmology to emergent consciousness and modern development crisis, implying real, essential and urgently needed progress. (shrink)

Virtuosity in music performance is often associated with fast, precise, and efficient sound-producing movements. The generation of such highly skilled movements involves complex joint and muscle control by the central nervous system, and depends on the ability to anticipate, segment, and coarticulate motor elements, all within the biomechanical constraints of the human body. When successful, such motor skill should lead to what we characterize as fluency in musical performance. Detecting typical features of fluency could be very useful for technology-enhanced learning (...) systems, assisting and supporting students during their individual practice sessions by giving feedback and helping them to adopt sustainable movement patterns. In this study, we propose to assess fluency in musical performance as the ability to smoothly and efficiently coordinate while accurately performing slow, transitionary, and rapid movements. To this end, the movements of three cello players and three drummers at different levels of skill were recorded with an optical motion capture system, while a wireless electromyogrphy (EMG) system recorded the corresponding muscle activity from relevant landmarks. We analyze the kinematic and coarticulation characteristics of these recordings separately and then propose a combined model of fluency in musical performance predicting music sophistication. Results suggest movements from expert performers' are characterized by consistently smooth strokes and scaling of muscle phasic coactivation. The explored model of fluency as function of movement smoothness and coarticulation patterns was shown to be limited by the sample size but serves as a proof of concept. Results from this study show the potential of a technology-enhanced objective measure of fluency in musical performance, which could lead to improved practices for aspiring musicians, instructors, and researchers. (shrink)

The length of resolution proofs is investigated, relative to the model-theoretic measure of Herband complexity. A concept of resolution deduction is introduced which is somewhat more general than the classical concepts. It is shown that proof complexity is exponential in terms of Herband complexity and that this bound is tight. The concept of R-deduction is extended to FR-deduction, where, besides resolution, a function introduction rule is allowed. As an example, consider the clause P Q: conclude P) (...) Q, where a, f are new function symbols extending Herbrand's universe. P ()Q from Q) and Skolemization). By modification of a construction of Statman it is shown that FR-deductions may be nonelementarily shorter than R-deductions . Finally it is proved that FR-deduction has weaker effects only if clausal logic is restricted to Horn logic. (shrink)

In both the philosophical and visual sense, ‘seeing is believing’ does not apply to nanotechnology, for there is nothing even remotely visible to create proof of existence. On the atomic and molecular scale, data is recorded by sensing and probing in a very abstract manner, which requires complex and approximate interpretations. More than in any other science, visualization and creation of a narrative becomes necessary to describe what is sensed, not seen. Nevertheless, many of the images generated in science (...) and popular culture are not related to data at all, but come from visualizations and animations frequently inspired or created directly from science fiction. Likewise, much of this imagery is based on industrial models and is very mechanistic in nature, even though nanotechnology research is at a scale where cogs, gears, cables, levers and assembly lines as functional components appear to be highly unlikely. However, images of mechanistic nanobots proliferate in venture capital circles, popular culture, and even in the scientific arena, and tend to dominate discourse around the possibilities of nanotechnology. The authors put forward that this new science is ultimately about a shift in our perception of reality from a purely visual culture to one based on sensing and connectivity. (shrink)

Legal scholarship exploring the nature of evidence and the process of juridical proof has had a complex relationship with formal modeling. As evident in so many fields of knowledge, algorithmic approaches to evidence have the theoretical potential to increase the accuracy of fact finding, a tremendously important goal of the legal system. The hope that knowledge could be formalized within the evidentiary realm generated a spate of articles attempting to put probability theory to this purpose. This literature was both (...) insightful and frustrating. Much light was shed on the legal system, but it also quickly became evident that the tools of probability theory were in many ways ill-constructed for the task. Fundamental incompatibilities between the structure of legal decision making and the extant formal tools were identified, and it became evident that many of the purported explanations of legal phenomena were internally inconsistent. As a consequence, interest in this type of formal modeling declined, and attention was directed toward different kinds of explanations of the phenomena. Perhaps under the influence of a recent trend toward various types of formal modeling in legal scholarship, a recent burst of articles, rather than attempting to explain the macro structure of trials, which was the previous object of interest, attempts to quantify the probative value of various items of evidence in ways consistent with the formal features of various probability theories, and then to study decision making from that perspective. For example, the value of evidence is often purported to be its likelihood ratio, that is, the probability of discovering or receiving the evidence given a hypothesis (e.g., the defendant did it) divided by the probability of discovering or receiving the evidence given the negation of the hypothesis (the defendant didn't do it). Alternatively, the value of evidence is purported (more contextually) to be the information gain it provides, defined as the increase in probability it provides for a hypothesis above the probability of the hypothesis based on the other available evidence. Both conceptions then assume that all of the various probability assessments conform or ought to conform to the dictates of Bayes' theorem (that maintains consistency among such assessments); empirical studies are then done testing the extent to which this is so and proposing how the law can increase the probability that it is so. The general criticisms of using Bayes' theorem as a formal model of juridical proof are well known and were integral to the last wave of interest in formal modeling of the evidentiary process. This paper thus for the most part puts aside that more general issue, and focuses specifically on mathematical modeling of the value of particular items of evidence. The paper demonstrates that formal modeling has only limited value in explaining the value of legal evidence, much more limited than those constructing and discussing the models assume, and thus that the conclusions they draw about the value of evidence are unwarranted. This is done through a discussion of four recent examples that attempt to quantify evidence relating to, respectively, carpet fibers, infidelity, DNA random-match evidence, and character evidence used to impeach a witness. This article thus makes two contributions. First, and most importantly, it is another demonstration of the complex relationship between algorithmic tools and legal decision making. Second, at a minimum it points out serious pitfalls for analytical or empirical studies of juridical proof. (shrink)

I examine Popper’s claims about Newton’s use of induction in Principia with the actual contents of Principia and draw two conclusions. Firstly, in common with most other philosophers of his generation, it appears that Popper had very little acquaintance with the contents and methodological complexities of Principia beyond what was in the famous General Scholium. Secondly Popper’s ideas about induction were less sophisticated than those of Newton, who recognised that it did not provide logical proofs of the results obtained using (...) it, because of the possibilities of later, contrary evidence. I also trace the historical background to commonplace misconceptions about Newton’s method.Author Keywords: Newton; Popper; Induction; Principia; Kepler’s laws. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Authority, Reading, Reflexivity: Pierre Bourdieu and the Aesthetic Judgment of KantKoenraad Geldof (bio)Translated by Alex Martin (bio)1. AuthorityFor some time now, Pierre Bourdieu has been a true author 1 —a producer, in other words, of an impressive number of theoretical and analytical discourses in a wide variety of research fields. 2 Whether in anthropology or ethnology, in the sociology of institutions or of the structure and workings of the (...) market of symbolic goods, or in the sociology of literature, the name Bourdieu imposes itself: it establishes authority. Contributing to this is another particularity quite typical of authority: it gets itself talked about; it generates, at accelerated speed, discourses that in one way or another (re-)produce and thus reinforce the authority of the author who was their point of departure and origin. The international proliferation of articles dealing with one aspect or another of Bourdieu’s thought, 3 essay collections, 4 and introductions to Bourdieu’s work 5 testifies to this. The intensity of metadiscursive production not only gives material form to the intrinsic analytic pertinence and richness of the discourse on which it comments—these are evident—but also shows, independent of the pieces’ intentions, that the proliferation is situated in a process of (re-)production of authority, that it is an integral part of a canonizing effort. The articles and introductions problematize, implicitly or explicitly, the direct accessibility of Bourdieu’s discourse. They make the oeuvre into a complex totality, difficult and diversified to the extreme, whose coherence can be revealed only with the help of a gradual, exegetic, and essentially submissive reading practice.To put it another way, and to examine the question more closely, the metadiscursive (re-)production of the authority of the author we call Pierre Bourdieu reveals itself to be [End Page 20] ambivalent: it is at once innovative—it invites the uninitiated to get to know a fascinating new universe of thought—and authoritarian, indeed even conservative. An introduction is never a simple mediation of discourse between an author and his or her potential readers: it provides a sort of user’s guide, it (re-)produces a reading habitus, an ideology of the correct interpretation the reader must learn and interiorize to be able to distinguish orthodox from false, heterodox, readings. Between the metadiscourse/commentary and the authority it discusses, a relationship of tautological circularity is established that surreptitiously transforms the discourse read into a legend (into what must be read): the possibility of an autonomous reading practice is erased by a pedagogy of legendary decoding. 6The authority of an author, that of Bourdieu for example, is consequently implicated in a dynamic of opening (the proliferation of metadiscourses) and closing (the tendency to discipline reading, to make it obey certain norms of legibility). It would be erroneous, however, to attribute the paradoxical aspect of authority simply to the ambivalent effect of metadiscourses/commentaries. This is because authors who establish authority can only rarely escape the canonizing efforts of which they are the object: in many cases, authors themselves participate in the production and reproduction of a mode of reading and interpretation judged to be legitimate. The fact that no authority can completely and exhaustively control its discursive and institutional space does not in any way contradict, therefore, that other distinctive and essential feature of authority: its intrinsic animation by a desire to control, by the desire to see itself respected. Authority wants to become Law and be recognized as such. Thus the tendency of certain authors to become actively involved in the modes of appropriation of their name and their discourses, to endeavor to make the reading of the author’s work not simply an act of connaissance but also a proof of reconnaissance. Authority detests being insufficiently or poorly known. Let us take the case of Bourdieu. As an author, he is first and foremost the one to whom we owe Raisons pratiques, La distinction, Homo academicus and Les règles de l’art, the producer, in other words, of a series of momentous works that, upon publication, established authority. Furthermore, Bourdieu is also the author of a considerable number of accompanying discourses, dominated by the... (shrink)

Euclidean diagrammatic reasoning refers to the diagrammatic inferential practice that originated in the geometrical proofs of Euclid’s Elements. A seminal philosophical analysis of this practice by Manders (‘The Euclidean diagram’, 2008) has revealed that a systematic method of reasoning underlies the use of diagrams in Euclid’s proofs, leading in turn to a logical analysis aiming to capture this method formally via proof systems. The central premise of this paper is that our understanding of Euclidean diagrammatic reasoning can be fruitfully (...) advanced by confronting these logical and philosophical analyses with the field of cognitive science. Surprisingly, central aspects of the philosophical and logical analyses resonate in very natural ways with research topics in mathematical cognition, spatial cognition and the psychology of reasoning. The paper develops these connections, concentrating on four issues: (1) the cognitive origins of Euclidean diagrammatic reasoning, (2) the cognitive representations of spatial relations in Euclidean diagrams, (3) the nature of the cognitive processes and cognitive representations involved in Euclidean diagrammatic reasoning seen as a form of visuospatial relational reasoning and (4) the complexity of Euclidean diagrammatic reasoning for the human cognitive system. For each of these issues, our analysis generates concrete experiment proposals, opening thereby the way for further empirical investigations. The paper is thus a prolegomenon to a research program on Euclidean diagrammatic reasoning at the crossroads of logic, philosophy and cognitive science. (shrink)

