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

The guiding idea behind formalism is that mathematics is not a body of propositions representing an abstract sector of reality but is much more akin to a game, bringing with it no more commitment to an ontology of objects or properties than ludo or chess. This idea has some intuitive plausibility: consider the tyro toiling at multiplication tables or the student using a standard algorithm for differentiating or integrating a function. It also corresponds to some aspects of the practice (...) of advanced mathematicians in some periods—for example, the treatment of imaginary numbers for some time after Bombelli's introduction of them, and perhaps the attitude of some contemporary mathematicians towards the higher flights of set theory. Finally, it is often the position to which philosophically naïve respondents will gesture towards, when pestered by questions as to the nature of mathematics. (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.

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)

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)

Constitutivists have tried to answer Enoch’s “schmagency” objection by arguing that Enoch fails to appreciate the inescapability of agency. Although these arguments are effective against some versions of the objection, I argue that they leave constitutivism vulnerable to an important worry; namely, that constitutivism leaves us alienated from the moral norms that it claims we must follow. In the first part of the paper, I try to make this vague concern more precise: in a nutshell, it seems that constitutivism cannot (...) provide an adequate account of the relation between the constitutive norms of agency and the particular ends the agent pursues. I then provide a broad outline of an interpretation of Kant’s formalism that is immune to this objection. I conclude that constitutivism is best understood as the upshot of a formalist view of categorical practical principles. (shrink)

The structure of the rate of variation of the atomic energy for an arbitrary stationary motion of the atom in interaction with a quantum electromagnetic field is investigated. Our main purpose is to rewrite the formalism in Zhu et al. and to deduce the general expressions of the Einstein A coefficients of an atom on an arbitrary stationary trajectory. The total rate of change of the energy and Einstein coefficients of the atom near a plate with finite temperature or (...) acceleration are also investigated. (shrink)

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)

In a narrow sense, the formalistic conception of art cannot range and explain a wide variety of contemporary art phenomena. In a wide sense, formalism – rejecting the reductionist definition of the essence of art, involving the conceptual dimension of the artwork, and expanding the dimension of sensibility – becomes a productive philosophical art theory allowing one to explain and investigate some part of the contemporary art (modern and postmodern) phenomena. In the expanded conception, the basic principles of aesthetic (...) formalism include a wide scale of the principles of the creation and comprehension of the artwork – from an abstract form (perceived by senses) to a conceptual form. The article gives an illustration of these principles discussing abstractionism and surrealism as two different types of the contemporary formalistic visual art. (shrink)

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)

Hegel’s aesthetic ideal is the perfect integration of form and content within a work of art. This ideal is incompatible with the predominant 20th-century principle of formalist criticism, that form is the sole important factor in a work of art. Although the formalist dichotomy between form and content has been criticized on philosophical grounds, that does not suffice to justify Hegel’s ideal. Justifying Hegel’s ideal requires detailed art criticism that shows how form and content are, and why they should be, (...) integrated in good works of art. This essay provides some of this criticism. By focusing on the work of the contemporary artist, Robert Turner, this criticism further suggests that Hegel’s aesthetic ideal is still relevant. Moreover, the nature of Turner’s work suggests that art is still relevant in our day in ways Hegel did not expect. (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)

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)

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)

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)

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)

The traditional exposition of Kantian ethical theory in the business ethics literature is abstract, esoteric, and impractical compared to the more usable presentations of utilitarianism. This situation can be improved by identifying and describing the conceptual dimensions of formalistic ethical reasoning, as contained in the interplay between case and principle, with examples from the business/society literature.

This essay presents the Aestheticism of the 19th century as the foundational movement of modernist-formalist aesthetics of the 20th century. The main principle of this movement is what I denominate “productive opacity”. Aestheticism has not been recognized as a philosophical aesthetic theory. However, its definition of artwork as an exclusive kind of form—a deep, opaque form—is among the most precise ever given in the discipline. This essay offers an interpretation of aestheticism as a formalist theory, referred to here as “deep (...) formalism”, focusing on the thinking of leading aestheticists, Walter Pater, Oscar Wilde, and James Whistler. These three thinkers defined artwork as a form saturated with an inextricable content, viz. opaque form. (shrink)

LEGO tells about not just LEGO architecture but architecture generally: its objects, its aesthetic properties, and how people judge them. To illustrate how thinking about LEGO can help people with such matters, this chapter considers some scenarios. These scenarios illustrate two very different ways of thinking about architecture. On the one hand, people might think architectural objects (more commonly, "works of architecture"), like buildings, bridges, and aqueducts, have forms that stand on their own, and which thereby do not depend on (...) historical, environmental, or any other contexts. On the other hand, they might think that architectural objects are best understood (maybe only understood) if they have one or more kinds of contextual information. From a different angle, some might see formalism as a better fit with original LEGO world design than with real‐world architectural design. Proponents of formalism take note of what makes architectural objects distinctive. (shrink)

Introductory Remarks IN A MAJOR WORK planned for the near future I will attempt to develop a non-formal ethics of Values on the broadest possible basis of ...

