Le but de cet article est d’étudier la référence à l’espace et au temps dans le problème du fondement des mathématiques, au cours de la période 1880-1935. Après avoir évoqué la problématique kantienne, qui reste présente dans la controverse entre Brouwer et Hilbert, nous discutons de la référence au temps dans l’intuitionisme et dans le programme formaliste pour montrer comment, dans les deux cas mais de façon différente, la référence au temps introduit des restrictions sur ce qui peut être considéré (...) comme une démonstration mathématique. Nous évoquons ensuite trois tentatives, Frege, le Hilbert d’avant le programme formaliste et Gentzen, pour éliminer la référence au temps et ne fonder les mathématiques que sur l’intuition de l’espace. (shrink)

The calculus DNL results from the non-associative Lambek calculus NL by splitting the product functor into the right (⊲) and left (⊳) product interacting respectively with the right (/) and left () residuation. Unlike NL, sequent antecedents in the Gentzen-style axiomatics of DNL are not phrase structures (i.e., bracketed strings) but functor-argument structures. DNL − is a weaker variant of DNL restricted to fa-structures of order ≤ 1. When axiomatized by means of introduction/elimination rules for / and , it (...) shows a perfect analogy to NL which DNL lacks. (shrink)

In 1930, Johann von Neumann, together with Rudolf Carnap and Arend Heyting, participated in a conference held in Königsberg, called “Second Seminar on the Epistemology of Exact Sciences”. The idea behind the reunion of these three researchers was to compose a fairly faithful picture of the three main foundational programs of mathematics at the time: formalism, logicism, and intuitionism. The main objective of this paper is to propose an analysis of the text “The Formalist Foundation of Mathematics” presented by (...) von Neumann in this conference. We show how, from von Neumann's perspective, the problem of the consistency of mathematical theories results from the transformation of a genuinely philosophical question into a genuinely logical-mathematical question and how a proof of consistency could be seen as the solution to the foundational problem. In the final part of the text, we present some considerations about Gerhard Gentzen’s consistency proofs and their importance to contemporary proof theory. (shrink)

How natural is natural deduction?– Gentzen's system of natural deduction intends to fit logical rules to the effective mathematical reasoning in order to overcome the artificiality of deductions in axiomatic systems (¶ 2). In spite of this reform some of Gentzen's rules for natural deduction are criticised by psychologists and natural language philosophers for remaining unnatural. The criticism focuses on the principle of extensionality and on formalism of logic (¶ 3). After sketching the criticism relatively to the (...) main rules, I argue that the criteria of economy, simplicity, pertinence etc., on which the objections are based, transcend the strict domain of logic and apply to arguments in general (¶ 4). (¶ 5) deals with Frege's critique of the concept of naturalness as regards logic. It is shown that this concept means a regression into psychologism and is exposed to the same difficulties as are: relativity, lack of precision, the error of arguing from `is' to `ought' (the naturalistic fallacy). Despite of these, the concept of naturalness plays the role of a diffuse ideal which favours the construction of alternative deductive systems in contrast to the platonic conception of logic (¶ 6). (shrink)

The logical formalism of abductive reasoning is still an open discussion and various theories have been presented about it. Abduction is a type of non-monotonic and defeasible reasonings, and the logic containing such a reasoning is one of the types of non-nonmonotonic and defeasible logics, such as inductive logic. Abduction is a kind of natural reasoning and it is a solution to the problems having this form "the phenomenon of φ cannot be explained by the theory of Θ" and (...) we need to search for (or to adduce) an explanation for φ. Taking α as the explanation and <Θ,φ> as the abductive problem, Θ∪{α}⊨φ is the logical format of a strong abductive reasoning. In this article, the proof theory for first-order abductive calculus is proposed based on Gentzen-type first-order proof theory. Considering its dual, i.e., the semantic tableaux, and without the need to translate the formulas, this calculation is sound and complete. After examining the structural rules of this calculus, some metalogical considerations and open problems such as consistency, completeness and undecidability of this account have been addressed. (shrink)