IntroductionIn May 2006, the small group of doctoral students working on ecophilosophy at the Higher Institute of Philosophy at K.U.Leuven invited the Dutch environmental philosopher Martin Drenthen to a workshop to discuss his writings on the concept of wilderness, its metaphysical and moral meaning, and the challenge social constructivism poses for ecophilosophy and environmental protection. Drenthen’s publications on these topics had already been the subject of intense discussions in the months preceding the workshop. His presentation on the workshop and the (...) three critical responses by Wim Bollen, Glenn Deliège, and Richard Kover are published here in a thoroughly revised form on the basis of further and ongoing discussions about the issues involved. Three further contributions to the discussion are added: one by Nathan Edward Kowalski, a young Canadian philosopher who studied in Leuven and wrote his as yet unpublished dissertation on “Evil in Nature,” a second one by Kingsley Goodwin who is currently writing a dissertation on deep ecology at University College in Dublin and, finally, my own unsystematic thoughts inspired by Martin Drenthen’s provocative and stimulating hermeneutics of our postmodern understanding of nature and wilderness, as well as the diverse critical responses to Drenthen’s views by the other contributors. I owe gratitude to all of them for enriching my own ecophilosophical thought and enlarging its horizon although as my fragmentary reflections show at the cost of yet greater uncertainty and more questions about the possibilities of arguing for a radical environmentalist position. Such a position would somehow connect or ground our moral obligation to preserve genuine non-human otherness, wild and free nature, in the non-human meaning of such nature and in our essential need to be in direct and concrete contact with that field of wild non-human meaning because we ourselves belong to nature, because we are not only and purely civilized but are also and most fundamentally wild and free. Three fundamental questions need to be distinguished: the question about our moral obligations regarding nature and its preservation, the metaphysical question about the nature of nature or the meaning of nature, and the question of the metaphysical grounding of our morals in general, and of our moral obligation regarding nature in particular, in the nature or meaning of nature. The complex intertwinement between these three questions is at the heart of Martin Drenthen’s reflections.The authors contributing to the present discussion approach the various issues from markedly different philosophical backgrounds and concerns: critical theory, anthropology, analytic philosophy, phenomenology, hermeneutics, and deep ecology are all to some degree or other brought to bear on the question of what wilderness means for us, why we are fascinated by it and what it actually and in the first place is.Another interesting difference between the five respondents to Martin Drenthen’s wilderness-philosophy relates to the place from which they write: while the Canadians Richard Kover and Nathan Edward Kowalski have wilderness of some sort still close at hand, the Belgians Wim Bollen and Glenn Deliège write from a place which has been thoroughly domesticated, cultivated, and industrialised for a long time. Martin Drenthen himself, of course, writes from the same kind of place as these last two, which explains why he only refers to the wilderness that he has read about in novels, seen in movies, or experienced as recreated in the reserves of the Netherlands. The Irish Kingsley Goodwin writes perhaps from a third, intermediate place that is not wild anymore but still much more natural than Belgium or the Netherlands are.The differences in philosophical approach and concern combined with the difference of place generate a rich tapestry of perspectives and voices wrestling with the fundamental questions of our perceived homelessness and alienation from nature, our cultural discontent, our longing for and need for radical alterity, of immanence and transcendence, of who we are and where we belong, and more concretely of whether there is more to ground our efforts to preserve nature than enlightened self-interest and subjective preference. Martin Drenthen deserves praise for inaugurating this fruitful discussion with his intriguing and highly paradoxical effort to hybridize a postmodern deconstruction of the idea and ideal of wilderness as unspoilt and true nature on the one hand with a new metaphysics of wilderness on the other. Arne Naess, who is generally regarded as the founding father of deep ecology, pointed out the positive and stimulating role that vagueness in the expression of ideas can play in political and philosophical debate. Such general and vague but highly suggestive expressions as ‘all life is one’ are not empty at all, as is often claimed; they are rather full of possibilities of ‘precisation’ and differentiation. Drenthen’s wilderness-philosophy is certainly not vague to the degree of ‘all life is one,’ but his expositions are similarly pregnant with a wealth of further questions, precisations, distinctions, and conceptual clarifications. Good examples of different kinds are to be found in the contributions of Nathan Edward Kowalski and Richard Kover. Kowalski points out that Drenthen’s notions of culture and wilderness need to be made more precise by distinguishing them from the notions of civilization and wildness such that wild cultures become a conceptual possibility. Richard Kover takes the lead from Drenthen’s question about the reason for our postmodern fascination for wilderness to confront it with our primordial fascination in terms of the dependence in the evolution of our subjectivity and our consciousness on the focused perceptual attention on wild forms.Glenn Deliège situates Martin Drenthen’s wilderness-philosophy in its original context of the Dutch debate on the preservation of nature, in particular on the new wilderness paradigm. According to this paradigm, the nature to be preserved should be true and pure, truly natural and wild nature, that is nature outside of and free from the cultivating and domesticating influence of humans. In the Netherlands, however, such nature no longer exists; it can only be preserved after it has first been artificially recreated. Deliège gives an informative summary account of the highly charged three-cornered debate between the adepts of this wilderness-paradigm, the proponents of the more traditional form of conservation of remnants of the typical European landscapes created by pre-industrial farming practices and the modern technocratic functionalists. This debate makes obvious that there are quite different views of the nature of nature and its value. Enter social constructivism: the contested nature of nature is nothing more than a social construct, an idea in a collective mind. The contestation itself is a struggle for authority and power between different such social constructs, any claim to scientific objectivity and objective truth for one’s image of nature is an arrogation of a truth which doesn’t exist and is potentially oppressive by silencing other views. Nature then is up for the grabs of politics, political negotiation, and compromise between various interest groups who all want to get some benefit from nature, some use-value.The problem with Drenthen’s effort to articulate a normative concept of nature which cannot be deconstructed in this way is the extreme formality and emptiness of his metaphysical concept of wilderness as that which is always ‘beyond’, always ‘more’ and which can never be exhausted by any image and construction of nature. Any more concrete determination is mere appearance; the thing in itself is a border concept of that which in fact cannot be grasped conceptually. Glenn Deliège is critical of this formality and emptiness from which it will be impossible to derive any concrete guidance for our conservation practices. He then goes on to defend the Arcadian tradition of our appreciation of nature as giving us a historically well-grounded orientation for those practices. Of course that Arcadian tradition is obviously a cultural construction but that doesn’t make it completely arbitrary and contingent since it is itself ultimately grounded in and open to modification by our concrete sensuous experiences of nature.Deliège here raises the very important issue of the relation between our concrete perceptual and bodily experiences in and with nature and our concepts and metaphysical images of nature. Drenthen himself approaches this issue from the hermeneutical perspective of our need to make sense of our experiences and to articulate their meaning. His confrontation with social constructivism and his effort to salvage something from its relativizing acid lead him, however, into the direction of a highly abstract and empty concept of wilderness.Richard Kover shows that perception is, indeed, the key to a radical critique of social constructivism. He does this with the help of the ecological anthropology of Paul Shepard, the structuralist anthropology of Claude Levi-Strauss and the phenomenology of Maurice Merleau-Ponty. It needs to be said here that Kover takes his leave from an unpublished paper by Martin Drenthen with the title “How to appropriate wilderness appropriately?” – the paper is accessible though on Drenthen’s web-site - in which Drenthen comments and reflects upon three contemporary wilderness-tales, one of them being Werner Herzog’s much-discussed documentary on Timothy Treadwell, an American environmentalist drop-out who lived thirteen summers among wild grizzlies in the Alaskan wilderness before being killed together with his girlfriend by a grizzly bear.Kover tries to show that we are not imprisoned in the mirror-cabinet of our mind or of our human sociality. To make his case, he starts from Paul Shepard’s thoughtful observation that the evolutionary emergence of the mind did not remove us from the natural world but actually connected us deeper with it. Through our mind we evolved more fully into it because our mind is primordially in our senses and in the ability to articulate a perceptual world. The natural world in the form of the system of natural species and their relations is the primary model for perceptual discriminations, for the fundamental discovery of order with its twin-aspects of distinction and discontinuity on the one side and relations and continuity on the other side. Our cognition is essentially perceptually based, our subjectivity is primordially informed by our inherence in the sensible world. Perceptual cognition is inherently intentional and relational, it is not an autonomous mental construction, it is rather in the world and informed by that which is perceived. The other is always intertwined with the subject. We are always already hooked up with the world, with the others, before we start reflecting, thinking, and conceptualizing. This is one of the major themes in Shepard’s work: what this insight means with regard to our relations to non-human otherness, in particular to our perceptual encounter with wild animals. In our encounter with non-human otherness, we are confronted with an essentially ambiguous sense of both affinity and disjuncture. Wild animals demonstrate to us that we are not homeless or alien to this world, that we fundamentally belong to it and are part of it. Their difference, however, shows us that there are different perspectives and ways of being in the world. “Thus the encounter with non-human others inform humans that we are both of the world and simultaneously not the world and there are horizons of significance beyond our own perspective.”Nathan Edward Kowalski comes to a similar conclusion when he situates our morality in the larger moral order of wild nature. According to Kowalski it is indeed a moral order though not our own, an Other to our morality but not simply a negative one. It is, as Drenthen says, a highly paradoxical amoral morality. But what can it mean to speak of morality here? For Kowalski it means that the issue is not merely the theoretical recognition and acknowledgment of axiological transcendence, of non-human values out there, but whether we will appropriately conform to and comport ourselves towards that axiological transcendence so that we can find a human homeland in wild nature instead of turning it into a wasteland. The question then is, how to fit our own human morality into this larger order and greater reality, to subordinate it to that greater reality as a part of it and encompassed but not denied by it so that we can be moral and natural at the same time, tender carnivores hunting the sacred game. Indigenous people follow a moral protocol in their relationship with non-human others which acknowledge and respects their autonomy, their alterity, and the sacred mystery of their different ways of being. They are not befriended but some of them are hunted and eaten – as an exchange in the gift-economy of nature.The problem of the homelessness and alienation from nature is discussed by Wim Bollen from the radically different vantage point of critical theory. In Adorno’s and Horkheimer’s Dialectic of Enlightenment the human story unfolds as a tragic effort to liberate ourselves from nature and to become fully human through rational control when in the end this effort turns against ourselves and instead of liberation and human flourishing we get social domination and repression of our natural needs. The frustrations and mutilations imposed upon us by a repressive civilization give rise to the regressive desire for a return to a natural existence. Referring to the myth of Odysseus and Circe as interpreted by Adorno and Horkheimer, Wim Bollen equates such a natural existence with animal life. In the light of contemporary knowledge about primitive existence, however, this equation is highly problematic. Palaeolithic foragers were already fully human in all respects. A life without civilization need not be an animal-existence.According to Adorno/Horkheimer, our subjectivity and autonomy have to be wrested from animal-nature through an exertive struggle that cannot but proceed by a threefold strategy of domination: the domination of non-human otherness, of human others, and of our own inner nature. The siren song of nature consists in the temptation to give up that struggle for the promise of an unreflective, non-alienated natural existence without toil.When Odysseus’ men give in to Circe’s seduction they are, however, not transformed into wild animals but into swine. According to Bollen this signifies that we cannot re-wild ourselves, that we remain imprisoned in civilization, either as master-subjects or as domesticated animals. Bollen, by the way, like Drenthen, does not distinguish conceptually between civilization and culture. Since re-wilding for Bollen means the impossibility of returning to animal-existence, which is actually the impossibility of becoming another species since biologically we are animals already, to be imprisoned in civilisation means nothing else than to be imprisoned in our humanity. The fact that we cannot become wild animals again – according to Paul Shepard, however, we are still wild animals – is indeed, as Bollen stresses, a trivial form of alienation. A more meaningful concept of alienation which can be put to critical use must refer to a culturally and historically determined frustration and deficit inside the cultural order that is to be overcome not by an impossible regression but by progressive politics leading beyond the three intertwined forms of domination. Not back to nature but forward towards a reconciliation of culture with nature! Such a reconciliation would neither collapse culture into the immanence of nature nor submerge nature into the transcendence of culture. According with the three forms of domination it would be a three-dimensional form of reconciliation.Odysseus’ men did not got what they thought were promised them: the complete liberation from the burden of being autonomous subjects, of being civilized, but they got something worthwhile, namely a certain from of pleasure, a limited and provisional relief and escape. Bollen sees a critical potential in these piggish pleasures after all and consequently even in the siren song of nature since these pleasures were only experienced because Odysseus’ men gave in to it. But one has to give in to this siren song critically if it is to unfold its liberating potential. The problem with pleasure though in bourgeois society is that it is necessarily betrayed, instrumentalized and sold.Kingsley Goodwin’s paper starts by defending deep ecology against the charge of asserting a naïve and dogmatic metaphysics of nature. For convenience sake, but not quite unproblematically, he identifies deep ecology with the environmental thought of Arne Naess. A metaphysics of nature for Naess is always part of a personal worldview. Worldviews which are partly inspired by ecology Naess calls ecosophies. Naess always stresses the personal character of such a worldview and the irreducible plurality of them. It used to be a common misunderstanding of the platform of deep ecology which Naess conceived together with George Sessions that it expressed in a condensed form the metaphysical beliefs of deep ecology. Naess’ intention, however, was to try to articulate in a general fashion the basic viewpoints on which the supporters of the deep ecology movement agreed. It would be possible then to derive these viewpoints from quite different worldviews. In light of this, I think, that Goodwin’s characterisation of the platform as “pragmatic and activist” is not quite appropriate.Goodwin goes on to compare Drenthen’s conception of wilderness and wildness with the similar position developed by Neil Evernden in his book The Social Creation of Nature. He like Naess in his personal ecosophy stresses the crucial role of perceptual experience of wild otherness to find an antidote to the relativity and contingency of our interpretations of nature. Evernden refers to studies on child development to argue for a non-dualistic and relational conception of the self, similar to Naess’ relational ontology. According to Goodwin, Drenthen, in spite of his hermeneutical approach remains too much indebted to the Western subject/object-dualism. The object, in this case wild nature, is pure otherness.With regard to a compost heap, Goodwin shows that we are not separated from nature by an inseparable gulf, that nature is close at hand, and that we can experience and explore it, appreciate the complex play of identity and difference it consists of, the exchanges and transformations and that we can identify with the organisms involved in these exchanges.*The proceeding remarks only highlight some of the prominent themes and considerations in the various contributions to this topical issue of Ethical Perspectives. Hopefully the reader will feel invited and enticed to read the articles themselves, starting, preferably of course, with the essay by Martin Drenthen, who sparked off this lively discussion on the moral meaning of wild nature. I am honoured and pleased to be able to introduce these rich explorations by such talented young environmental philosophers who are eager to develop environmental philosophy further and lead it into new directions. They all have to wrestle with and find answers to the challenge postmodern deconstruction and social constructivism pose for environmental philosophy. In the face of that challenge, it is even more difficult than it already was to provide solid and convincing arguments for the defence of wild and free nature beyond merely pragmatic arguments in terms of its utilitarian value for us. The reason for that defence is not clear-cut and proves to be just as evasive as the moral meaning of nature itself. But neither that reason nor that meaning are mere phantoms and mirages that vanish when we approach them and try to grasp them. They are more substantial, even if they will never have the solidity of a mathematical proof or a scientific measurement. Their locale is the lifeworld which from the point of view of scientific and logical precision and clarity is a messy place where subject and object, reason and heart, fact and value, experience and thought and, yes, nature and culture cannot be clearly separated. This has important consequences for environmental philosophy and its methodology. Conceptual analysis and logical argumentation have to become sensitive to the richness of experience in the lifeworld and to the diverse efforts of its articulation: i.e., they have to open themselves up and maybe even to subordinate themselves to phenomenological and hermeneutical approaches. For me personally this is one of the most important conclusions I draw from the discussion presented here. (shrink)

