I reconcile the spatiotemporal location of repeatable artworks and impure sets with the non-location of natural numbers despite all three being varieties of abstract objects. This is possible because, while the identity conditions for all three can be given by abstraction principles, in the former two cases spatiotemporal location is a congruence for the equivalence relation featuring in the relevant principle, whereas in the latter it is not. I then generalize this to other ‘physical’ properties like shape, mass, and (...) causal powers. (shrink)

Mathematics is everywhere and yet its objects are nowhere. There may be five apples on the table but the number five itself is not to be found in, on, beside or anywhere near the apples. So if not in space and time, where are numbers and other mathematical objects such as perfect circles and functions? And how do we humans discover facts about them, be it Pythagoras’ Theorem or Fermat’s Last Theorem? The metaphysical question of what numbers are and the (...) epistemological question of how we know about them are central to the philosophy of mathematics. These and related philosophical questions are of particular interest because of mathematics’ unusual status. Mathematics is exceptional in that, on the one hand, it appears unhesitatingly true—no one doubts that 2 + 3 = 5—but on the other, as just noted, it is not about the physical world. This ambivalent status is what gives the philosophy of mathematics its special interest. The philosophy of mathematics is also one of the oldest academic fields, more or less coeval with philosophy itself. But contemporary philosophy of mathematics is rather different from its pre-twentieth-century antecedents, largely for three reasons. The first is that since the seventeenth century, mathematics has become integral to science. Science has over the past few centuries become increasingly mathematical, and indeed the fundamental science of nature, physics, is today recognised as a branch of applied mathematics. The second is that mathematics underwent a transformation in the course of nineteenth century: having started the century as a rather traditional-looking science of quantity it emerged a hundred years later a radically transformed abstract theory of structure. The final factor in the transformation of the philosophy of mathematics is the rise of modern logic. Developed by Frege, Cantor and others in the late nineteenth century, modern logic pervades contemporary mathematics, philosophy and computer science, and has had an immeasurable effect on the philosophy of mathematics. These volumes will collect the major works in this major field, with a focus on the last few decades. The anthology will include technical work, which interprets philosophically significant mathematical results or subfields of mathematics, as well as purely philosophical writing, aimed at those without advanced mathematics. The collection should be of interest to both philosophers and mathematicians, as well as to anyone who is susceptible to wondering what the main intellectual tool used in science, economics and finance, and indeed everyday life is ultimately about. (shrink)

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and (...) compares these options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

First, I offer a short overview on the classical occidental philosophy as propounded by the ancient Greeks and the natural philosophies of the last 2000 years until the dawn of the empiricist logic of science in the twentieth century, which wanted to delimitate classical metaphysics from empirical sciences. In contrast to metaphysical concepts which didn’t reflect on the language with which they tried to explain the whole realm of entities empiricist logic of science initiated the end of metaphysical theories (...) by reflecting on the preconditions for foundation and justification of sentences about objects of investigation, i.e. a coherent definition of language in general, which was not the aim of classical metaphysics. Unexpectedly empiricist logic of science in the linguistic turn failed in the physical and mathematical reductionism of language and its use in communication, as will be discussed below in further detail. Nevertheless, such reflection on language and communication also introduced this vocabulary into biology. Manfred Eigen and bioinformatics, later on biolinguistics, used ‘language’ applied linguistic turn thinking to biology coherent to the logic of science and its formalisable aims. This changed significantly with the birth of biosemiotics and biohermeneutics. At the end of this introduction it will be outlined why and how all these approaches reproduced the deficiencies of the logic of science and why the biocommunicative approach avoids their abstractive fallacies. (shrink)