In this paper we introduce hypersequent-based frameworks for the modelling of defeasible reasoning by means of logic-based argumentation and the induced entailment relations. These structures are an extension of sequent-based argumentation frameworks, in which arguments and the attack relations among them are expressed not only by Gentzen-style sequents, but by more general expressions, called hypersequents. This generalization allows us to overcome some of the known weaknesses of logical argumentation frameworks and to prove several desirable properties of the entailments that (...) are induced by the extended frameworks. It also allows us to incorporate as the deductive base of our formalism some well-known logics, which lack cut-free sequent calculi, and so are not adequate for standard sequent-based argumentation. We show that hypersequent-based argumentation yields robust defeasible variants of these logics, with many desirable properties. (shrink)

Les divers points de vue relatifs au concept mathématique de l’infini sont ordonnés en série croissante d’après le degré où l’on reconnaît ce concept en ses diverses complications. Cette série est divisée en trois groupes : la mathématique du fini, la « conception constructive », et la « conception en soi » de l’infini. D’après cette série, l’on explique le programme d"Hilbert, qui est de prouver que la mathématique est libre de contradiction, et l’on rapporte brièvement les méthodes qui sont (...) en question pour administrer cette preuve. (shrink)

According to long-established narratives, legal thinking in Germany, France and the U.S.A. was shaped by formalist legal cultures for the most part of the nineteenth century until the respective legal sciences embraced their social responsibility in the early twentieth century. Recently, legal historians have begun to question these narratives. In separate analyses, they have shown that the critics of ‘Legal Formalism’ exerted a lasting influence on historical research since the early twentieth century, thereby fostering a deeply charged understanding of (...) nineteenth century jurisprudence. What went largely unnoticed, however, was the transnational dimension of this false narrative. Due to the transnational ties of the critics of ‘Legal Formalism’, a powerful misrepresentation of nineteenth century Western legal thought emerged that still influences contemporary global legal history. Dismantling the prejudices surrounding the Western jurisprudence of the nineteenth century causes crucial challenges for conceptualising nineteenth century Western legal thought. (shrink)

Rosenberg concludes with a critical reassessment of widely accepted views regarding the relationships among natural languages, mathematical formalisms, and philosophical commitments. The culmination of twenty years' reflection, Beyond Formalism is an original and sophisticated book of importance to both philosophers and linguists.

This book revisits the theory of the sources of international law from the perspective of formalism.

The ramified legacy of Bernard Berenson’s writings within 20th century art historiography is both celebrated and maligned. In an effort to help reconcile this situation, this essay argues for the partial validity of Berenson’s peculiar version of art historical formalism by detailing its historical connections to the Pragmatist philosophy and psychology of William James and by analytically correlating Berenson’s arguments with recent work in aesthetics and the philosophy of perception. The essay examines the specific example of Berenson’s analysis of (...) Giotto’s paintings and leverages a Pragmatist interpretation of Berenson’s writings to frame Berenson’s known connections with other writers: including Adolf Hildebrand, Giovanni Morelli, and Walter Pater. In conclusion, the failure, potential, reception, and legacy of Berenson’s art historical scholarship are assessed in relation to Pragmatist ideas. Der vielfältige Einfluss von Bernard Berensons Schriften auf die Kunstgeschichte des 20. Jahrhunderts wird ebenso geschätzt wie verachtet. In dem Bestreben, diese Divergenzen zu berichtigen, versucht dieser Beitrag den Nachweis für die bedingte Gültigkeit von Berensons eigentümlicher Variante eines kunsthistorischen Formalismus zu erbringen, indem er einerseits seine historischen Verbindungen zur pragmatischen Philosophie und Psychologie von William James aufzeigt und indem er andererseits Berensons Argumentation ins Verhältnis zu aktuellen Debatten der wahrnehmungstheoretischen Ästhetik und Philosophie setzt. Am Beispiel von Berensons Interpretation der Gemälde Giottos führt dieser Essay eine pragmatische Analyse von Berensons Schriften durch und stellt diese ins Umfeld seiner Kontakte zu anderen Autoren: darunter Adolf Hildebrand, Giovanni Morelli und Walter Pater. Schließlich soll das Scheitern, das Potenzial, die Rezeption und das Erbe von Berensons kunsthistorischen Studien unter pragmatischen Gesichtspunkten bewertet werden. (shrink)