We propose a rudimentary formal framework for reasoning about recursion equations over inductively generated data. Our formalism admits all equational programs , and yet singles out none. While being simple, this framework has numerous extensions and applications. Here we lay out the basic concepts and definitions; show that the deductive power of our formalism is similar to that of Peano's Arithmetic; prove a strong normalization theorem; and exhibit a mapping from natural deduction derivations to an applied λ -calculus, à la (...) Curry–Howard, which provides a transparent proof of Gödel's result that the provably recursive functions of PA are definable by primitive recursion in finite types. Also of interest is the extent, uncommon among similar constructions, to which our mapping is oblivious to first order terms, including first order quantification and equality. Additional applications and extensions of the method will be presented in sequel papers. In particular, the methodology introduced here lends itself to systematic links between natural sub-formalisms and computational complexity classes. Since these sub-formalisms are largely transparent, allowing one to reason formally in the full system while leaving to automatic software tools the task of verifying that restrictions are complied with, the framework is particularly attractive for the development of Feasible Mathematics. (shrink)

The following is excerpted from Sean Gurd’s translation of Euripides’ Hippolytus published with Uitgeverij this year. Though he was judged “most tragic” in the generation after his death, though more copies and fragments of his plays have survived than of any other tragedian, and though his Orestes became the most widely performed tragedy in Greco-Roman Antiquity, during his lifetime his success was only moderate, and to him his career may have felt more like a failure. He was regularly selected to (...) have his plays performed in the annual festival of Dionysus at Athens. But the plays were performed in a competition, and in the competitions he lost and lost, winning first place only five times in a career that spanned more than forty years. His corpus reads like the remains of a life spent trying to get a few things right and never being quite satisfied: scenes and schematized combinations of elements recur again and again, each time slightly changed. Hippolytus itself is an example: it contains two scenes in which a sick or dying character lies on a couch or a stretcher and is surrounded by silence and grief; he had experimented with such scenes in Alcestis, and would work with them again in Herakles and Orestes. Nor, judging from appearances, was he immune to the desire to score a hit, or to satisfy the audiences and the judges at the competition. Hippolytus is a rewrite of another Hippolytus, performed several years earlier, which scandalized the audience and certainly did not take first place. In that earlier Hippolytus (called by later writers Hippolytus Veiled), Euripides fashioned a Phaedra who was neither ashamed of loving her stepson nor incapable of taking steps to satisfy her desire. The forwardness of her approach to Hippolytus so outraged him that he covered his head with a robe (hence the play’s later title). Euripides liked to write energetic and self-confident women like this. He staged his Medea in 431, just three years before he revised Hippolytus, and in it the jilted title character murders her children in revenge for Jason’s new marriage to a more politically advantageous wife. Another flop: the play came in third of three. Evidently the judges did not like their women showing agency, especially when the consequences were transgressive. “The ornament of women,” wrote another great Athenian a little earlier, “is silence.”i Euripides’ Medea and his first Phaedra had other decorations. So three years after Medea crashed and burned and some time after the first Hippolytus suffered the same fate, he produced another Hippolytus play with a different setting and story. How could this one not have been a reaction to the failure of the first? “Second thoughts are always better,” he has one of his characters say, as though he were offering an apology for his first thoughts on the same story. In this new version, Phaedra is suicidally ashamed of her love, choosing to die rather than have it discovered, and willing to bring down her beloved Hippolytus in the service of the same end. Now it is Phaedra herself who covers her head in shock and horror at her desire. Only an intermediary, a slave of the house and an old crone with her fingers in all sorts of illicit pots, communicates Phaedra’s secret – against orders – and in consequence brings about the demise of two resolutely moral figures. It worked. For his tale of two well-born nobles with an unswerving commitment to moral action, destroyed by the meddling of a slave (and by the hostility of some particularly uncaring divinities), the judges crowned Euripides winner. It was the first time he had won, after almost thirty years of work. *** What Hippolytus wants more than anything else, as he protests with outrage just before leaving Trozen for the last time, is the benefit of due process – oaths, witnesses, an investigation that would reveal who really lusted after whom. Ironic, given that in the first Hippolytus, as far as we can make out, he did get a trial – but too late, and Theseus’ successful processing of Hippolytus failed to prevent his death. I cannot help hearing in the second Hippolytus’ complaints the anger of his authorial semblable who lost year after year by a process of judging that might have seemed brutal to him. Judges took no cognizance of how he worked, how long or hard or why; they just voted, and Euripides lost. But then I hear, in the fate of the first Hippolytus, an even grimmer commentary. Processed or not, we lose. We run aground. At the very least, this might cast a darker shadow on the business of watching or reading a play: in that twilight, things move too fast to make a fair assessment and, in any case, the reasonably high definition of what you see and hear belies the truth of indistinction, the multiplicities just below the surface. Watching is a constrained activity, a form of captivity, but reading is plastic. This translation was begun to supply the script for an audio version of the play, recorded in the spring of 2012, and here and there it sacrifices literalness for sayability and the voices and passions of the actors on that occasion. But it was finished to be read, and engineered to create a reading caught between the poles of specificity and non-specificity that the play itself conjures and relies on at its generative core. Thus two texts: the one clear, perhaps overprecise, respectably modern in its presentation, and the other ancient, a reasonable approximation of the classical reading process, and indistinct in vital ways. Like early modern dramatic texts, the exact provenance and original function of the scripts that have made it down to us are unknown. We do know that the ultimate survival of Hippolytus was thanks to its wide-spread use as a school text. But its relation to the first performance in 428 BCE is a matter only of speculation. Euripides presumably had a fair copy before production began; it was probably modified during rehearsals. The actors may have had copies, too (probably the young men who served in the chorus did not). The chor?gos, a wealthy Athenian who paid for the production, might have been presented with or just made sure to get his hands on a high-quality text, to commemorate the play’s success. Fans might have been able to buy copies. Any one of these sources might have provided the ultimate progenitor of our text. Several of them might have, in fact, if later editors compared and conflated traditions. The plays toured extensively in the Greek-speaking world, and were restaged as repertoire for centuries afterwards. In the process they were subjected to enough modifications by producers and actors that in the later fourth century the Athenian statesman Lycurgus had a law passed prohibiting performances at Athens from deviating from the official “city” text of the tragedies (which he had commissioned). It is unlikely that the city text was the last word; but it marks an important moment in Athens’ role in their dissemination, since after this the major sites of textual curatorship moved overseas, to Alexandria, Pergamum, Rome, Antioch, Constantinople. Any single text of any single play is thus no more than a moment of clarity and stability, crossed and subtended by multiple vectors of change, including that of textual flux (from Euripides’ first drafts to the latest printed text) and that of Euripides’ life work as a writer, in the light of which the Hippolytus is just one attempt to produce something exactly right, to win the prize or finally satisfy himself or prove to himself that all this longing for perfection was a waste of time. I am intrigued by the possibility that a single text might somehow serve as a lens onto this wider zone of indistinction, representing a small region clearly but nonetheless containing a blurred image of the whole, like a Leibnizian monad. After all, no text is a witness: it is, rather, a perception and a performance, and what every Hippolytus must perform is the awkward overlay of singular and plural, social practise and subliminal drive, that structures the tragedy. Modern critical editions can do this by juxtaposing a (relatively) clean and simple text with a textual apparatus reporting variants and theorizing the relationship between textual instantiations. A translation lacks the resolution to catch the fine variations which feed into the construction of an apparatus. But there are other means to hand. Guessing at the exact words of the original text is always just that – guessing – though Hippolytus is less problematic than plays like Iphigenia at Aulis. But we can make a pretty secure guess at what such a text looked like. Texts read by fifth-century readers were fundamentally different from what is presented in the best medieval MSS and printed in modern publications. The texts of Euripides’ time and for many centuries afterwards did not indicate word division with spaces (or by any other means) and were extremely sparing in punctuation, often avoiding it altogether. There was only one “case” of letters – what we call “capitals.” Musical sections were inscribed as though they were prose, without line endings to reflect their rhythm or their rhetorical structure. There was no indication of the speakers within a dramatic text, either. The earliest surviving dramatic texts, which come from over a century after Euripides’ death and a time which was much more elaborate and precise in its textual culture, indicated speaker-change with a little line or paragraphos below the line and a double-point [:] at its end. This convention almost certainly does not date back to Euripides’ own time; it may not even date to Lycurgus. In any case, such indications were understood to be readers’ marks: they did not have the same authority as the text and they were subject to readers’ revisions. At least ideationally, if not in concrete and distinguishable fact, a dramatic text was understood as containing no indication of speaker-change (and certainly none of who was saying what). The reader had to figure that out for him-/herself. The clear, legible, easy texts we are accustomed to reading are the fabrications of medieval and modern reading cultures. The first text, then, presents the translation as the text would have been in the mid-fourth century, perhaps decades after Euripides’ death. The second text processes Hippolytus, makes sense of it and makes it “readable.” That is: too distinct, dangerously so. It gets in your way, conjures fantoms, tries to make you see. Fragments of the Hippolytus Veiled, otherwise almost entirely lost, are juxtaposed with similar passages in the text – not to suggest a reconstruction, but a dialogue and a complex set of reassessments. I have not tried to represent the full variety of textual variants, but I have signalled where, for one reason or another, the text is soft and different editions propose excisions or changes. The two texts must be read together: you, reader, form the bridge between the distinctly seen and the indistinct invisible. You, in the end, will be its mask. *** illsandhateddiseaseswhatcanidoforyouwhatshouldinotdohereisthelightandthebrightairyoursickbedisnowout s idethehouseyoureverywordwastocomeherebutyouwillrushbacktoyourbedroomagainyoufadequicklyyoudelig htinnothingyoudislikewhatyouhaveyoulovewhatyoudontitsbettertobesickthantendthesickthefirstissimpleto the secondattachesheartspainandhandsworkeveryhumanlifeispainfulthereisnoendtotoildarknesshidesbehindclo udswhateverisdearerthanlifeweproveunhappyloversofwhatshineshereforlackofknowledgeofanotherlifethere isnoproofforthingsbeneaththeearthonlystoriessustainus liftmybodystraightenmyheadthebondsofmylimbsareloosenedtakemyhandsandmypalearmsthishatistooheavy formyheadtoweartakeitoffspreadmyhairovermyshoulders couragechilddonttossandturnsoviolentlyyouwillbearyoursicknessmoreeasilywithpeaceandanoblemindallmort alssuffer aiaiiwanttodrinkpurewaterfromadewyspringiwanttoliebackandrestunderthetreesinsomegrassymeadowchildw hycrydontsaythesethingsnearthecrowdhurlingwordsmountedonmadness sendmetothemountainiwillgototheforestwherebeastkillingdogspressthespotteddeergodsilongtoshouttothedo gstoshootthethessalianjavelinpastmyblondhairtoholdthebarbeddartinmyhandwhythisanxietychildwhatdoyouc areabouthuntingwhydoyoulustforflowingspringstheresahillwithwaterjustnexttothetowerwecangetyouadrinkt h ereartemismistressofthesaltlakeandthecoursethunderingwithhorseshoovesiwishiwereonyourplainsbreakings t uds whythrowthesefrenziedwordsaboutjustnowyouweresettingoutforthemountaintohuntandnowyoulongforhorses onthewavelesssand thesethingsneedanoracletotellwhichgodreinsyouinanddrivesyoufromyoursenseschild whathaveidonehowfarhaveibeendrivenfromgoodthoughtsiwascrazyiwascursedbysomepowerpheupheualasn ursecovermyheadagainimashamedofwhativesaidcovermeatearmovesdownfrommyeyeanditembarrassesm eithurtstostraightenourmindanditisterribletobeinsanebesttodiebeforeyoubecomelucidagain illcoveryoubutwhenwilldeathcovermeimoldivelearnedalotmortalsshoulddrinkoffriendshipmoderatelynotfrom th edeepestmarrowofthesoulamindslovecharmsshouldbeeasytoundoorthrustawayortiemoretightlyitisaverydiffic ultweightforonesoultolaborovertwopeopleasiamwrackedbypainforhertoomuchdisciplinecausesmoreharmthan pleasureandwarswithhealthipraisetoomuchlessthannothinginexcessandthewiseagreewithme *** Scene 1 Enter Phaedra and the Nurse; the former is recumbent on a couch while the latter bustles about her with gestures that might be those of a caregiver or a guard. NURSE Ills and hated diseases! What can I do for you? What should I not do? Here is the light, and the bright air; your sick bed is now outside the house. Your every word was to come here, but you will rush back to your bedroom again. You fade quickly. You delight in nothing. You dislike what you have. You love what you don’t. It’s better to be sick than tend the sick – the first is simple. To the second attaches heart’s pain and hand’s work. Every human life is painful. There is no end to toil. Darkness hides behind clouds whatever is dearer than life. We prove unhappy lovers of what shines here for lack of knowledge of another life. There is no proof for things beneath the earth. Only stories sustain us. PHAEDRA Lift my body, straighten my head: the bonds of my limbs are loosened. Take my hands and my pale arms. This hat is too heavy for my head to wear. Take it off, spread my hair over my shoulders. NURSE Courage, child; don’t toss and turn so violently. You will bear your sickness more easily with peace and a noble mind. All mortals suffer. PHAEDRA Aiai. I want to drink pure water from a dewy spring. I want to lie back and rest under the trees in some grassy meadow. NURSE Child, why cry? Don’t say these things near the crowd, hurling words mounted on madness. PHAEDRA Send me to the mountain. I will go to the forest where beast-killing dogs press the spotted deer. Gods! I long to shout to the dogs, to shoot the Thessalian javelin past my blond hair, to hold the barbed dart in my hand. NURSE Why this anxiety, child? What do you care about hunting? Why do you lust for flowing springs? There’s a hill with water just next to the tower; we can get you a drink there. PHAEDRA Artemis, mistress of the salt lake and the course thundering with horse’s hooves: I wish I were on your plains breaking studs! NURSE Why throw these frenzied words about? Just now you were setting out for the mountain to hunt – and now you long for horses on the waveless sand. These things need an oracle to tell which god reins you in and drives you from your senses, child. PHAEDRA What have I done? How far have I been driven from good thoughts? I was crazy, I was cursed by some power. Pheu pheu. Alas. Nurse, cover my head again. I’m ashamed of what I’ve said. Cover me – a tear moves down from my eye and it embarrasses me. It hurts to straighten your mind and it is terrible to be insane. Best to die before you become lucid again. NURSE I’ll cover you. But when will death cover me? I’m old; I’ve learned a lot. Mortals should drink of friendship moderately, not from the deepest marrow of the soul. A mind’s love-charms should be easy to undo, or thrust away, or tie more tightly. It is a very difficult weight for one soul to labor over two people, as I am wracked by pain for her. Too much discipline causes more harm than pleasure and wars with health. I praise “too much” less than “nothing in excess,” and the wise agree with me. ***. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Imagining Karma, Ethical Transformation in Amerindian, Buddhist, and Greek RebirthA. L. HermanImagining Karma, Ethical Transformation in Amerindian, Buddhist, and Greek Rebirth. By Gananath Obeyesekere. Berkeley: University of California Press, 2002. 448 pp.Gananath Obeyesekere, professor emeritus of anthropology at Princeton University, is probably one of the world's greatest living anthropologists. The proof of that assertion lies in this his latest work on comparative anthropology, a study of the (...) concept and practice of rebirth—"reincarnation" as he calls it—from ancient Greece and India to modern West Africa, Melanesia, and North America. In what follows let me briefly set forth what I understand his central thesis to be, the evidence for it, and the conclusions to be drawn from it. I will end with two very modest problems connected with this altogether intriguing and brilliant investigation into the doctrine of rebirth.As a modified structural anthropologist, Gananath Obeyesekere rejects the rationalist model structuralism of Claude Levi-Strauss. He follows instead the lead of two of his empiricist predecessors, Max Weber and Fernand Braudel, pointing out that structures, types, or models "are constructions of scholars, not universals waiting to be discovered" (p. 351). Consequently, as this book aptly proves, Obeyesekere's own "imagined" models are based on parole data following the actual language of the cultures being studied.Obeyesekere thus begins his book with an elementary rebirth structure based on such parole. He refers to this primitive, first structure as "rebirth eschatology," which entails the rebirth of a deceased's soul into the body of a newborn relative of the deceased but without any determining ethical component as found in the more sophisticated and complex "karmic eschatologies." Such ethically neutral kin group rebirths are found among certain small-scale societies in West Africa, the Trobriand Islands, the Northwest American coast tribes, and the Inuits (Eskimos). Obeyesekere [End Page 303] draws extensively on field studies of these groups to bolster his conclusions about them. This simple rebirth eschatology is non-Indic in origin, Obeyesekere claims, since no evidence of influences have been found between these practitioners of rebirth eschatology and work relating to the Indic religions (Hinduism and Buddhism). The entry of the soul into postdeath life, either in this world or the world of the ancestors, depends not at all, consequently, on one's this-worldly behavior such that "morality does not affect the fate of the soul after death" (p. 74).Obeyesekere turns next to the "transformation" of this rebirth eschatology. It occurs with the introduction of the concept of ethicization, "the processes whereby morally right or wrong action becomes a religiously right or wrong action that in turn affects a person's destiny after death" (p. 75). It is Obeyesekere's contention that when ethicization is systematically introduced into any rebirth eschatology the latter must logically transform itself into a so-called "karmic eschatology." In other words, this second topological model or conceptual structure, karmic eschatology, the new ethicized rebirth eschatology, says that the otherworld for the dead is not simply a way station for departed souls waiting to be reborn, but rather that otherworld is now a place of retribution and punishment. Kinship affiliation alone no longer determines who goes where after death but moral action does. My own guess is that rebirth eschatology was introduced to answer the question of why the grandchild resembles the deceased grandparent. Karmic eschatology, on the other hand, was probably introduced to solve a more complex and sophisticated puzzle: Why is it, if there really is justice in the universe, that the wicked never get appropriately punished and the good never get sufficiently rewarded? At this stage of transformation, the law of karma, or some other principle of universal justice, now says that everyone gets what's coming to them, either in this life or the next, and that rebirth is necessary to make that principle of justice workable.1 In arguing that "rebirth cannot be divorced from ethics; it looks as if it is generated from ethics" (p. 82), Obeyesekere would seem to concur that karmic eschatology was introduced to answer the question: Why do the wicked seem to prosper and the good appear to suffer?Following... (shrink)