Du Châtelet holds that mathematical representations play an explanatory role in natural science. Moreover, she writes that things proceed in nature as they do in geometry. How should we square these assertions with Du Châtelet’s idealism about mathematical objects, on which they are ‘fictions’ dependent on acts of abstraction? The question is especially pressing because some of her important interlocutors (Wolff, Maupertuis, and Voltaire) denied that mathematics informs us about the properties of material things. After situating Du Châtelet in this (...) debate, this chapter argues, first, that her account of the origins of mathematical objects is less subjectivist than it might seem. Mathematical objects are non-arbitrary, public entities. While mathematical objects are partly mind-dependent, so are material things. Mathematical objects can approximate the material. Second, it is argued that this moderate metaphysical position underlies Du Châtelet’s persistent claims that mathematics is required for certain kinds of general knowledge, including in natural science. The chapter concludes with an illustrative example: an analysis of Du Châtelet’s argument that matter is continuous. A key premise in the argument is that mathematical representations and material nature correspond. (shrink)

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the (...) physical world and are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

In Metaphysics A, Aristotle offers some objections to Plato’s theory of Forms to the effect that Plato’s Forms would not be explanatory in the right way, and seems to suggest that they might even make the explanatory project worse. One interesting historical puzzle is whether Aristotle can avoid these same objections to his own theory of universals. The concerns Aristotle raises are, I think, cousins of contemporary concerns about the usefulness and explanatoriness of abstract objects, some of which have (...) recently been receiving attention in the philosophy of mathematics. After discussing Aristotle’s objections and their contemporary cousins, the paper discusses some of the main available lines of response to these sorts of challenges, before concluding with an examination of whether these responses could assist Plato or Aristotle in responding to these challenges. (shrink)

I consider the field of aesthetics to be at its most productive and engaging when adopting a broadly philosophically informative approach to its core issues (e.g., shaping and testing putative art theoretic commitments against the relevant standard models employed in philosophy of language, metaphysics, and philosophy of mind) and to be at its most impotent and bewildering when cultivating a philosophically insular character (e.g., selecting interpretative, ontological, or conceptual models solely for fit with pre-fixed art theoretic commitments). For (...) example, when philosophical aesthetics tends toward insularity, we shouldn’t be surprised to find standard art-ontological categories incongruous with those standardly employed in contemporary metaphysics. Of course, when contemporary metaphysics tends to ignore aesthetic and art theoretic concerns, perhaps we likewise shouldn’t be surprised to find the climate of contemporary metaphysics inhospitable for a theory of art. While this may seem to suggest at least a prima facie tension between our basic art theoretic commitments considered from within philosophical aesthetics and our standard ontological commitments considered from without, I think any perceived tension or antagonism largely due to metaphysicians and aestheticians (at least implicitly) assuming there to be but two available methodological positions with respect to the relationship between contemporary metaphysics and philosophical aesthetics (in the relevant overlap areas). I call these two opposing views the Deference View and the Independence View. I argue that either view looks to lead to what I call the Paradox of Standards. (shrink)

Indonesia has recently faced problems in various aspects of life. The results of a social media survey in Indonesia in early 2021 that the biggest threat to the Pancasila ideology is communism and other western ideologies. Communism has a dark history in the life of the Indonesian people. It shows the problem of thinking and philosophical views of the Indonesian people. This research is textbook research that aims to analyze philosophy books, namely mathematics education philosophy textbooks written with (...) a Western cultural background by Paul Ernest. This book was chosen as the object of analysis because this book is the essential reference textbook for the philosophy of mathematics education in universities providing prospective mathematics teachers in Indonesia. The research results show that this book contributes to developing Western thoughts that are contrary to Pancasila. It needs to eliminate the most critical thing in this book. It is the atheistic and individualistic philosophy underlying the ideology of mathematics education because this is not in line with the theistic collectivistic Pancasila. Therefore, this research recommends developing a textbook on the philosophy of mathematics education. It should be under the Pancasila ideology or written based on the Pancasila ideology. (shrink)

Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of two heights, for (...) example, is a perceivable and measurable real relation between properties of physical things, a relation that can be shared by the ratio of two weights or two time intervals. Ratios are an example of continuous quantity; discrete quantities, such as whole numbers, are also realised as relations between a heap and a unit-making universal. For example, the relation between foliage and being-a-leaf is the number of leaves on a tree,a relation that may equal the relation between a heap of shoes and being-a-shoe. Modern higher mathematics, however, deals with some real properties that are not naturally seen as quantity, so that the “science of quantity” theory of mathematics needs supplementation. Symmetry, topology and similar structural properties are studied by mathematics, but are about pattern, structure or arrangement rather than quantity. (shrink)