When discussing Wilfrid Sellars’s philosophy, very little work has been done to offer a developmental account of his systematic views. More often than not, Sellars’s complex views are presented in a systematic and holistic fashion that ignores any periodization of his work. I argue that there is a metaphilosophical shift in Sellars’s early philosophy that results in substantive changes to his conception of language, linguistic rules, and normativity. Specifically, I claim that Sellars’s shift from a formalist metaphilosophy to one more (...) closely aligned with psychology allows for the construction of a normative conception of language. My central claim is that without his abandonment of earlier metaphilosophical commitments, Sellars could not hold what I call an external conception of normativity. It is this move away from a formalist notion of philosophy that allows Sellars to construct a normative picture of language. I conclude that because this substantive shift in philosophical commitments results from changes in Sellars’s metaphilosophical views, there is insight to be found in a meticulous periodization of his work. (shrink)

Mathematical formalism is the the view that numbers are "signs" and that arithmetic is like a game played with such signs. Frege's colleague Thomae defended formalism using an analogy with chess, and Frege's critique of this analogy has had a major influence on discussions in analytic philosophy about signs, rules, meaning, and mathematics. Here I offer a new interpretation of formalism as defended by Thomae and his predecessors, paying close attention to the mathematical details and historical context. (...) I argue that for Thomae, the formal standpoint is an _algebraic perspective_ on a domain of objects, and a "sign" is not a linguistic expression or mark, but a representation of an object within that perspective. Thomae exploits a shift into this perspective to give a purely algebraic construction of the real numbers from the rational numbers. I suggest that Thomae's chess analogy is intended to provide a model for such shifts in perspective. (shrink)

ABSTRACT In a recent essay, Sören Stenlund tries to align Wittgenstein’s approach to the foundations and nature of mathematics with the tradition of symbolic mathematics. The characterization of symbolic mathematics made by Stenlund, according to which mathematics is logically separated from its external applications, brings it closer to the formalist position. This raises naturally the question whether Wittgenstein holds a formalist position in philosophy of mathematics. The aim of this paper is to give a negative answer to this question, defending (...) the view that Wittgenstein always thought that there is no logical separation between mathematics and its applications. I will focus on Wittgenstein’s remarks about arithmetic during his middle period, because it is in this period that a formalist reading of his writings is most tempting. I will show how his idea of autonomy of arithmetic is not to be compared with the formalist idea of autonomy, according to which a calculus is “cut off” from its applications. The autonomy of arithmetic, according to Wittgenstein, guarantees its own applicability, thus providing its own raison d’être. RESUMO Em um recente artigo, Sören Stenlund procura alinhar a abordagem de Wittgenstein em relação aos fundamentos e à natureza da matemática com a tradição da matemática simbólica. A caracterização da matemática simbólica feita por Stenlund, de acordo com a qual a matemática é logicamente separada de suas aplicações externas, a aproxima da posição formalista. Isto naturalmente levanta a questão de se Wittgenstein defende uma posição formalista em filosofia da matemática. O objetivo deste artigo é dar uma resposta negativa a esta questão, ao defender que Wittgenstein sempre pensou não haver separação lógica entre a matemática e suas aplicações. Atenção especial será dada às observações de Wittgenstein sobre a aritmética pertencentes ao seu período intermediário, pois é neste período que uma leitura formalista de seus escritos é mais sedutora. Mostrarei como sua ideia de autonomia da aritmética não deve ser comparada à ideia formalista de autonomia, segundo a qual um cálculo é “cortado” de suas aplicações. A autonomia da aritmética, para Wittgenstein, garante ela própria sua aplicabilidade, provendo, assim, sua própria raison d’être. (shrink)

This volume examines the elements of formalism and decisionism in Russian legal thinking and, also, the impact of conservatism on the interplay of these elements. This combination leads to internal contradictions in theorizing about law and rights in Russian legal culture.

A comprehensive historical overview of formalist ideas in the philosophy of mathematics.