The volume starts with the paper of Lynn Maurice Ferguson Arnold, former Premier of South Australia and former Minister of Education of Australia, concerning the Exposition Internationale des Arts et Techniques dans la Vie Moderne (International Exposition of Art and Technology in Modern Life) that was held from 25 May to 25 November 1937 in Paris, France. The organization of the world exhibition had placed the Nazi German and the Soviet pavilions directly across from each other. Many papers are devoted (...) to the interpretation of this opposition. Arnold’s paper considers the differences in the two totalitarian states’ architectural and design discourse styles. Although each of them communicated a totalitarian language of purposes, permissions, and boundaries, they essentially differed in the styles of discourse represented by the architecture and design of their respective pavilions. They were opposites of each other and the liberal ideals they contested. The internationalist viewpoint reflecting the multi-ethnic mix of the USSR is contrasted to the ein Volk homogeneity represented in the German pavilion. The Soviet pavilion opted for a utopian future to be arrived at by “benign” leadership, whereas the German pavilion anchored itself in the myth of Teutonic history with the nostalgic pride protected by the swastika-bearing eagle. Jean-Yves Béziau is best known as a logician and founder of Universal Logic. However, in this paper, he poses a new philosophical question: why philosophical discourse does not use images? The paper begins with a general analysis of different types of philosophical discourses. Then, he focuses on why images have been and are still rejected in philosophical discourse. He explains various ways to use images in a fruitful way to develop philosophical thinking and discourse and illustrates his view by providing several examples. He concludes with a promising programmatic declaration about founding a new journal entitled World Journal of Pictorial Philosophy to stimulate the usage of images for developing philosophical thinking. Although complaints about the obscurity of many philosophers’ discourse are widespread, Tatiana Denisova, ex-professor of the Surgut University and a research associate of the University of the Aegean undertakes a positive attitude to this problem. She explains that the reasons for the obscurity of philosophical texts, the subsequent complexity in communicating philosophers’ meanings and their eventual incomplete understanding is not a sign of their inferiority. On the contrary, it is a sign of the fruitfulness of philosophical discourse, which can generate new meanings. Thus, the darkness of philosophical discourse is like the life-giving chaos, and the obscurity that it inevitably contains can be the keeper of implicit meanings and even their generator. Katarzyna Gan-Krzywoszyńska and Piotr Leśniewski, authors of a monograph on Polish philosopher Kazimierz Ajdukiewicz (1890 – 1963), make a comparative analysis of his educational style with that of Paulo Reglus Neves Freire (1921 – 1997). The authors highlight that these scholars have distinctly different backgrounds. The former was an educator, philosopher and a leading advocate of critical pedagogy connected with the dialogical Latin-American tradition. The latter was a philosopher and logician, a notable representative of the Lwów–Warsaw school of logic with significant contributions to semantics, model theory and the philosophy of science. Despite their apparent difference, the authors identify some striking similarities regarding their attitudes towards education; notably, their approaches are essentially dialogical. Gilah Yelin Hirsch is a multidisciplinary artist who works as a painter, writer, curator, educator, and filmmaker. In her experiential paper, he explains how and why she uses four channels of communication in her creative expression: writing, painting, filmmaking, and teaching, as dialogic inquiry. She considers that these channels compose a continuum of discourse styles, each reaching a different facet of kaleidoscopic consciousness. She claims that her choice of medium is prompted by a need to communicate her insights profoundly. Thus, for instance, she created painted allegories to express narratives based on dreams and visions emerging from her subconscious. Art and healing is another Tibetan-rooted type of discourse practised by the author. Creating and observing a healing image produces a positive psychophysiological change in both the artist and the viewer. The discourse here is tripartite between creator, image, and viewer. Filmmaking is another type of discourse she practices between the filmmaker/artist as shaman or healer and the viewer as respondent or participant. These films are meant to be experienced frame by frame, physically and emotionally in both the body and mind. Jocelyn Ireson-Paine is a cartoonist and programmer. His paper offers an original attempt to use category theory in cartooning. Category theory is a branch of mathematics that provides concepts and methods to describe general abstract structures in terms of a labelled directed graph called category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms or transformations or mappings). The author explores different ways to define “style” in art using category theory and a relational view of art. Moreover, he examines how “translation” (transformation) between styles could be defined and reveals the difficulties of how it could be implemented in a computer using special software. Jens Lemanski is a philosopher who has revived research in Arthur Schopenhauer’s legacy by exploring his work on mathematical evidence, logic diagrams, and problems of semantics. In this paper, he advances a new approach to eristic as an art of protecting oneself from the one who deliberately violates norms of discourse ethics to win an argument. The author attributes the origins of this view to Schopenhauer, suggesting a new reading of his work. According to the author, eristic is a prohibitive technique that takes effect when the norms of discourse ethics are transgressed and violated. Thus, eristic is viewed as a discipline of Enlightenment philosophy and a correlate of discourse ethics. Roshdi Rashed is an authority on the history of Arabic mathematical sciences. Proceeding from Gilles-Gaston Granger’s definition of mathematical style, he studies the question of whether a mathematical work can be characterized by a single style or by a multitude of styles. This question is explored within the style of a single work, namely Menelaus’s Sphaerica, and through the study of the development of a single problem over time, namely the isoperimetric problem. He indicates Menelaus’s divergence from the Euclidean style of (plane and stereometric) geometry caused by the drop of the Parallel Postulate, which associated it with hyperbolic geometry. Hence, the author concludes that Menelaus’s Sphaerica incorporates the well-known Euclidean style with a variation of the non-Euclidean one. On the other hand, examining the isoperimetric problem reveals another interesting historical picture. A succession of different styles (cosmological, geometric, infinitesimalistic, style of the calculus of variations, of synthetic geometry) can be observed due to the transformation of the research object over time. Boris Shalyutin, a Russian social philosopher, and Ombudsman for Human Rights in the Kurgan Region, explores the origins of discourse in combination with the birth of society and law. He claims that legal discourse marks the historical emergence of discourse in general. Homo Juridicus generated Homo Sapiens, which have created new spheres of discourse, notably moral discourse and further philosophical, political, and scientific discourse. Petros Stefaneas, a logician, computer scientist and novelist, suggests an alternative to the traditional narratology approach for studying (interactive) social media discourse. He claims that style describes how the parts of a narrative are blended into a whole. The author relies upon Goguen’s and Harrel’s concept of style as a choice of blending principles and transfers to the study of the social media narratives elements from the methodology of studying Web-based collaborative search for mathematical proofs like those implemented within the Polymath project. The last paper belongs to Ghil‘ad Zuckermann, a linguist, language revivalist, and proponent of a model of the emergence of Israeli Hebrew, according to which Hebrew and Yiddish were the primary sources of Modern Hebrew. This paper explores the fascinating and multifaceted Yiddish language and its survival in Israeli. Yiddish is characterized by a unique style that embeds psycho-ostensive expressions throughout its discourse. The author highlights the cross-fertilization between Hebrew and Yiddish as it manifests itself in any aspect of the Israeli language. He claims that Yiddish survives beneath Israeli phonetics, phonology, discourse, syntax, semantics, lexis, and even morphology, although traditional and institutional linguists have been most reluctant to admit it. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Ideas 64.4 (2003) 599-618 [Access article in PDF] The "Herbert Butterfield Problem" and its Resolution Keith C. Sewell Dordt College Herbert Butterfield (1900-1979) 1 published The Whig Interpretation of History in 1931, a year after he became a Lecturer in the University of Cambridge. 2 He became Professor of Modern History in the university in 1944, the same year in which he published The (...) Englishman and His History. 3 This discussion addresses what J.G.A. Pocock referred to as das Herbert Butter-fieldproblem—that is, the problem of ascertaining the precise relationship between the arguments presented in these volumes. 4The Whig Interpretation of History of 1931 (hereafter, The Whig Interpretation) was a sustained critique of the motivation, methods and fallacious conclusions of the "Whig" practice of staging historiographical narratives anachronistically so as to produce a ratification of the present, or justification of a position currently espoused by the author. 5 The term "Whig," as Butterfield used it in 1931, referred to the nineteenth-century school of historiography that praised "progress," "protestantism," and "liberty" and that generally attributed the triumph of such principles to the beliefs and activities of generations of Whig politicians. 6 [End Page 599]The Whig Interpretation argued that the judgments concerning both progressive and reactionary individuals and movements in Whig historiography were anachronistic, in that they were based on general propositions held by the Whig historian in the present. To correct this error Butterfield argued that historiography should not be based upon any general propositions. Basing historiographical narratives on general propositions purporting to be of universal and unchanging validity was seen as inevitably placing them on an anachronistic basis. In The Whig Interpretation Butterfield required the historian to represent past "life itself " but to avoid doing so on the basis of a view or "interpretation" of past human experience and action. Historiographical narration should be free of any interpretative standpoint and equally open to any possible subsequent interpretation. For Butterfield "the historian carries us away from the world of general ideas. It is not for him to give a philosophical explanation of what happens in time and space"; "any history that he writes ought to be as capable of varied philosophical interpretation as life itself seems to be... the historian's explanation... is not a piece of general reasoning," and "if at any point we need further elucidation all that he can do is to take us into greater detail." 7 It seems appropriate to refer to this as Butterfield's "neutrality postulate." Butterfield subsequently used the term "technical history" to denote this methodological concept. 8As might be expected, such a neutrality postulate was replete with problems, however valid Butterfield's strictures towards Whig historiographical anachronism. How is the historian to abridge for the purposes of narration and explanation without interpreting? Butterfield argued that we are right to demand of the historian that "he shall not change the meaning and purport of the historical story in the mere act of abridging it." No theory should intrude. As the detailed complexity is abridged, there should be no loss of "the purport and tenor of the whole." 9 Abridgment was to be accomplished without interpretative perspective, even though sympathy and imagination are indispensable. 10 Such was Butterfield's prescribed antidote. Yet the exercise he recommended requires more than a specific statement of the concrete, the individual, and the particular. It also requires a general idea of the "meaning" of "the whole" that the historian is called upon not to distort. Indeed, Butterfield drew attention to the basis of his historical thinking when he asserted that while historical research does not provide proof of the existence of a providential order, it nevertheless provides evidence of how profound and mysterious are its workings. 11In The Whig Interpretation no attempt was made to reconcile the advocacy of a non-interpretative neutrality dealing exclusively with the individual, the [End Page 600] concrete and the particular, and Butterfield's actual belief in a providential order to which the historical process is subject. After 1931... (shrink)

