This paper presents a new reconstruction of Wittgenstein’s famous (and controversial) rule-following arguments. Two are the novel features offered by our reconstruction. In the first place, we propose a shift of the central focus of the discussion, from the general semantics and the philosophy of mind to the philosophy of mathematics and the rejection of the notion of a function. The second new feature is positive: we argue that Wittgenstein offers us a new alternative notion of a rule (to replace (...) the rejected functions), a notion reminiscent of Category Theory’s notion of a morphism. (shrink)
This paper offers a new interpretation for Wittgenstein`s treatment of mathematical identities. As it is widely known, Wittgenstein`s mature philosophy of mathematics includes a general rejection of abstract objects. On the other hand, the traditional interpretation of mathematical identities involves precisely the idea of a single abstract object – usually a number –named by both sides of an equation.
Cet article poursuit un double but : d’une part, situer dans le parcours philosophique de Wittgenstein une partie de l’histoire du « problème du champ visuel », thème clé de sa période intermédiaire ; d’autre part, mettre en lumière sa critique de l’idée d’un champ visuel (et celle de l’idée d’un objet interne). Nous croyons que ses arguments sont nouveaux, pénétrants, et ainsi leur intérêt dépasse les limites d’un exposé purement exégétique.
This paper offers a reconstruction of Wittgenstein's discussion on inductive proofs. A "algebraic version" of these indirect proofs is offered and contrasted with the usual ones in which an infinite sequence of modus pones is projected.
This paper offers an overview of various alternative formulations for Analysis, the theory of Integral and Differential Calculus, and its diverging conceptions of the topological structure of the continuum. We pay particularly attention to Smooth Analysis, a proposal created by William Lawvere and Anders Kock based on Grothendieck’s work on a categorical algebraic geometry. The role of Heyting’s logic, common to all these alternatives is emphasized.
O presente artigo tem como tema as extensas discussões de Wittgenstein sobre uma das formas mais simples e elementares de infinitude em matemática: as dízimas periódicas. Tentamos organizar os vários argumentos do autor em uma única exposição continuada. No final do artigo, introduzimos, ainda que de forma breve, o famoso argumento sobre “execução de regras” de Wittgenstein, bem como a idéia de interpretações nãostandard de processos infinitos.
Wittgenstein’s philosophy of mathematics involves two highly controversial theses: the idea that mathematical propositions are not about (abstract) objects and the idea that no mathematical conjecture is ever answered as such, because the advent of the proof always determines a semantical shift of the meanings of the terms involved in the conjecture. The present article offers a reconstruction of Wittgenstein’s arguments supporting these theses within a very restricted setting: Archimedes’ discovery of an algorithm for calculating the number Pi.
This article discusses some of Chateaubriand’s views on the connections between the ideas of formalization and infinity, as presented in chapters 19 and 20 of Logical Forms. We basically agree with his criticisms of the standard construal of these connections, a view we named “formal proofs as ultimate provings”, but we suggest an alternative way of picturing that connection based on some ideas of the late Wittgenstein.
Este artigo oferece uma nova reconstrução para os argumentos do famoso segundo capítulo de Word and Object de Quine e sua idéia da Tradução Radical . De acordo com essa abordagem, o maior alvo de Quine é a noção de composicionalidade como sendo o elemento fundamental para qualquer teoria do significado. Em poucas palavras, não poderia haver nenhuma “teoria do significado”, para Quine, simplesmente porque a noção de composicionalidade deveria ser rejeitada como a concepção central da semântica. Além disso, tomamos (...) o cuidado de diferenciar argumentos empíricos de argumentos a priori de natureza modal. Esses últimos constituem-se no que propomos chamar de Teorema do Automorfismo de Quine , a idéia de que há maneiras alternativas de se reconstruir a estrutura gramatical de qualquer língua incluindo predicação e que poderiam manter invariantes todas as nossas predisposições para comportamento verbal sob quaisquer estados de coisas, atuais ou meramente possíveis. Em nosso entender, é esse teorema que determina fundamentalmente a rejeição da composicionalidade por Quine e, assim, de todas as teorias do significado. (shrink)
This article offers an explanation of perhaps Wittgenstein’s strangest and least intuitive thesis – the semantical mutation thesis – according to which one can never answer a mathematical conjecture because the new proof alters the very meanings of the terms involved in the original question. Instead of basing our justification on the distinction between mere calculation and proofs of isolated propositions, characteristic of Wittgenstein’s intermediary period, we generalize it to include conjectures involving effective procedures as well.
The main issue André Porto raises in his paper concerns the use of dot notation to indicate an infinite set of hypotheses. Whereas I agree that one cannot extract a unique infinite expansion from a finite initial segment, in my response I argue that this holds for finite expansions as well. I further explain how my remarks on infinite proof structures are neither motivated by the impact of Gödel’s incompleteness theorems on Hilbert’s program, nor by a negative view of (...) strict finitism.O problema central que André Porto discute em seu artigo diz respeito ao uso da notação de pontos para indicar um conjunto infinito de hipóteses. Mesmo estando de acordo não ser possível extrair uma expansão infinita a partir de um segmento inicial finito, em minha réplica argumento que isto vale igualmente para expansões finitas. Explico também que minhas observações sobre estruturas de prova infinitas não são motivadas pelo impacto dos teoremas de incompletude de Gödel no programa de Hilbert, e tampouco por uma visão negativa do finitismo estrito. (shrink)
Esta resenha versa sobre a obra Cidadania e Democracia Deliberativa de Catherine Audard que se refere à uma articulação rawlsiana entre uma teoria da justiça e as teses centrais do seu liberalismo político sobre o pluralismo razoável, democracia deliberativa, cidadania participativa, razão pública, direito dos povos e multiculturalismo.
