We first discuss some technical questions which arise in connection with the construction of undecidable propositions in analysis, in particular in connection with the notion of the normal form of a function representing a predicate. Then it is stressed that while a function f(x) may be computable in the sense of recursive function theory, it may nevertheless have undecidable properties in the realm of Fourier analysis. This has an implication for a conjecture of Penrose's which states that classical physics is (...) computable. (shrink)

We investigate systems of ordinary differential equations with a parameter. We show that under suitable assumptions on the systems the solutions are computable in the sense of recursive analysis. As an application we give a complete characterization of the recursively enumerable sets using Fourier coefficients of recursive analytic functions that are generated by differential equations and elementary operations.

A summation of the work of one of the most influential and provocative interpreters of science, it aims at saving what is good and valuable in modernity and ...

From the book: What is to be done with political ecology? Nothing. What is to be done? Political ecology!

I argue against the interpretation of propositions as intentions and proof-objects as fulfillments proposed by Heyting and defended by Tieszen and van Atten. The idea is already a frequent target of criticisms regarding the incompatibility of Brouwer’s and Husserl’s positions, mainly by Rosado Haddock and Hill. I raise a stronger objection in this paper. My claim is that even if we grant that the incompatibility can be properly dealt with, as van Atten believes it can, two fundamental issues indicate that (...) the interpretation is unsustainable regardless: (1) it is hard to determine, without appealing to propositional intentions on pain of circularity, what intention a proof-object should be understood as a fulfillment of; (2) due to a difficult fulfillment dilemma, it is unclear, at best, what the object of an intention corresponding to a proposition is. (shrink)

Throughout his career, Schelling assigns knowledge of the absolute first principle of philosophy to intellectual intuition. Schelling's doctrine of intellectual intuition raises two important questions for interpreters. First, given that his doctrine undergoes several changes before and after his identity philosophy, to what extent can he be said to “hold onto” the same “sense” of it by the 1830s, as he claims? Second, given that his doctrine of intellectual intuition restricts absolute idealism to what he calls a “science of reason”, (...) which he says cannot prove the absolute’s existence, what other doctrine does he require in order to prove this? I will answer these questions by tracing the shifts in Schelling’s doctrine of intellectual intuition from the 1790s to the 1830s and drawing out its evolving methodological role within his science of intelligibility. (shrink)

This paper argues that we should solve paradoxes for propositions (such as the Russell–Myhill paradox) in essentially the same way that we solve Russellian paradoxes for sets. That is, the standard, iterative approach to sets is extended to include properties, and then the resulting hierarchy of sets and properties is used to construct propositions. Propositions on this account are structured in the sense of mirroring the sentences that express them, and they would seem to serve the needs of philosophers of (...) language and metaphysicians who rely on such propositions. The resulting account has limitations, comparable to those faced by the iterative approach to sets, but it is argued to be in important ways preferable to others that have been proposed. (shrink)

É possível constatar, mesmo em uma leitura superficial, que Kant tem o propósito de empreender em seu opúsculo de 1795, À Paz Perpétua, uma defesa dos princípios normativos do direito em todas as esferas da vida pública. Isso se evidencia na tentativa de desenvolver uma teoria da paz erigida sobre uma teoria tríplice do direito público, dividida nos âmbitos do direito estatal, das gentes e cosmopolita. Mas, se a questão é, por um lado, adequar os princípios puros da doutrina do (...) direito à teoria da paz, por outro, o problema com o qual Kant tem de lidar é essencialmente histórico. Não se trata apenas de justificar, em um contexto abrangente, a normatividade dos princípios do direito, mas também de demonstrar a sua aplicabilidade. Por conseguinte, em seu empreendimento, Kant encontra-se diante do desafio de demonstrar a inconsistência da “expressão corrente”, com a qual teve que lidar antes, que diz que “isto pode ser correto na teoria, mas nada vale na prática”. Tal preocupação pode ser observada direta e indiretamente, em várias passagens, no decorrer do texto, explicitando-se ,sobretudo, na discussão apresentada no Suplemento Primeiro no qual a questão suscitada pelo problema da teoria e prática, acerca das condições de possibilidade de realização do ideal da paz, é discutida no contexto de sua filosofia da história e também no Apêndice I, no qual, através de uma reflexão sobre o consenso entre moral e política, o que se põe em questão é a possibilidade da concordância entre teoria e prática no contexto da filosofia e da prática política. Ocupar-me-ei nesse texto apenas em tratar da questão sob esse segundo aspecto, tentando destacar que a estratégia kantiana para superar o problema da teoria e prática, no âmbito da filosofia política, depende do desenvolvimento de uma “renovada” doutrina da prudência. Como observa Castillo1, “[p]ara provar que a teoria jurídica da paz se aplica na teoria e na prática ao mesmo tempo”, Kant “deve fazer um exame crítico do conceito de prudência do Estado”2 . Esta análise deve nos permitir concluir, em primeiro lugar, que, bem diferente da visão antes difundida de que a filosofia política de Kant é expressão de um moralismo “que é certamente sincero, mas basicamente ingênuo”3, o objetivo de Kant, na verdade, nunca foi o de se afastar da realidade concreta da política em nome de um ideal abstrato de virtude política. Sua preocupação foi, em vez disso, a de ajustar os co. (shrink)