continent. 1.2 (2011): 78-91. This article consists of three parts. First, I will review the major themes of Quentin Meillassoux’s After Finitude . Since some of my readers will have read this book and others not, I will try to strike a balance between clear summary and fresh critique. Second, I discuss an unpublished book by Meillassoux unfamiliar to all readers of this article, except those scant few that may have gone digging in the microfilm archives of the École normale (...) supérieure. The book in question is Meillassoux’s revised doctoral dissertation L’Inexistence divine (or The Divine Inexistence ), with its seemingly bizarre vision of a God who does not yet exist but might exist in the future. Without literally accepting this view, I will claim that it is philosophically interesting in ways that even a hardened sceptic might be able to appreciate. Third and finally, I will speculate on the possible future of Meillassoux’s speculative materialism itself. And here I mean its future development not by Meillassoux, but by those readers who might be inspired by his book. Plato could never have predicted the emergence of Aristotle’s philosophy, despite the obvious debt of the latter to the former. Nor could Descartes have predicted Spinoza and Leibniz, nor Kant the German Idealists, and neither could Husserl in 1901 have foreseen the later emergence of Heidegger. How are the works of interesting philosophers transformed by later thinkers of comparable importance? While it may seem that there are countless ways to do this, I think there are only two basic ways in which this happens: you can radicalize your predecessors, or you can reverse them. I will close this article with a few words about these two methods, and try to imagine how Meillassoux might be radicalized or reversed by some future admirer. My view is that the more important thinkers are, the easier they are to radicalize or reverse. This helps explain why the great philosophers of the West have so often appeared in clusters, succeeding one another at relatively brief intervals during periods of especial ferment. 1. After Finitude After Finitude is unusually short for such an influential book of philosophy: running to just 178 pages in the original French, and an even more compact 128 pages in the English version, despite the introduction of roughly eight pages of new material for the English edition. Rather than summarizing Meillassoux’s book in the order he intended, I will focus on six points that strike me as the pillars of his debut book. Along the way, I will offer a few criticisms as well. The first pillar of the book is Meillassoux’s own term “correlationism.”1 Although he introduces this term as the name for an enemy, it is striking that Meillassoux remains impressed by correlationism much more than his fellow speculative realists are. This continued appreciation for his great enemy influences the shape of his own ontology. Is there a world outside our thinking of it, or does the world consist entirely in being thought? Traditionally, this dispute between realism and idealism has been dismissed in continental philosophy as a “pseudo-problem,” in a strategy pioneered by Husserl and extended by Heidegger. We cannot be realists, since following Kant we have no direct access to things-in-themselves. But neither are we idealists, since the human being is always already outside itself, aiming at objects in intentional experience, deeply engaged with practical implements, or stationed in some particular world-disclosing mood. The centuries-old dispute between realism and idealism is dissolved by saying that we cannot think either real or ideal in isolation from the other. There is neither human without world nor world without human, but only a primordial correlation or rapport between the two. This is what “correlationism” means: philosophy trapped in a permanent meditation on the human-world correlate, trying to find the best model of the correlate: is it language, intentionality, embodiment, or some other form of correlation between human and world? Among other problems, this generates some friction between philosophy and the literal meaning of science. When cosmologists say that the universe originated 13.5 billion years ago, they do not mean “13.5 billion years ago for us ,” but literally 13.5 billion years ago, well before conscious life existed, and thus at a time when there was no such thing as a correlate. Meillassoux also coins the term “ancestrality” (10) for the reality that predated the correlate, and later expands this term to “dia-chronicity,” (112) to refer to events occurring after the extinction of human beings no less than to those occurring before we existed. Up to this point, Meillassoux’s focus on ancestral entities existing prior to consciousness might seem like a straightforward realist who wants to unmask correlationism as just another form of idealism. Yet Meillassoux also admires the correlationist maneuver, which can obviously be traced back to Kant. Unlike a thinker such as Whitehead, Meillassoux feels no nostalgia for the pre-critical realism that came before Kant: “we cannot but be heirs of Kantianism,” he says (29). What impresses Meillassoux about correlationism is something both simple and familiar. If we attempt to think a tree outside thought, this is itself a thought . Any form of realism which thinks it can simply and directly address the world the way it is fails to escape the correlational circle, since the attempt to think something outside of thought is itself nothing other than a thought, and thereby collapses back into the very human-world correlate that it pretends to escape. For Meillassoux this step, suggested by Kant but first refined by the ensuing figures of German Idealism, marks decisive forward progress in the history of philosophy that must not be abolished. Any attempt to break free from the correlate must first acknowledge its mighty intellectual power. Realist though he may seem, Meillassoux’s works are filled with praise of such figures as Fichte and Hegel, not of so-called “naïve realists.” It is also the case that for Meillassoux, not all correlationisms are the same. The second pillar of his book is a distinction between various positions that I have termed “Meillassoux’s Spectrum,” though of course he is never so immodest as to name it after himself. He distinguishes between at least six different possible positions, and perhaps we could add even subtler variations if we wished. But in its simplest form, Meillassoux’s Spectrum allows for just four basic outlooks on the question of realism vs. anti-realism. Three of these are easy to understand, since we have already been discussing them. At one extreme is so-called “naïve realism,” which holds that a world exists outside the mind, and that we can know this world. Meillassoux rejects this naïve realism as having been overthrown by Kant’s critical philosophy. At the other extreme is subjective idealism, in which nothing exists outside the mind. For to think a dog outside thought immediately turns it into a thought, and therefore there cannot be anything outside; the very notion is meaningless. In between these two is what we have called correlationism. And here comes a crucial moment for Meillassoux, since he distinguishes between the two forms of “weak” and “strong” correlationism, and chooses the strong form as the launching pad for his own philosophy. Weak correlationism is easy to explain, since we all know it from the philosophy of Kant. The things-in-themselves can be thought but not known. They certainly must exist, since there cannot be appearances without something that appears. And we can think about them, which idealism holds to be impossible. They are simply unknowable due to the finitude of human thought. Strong correlationism is the new position introduced by Meillassoux (though he sees it at work in numerous twentieth century thinkers), midway between weak correlationism and subjective idealism. The major difference between the three positions is as follows. Weak correlationism says: “The things-in-themselves exist, but we cannot know them.” The subjective idealist says: “This is a contradiction in terms, since when we think the things-in-themselves, we already turn them into thoughts.” But the strong correlationist says: “Just because ‘things-in-themselves’ is a meaningless notion does not mean that they cannot exist. No one has ever traveled to the world-in-itself and come back to make a report on it. Thus, the fact that we cannot think things-in-themselves without contradiction does not prove that they do not exist anyway. There may be things-in-themselves, we simply are not capable of thinking them without contradiction form within the correlational circle.” This step is crucial for Meillassoux, since strong correlationism is the position he attempts to radicalize into his own new standpoint: speculative materialism. As I see it, this step of the argument fails. Strong correlationism cannot avoid collapsing into subjective idealism, since the statements of the strong correlationist are rendered meaningless from within. All three of the other positions in the Spectrum make perfectly good sense even for those who disagree with them. The naïve realist says that things-in-themselves exist and we can know them; the meaning of this statement is clear. The weak correlationist can say that things-in-themselves exist but lie forever beyond our grasp; this too makes perfect sense, even though the German Idealists try to show a contradiction at work here. We can also understand the claim of the subjective idealist that to think anything outside thought turns it into a thought, and that for this reason we cannot think the unthought. The strong correlationist, alone among the four, speaks nonsense . This person says “I cannot think the unthought without turning it into a thought, and yet the unthought might exist anyway.” But notice that the final phrase “the unthought might exist anyway” is fruitless for this purpose. For we have already heard that to think any unthought turns it into a thought. But now the strong correlationist wants to do two incompatible things simultaneously with this unthought. On the one hand, he neutralizes the unthought by showing that it instantly changes into just another thought. But on the other hand, he wants to appeal to the unthought as a haunting residue that might exist outside thought, thereby undercutting the absolute status of the human-world correlate found in idealism. But this is impossible. If you accept the argument that thinking the unthought turns it into a thought, you cannot also add “but maybe there is something outside that prevents this conversion from being absolutely true,” because this “something outside” is immediately converted into nothing but a thought for us. In short, Meillassoux here seems to be offering a kind of Zen koan: his “strong correlationism” is reminiscent of the gateless gate, or the sound of one hand clapping, or the command to punch Hegel in the jaw when meeting him on the road. We cannot at the same time both destroy the realist challenge of the things-in-themselves in order to undercut realism and reintroduce that very realist sense in order to undercut idealism. In a world where everything is instantly converted into thought, we cannot claim that there might be something extra-mental anyway, because this “might be something” is itself converted into a thought by the same rules that condemned dogs, trees, and houses to the idealist prison. This brings us to the third pillar of Meillassoux’s argument, which is the key to all the rest: the necessity of contingency. His strategy is to transform our supposed ignorance of things-in-themselves into an absolute knowledge that they exist without reason, and that the laws of nature can change at any time for no reason at all. In this way the cautious agnosticism of Kantian philosophies is avoided, but so is the collapse of reality into thought as found in German Idealism. Meillassoux does try to prove the existence of things-in-themselves existing outside thought; he simply holds that they must be proven after passing through the rigors of the correlationist challenge, not just arbitrarily decreed to exist in the manner of naïve realism. As he puts it, “Everything could actually collapse: from trees to stars, from stars to laws, from physical laws to logical laws; and this not by virtue of some superior law whereby everything is destined to perish, but by virtue of the absence of any superior law capable of preserving anything, no matter what, from perishing” (53). If idealism thinks that the human-world correlate is absolute, for Meillassoux it is the facticity of the correlate that is absolute. He tries to show this with a nice brief dialogue between five separate characters (55-59) which is covered in detail in my forthcoming book,2 but which I will simplify here for reasons of time. In this simplified version, we first imagine a dogmatic realist arguing with a dogmatic idealist. The realist says that we can know the truth about the things-in-themselves; the idealist counters that we can only the truth about thought, since all statements about reality must be turned into statements concerning our thoughts about reality. Here the correlationist enters and proclaims that both of these positions are equally dogmatic. For although we have access to nothing but thoughts, we cannot be sure that these thoughts are all that exist; there could be a reality outside thought, there is simply no way to know for sure. And this latter position is the one that Meillassoux attempts to transform from an agnostic, skeptical point into an ontological claim about the contingency of everything. Consider it this way. How does the correlationist defeat the idealist? The idealist holds that the existence of anything outside thought is impossible. The correlationist, by contrast, holds that something might exist outside the human-world correlate. But this “something might” has to be an absolute possibility. It cannot mean that “something outside thought might exist for thought ,” because that is what the idealist already says. No, the correlationist must mean that something might exist outside thought quite independently of thought. In other words, the correlationist says that idealism might be wrong, and this means it is absolutely true that idealism might be wrong. Thus, correlationism is no longer just a skeptical position. It holds that all the possibilities of the world are absolute possibilities. We have absolute knowledge that any of the possibilities about the existence or non-existence of things-in-themselves might be true, and this means that correlationism flips into Meillassoux’s own position: speculative materialism. As Meillassoux sees it, there are only two options here. Option A is to absolutize the human-world correlate, which is what the idealist does: there absolutely cannot be anything outside thought. Option B, by contrast, is to absolutize the facticity of the correlate: its character of simply being given to us, without any inherent necessity. The correlationist cannot have it both ways by saying: “there absolutely might be something outside thought, yet maybe this is absolutely impossible.” In other words, once we escape dogmatism we can only be idealists or speculative materialists, not correlationists. The human-world correlate is merely a fact, not an absolute necessity. But this facticity itself cannot be merely factical: it must be absolute. Here Meillassoux coins the French neologism factualité , which has been suitably translated into the English neologism “factiality.” (7, 122-3) Factialty means that for everything that exists, it is absolutely possible that it might be otherwise, not just that we cannot know whether or not it might be otherwise. Just as Kant transformed philosophy into a meditation on the categories governing human finitude, Meillassoux wishes to turn philosophy into a meditation on the necessary conditions of factiality, which he calls “figures”—a new technical term for him. (80) One such figure is that the law of non-contradiction must be true, and for an unusual reason. Since everything is proven to be contingent, nothing that exists can be contradictory, for whatever is contradictory has no opposite into which it might be transformed, and thus contingency would be impossible.3 Another such figure is that there must be something rather than nothing: for since contingency exists, something must exist in order to be contingent. It is a daring high-wire act, one that sacrifices realism to the correlational circle in order to rebuild it from out of its own ashes. Some might conclude that the lack of reason in things is a byproduct of the ignorance of finite humans, Meillassoux is making precisely the opposite point. For in fact, the doctrine of finitude usually leads directly to belief in a hidden reason. The fact that it lies beyond human comprehension merely increases our belief in this arbitrarily chosen concealed ground. By defending anew the concept of absolute knowledge Meillassoux evacuates the world of everything hidden. The reason for things having no reason is not that the reason is hidden, but that no reason exists. Thus, even while insisting on the necessity of non-contradiction, he rejects the other Leibnizian principle: sufficient reason. Everything simply is what it is, in purely immanent form, without deeply hidden causes. Or as Meillassoux puts it: “There is nothing beneath or beyond the manifest gratuitousness of the given—nothing but the limitless and lawless power of its destruction, emergence, or persistence” (63).The world is a “hyper-chaos”(64). But this is not the same thing as flux. For the chaos of the world is such that stability might occur just as easily as constant, turbulent change. Let’s now digress a bit, and return to the question of ancestrality, which Meillassoux transforms later in the book into “dia-chronicity.” Correlationism holds that all talk of a world outside the correlate is immediately recuperated by the correlate. The phrase “13.5 billion years ago” becomes “13.5 billion years ago for us ,” and the phrase “the universe following the extinction of humans” becomes “the universe following the extinction of humans for humans .” But notice that whether we talk about the world before or after humans, in both cases it is time that is used to challenge the correlate. Meillassoux has no interest in challenges that might be posed by space. For example, what about a vase in a lonely country house that topples to the floor and smashes when no one is there to watch it? Isn’t this also a challenge to correlationism, no less than the Big Bang or the heat death of the universe long after humans have vanished? In an eight-page supplement to the English translation of After Finitude ,4 possibly in response to my own 2007 review of the French original,5 Meillassoux bluntly denies that space is of any relevance to the question. Spatial distance is a merely harmless challenge to the human-world correlate. After all, even though no one is there in the lonely country house to witness the shattering of the vase, we can say that had there been an observer , that observer would have witnessed the toppling and destruction of the vase. For this event still occurs in a world in which the human-world correlate already exists, whereas the diachronicity of events both before and after the existence of humans makes it impossible to say that had there been an observer they would have witnessed the Big Bang occurring in such and such a fashion. However, it seems to me that Meillassoux merely asserts that the temporal simultaneity of our existence with that of the vase in the lonely country house is enough to render it harmless. It is true that the house does not exist prior to the correlate, but nonetheless it exists outside the correlate, and that is enough to make the same challenge. It is difficult to see why the “had there been an observer” maneuver succeeds in the case of a vase in the countryside in April 2011 but fails in the case of the Big Bang. This is not just a matter of nitpicking Meillassoux’s argumentative style: the fact that he bases his argument on time has at least two important consequences for his position. For in the first place, even though Meillassoux insists that the laws of nature are absolutely contingent, this turns out to be true only in a temporal sense. That is to say, it is a paradoxical feature of Meillassoux’s philosophy that he does allow for the existence of laws of nature, and simply believes that they can change at any moment without reason. Within any given moment, laws of nature do exist. He never suggests that different parts of the universe can have different laws at the same time, nor does he have any interest in the laws of part/whole composition that take place within any given instant. Could it be the case that rather than being made of gold atoms, a small chunk of gold could be made of silver atoms, cotton, horses, or that this same small piece of gold could be made of gigantic vaults filled with even more gold? These are not topics that draw Meillassoux’s attention, since he is focused solely on how the laws of nature might change or endure from one moment to the next . Another implication for Meillassoux’s system is that his concept of things-in-themselves turns out to be to be inadequate. For when he proves that things-in-themselves can exist without humans, this turns out to be true only in a temporal sense as well. Namely, things-in-themselves existed ten billion years ago, and they will continue to exist after all humans have succeeded in exterminating themselves. However, being able to exist before our births and after our deaths is just one small part of what it means to be a thing-in-itself. The more important part is that even if a thing is sitting on a table right now, in front of me, even if I stroke it lovingly or press my face up against it directly, I am still dealing only with a phenomenal version of the thing; the thing-in-itself continues to withdraw from all access. Yet no such thing is acknowledged by Meillassoux. For him finitude is a disaster, and absolute knowledge is in fact possible. Meillassoux’s thing-in-itself exists in independence only of the human lifespan , not of human knowledge. The fifth pillar of Meillassoux’s argument is his use of Cantor’s transfinite mathematics to show that even if the laws of nature are contingent, they need not be unstable, and thus we cannot use the apparent stability of nature to disprove his metaphysics of absolute contingency. What Cantor showed is that there are different sizes of infinity, and that all these infinities cannot be totalized in a single infinite number of infinities. Meillassoux sees this as crucial, since it allows him to discredit any “probabilistic” argument against his theory. The probabilistic argument (as defended quite clearly by Jean-René Vernes)6 would say this: given that the laws of nature seem so stable, it it is extremely improbable that there is no hidden reason for their remaining so stable. As Meillassoux sees it, probability is of value only when we can index an accessible total of cases. These can even be infinite: for example, there are an infinite number of points where a rope can break when stretched tight, but this does not stop us from calculating probabilities for various sections of the rope to break. By contrast, there is no way to sum up the number of possible laws of nature. For here there is no way to totalize; we cannot stand outside of nature and calculate the possible number of laws so as to determine the probabilities that any one of them might change. Therefore, although we can speak of probability when dealing with intraworldly events such as elections, horse races, and coin-flips, we cannot use the words “probable” or “improbable” when describing alterations at the level of nature as a whole. Rather than commenting on the validity of this argument and its use of Cantor, let me simply note that it once again creates a dualistic ontology. We already saw that Meillassoux treats time differently from space. In analogous fashion, he now treats the level of world differently from that of intraworldly events. The emergence of worlds is purely contingent and virtual and governed by no probability at all, while events within the world necessarily follow laws (even if these laws can change at any moment without reason), and thus their probabilities can be calculated. It is a strategy deeply reminiscent of Badiou’s (2005) own dualism between the normal “state of the situation” and the rare and intermittent “event.” The sixth and final pillar of Meillassoux’s book can be dealt with briefly, since we have already touched on it elsewhere. It comes at the very beginning of the book, when Meillassoux says that we must revive the distinction between primary and secondary qualities, and that the primary qualities are the ones that can be mathematized. He admits that he has not yet published a proof of this idea, though in fact it is already known as one of his primary doctrines. And here we encounter the familiar problem with Meillassoux’s inadequate conception of things-in-themselves. “Primary qualities” refers to those qualities that a thing has independently of its relations with us or anything else. But if the primary qualities can be mathematized, this means that they are not entirely independent of us, since our knowledge can get right to the bottom of them. The mathematized qualities of things are independent of us only in Meillassoux’s sense that they will still have those qualities even when all humans are dead. But to repeat, autonomy from the human lifespan is not the same as autonomy from human access. Here once more Meillassoux is concerned only with independence from the human-world correlate across time, not in any given instant. 