Economic formalism crowds out the analysis of change and adjustments to change under capitalism. The style of analytical narrative that was practiced by the first generation of neoclassical economists, in contrast, is more productive of genuine economic understanding. Despite Daniel Haus‐man's challenging argument to the contrary, I maintain that Joseph Stiglitz's work is formalist at its core. While I agree with Robert Heilbroner's critique of contemporary economics, there is a limited sense in which nonformalist economics can rely on universalistic (...) assumptions. And Thomas Mayer has provided useful guidelines for focusing nonformalist analysis on real‐world economic problems. (shrink)

Quantum Mechanics can be viewed as a linear dynamical theory having a familiar mathematical framework but a mysterious probabilistic interpretation, or as a probabilistic theory having a familiar interpretation but a mysterious formal framework. These points of view are usually taken to be somewhat in tension with one another. The first has generated a vast literature aiming at a “realistic” and “collapse-free” interpretation of quantum mechanics that will account for its statistical predictions. The second has generated an at least equally (...) large literature aiming to derive, or at any rate motivate, the formal structure of quantum theory in probabilistically intelligible terms. In this paper I explore, in a preliminary way, the possibility that these two programmes have something to offer one another. In particular, I show that a version of the measurement problem occurs in essentially any non-classical probabilistic theory, and ask to what extent various interpretations of quantum mechanics continue to make sense in such a general setting. I make a start on answering this question in the case of a rudimentary version of the Everett interpretation. (shrink)

Legal decisions and theories are frequently condemned as formalistic, yet little discussion has occurred regarding exactly what the term "'formalism" means. In this Article, Professor Schauer examines divergent uses of the term to elucidate its descriptive content. Conceptions offormalism, he argues, involve the notion that rules constrict the choice of the decisionmaker. Our aversion to formalism stems from denial that the language of rules either can or should constrict choice in this way. Yet Professor Schauer argues that this (...) aversion to formalism should be rethought: At times language both can and should restrict decisionmakers. Consequently, the term "'formalistic" should not be used as a blanket condemnation of a decisionmaking process; instead the debate regarding decision according to rules should be confronted on its own terms. (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).

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.

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)

Musical formalism, which strictly limits the type of thing any description of the music can tell us, is ill-equipped to account for contemporary performance practice. If performative interpretations are in a position to tell us something about musical works—that is if performance is a kind of description, as Peter Kivy argues—then we have to loosen the restrictions on notions of musical relevance to make sense of performance. I argue that musical formalism, which strictly limits the type of thing (...) any description of the music can tell us, is inconsistent with Kivy's quite compelling account of performance. This shows the difficulty that actual performances pose to overly rigid conceptions of music. Daniel Barenboim unannounced performance of Wagner in Israel in 2001 shows that the problem of the boundaries of musical relevance is no mere philosophical puzzle. It is a pressing problem in the musical public sphere. (shrink)

Carnap's position on mathematical truth in The Logical Syntax of Language has been attacked from two sides: Kreisel argues that it is formalistic but should not be, and Friedman argues that it is not formalistic but needs to be. In this paper I argue that the Carnap of Syntax does not eliminate our ordinary notion of mathematical truth in favour of a formal analogue; so Carnap's notion of mathematical truth is not formalistic. I further argue that there is no conflict (...) between Carnap's use of informal notions and his principle of tolerance; so Carnap's definition of mathematical truth need not be formalistic. CiteULike Connotea Del.icio.us What's this? (shrink)