The combination of increased availability of large amounts of fine-grained human behavioral data and advances in machine learning is presiding over a growing reliance on algorithms to address complex societal problems. Algorithmic decision-making processes might lead to more objective and thus potentially fairer decisions than those made by humans who may be influenced by greed, prejudice, fatigue, or hunger. However, algorithmic decision-making has been criticized for its potential to enhance discrimination, information and power asymmetry, and opacity. In this paper, we (...) provide an overview of available technical solutions to enhance fairness, accountability, and transparency in algorithmic decision-making. We also highlight the criticality and urgency to engage multi-disciplinary teams of researchers, practitioners, policy-makers, and citizens to co-develop, deploy, and evaluate in the real-world algorithmic decision-making processes designed to maximize fairness and transparency. In doing so, we describe the Open Algortihms project as a step towards realizing the vision of a world where data and algorithms are used as lenses and levers in support of democracy and development. (shrink)

The aim of the paper is to argue that all—or almost all—logical rules have exceptions. In particular, it is argued that this is a moral that we should draw from the semantic paradoxes. The idea that we should respond to the paradoxes by revising logic in some way is familiar. But previous proposals advocate the replacement of classical logic with some alternative logic. That is, some alternative system of rules, where it is taken for granted that these hold without exception. (...) The present proposal is quite different. According to this, there is no such alternative logic. Rather, classical logic retains the status of the ‘one true logic’, but this status must be reconceived so as to be compatible with (almost) all of its rules admitting of exceptions. This would seem to have significant repercussions for a range of widely held views about logic: e.g. that it is a priori, or that it is necessary. Indeed, if the arguments of the paper succeed, then such views must be given up. (shrink)

Giordano Bruno's notorious public death in 1600, at the hands of the Inquisition in Rome, marked the transition from Renaissance philosophy to the Scientific Revolution of the seventeenth century. In his philosophical works he addressed such delicate issues as the role of Christ as mediator and the distinction, in human beings, between soul and matter. This volume presents new translations of Cause, Principle and Unity, in which he challenges Aristotelian accounts of causality and spells out the implications of Copernicanism (...) for a new theory of an infinite universe, and of two essays on magic, On Magic and A General Account of Bonding, in which he interprets earlier theories about magical events in the light of the unusual powers of natural phenomena. (shrink)

L'origine dell'anima umana secondo Dante.--La dottrina d'Alberto Magno sull'Inchoatio formae.--Alberto Magno e san Tommaso.--La posizione di Alberto Magno di fronte all'averroismo.--L'anima umana secondo Sigieri.--Anima e corpo nel pensiero di san Tommaso.--L'aristotelismo della scolastica e i francescani.--Individualità e immortalità nell'averroismo e nel tomismo.