2. L’Inexistence divine In 1997, the same year in which he turned thirty years old, Meillassoux earned his doctorate at the École normale supériuere with a brazen dissertation entitled L’Inexistence divine ( The Divine Inexistence ). The work was substantially revised in 2003. But even then, with typical fastidiousness, Meillassoux decided that the work was not yet ready for press. It has now been scrapped in favor of some future, multi-volume work bearing the same title. While writing my book on Meillassoux for Edinburgh University Press, I was permitted to translate excerpts from this unpublished work for use as an appendix in my own book; in total, the appendix contains approximately twenty percent of Meillassoux’s 2003 manuscript, the first time any of it will be published in any language. Nonetheless, a portion of the argument was already tested in the article “Spectral Dilemma,” published in English in the journal Collapse (2008: 261-75). There the philosophical motives for the virtual God are already made clear. What troubles us most are early deaths, brutal deaths, deaths of especial injustice– the sorts of deaths in which the brutal twentieth century was so abundant. And here, neither the atheist nor the believer can help us. The atheist can offer nothing but a sad and cynical resignation when reflecting on the victims of these terrible crimes. The believer does little better, being unable to explain how God could have allowed such things to happen, due to the famous intractability of the problem of evil. The solution offered to this dilemma by Meillassoux is bold, and all the more so given that he emerges from such a deeply Leftist, materialist, and unreligious background. His solution is that God does not yet exist, and therefore is not blameworthy for these catastrophes. Given that everything is contingent in Meillassoux’s philosophy, this God and divine justice might never exist, but they can at least exist as an object of hope. Let’s begin by jumping to the end of L’Inexistence divine , where the alternatives are laid out so nicely. There are four basic attitudes that humans can have towards God, Meillassoux says. First, we can believe in God because he exists. This is the classical theist attitude, rejected for the simple reason that it would be amoral and blasphemous to believe in a God who allows children to be eaten by dogs, to use Dostoevsky’s example. Second, we can disbelieve in God because he does not exist: the classical atheist attitude. But this leads to sadness, cynicism, and a sneering contempt for the greatness of human capacity. The third option, rather more complex, is to disbelieve in God because he does exist: in other words, to exist in rebellion against God as the one who must be blamed for the evils of the earth. The examples here might range from Lucifer himself, to the more human figure of Captain Ahab in Melville’s Moby-Dick , to Werner Herzog’s even more recent catchphrase, “Every man for himself, and God against all.” That leaves only the fourth option: believing in God because he does not exist. Meillassoux closes his book by saying that the fourth option has now been tried (namely, in the course of his own book), and that now that all four have been specified, we must choose. The first reaction to this theory of the inexistent God will be laughter. Few readers will ever be literally convinced by it, and probably none will immediately be convinced by it. But if we ask ourselves why we laugh, the answer is because it sounds so improbable that an inexistent God might suddenly emerge and resurrect the dead. It obviously sounds more like a gullible theology than a rigorous piece of philosophical work. Yet two things need to be kept in mind. First, Meillassoux’s theories are hardly more unlikely than those of great philosophers of the past such as Plato, Plotinus, Avicenna, Malebranche, Spinoza, Leibniz, Nietzsche, or Whitehead. We read the great philosophers not because their systems are plausible in commonsense terms that can be measured by the laws of probability. Instead, we read them precisely because they shatter the existing framework of common sense and open up new window on the universe. Second, and even more importantly, Meillassoux has already rejected probability as a valid measuring stick in philosophy. Or rather, he accepts probability in the intra-worldly realm (where it is linked with potentiality), and rejects it at the level of the world itself (where potentiality is replaced with what he calls virtuality). The virtual God can appear at any moment for no reason at all, just as any other new configuration of laws of nature can appear: in a manner that the laws of probability cannot calculate. Responding to those who might ridicule the idea of a sudden emergence of God and a resurrection of the dead, Meillassoux cites Pascal, who asserts that the resurrection of the dead would be far less incredible than the fact that we were born in the first place. This shifts philosophy onto new ground. Rather than concerning ourselves with what is likely to happen in the world as we know it, we focus instead on the most important things that could happen. For this reason, the expected objection that a virtual God is no more likely to appear than a virtual unicorn or a virtual flying spaghetti monster misses the point. Unicorns and spaghetti monsters could also appear, just like any other non-contradictory thing. But these would just be novel bizarre entities among others, not the heralds of completely new worlds. For Meillassoux, the emergence of matter, life, and thought have been the three truly amazing advents of the world so far, each of them dependent on the advent(s) preceding them. As he sees it, there can be no greater intraworldly entity than the human beings who already exist, since nothing in the world is better than the absolute knowledge of which humans alone are capable. This means that the next great advent must be something that perfects human beings rather than superseding them. And this can only be the world of justice, in which the dead are resurrected and their horrible deaths partially cancelled (Meillassoux never considers the possibility of a God who would literally erase the pre-divine past so that it never happened at all). The only immortality worth having is an immortality of this life, not an existence in some ill-defined afterworld. Human existence, he holds, must always be governed by a “symbol” that gives us the “immanent and comprehensible inscription of values in a world.” And just as cosmic history made the three great contingent leaps of matter, life, and thought, with a leap to justice as the only one still to come, a similar structure occurs within human culture and its symbols, which consist so far of the cosmological, naturalistic, and historical symbols, with a “factial” symbol still to come. We can review each of these symbols briefly. The cosmological symbol refers to the ancient dualism between the terrestrial and celestial spheres. Here below everything is conflict, corruption, and decay; but in the heavens nothing is perishable, all movement is circular, and everything is arranged in mutual harmony. This symbol is ended by modern physics when Galileo discovers such blemishes as sunspots and craters on the moon, and when Newton integrates both celestial and terrestrial movement into a single gravitational law. Next comes the naturalistic or romantic symbol, in which perfection comes not from the sky but from nature itself. The world is filled with pretty flowers (Meillassoux claims that the ancients never discussed the beauty of flowers until Plotinus in the third century) and with living creatures naturally moved by pity, at least until society corrupts them. This symbol collapses in the face of reality as we know it, since pity is no more common than war, corruption, and violence. This brings us to the historical symbol, which only now is passing away. Bad things may happen, but history has an inner logic of its own, such that everything works out in the end. The ultimate form of the historical symbol is the economic symbol, whether in a Marxist or neo-liberal form. Just as the Marxist holds that the inner economic logic of the capitalists will inexorably lead them to self-destruction, the neo-liberal assumes that the sum total of individual selfish actions will lead, in the long run, to the greatest possible good. We worship the economy and let it guide history for us, just as the ancients worshipped celestial bodies and held them to be free from blemish. The final remaining symbol is the factial symbol, which Meillassoux hopes will now emerge. Factiality, we recall, is his term for the absolute contingency of everything that exists. Once we have grasped this absolute contingency, we are free to expect the dramatic advent of the coming fourth World: the world of justice, inaugurated by a virtual God and even mediated by a messianic human figure. There is the added feature, however, that this messiah must abandon all claims to special status once the messianic realm of justice is achieved. The messianic figure will then be no more special than any person on the street, since a reign of human equality will have arisen. Although this focus on human being might seem like a return to standard humanism, Meillassoux holds that human pre-eminence has never truly been maintained. Previously, humans have been treated as special only because they contemplate the Good, because they resemble their omnipotent creator, or because they happen to be the temporary victors in a cruel Darwinian death-match between millions of living species. For Meillassoux, by contrast, humans have value because they know the eternal. But it is not the eternal that is important, since this merely represents the blind, anonymous contingency of each thing. What is important is not knowledge of the eternal , but knowledge of the eternal. We should not admire Prometheus for stealing fire from the gods; Prometheus is simply as bad as all the gods, no matter how much he increased our power. Feuerbach and Marx were wrong to say that God is a projection of the human essence, since for Meillassoux the usual concept of God represents the degradation of the human essence. If the traditional God was allowed to inflict plagues and tsunamis on the human race, the Promethean human of the twentieth century simply assumes the right to inflict death camps and atomic fireballs instead. In this respect, we have simply begun to imitate the degradation of humanity that was formerly invested in an omnipotent and arbitrary God. In response to charges that absolute contingency might lead to political quietism, Meillassoux counters that the World of justice would mean nothing unless we had already hoped for it beforehand. A World of justice that came along at random would merely be an improved third World of thought: indeed, a perfect one. But it would have satisfied no craving, and would therefore have no redemptive power. For this reason, we must actively hope for the fourth World of justice for such a fourth World ever to arise. Not only justice, but beauty is dependent on such hope: for Meillassoux, who is here somewhat dependent on Kant, beauty means an accord between our human symbolization and the actual world, which could never be present in a World of the blessed any more than justice could. And just as a messianic figure is needed to incarnate our hope and then abandon power once the World of justice is realized, it is the figure of the child whose fragile contingency shows us a dignity and a demand for justice beyond all power. 3. Meillassoux Radicalized or Reversed Given the promising reception of Meillassoux’s first book, it would not be groundless to engage in early speculation about what it might take to earn him a place in the history of philosophy. Maybe this will never happen—who knows?—but quite possibly it will: his lucid argumentative methods and sheer philosophical imagination at least make him a good candidate to be read well into the future, especially following further elaboration in print of his mature system. Philosophy is often practiced as thought it were nothing more than the amassing of “knockdown arguments.” But this is no more insightful than saying that good architecture is the amassing of steal beams. It is true that poorly constructed building cannot stand for long, but sound construction is merely the first, indispensable step in building. In fact, I am inclined to say that what really makes a philosopher important is not being right, but being wrong . I mean this in a very specific sense. I once heard the interesting remark about twentieth century culture that “you have to remember that the sixties really happened in the seventies.” That is to say, it was in the 1970’s rather than the more honored 1960’s that civil rights, free love, long hair, and the rock and roll drug culture really took root. With respect to the history of philosophy, we might just as easily say: “you have to remember that Plato really happened in Aristotle,” that “Kant really happened in Hegel” or “Hume really happened in Kant,” or that “Husserl’s phenomenology first achieved its truth in Heidegger.” One becomes an important philosopher not by being right, but by attracting rebellious admirers who tell you that you are wrong , even as their own careers silently orbit around your own. To recruit faithful disciples may be comforting and flattering, but the greatest thinkers have generally had to experience refutation at the hands of their most talented heirs. For this reason, I would propose that we size up the magnitude of living thinkers not by deciding how many times they are right and wrong, but by asking instead: who would take the trouble to refute this author? For this reason I do not ask: “Is Meillassoux right?”, since I do not believe in the virtual God myself, nor am I convinced by any important aspect of Meillassoux’s philosophy. Instead, I ask if there are interesting ways to overturn him. Only by being overturned, by no longer remaining a contemporary, does one become a classic. Let’s begin with a simple model of refutation, which can be refined further at a later date once the basic point is established. One kind of refutation simply consists in saying: “This author is a complete idiot.” The refuter now walks away in celebration, and no link between the present and the future is built; all is reduced to rubble. But this sort of mediocre triumphalism is generally practiced by those who achieve little of their own, and is not especially interesting. Much more interesting is the sort of refutation that does not take its target to be a complete idiot. I would like to suggest that there are just two basic ways in which this can be done: radicalization and reversal . It has not escaped my notice that this is a fairly good match for the Deleuzian distinction between irony and humor. Whereas irony critiques and adopts the opposite principle of what it attacks, humor accepts what it confronts but pushes it into highly exaggerated form. The ironist is like the worker who sows chaos by rebeling and contradicting the boss, while the humorist is like the worker who follows orders to an absurdly literal degree, with equally chaotic results. Let’s start with a few examples. In Aristotle’s treatment of Plato, and Heidegger’s of Husserl, we find reversal. Plato’s eidei are transformed by Aristotle into mere secondary substances, and the individual worldly things despised by Plato become what is primary. For Husserl what is primary is whatever is present to consciousness, while for Heidegger this is precisely what is secondary, since the primary stuff of the world withdraws from any form of presence at all. As for radicalization, it is most easily found in the transformation of Kant by German Idealism: “Kant was right to wall off the things-in-themselves from human access, and simply should have realized that the thought of the Ding an sich is also a thought, and thereby the noumena are just special cases of the phenomena,” with much following from this discovery. It would also be easy to read Spinoza as a radicalizer of Descartes, and Berkeley and Hume as radicalized versions of Locke. Perhaps the distinction is now sufficiently clear. Admiring refutations are not those that say “Professor X is an idiot,” which is merely the flip side of the eager disciple’s fruitless “Professor X got everything right.” Instead, it will be some variant of one of the following two options: “Professor X is important, but got it backwards,” or “Professor X is important, but didn’t push things far enough.” In the history of philosophy these two latter cases have often been painful in purely human terms: Aristotle expresses sadness at refuting Plato, Kant is openly annoyed at Fichte, and Husserl feels betrayed and used by Heidegger. Rude handling from later figures almost seems to be the sine qua non of being a great philosopher. Now, it has already been claimed that Meillassoux is an emerging philosopher of the first importance, and by no less a figure than Alain Badiou: “It would be no exaggeration to say that Quentin Meillassoux has opened up a new path in the history of philosophy…” (Preface, vii). But rather than taking Badiou’s word for it, or rejecting his word, we might experiment by asking how Meillassoux could be radicalized or reversed. Are there interesting ways of doing this that might launch whole new schools of philosophy, unexpected or even condemned by Meillassoux himself? While no one can see the future, the present is poor when it is not riddled with virtual futures. The relation between philosophers and their predecessors and successors is always somewhat complicated, of course. But generally there is one central divergence at stake, which might be taken as the key to all the others. On this basis we could say that new thinkers primarily radicalize or primarily reverse the main ideas of their chief philosophical forerunner. There may be specific historical conditions and perhaps even personality traits connected with these two types, but this question can be left aside for now. More important for us is that radicalizers will generally be followed by reversers, and vice versa. Consider the textbook example of a reversal in the history of philosophy: Kant’s Copernican Revolution, which inverts the so-called dogmatic tradition that addresses the world itself, and makes the world revolve instead around the conditions by which it is known. While it is not completely impossible that Kant’s successors might have re-reversed this principle back into a new and stronger dogmatic realism, conditions were premature for such a move. Anyone doing this too early would likely have been an angry anti-Kantian reactionary rather than an original thinker in command of a genuinely new realist principle. The far more likely outcome is the one that actually happened: Kant’s reversal of his predecessors was viewed as incomplete, or as retaining lamentable bits of the traditional view, which despite his admirable breakthrough he was unable to shake off. This was the view of German Idealism, anyway. In similar fashion, Spinoza could also be viewed as a radicalizer of Descartes, who is equally accused of preserving various Scholastic dogmas in an otherwise radical project of philosophical reversal. The point is this: reversals in the history of thought tend to be followed soon thereafter by radicalizations of those reversals. The same may hold true in reverse: radicalizations might generally be followed by reversals, given that it is not always possible to be more radical than the radicals have already been. Consider the case of Husserl, who radicalizes Brentano’s early vagueness about what lies beyond immanent objectivity, and Twardowski’s assertion that there must be an external object lying outside the intentional content, by collapsing everything into the intentional sphere: there is no difference between the Berlin in my consciousness and the actual Berlin that is home to millions of people. It is difficult to see how one could be even more radical than Husserl’s idealist turn here. And thus the road is paved to Heidegger’s reversal of classical phenomenology, in which the key point is what lies deeper than any presence to consciousness: the Sein whose power and obscurity cannot be made exhaustively present, but only sends itself in historical epochs. In similar fashion we might also read Leibniz as a reverser of Spinoza’s radicalization, retrieving a strong sense of individual substance and a certain validity of what the Scholastics had said. Returning to Meillassoux, we might ask which kind of philosopher he is: a radicalizer or a reverser? At present, Meillassoux looks to me like a radicalizer (though for now his future remains shrouded in mist). He takes the correlationist tradition, which allows us to speak only of the relation between human and world, and tries to raise it into an even more extreme claim about the absolute contingency of everything. But whereas German Idealism did this by trying to collapse the distinction between thought and world entirely into the “thought” side, Meillassoux does it by trying to shift the non-absolute contingency of the thought-world correlate from epistemology to ontology. It is no longer a question of the inability of human knowledge to know what lies outside the correlate, but the inability of reality itself to be rooted in any definite laws. Furthermore, if we look at the various features of Meillassoux’s philosophy identified earlier tonight, all but one are already so radical that there is no obvious way to push them further. The one exception would be his claim that the world as a whole can change for no reason at any moment, coupled with the inconsistent claim that within a given world there are laws of nature that everything must follow. If gravitational attraction between all masses is a current law in our world, then for Meillassoux there can be no exceptions to this law for as long as it remains in force. A toppled vase will fall to the floor every time for sure,unless there is a cosmic change by which the laws of nature as a whole have altered. (This is reminiscent of the late medieval distinction between the absolute and ordained power of God, according to which God has the power to set or change the laws of nature, but not to contravene those laws locally once they are set.) On this point, to radicalize Meillassoux would simply be to say: there are no laws of nature even in the local sense. Everything that happens, even in the world here and now, is purely contingent and not governed by even a trace of law. And while this would be a more consistent development of Meillassoux’s thoughts on contingency, it is difficult to see how it could lead to a new philosophy. Instead, the admiring successors of Meillassoux are more likely to reverse one of his already sufficiently radical points. At least four candidates come to mind: *First, we have seen that Meillassoux thinks correlationism is challenged by a time before or after consciousness, but not by a space lying outside it. Perhaps this could be reversed into saying that spatial exteriority is the really crucial point. The arguments on this point are perhaps the least convincing in After Finitude (and do not even occur in the original French edition), and therefore it might be a candidate for the “blind spot” of which no philosopher is ever free. *Second, Meillassoux uses Cantor to claim that the contingency of laws of nature would not entail that they are unstable. A successor of Meillassoux might claim that it does make them unstable, and celebrate this fact. This person would then have to explain why common sense seems to encounter a relatively stable world despite its truly rampant instability. Whereas Meillassoux’s problem is to show how stability might exist despite contingency, this successor’s problem would be slightly different: to show why actual, full-blown instability might have the appearance of stability. *Third, Meillassoux claims that the primary qualities of things are those that can be mathematized. He might be reversed by a successor who says the opposite: the mathematizable qualities are the secondary ones, and the primary ones are those that elude symbolic formulation. While this is a perfectly valid possible objection to Meillassoux, it is one that is made in advance by some of his predecessors and is still made by some of his peers, making it less interesting for futurology than some of his other points. *Fourth and finally, whereas Meillassoux claims that God does not exist but might exist in the future, a successor might argue even more bizarrely that God has always existed but might vanish in the future. Let’s arbitrarily select the first of these possibilities, and imagine briefly where it might lead, if pursued in the future by admiring detractors. Meillassoux comes from the circle of Badiou, and some of Badiou’s most ardent admirers are found in Latin America. So, let’s imagine that towards mid-century some ingenious reversers of Meillassoux emerge in that portion of the Spanish-speaking world. Just for fun, let’s call them Castro and Chávez. And in order to avoid any confusion with the present-day politicians of those names, we will stipulate that Meillassoux’s great successors are both women. The philosopher Castro (we will suppose she comes from Peru) reverses Meillassoux’s argument that the ancestral or diachronic are what most threaten the human-world correlate. Instead, she claims that the diachronic does not threaten the correlate at all, and that we must instead look at space as what ruins the correlate and demands a strange new realism. What would such a philosophy look like? In order to determine this, we might ask what price Meillassoux pays for doing it the opposite way. As I see it, he pays in two separate ways. One is that laws of nature for him are contingent over time . The laws of nature apply to the universe as a whole at any given moment, and would be changed globally if they are ever changed at all. The second price he pays is that Meillassoux has no mereology , or theory of parts and wholes. Everything for him is on the level of the given, or immanent in experience, with the sole proviso that the laws governing this immanence might change without notice at any given moment. In reversing Meillassoux, Castro makes the following claims in the preface to her stunning debut book of 2045, The Cosmos and its Neighborhoods , rapidly translated from Spanish into all the languages of the world: Despite his brilliant analysis of the contingency of laws of nature over time, Meillassoux gets two important assumptions wrong. First, he allows for only one set of contingent laws to govern nature as a whole. Second, he allows laws to govern only the world that is immanent in experience, and thereby fails to explore the contingency among part-whole relations. In this book I will argue, first, that the laws of nature vary in any given instant between one region of the universe and the next; and second, that the world is made up of layers of parts and wholes that are also contingent with respect to one another. Those are the words of Castro. This may sound like a hopeless free-for-all of chaos, yet the book somehow succeeds in drawing some compelling deductions about how laws must vary from one place or level in the world to the next. Trapped in the limited horizon of 2011, and not yet inspired by the heavily balkanized political and technological situation of 2050 that somehow lends additional credence to Castro’s vision, we can only vaguely grasp what such a philosophy might look like. After this reversal of Meillassoux by Castro, the usual pattern leads us to expect a radicalization by Chávez, a young Argentine student of Castro. How could the already strange theories of Castro be radicalized? Perhaps as follows, in a disturbing new book entitled The Implosion of the Neighborhoods , which argues as follows: Castro was right to shift the Meillassouxian framework of contingency from time to space. However, in this respect she retained a surprisingly traditional opposition between the two. In this book I will show that time and space collapse into one another. This may sound too much like the discredited four-dimensional block universe of twentieth century physics and philosophy. However, the four-dimensional universe is a model biased in favor of space, merely adding an extra dimension to the commonsense spatial continuum while stipulating that the serial passage of time is an illusion. In this book I will argue instead for a one-dimensional space-time modeled after our experience of time, in which there is no simultaneous co-existence at all between different parts of the universe, or ‘neighborhoods’ as my esteemed teacher Castro has called them. Instead, the various portions of the universe link to one another by succession rather than by coexistence. Buenos Aires, New York, and Amsterdam do not exist simultaneously in the same landscape, but one after the other in the mind of some observer, and this observer can only be an observer much larger than any human. Against Meillassoux’s notion of a virtual God that does not exist now but might exist in the future, I will argue for an actual God that surveys the universe in sequence, thereby generating the illusion of spatial diversity and even the illusion of individual minds located within that diversity. Once this divine observer dies, the universe as a whole must perish. Again, these ideas are so bizarre that we of 2011 can barely comprehend them, just as Aristotle would have had a difficult time grasping the theories of Descartes. We could then perhaps imagine a further reversal of this theory, emanating from the intellectually resurgent Philippines of the twenty-second century. The Filipino School might argue that the universe is already dead, given the collapse of its spatial richness into the serial observations of a flimsy and mortal God. The virtual universe does not yet exist, but might exist in fully spatial form in the future, and this would require the death of God and the resulting liberation of God’s succession of images as independent, spatially situated realities. With a bit of sharpening, we might be able to make all of these imaginary thinkers more intuitively clear. Along with the history of philosophy, there might arise a new discipline generating imaginary futures for philosophy. The richness of Meillassoux’s system comes not from the fact that he is plausibly right about so many things, but because his philosophy offers such a treasury of bold statements ripe for being radicalized or reversed. He is a rich target for many still-unborn intellectual heirs, and this is what gives him the chance to be an important figure. NOTES 1. Quentin Meillassoux, After Finitude . Trans. R Brassier. (London: Continuum, 2008.) Page 5. The word “correlationism” does not appear in his doctoral thesis. As Meillassoux informed me in an email of February 8, 2011, he first coined this term in 2003 or 2004, while editing for publication a lecture he had given at the École normale supérieure on a day devoted to the theme of “Philosophy and Mathematics,” an event including Alain Badiou as one of the participants. 2. Graham Harman, Quentin Meillassoux: Philosophy in the Making . (Edinburgh: Edinburgh University Press, forthcoming 2011). 3. In an email of December 6, 2010, Meillassoux clarifies that in After Finitude he only deduces the impossibility of a “universal contradiction,” not of a determinate contradiction. In the same email he suggests that he can also prove the latter, though the proof is somewhat lengthier than the one found in After Finitude . 4. After Finitude (18-26), in the passage falling between the two sets of triple asterisks. These pages were sent by Meillassoux to translator Ray Brassier (in French) during the translation process, and do not appear in the original French version of the book. 5. Graham Harman, “Quentin Meillassoux: A New French Philosopher.” Philosophy Today 51.1 (2007): Pages 104-117. The passage where I raise the question of space can be found in the first column of page 107. 6. Vernes is first cited on p. 95 of After Finitude . See Jean-René Vernes, Critique de la raison aléatoire, ou Descartes contra Kant . (Paris: Aubier, 1982).  . (shrink)