In lieu of an abstract, here is a brief excerpt of the content:The Journal of Aesthetic Education 38.3 (2004) 19-35 [Access article in PDF] Moderate Formalism As a Theory of the Aesthetic Glenn Parsons Art history and art criticism explore, classify, and critique artworks from a number of perspectives. Their cultural, political, and moral significance are all of interest in this regard. This variety of perspectives notwithstanding, one way of considering artworks retains a central position for these disciplines. Despite (...) perennial shifting in conceptions of the purpose and nature of art, the aesthetic appreciation of art remains at the center of our practices for classifying and evaluating it. Without claiming that the aesthetic dimension of art is the only proper focus of its appreciation or that the aesthetic is a very clearly understood notion, we may say that some degree of contact with the aesthetic is what makes our consideration of an artwork an appreciation of it as art, rather than as historical artifact, manifestation of cultural ideology, and so on.In light of the central position of aesthetic appreciation in art history and criticism, an important concern regarding aesthetic education is what sort of knowledge, if any, is required for appropriate aesthetic appreciation of an artwork. Is knowledge of the historical context of an artwork required for appropriate aesthetic appreciation of it, or is its historical dimension a separate matter, perhaps worthy of appreciation in its own right but not relevant to the appreciation of the artistic achievement of the work? Virtually all modern theories of the aesthetic agree that in order to appreciate an artwork aesthetically, we need to perceive attentively certain of its perceptual properties, such as its shape and color. They differ, however, on whether such attentive perception is sufficient for appropriate aesthetic appreciation. On the one hand we have cognitivist views, which assert that, in addition to this sort of attentive perception, certain kinds of knowledge about the object of appreciation are required for appropriate aesthetic appreciation. On the other hand we have non-cognitivist views, which deny that such knowledge is necessary for appropriate aesthetic appreciation. According to the latter, the aesthetic qualities of an artwork, such as its gracefulness, [End Page 19] are formal in the sense that they depend only upon those perceptual properties, such as color, shape, and size, that constitute the "sensory surface" of the work.1 These properties, apparent to the attentive observer, are the only properties of the object to which one need attend in order to appreciate its gracefulness. Other properties of the painting, such as its depicting German peasants or having been rendered by a communist, are irrelevant to its aesthetic qualities, and knowledge about them is not required for its appropriate aesthetic appreciation. Today this non-cognitivist position is generally called formalism.2In the twentieth century, philosophical aesthetics largely rejected such views in favor of models of aesthetic appreciation with a richer cognitive element. This movement away from formalism has a number of dimensions, including the efforts of feminists to establish the aesthetic value of activist art, the desire to make sense of well-established practices of art appreciation, a growing awareness of the role that beliefs play in our aesthetic responses, the effort to do justice to the aesthetic character of the natural environment, and conceptual difficulties internal to formalism itself.3 Recently, however, some philosophers have revived the spirit, if not the letter, of formalism. Acknowledging widespread criticism of early formalists, such as Clive Bell, they concede that these theorists overstated the case when they claimed that all aesthetic qualities depend on nothing beyond the sensory surface of the object.4 The new formalists, in contrast, claim only that some aesthetic qualities are of this character. They call this doctrine moderate formalism, in contrast to the extreme formalism defended by earlier formalists.5By adopting this more modest position, moderate formalists can accommodate many of the criticisms that led to the decline of extreme formalism. However, if it turns out that only a handful of relatively unimportant aesthetic qualities are formal in nature, then moderate formalism would hardly make an impressively "formalist" theory of the... (shrink)

A simple mathematical extension of quantum theory is presented. As well as opening the possibility of alternative methods of calculation, the additional formalism implies a new physical interpretation of the standard theory by providing a picture of an external reality. The new formalism, developed first for the single-particle case, has the advantage of generalizing immediately to quantum field theory and to the description of relativistic phenomena such as particle creation and annihilation.

. This paper endeavors to formalize imperatives that convey actions. Imperatives, unlike propositions, do not hold the value of true or false. Peter Vranas proposed an alternate logical formalism in the literature of imperative logic with three values, namely: Satisfaction (), Violation () and Avoidance (). Although this formalism takes into account the conditional imperatives, it does not address imperatives from the perspective of actions. According to Mīmāṁsā, the prime motive of an imperative is to carry out action (...) so as to reach the goal. This paper modifies the notion of Vranas' imperative logic using Mīmāṁsā principles. The resulting imperative formalism, focusing on action representation, is presented with syntax, deduction rules and semantic tables. The effectiveness of this representation is demonstrated in the area of artificial intelligence planning. (shrink)

The recent discussions on “The Right and the Good” show that the reconciliation between formalism and teleology is still a problem. Some of the contending parties lean more strongly towards formalism, others towards teleology; but neither side has, I believe, done equal justice to formalism and teleology. To review the whole discussion is beyond the scope of this paper. Instead, I propose to suggest in outline a theory which may possibly harmonize the essential tenets of both parties.