Fichte assigns ‘intellectual intuition’ a new meaning after Kant. But in 1799, his doctrine of intellectual intuition is publicly deemed indefensible by Kant and nihilistic by Jacobi. I propose to defend Fichte’s doctrine against these charges, leaving aside whether it captures what he calls the ‘spirit’ of transcendental idealism. I do so by articulating three problems that motivate Fichte’s redirection of intellectual intuition from being to acting: (1) the regress problem, which states that reflecting on empirical facts of consciousness leads (...) only to further facts and so cannot yield a first principle; (2) the rhapsody problem, which states that the categories form a haphazard set and so lack necessity unless they derive from a first principle; and (3) the nihilism problem, which states that a first principle cannot lie outside our cognition of it, lest it be the cause of our cognition and, being first, the cause of all our actions, reducing us to machines. Crucially, Fichte’s three motivating problems are in fact aspects of a single problem. Leaving any aspect unsolved spoils putative solutions to the other two. Consequently, Fichte requires a single unified solution to all three, which his doctrine of intellectual intuition provides. (shrink)

This article calls for a rethinking of critical sociology. Representing classical critical sociology, the Bourdieusian paradigm illustrated domination, but its negative foundation removed actors’ power, privileging sociological knowledge as capable of identifying (social) transcendental categories of thought. Latour’s constructivism challenged this privilege, giving actors the political power of aggregating collectives around their common concerns at the cost of emphasizing domination and critique. We propose a critical approach that evades a transcendental perspective reliant on pure negation, producing a more positive critical (...) sociology founded on processualist phenomenological and pragmatic perspectives, principally Merleau-Ponty and Dewey. Actors and researchers can collectively ruminate on their perspectives and concerns to challenge power structures and transform their worlds. At the center of this is the social-mien, a creative reimagination of habitus. A brief discussion of the protests and scholarship surrounding the Notre-Dame-des-Landes activists’ practice of democracy highlights the utility of the approach and serves as a model for future application. (shrink)

One of the most debated questions in psychology and cognitive science is the nature and the functioning of the mental processes involved in deductive reasoning. However, all existing theories refer to a specific deductive domain, like syllogistic, propositional or relational reasoning.Our goal is to unify the main types of deductive reasoning into a single set of basic procedures. In particular, we bring together the microtheories developed from a mental models perspective in a single theory, for which we provide a formal (...) foundation. We validate the theory through a computational model (UNICORE) which allows fine‐grained predictions of subjects' performance in different reasoning domains.The performance of the model is tested against the performance of experimental subjects—as reported in the relevant literature—in the three areas of syllogistic, relational and propositional reasoning. The computational model proves to be a satisfactory artificial subject, reproducing both correct and erroneous performance of the human subjects. Moreover, we introduce a developmental trend in the program, in order to simulate the performance of subjects of different ages, ranging from children (3–6) to adolescents (8–12) to adults (>21). The simulation model performs similarly to the subjects of different ages.Our conclusion is that the validity of the mental model approach is confirmed for the deductive reasoning domain, and that it is possible to devise a unique mechanism able to deal with the specific subareas. The proposed computational model (UNICORE) represents such a unifying structure. (shrink)

Plausibly, mathematical claims are true, but the fundamental furniture of the world does not include mathematical objects. This can be made sense of by providing mathematical claims with paraphrases, which make clear how the truth of such claims does not require the fundamental existence of mathematical objects. This paper explores the consequences of this type of position for explanatory structure. There is an apparently straightforward relationship between this sort of structure, and the logical sort: i.e. logically complex claims are explained (...) by logically simpler ones. For example, disjunctions are explained by their (true) disjuncts, while generalizations are explained by their (true) instances. This would seem as plausible in the case of mathematics as elsewhere. Also, it would seem to be something that the anti-realist approaches at issue would want to preserve. It will be argued, however, that these approaches cannot do this: they lead not merely to violations of the familiar principles relating logical and explanatory structure, but even to reversals of these. That is, there are cases where generalizations explain their instances, or disjunctions their disjuncts. (shrink)

General-elimination harmony articulates Gentzen’s idea that the elimination-rules are justified if they infer from an assertion no more than can already be inferred from the grounds for making it. Dummett described the rules as not only harmonious but stable if the E-rules allow one to infer no more and no less than the I-rules justify. Pfenning and Davies call the rules locally complete if the E-rules are strong enough to allow one to infer the original judgement. A method is given (...) of generating harmonious general-elimination rules from a collection of I-rules. We show that the general-elimination rules satisfy Pfenning and Davies’ test for local completeness, but question whether that is enough to show that they are stable. Alternative conditions for stability are considered, including equivalence between the introduction- and elimination-meanings of a connective, and recovery of the grounds for assertion, finally generalizing the notion of local completeness to capture Dummett’s notion of stability satisfactorily. We show that the general-elimination rules meet the last of these conditions, and so are indeed not only harmonious but also stable. (shrink)