Beyond Formalism is Jay Rosenberg’s attempt to articulate his dissatisfactions with the Kripkean “revolution” in the philosophy of language and to propose an alternative to it. According to Rosenberg, even though a “surprisingly large number of philosophers simply adopted the Kripkean ideas, images, and idioms root and branch”, he has been “inarticulately irritated by Kripke’s views for almost twenty years”. Rosenberg claims that Kripke’s semantics for proper names and natural kind terms is a misguided attempt to apply results in (...) formal logic, in particular the semantics of modal logic, to natural language. In order to get “beyond formalism,” Rosenberg offers his own theory of nominal reference that is “for human beings.” Rosenberg’s “human” theory should be able to take account of the functioning of referring expressions in “contexts of individual understanding and interpersonal communication” —something Kripke’s theory cannot do, according to Rosenberg. Rosenberg’s book divides into criticism of Kripke’s ideas on naming and necessity, on the one hand, and adumbration of Rosenberg’s own theory for “human beings” on the other, supplemented by discussions of such topics as Kripke’s puzzle about belief and the role of formal logic in the philosophy of language. (shrink)

In his remarkable paper Formalism 64, Robinson defends his eponymous position concerning the foundations of mathematics, as follows:Any mention of infinite totalities is literally meaningless.We should act as if infinite totalities really existed. Being the originator of Nonstandard Analysis, it stands to reason that Robinson would have often been faced with the opposing position that ‘some infinite totalities are more meaningful than others’, the textbook example being that of infinitesimals. For instance, Bishop and Connes have made such claims regarding (...) infinitesimals, and Nonstandard Analysis in general, going as far as calling the latter respectively a debasement of meaning and virtual, while accepting as meaningful other infinite totalities and the associated mathematical framework. We shall study the critique of Nonstandard Analysis by Bishop and Connes, and observe that these authors equate ‘meaning’ and ‘computational content’, though their interpretations of said content vary. As we will see, Bishop and Connes claim that the presence of ideal objects in Nonstandard Analysis yields the absence of meaning. We will debunk the Bishop–Connes critique by establishing the contrary, namely that the presence of ideal objects in Nonstandard Analysis yields the ubiquitous presence of computational content. In particular, infinitesimals provide an elegant shorthand for expressing computational content. To this end, we introduce a direct translation between a large class of theorems of Nonstandard Analysis and theorems rich in computational content, similar to the ‘reversals’ from the foundational program Reverse Mathematics. The latter also plays an important role in gauging the scope of this translation. (shrink)

Joe L. Kincheloe (1950-2008) was one of North America's leading critical pedagogy scholars. He defined post-formalist thought in terms of deconstruction, affectivity, and non-linearity. His deconstruction focused on the context of ideas, ideologies, and teaching. It was a form of sociological deconstruction, and as such, inspired by Derrida, but different from him as well. In effect, Kincheloe was trying to marry Derrida to Foucault by making deconstruction see power in thought, relationships, and the world. Kincheloe's 'turn to affect' was inspired (...) by feminism and radical pedagogy. It was 'affect' focused on (in)justice and the social practices of repression. His 'self-other' construct was inherently politicized by his identification of 'unfreedom' with capitalism and the assumption that this link determines affect. Kincheloe assumed that linear rationality was inadequate to understanding human needs and hopes. Freedom as dynamism was seen to be inherently non-linear. The prison of rationality (it can only repeat the same, over and over again) was the crux of his critique of Newtonian-Cartesian linearity. Kincheloe attempted to construct a concept of 'place'--such as the classroom. But it was a particular, concrete classroom and not an abstract or theoretical one. Here, the three concepts could come together. 'Place' is context, and to understand it, deconstruction is needed. 'Place' exists as it is felt and requires affectivity; it is eventful, alive, and dynamic. It requires non-linearity to be understood. Post-formalism, Pedagogy Lives (in memory of Kincheloe's contribution) encompasses each of the basic principles of Kincheloe's post-formal thought. (shrink)