This chapter contains sections titled: Empty Formalism in the Philosophy of Right Kantian Reply Strategies Acknowledgments Notes Abbreviations References.

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)

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)

Mīmāṁsā, an Indian hermeneutics provides an exhaustive methodology to interpret Vedic statements. A formalism namely, Mīmāṁsā Inspired Representation of Actions has already been proposed in a preliminary manner. This paper expands the formalism logically and includes Syntax and Semantics covering Soundness and Completeness. Here, several interpretation techniques from Mīmāṁsā have been considered for formalising the statements. Based on these, instructions that denote actions are categorized into positive and prohibitive unconditional imperatives and conditional imperatives that enjoin reason, temporal action (...) and goal. These instructions are evaluated to three values S, V and N. If the instruction is executed with the intention of goal, then it acquires the value of S; if the intention of goal is present and the instruction is not executed, then it assumes the value of V; and if there is no intention of goal, then the instruction is evaluated to N. This formalism has also been applied in different computational settings such as AI planning, Robotics, task analysis and classification of natural language instructions. An outline of these applications and the comparison of this formalism with other relevant theories are also presented in this paper. (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)

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

This dissertation is intended to provide a formalism for those generics that trigger nonmonotonic inferences. The formalism is to reflect intentionality and exception-tolerating features of generics, and has an emphasis on the axiomatization of generic reasoning that encodes nonmonotonicity. ;A modal conditional approach is taken to formalize the nonmonotonic reasoning in general at the level of object language. A serial of logic systems---MN, NID, NCUM, N STCUM---are constructed in an increasing strength of the characterized nonmonotonic inference relation. In (...) these systems, two binary modal operators ⩾ and > are introduced in their syntax, and a ⊛ function lifted from the traditional * function is deployed in their semantics. These systems are shown to be sound and complete with respect to certain classes of frames defined in the semantics. They are decidable as well. The nonmonotonic inference is argued to be a ternary relation "[phi], Gamma |∼ alpha", and is defined in the system NSTCUM. Many widely discussed nonmonotonic inference patterns such as Defeasible Modus Ponens, Defeasible Transitivity, the Penguin Principle etc. are justified. The specificity rule is proved to be a theorem of the system N STCUM. The impact of negated defaults on an inference is also investigated and accounted for. ;A canonical form to read off generics is proposed: All generic sentences with subject-predicate structure can be re-written into their canonical form S . If S is a plural noun phrase, it can be further refined to be . Normal objects are selected based on the "meaning" of the subject and predicate terms. The second parameter provides an aspect with respect to which certain objects of a kind are considered normal. Due to such a way to select normal objects, the drowning problem is solved. ;The inference behaviors of generics are axiomatized in the system G, which is a quantificational extension of the system NSTCUM. It is proved to be sound and complete with respect to the class of L⩾,G -frames. Those benchmark examples of generic inferences are examined in the system G. (shrink)

Social network research is widely considered atheoretical. In contrast, in this article I argue that network analysis often mixes two distinct theoretical frameworks, creating a logically inconsistent foundation. Relationalism rejects essentialism and a priori categories and insists upon the intersubjectivity of experience and meaning as well as the importance of the content of interactions and their historical setting. Formalism is based on a structuralist interpretation of the theoretical works of Georg Simmel. Simmel laid out a neo-Kantian program of identifying (...) a priori categories of relational types and patterns that operate independently of cultural content or historical setting. Formalism and relationalism are internally consistent theoretical perspectives, but there are tensions between them. To pave the way for stronger middle-range theoretical development, I disaggregate the two approaches and highlight the contradictions that must be addressed or resolved for the construction of any general and inclusive theory. (shrink)

Extreme formalism is a radical and important position in the aesthetics of inorganic nature. Zangwill offers a new formulation of what formal aesthetic properties are, according to which a formal aesthetic property of a thing is an aesthetic property that is determined merely by its appearance properties. An appearance property of a thing is the way it seems if perceived under certain conditions. With the notion of formal aesthetic properties formulated as such, extreme formalism, the claim that all (...) aesthetic properties of inorganic things are formal, says that all aesthetic properties of inorganic things are determined merely by their appearance properties. Recently, Zangwill defends extreme formalism by deflecting the famous counter-example of the cumulonimbus cloud given by Budd, and argues for extreme formalism by arguing against the idea of the existence of non-formal beauty in inorganic nature. In this paper, we argue that both his defense and positive arguments are not successful. If our arguments are right, the burden of proof is still on Zangwill’s side. (shrink)