This article discusses the meanings, origin, context, scope, and central intellectual claims of atheism in the Age of the European Enlightenment. It emphasizes debates about proofs of the existence of God and about the problem of categorical naturalism, that is, of whether or not the world we observe and its seeming design could be the product of unintelligent causes. It explores the philosophical origins of Enlightenment atheism both in prior heterodox and Epicurean thought, and, of even greater importance, in the (...) orthodox debates, scholarship, and mutual contestations that generated so many of the themes and often arguments of Enlightenment atheists. It pays special attention to the complexity of the relationship between philosophical skepticism and atheistic thought. Given the flowering of explicitly atheistic thought in the late French Enlightenment, the article looks closely at the work of Denis Diderot, the baron d’Holbach, and Jacques-André Naigeon. (shrink)

The article is devoted to philosophy of mathematics. Mathematical heuristics, being a complex of methods for solving the non-standard problems of mathematics (such problems which have no known algorithms to be solved), is the main subject of the research. As a specific mechanism for thinking, generating elements of guesswork needed as the basis of mathematical heuristics, the author considers intuition. In the work, the author uses Descartes’s, Poincaré’s, Hadamard’s and Piaget’s findings. Based on Descartes’s concept of rational intuition, the author (...) develops the concept of heuristic intuition. As a result, the author turns to the question of possibility of a complete translation of the user-derived mathematical statements in a discourse, in fact, that means a maximum depth of mathematical proof, i.e. its maximum rationalization. For this purpose, it is necessary to re-attract the intuition since it is able to transform the intuitive elements into the discourse ones. Therefore, from this point of view, the rationale is intuitively derived mathematical proof should be no more than a ‘multilayer‘ creative process. In general, the author, based on Poincaré’s research, proves that the essence of mathematical creativity is not to «sort out» and «choose». Referring to examples for illustration, the author reveals moments of «interference» of intuition, even in the process of solving school problems. Therefore, it is currently impossible to ignore the phenomenon of intuition and the results that have been historically derived a theory of knowledge in the study of creative mechanisms. (shrink)