Compares formalism and instrumentalism and evaluates their general claims. “Part of what is meant by formalism is this: The law provides sufficient basis for deciding any case that arises. There are no “gaps” within the law, and there is but one sound legal decision for each case.” The formalist also holds that law is traceable to an authoritative source. “…sound legal decisions can be justified as the conclusions of valid deductive syllogisms. Because law is believed to be complete (...) and univocal, all cases that arise can in principle be decided in this way. This is the formalistic model for legal justifications.” Moderate instrumentalism also accepts the notion that the law has sources and that past political events can determine what the law is for the future. However, they also maintain that the law has gaps. “Moderate instrumentalists and positivists alike embrace a source-based conception of the law as well as a formalistic model for legal justification; partly because of this combination, they reject the formalistic notion that law is complete and univocal. Instrumentalists, like positivists, emphasize that because the interpretation of authoritative legal texts and their application to cases are often controversial, reasonable arguments are often possible on both sides of a legal issue. Since there are no hard and fast rules for adjudicating such disputes, positivists conclude that law in such cases is indeterminate – not yet fully formed, needing judicial legislation. Instrumentalists most likely have a similar view of the law. They assume that law is determinate on an issue at a given time only if its identification and application are, roughly speaking, mechanical.” [Must keep in mind the distinction between determinate and conclusive.] “The instrumentalists make the decisive assumption that law is not determinate if it is controversial, for law is thought to be gappy and indeterminate only when reasonable legal arguments are possible on both sides of a legal question…This reasoning exposes a more fundamental assumption of formalism, instrumentalism, and positivism: Nondeductive reasoning is incapable of adequately establishing any conclusion.” This, though, is a radical thesis given that all scientific hypotheses are unproven deductively. [But it might be a claim restricted to legal reasoning. Moreover, it seems legitimate to wonder whether legal reasoning is really comparable to offering scientific hypotheses.] We should, however, reject the mechanistic model of legal and judicial reasoning that assumes a given major premise captured in authoritative legal texts that deductively produces conclusions in a situation to which a law applies. Take a law that requires some activity to be “economically efficient” but fails to specify economic efficiency. The text assumes that there is such a thing as economic efficiency, that some activities are so efficient and others not, that some judgments about economic efficiency are true and others false, and that the criteria of efficiency are determinable in principle. “[I]f there really are reasons for preferring one conception of economic efficiency to others in a given context, the assumption of the law in question would be true; if the court correctly identifies and applies that conception, it is simply carrying out its legal mandate. It would be faithful to the text, but it would not be limited to the four corners of the text and its literal implications.” Thus, a judge may employ non-mechanical reasoning in arriving at the meaning of legal requirements such as economic efficiency. Moral reasons are constitutive of the law. A judge must, in deciding the appropriate conc ption of economic efficiency, offer reasons for preferring that conception in the given context. (shrink)

This chapter contains sections titled: The Project of Formalism The Nature of Justification The Structures of Justification The Ground of Justification The Immanent Intelligibility of Law Conclusion References.

It is sometimes claimed that formalism and the logical incompatibility thesis together imply that fouls cannot be part of the game. Some philosophers think this proves that therefore strategic fouls are always morally wrong, but other philosophers think this result undermines formalism itself, since strategic fouls clearly are part of the game and are at least sometimes morally permissible. I show that formalism in fact does accommodate strategic fouls and that it is neutral about whether strategic fouls (...) are morally permissible or not. (shrink)

Mark Blaug played a central role in the development of the field of the methodology of economics, alongside his theoretical work and contributions to the history of economic thought. The purpose, in this article, is to focus on his contributions to the topic of 'formalism in economics', in relation to his methodological commentaries on the Popperian and Lakatosian approaches to the philosophy of science. In Blaugian spirit, the discussion is related to economic theory and draws on the history of (...) economic thought. The argument focuses on the troublesome interface between theoretical and applied economics in mainstream economics. The article includes, as a case study, an assessment of new behavioural economics in Popperian and Lakatosian terms. The conclusion is that such an appraisal exercise—i.e., whether the research programme is progressive or degenerative—is clouded by the interface between the form of empiricism promoted by Popper and Lakatos and the methodological framework of mainstream economics. No conclusion is feasible independent of methodological approach. (shrink)

This chapter contains sections titled: Formalism Enhanced Formalism Beyond Formalism.