In teaching jurisprudence, I typically distinguish between two different families of theories of adjudication—theories of how judges do or should decide cases. “Formalist” theories claim that the law is “rationally” determinate, that is, the class of legitimate legal reasons available for a judge to offer in support of his or her decision justifies one and only one outcome either in all cases or in some significant and contested range of cases ; and adjudication is thus “autonomous” from other kinds of (...) reasoning, that is, the judge can reach the required decision without recourse to nonlegal normative considerations of morality or political philosophy. I also note that “formalism” is sometimes associated with the idea that judicial decision-making involves nothing more than mechanical deduction on the model of the syllogism—Beccaria, for example, expresses such a view. I call the latter “Vulgar Formalism” to emphasize that it is not a view to which anyone today cares to subscribe. (shrink)

Infinite-dimensional manifolds modelled on arbitrary Hilbert spaces of functions are considered. It is shown that changes in model rather than changes of charts within the same model make coordinate formalisms on finite and infinite-dimensional manifolds deeply similar. In this context the infinite-dimensional counterparts of simple notions such as basis, dual basis, orthogonal basis, etc. are shown to be closely related to the choice of a model. It is also shown that in this formalism a single tensor equation on an (...) infinite-dimensional manifold produces a family of functional equations on different spaces of functions. (shrink)

Various parallels between Kant's critical program and Hilbert's formalistic program for the philosophy of mathematics are considered.

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)

Coordinate formalism on Hilbert manifolds developed in \cite{Kryukov}, \cite{Kryukov1} is further analyzed. The main subject here is a comparison of the ordinary and the string bases of eigenvectors of a linear operator as introduced in \cite{Kryukov}. It is shown that the string basis of eigenvectors is a natural generalization of its classical counterpart. It is also shown that the developed formalism forces us to consider any Hermitian operator with continuous spectrum as a restriction to a space of square (...) integrable functions of a self-adjoint operator defined on a space of generalized functions. In the formalism functional coordinate transformations preserving the norm of strings are now linear isometries rather than the unitary transformations. (shrink)

In this paper I present a formalist philosophy mathematics and apply it directly to Arithmetic. I propose that formalists concentrate on presenting compositional truth theories for mathematical languages that ultimately depend on formal methods. I argue that this proposal occupies a lush middle ground between traditional formalism, fictionalism, logicism and realism.

LEGO tells about not just LEGO architecture but architecture generally: its objects, its aesthetic properties, and how people judge them. To illustrate how thinking about LEGO can help people with such matters, this chapter considers some scenarios. These scenarios illustrate two very different ways of thinking about architecture. On the one hand, people might think architectural objects (more commonly, "works of architecture"), like buildings, bridges, and aqueducts, have forms that stand on their own, and which thereby do not depend on (...) historical, environmental, or any other contexts. On the other hand, they might think that architectural objects are best understood (maybe only understood) if they have one or more kinds of contextual information. From a different angle, some might see formalism as a better fit with original LEGO world design than with real‐world architectural design. Proponents of formalism take note of what makes architectural objects distinctive. (shrink)

One of the ways our legal system has avoided confronting this ugly reality is through a commitment to legal formalism. Legal formalism allows us to ignore the social determinants that my AUSA friend saw every day as he prosecuted federal drug cases. As my colleague Professor Michael Seidman has suggested, legal formalism, which has been effectively critiqued and displaced by legal realism in many other areas of law, continues to exercise considerable influence over the way we think (...) about criminal law. This formalist approach, in my view, has strongly affected the way we approach the drug problem. One consequence is that we continue to pursue an increasingly futile war on drugs and refuse to see the issue in its broader, realist dimension. A little realism on the subject of drugs, I suggest, would go a long way. There is much to be said for formalism in the criminal law. Formalism, with its commitment to fair procedures, clear rules, and restricted discretion, is a necessary part of any fair system of criminal law. The sanctions involved in the criminal system are too severe to permit them to be allocated in an open ended discretionary or regulatory manner. The criminal law's commitment to formalism is thus not a fault, but a strength. Discretionary regulatory schemes too often invite subjective judgments susceptible to abuse, prejudice, and favoritism. Formalist rules, by contrast, are built on the promise of treating likes alike. Precisely for this reason, however, we ought to reconsider whether the criminal approach makes sense when there is substantial evidence that the commitment to equality has been seriously compromised. Our dual commitments to equality and to the reduction of the human damage that drug abuse inflicts suggest that we should reduce our reliance on the criminal justice system. Alternative approaches, such as treatment and rehabilitation, promise to be both more effective and more fair. (shrink)