We consider the problem about the length of proofs of the sentences $\operatorname{Con}_S(\underline{n})$ saying that there is no proof of contradiction in S whose length is ≤ n. We show the relation of this problem to some problems about propositional proof systems.

One of the main motivations for having a compositional semantics is the account of the productivity of natural languages. Formal languages are often part of the account of productivity, i.e., of how beings with finite capaci- ties are able to produce and understand a potentially infinite number of sen- tences, by offering a model of this process. This account of productivity con- sists in the generation of proofs in a formal system, that is taken to represent the way speakers grasp (...) the meaning of an indefinite number of sentences. The informational basis is restricted to what is represented in the lexicon. This constraint is considered as a requirement for the account of productivity, or at least of an important feature of productivity, namely, that we can grasp auto- matically the meaning of a huge number of complex expressions, far beyond what can be memorized. However, empirical results in psycholinguistics, and especially particular patterns of ERP, show that the brain integrates informa- tion of different sources very fast, without any felt effort on the part of the speaker. This shows that formal procedures do not explain productivity. How- ever, formal models are still useful in the account of how we get at the seman- tic value of a complex expression, once we have the meanings of its parts, even if there is no formal explanation of how we get at those meanings. A practice-oriented view of modeling gives an adequate interpretation of this re- sult: formal compositional semantics may be a useful model for some ex- planatory purposes concerning natural languages, without being a good model for dealing with other explananda. (shrink)

Speakers’ perception of a visual scene influences the language they use to describe it—which objects they choose to mention and how they characterize the relationships between them. We show that visual complexity can either delay or facilitate description generation, depending on how much disambiguating information is required and how useful the scene's complexity can be in providing, for example, helpful landmarks. To do so, we measure speech onset times, eye gaze, and utterance content in a reference production experiment (...) in which the target object is either unique or non-unique in a visual scene of varying size and complexity. Speakers delay speech onset if the target object is non-unique and requires disambiguation, and we argue that this reflects the cost of deciding on a high-level strategy for describing it. The eye-tracking data demonstrate that these delays increase when speakers are able to conduct an extensive early visual search, implying that when speakers scan too little of the scene early on, they may decide to begin speaking before becoming aware that their description is underspecified. Speakers’ content choices reflect the visual makeup of the scene—the number of distractors present and the availability of useful landmarks. Our results highlight the complex role of visual perception in reference production, showing that speakers can make good use of complexity in ways that reflect their visual processing of the scene. (shrink)

The contemporary use of the term ‘complexity’ frequently indicates that it is considered a unified concept. This may lead to a neglect of the range of different theories that deal with the implications related to the notion of complexity. This paper, integrating both the English and the Latin traditions of research associated with this notion, suggests a more nuanced use of the term, thereby avoiding simplification of the concept to some of its dominant expressions only. The paper further (...) explores the etymology of ‘complexity’ and offers a chronological presentation of three generations of theories that have shaped its uses; the epistemic and socio‐cultural roots of these theories are also introduced. From an epistemological point of view, this reflection sheds light on the competing interpretations underlying the definition of what is considered as complex. Also, from an anthropological perspective it considers both the emancipatory as well as the alienating dimensions of complexity. Based on the highlighted ambiguities, the paper suggests in conclusion that contributions grounded in contemporary theories related to complexity, as well as critical appraisals of their epistemological and ethical legitimacy, need to follow the recursive feedback loops and dynamics that they constitute. In doing so, researchers and practitioners in education should consider their own practice as a learning process that does not require the reduction of the antagonisms and the complementarities that shape its own complexity. (shrink)