Formalism in the philosophy of mathematics has taken a variety of forms and has been advocated for widely divergent reasons. In Sects. 1 and 2, I briefly introduce the major formalist doctrines of the late nineteenth and early twentieth centuries. These are what I call empirico-semantic formalism, game formalism and instrumental formalism. After describing these views, I note some basic points of similarity and difference between them. In the remainder of the paper, I turn my attention (...) to Hilbert’s instrumental formalism. My primary aim there will be to develop its formalist elements more fully. These are, in the main, its rejection of the axiom-centric focus of traditional model-construction approaches to consistency problems, its departure from the traditional understanding of the basic nature of proof and its distinctively descriptive or observational orientation with regard to the consistency problem for arithmetic. More specifically, I will highlight what I see as the salient points of connection between Hilbert’s formalist attitude and his finitist standard for the consistency proof for arithmetic. I will also note what I see as a significant tension between Hilbert’s observational approach to the consistency problem for arithmetic and his expressed hope that his solution of that problem would dispense with certain epistemological concerns regarding arithmetic once and for all. (shrink)

Previous work has shown that spontaneous collapse of Fock states of identical fermions can be modeled as arising from random Rabi oscillations between two states. In this paper, a mathematical formalism is presented to incorporate this into many-body quantum field theory. This formalism allows for nonlocal collapse in the context of a relativistic system. While there is no absolute time-ordering of events, this approach allows for a coherent narrative of the collapse process.

Coordinate formalism on Hilbert manifolds developed in \cite{Kryukov} is reviewed. The results of \cite{Kryukov} are applied to the simpliest case of a Hilbert manifold: the abstract Hilbert space. In particular, functional transformations preserving properties of various linear operators on Hilbert spaces are found. Any generalized solution of an arbitrary linear differential equation with constant coefficients is shown to be related to a regular solution by a (functional) coordinate transformation. The results also suggest a way of using generalized functions to (...) solve nonlinear problems on Hilbert spaces. (shrink)

This paper objectively defines the three main contemporary philosophies of mathematics: formalism, logicism, and intuitionism. Being the three leading scientists of each: Hilbert (formalist), Frege (logicist), and Poincaré (intuitionist).

In this paper we isolate a notion that we call "formalism freeness" from Gödel's 1946 Princeton Bicentennial Lecture, which asks for a transfer of the Turing analysis of computability to the cases of definability and provability We suggest an implementation of Gödel's idea in the case of definability, via versions of the constructible hierarchy based on fragments of second order logic. We also trace the notion of formalism freeness in the very wide context of developments in mathematical logic (...) in the 20th century. (shrink)

The relationship between mathematical formalism, physical interpretation and epistemological appraisal in the practice of physical theorizing is considered in the context of Bohmian mechanics. After laying outthe formal mathematical postulates of thetheory and recovering the historical roots ofthe present debate over the meaning of Bohmianmechanics from the early debate over themeaning of Schrödinger's wave mechanics,several contemporary interpretations of Bohmianmechanics in the literature are discussed andcritiqued with respect to the aim of causalexplanation and an alternative interpretationis proposed. Throughout, the over-arching (...) aimis to exhibit the connections betweenmathematical, ontological and methodologicalquestions in physical theory and to reflect onthe rationality of physical theorizing in lightof the present case. (shrink)

This paper compares the outwardly similar structural formalisms of Marc- Antoine Laugier and Arthur Schopenhauer. Laugier purports to base his aesthetics on an historical argument from the “primitive hut”; but his preferences are really based on aversion to structurally and programmatically non-functional elements. His preferences show disregard for purely aesthetic considerations, such as pleasing proportions. Schopenhauer’s formalism is based on his cognitivist approach to aesthetics, according to which architecture is above all supposed to demonstrate relations between load and support, (...) but in spite of this shows greater sensitivity to sensory beauty. (shrink)

I draw lessons from experimental economics. I argue that the lack of mathematical formalism cannot be usefully thought as the cause of the underappreciation of contextual and generalizability considerations. Instead, this lack is problematic because it hinders a clear relationship between theory and quantitative predictions. I also advocate a pragmatic policy-focused approach as a partial remedy to the generalizability problem.