In a famous text Descartes has written this: Whenever the thought of God's supreme power occurs to me, I cannot help feeling that he might easily, if he so wished, make me go wrong even in what I think I see most clearly with my mind's eye. On the other hand, whenever I turn to the matters themselves which I think I perceive very clearly, I am so convinced by them that I burst out: ‘let who will deceive me, he (...) can never bring it about that I should be nothing at the time of thinking that I am something, nor that it be true that I never existed if it is true that I exist now; nor even that two and three together make more or less than five, or any such thing in which I see manifest contradiction’. (shrink)
1. Legend has it that as Mozart lay dying, a stranger dressed in black entered the room. Without saying word, he walked to the death-bed, removed the manuscript sheets of the Requiem on which the composer had been working until his final hours, and departed. This was not as you might have thought an envoy from beyond—but the servant of a certain Viennese nobleman, Count Walsegg zu Stuppach. The Count was in the habit of commissioning music anonymously, and having it (...) played in his palace as though it were his own. In extremis he was collecting the score for a forthcoming soirée. (shrink)
The central point of this essay is to demonstrate the incommensurability of ‘Darwinian fitness’ with the numeric values associated with reproductive rates used in population genetics. While sometimes both are called ‘fitness’, they are distinct concepts coming from distinct explanatory schemes. Further, we try to outline a possible answer to the following question: from the natural properties of organisms and a knowledge of their environment, can we construct an algorithm for a particular kind of organismic life-history pattern that itself will (...) allow us to predict whether a type in the population will increase or decrease relative to other types? Introduction Darwinian fitness Reproductive fitness and genetical models of evolution The models of reproductive fitness 4.1 The Standard Viability Model 4.2 Frequency-dependent selection 4.3 Fertility models 4.4 Overlapping generations Fitness as outcome 5.1 Fitness as actual increase in type 5.2 Fitness as expected increase in type 5.2.1 Expected increase within a generation 5.2.2 Expected increase between generations 5.2.3 Postponed reproductive fitness effects The book-keeping problem Conclusion. (shrink)
Her heart shared between Iran and Italy, while Asma Mehan makes peace with lockdown in her current home in Porto, Portugal, her thoughts stray to those familiar places. In those distant horizons there is heartache, but also the unexpected promise of different futures.
ABSTRACT: In this essay I characterize arguments by analogy, which have an impor- tant role both in philosophical and everyday reasoning. Arguments by analogy are dif- ferent from ordinary inductive or deductive arguments and have their own distinct features. I try to characterize the structure and function of these arguments. It is further discussed that some arguments, which are not explicit arguments by analogy, nevertheless should be interpreted as such and not as inductive or deductive arguments. The result is that (...) a presumed outcome of a philosophical dispute will have to be reconsidered. (shrink)
Occasions of Identity is an exploration of timeless philosophical issues about persistence, change, time, and sameness. Andre Gallois offers a critical survey of various rival views about the nature of identity and change, and puts forward his own original theory. He supports the idea of occasional identities, arguing that it is coherent and helpful to suppose that things can be identical at one time but distinct at another. Gallois defends this view, demonstrating how it can solve puzzles about persistence dating (...) back to the Ancient Greeks, and investigates the metaphysical consequences of rejecting the necessity and eternity of identities. (shrink)
Porto Alegre est désormais un symbole politique, un symbole de la résistance à Davos pour un nouvel ordre mondial fondé sur l’ « inclusion sociale ». D’une ville méconnue en dehors des frontières de l’Amérique du Sud, elle est devenue un lieu de résistance contre la mondialisation, contre l’emprise des idées dites « néolibérales », comme si elle incarnait un autre monde..
In front of deterritorialisation / reterritorialisation process one which characterize the globalization, we try to dread the elements of continuance and break which connect among them the Forums of Porto Alegre and Genoa; these two big mobilizations of citizens of world constitute, indeed, the most relevant repository in a debate on the city. By tempting these comparisons, we build an approach of the new cycle of internationalist fights which puts production and constitution of urbanterritories as main stakes in the (...) future developments of the movement. (shrink)
Many word forms in natural language are polysemous, but only some of them allow for co-predication, that is, they allow for simultaneous predications selecting for two different meanings or senses of a nominal in a sentence. In this paper, we try to explain (i) why some groups of senses allow co-predication and others do not, and (ii) how we interpret co-predicative sentences. The paper focuses on those groups of senses that allow co-predication in an especially robust and stable way. We (...) argue, using these cases, but focusing particularly on the multiply polysemous word ‘school’, that the senses involved in co-predication form especially robust activation packages, which allow hearers and readers to access all the different senses in interpretation. (shrink)
In this challenging study, André Gallois proposes and defends a thesis about the character of our knowledge of our own intentional states. Taking up issues at the centre of attention in contemporary analytic philosophy of mind and epistemology, he examines accounts of self-knowledge by such philosophers as Donald Davidson, Tyler Burge and Crispin Wright, and advances his own view that, without relying on observation, we are able justifiably to attribute to ourselves propositional attitudes, such as belief, that we consciously hold. (...) His study will be of wide interest to philosophers concerned with questions about self-knowledge. (shrink)