I begin by outlining Du Châtelet’s ontology of mathematical objects: she is an idealist, and mathematical objects are fictions dependent on acts of abstraction. Next, I consider how this idealism can be reconciled with her endorsement of necessary truths in mathematics, which are grounded in essences that we do not create. Finally, I discuss how mathematics and physics relate within Du Châtelet’s idealism. Because the primary objects of physics are partly grounded in the same kinds of acts as yield mathematical (...) objects, she thinks we are sometimes licensed in drawing conclusions about physical things from mathematical premises. (shrink)

This paper deals with Natorp’s version of the Marburg mathematical philosophy of science characterized by the following three features: The core of Natorp’s mathematical philosophy of science is contained in his “knowledge equation” that may be considered as a mathematical model of the “transcendental method” conceived by Natorp as the essence of the Marburg Neo-Kantianism. For Natorp, the object of knowledge was an infinite task. This can be elucidated in two different ways: Carnap, in the Aufbau, contended that (...) this endeavor can be divided into two distinct parts, namely, a finite “constitution” of the object of knowledge and an infinite incompletable empirical description. In contrast, and more in the original spirit of Cohen and Natorp, the physicist and philosopher Margenau in The Nature of Physical Reality (Margenau. 1950) conceived the infinity of this “Aufgabe” as an infinite dialectical process, in which relative “data” and “conceptual constructs” determine each other. This dialectical process eliminates the dichotomy between Anschauung and Begriff that distinguished the Marburg Neo-Kantianism from Kantian orthodoxy, namely, the abandonment of the difference between intuition and concept. Finally, the paper deals with the non-Archimedean geometrical systems that played a central role in Natorp’s defence of Cohen’s “infinitesimal” metaphysics. (shrink)

Callard (2007) argues that it is metaphysically possible that a mathematical object, although abstract, causally affects the brain. I raise the following objections. First, a successful defence of mathematical realism requires not merely the metaphysical possibility but rather the actuality that a mathematical object affects the brain. Second, mathematical realists need to confront a set of three pertinent issues: why a mathematical object does not affect other concrete objects and other mathematical objects, what counts as a mathematical object, and (...) how we can have knowledge about an unchanging object. (shrink)

The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume’s Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one-to-one: (HP) #xx=#yy iff xx≈yy. The principal aim of this article is to use the notion (...) of grounding to develop this sort of abstractionism. The appeal to grounding enables a unified response to the two main challenges that confront abstractionism. First, we must explicate the metaphor of meta- physical “cheapness.” Second, we must rebut the “bad company” objection, which rejects abstraction principles like (HP) as tarnished by their similarity to inconsistent principles like Frege’s Basic Law V. By enforcing a simple requirement that all abstraction be properly grounded, we propose a unified solution to these two hard, and prima facie unrelated, problems. On our view, grounded abstraction simultaneously ensures “cheap” abstracta and permissible abstraction. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

A basic thesis of Neokantian epistemology and philosophy of science contends that the knowing subject and the object to be known are only abstractions. What really exists, is the relation between both. For the elucidation of this “knowledge relation ("Erkenntnisrelation") the Neokantians of the Marburg school used a variety of mathematical metaphors. In this con-tribution I reconsider some of these metaphors proposed by Paul Natorp, who was one of the leading members of the Marburg school. It is shown that (...) Natorp's metaphors are not unrelated to those used in some currents of contemporary epistemology and philosophy of science. (shrink)

The type-level abstraction is a formal way to represent molecular structures in biological practice. Graphical representations of molecular structures of biological objects are also used to identify functional processes of things. This paper will reveal that category theory is a formal mathematical language not only to visualize molecular structures of biological objects as type-level abstraction formally but also to understand how to infer biological functions from the molecular structures of biological objects. Category theory is a toolkit to understand biological knowledge (...) at the type-level formally, not individual token-level, as well as typical heuristic strategies in molecular biology. (shrink)

Any philosophy of science ought to have something to say about the nature of mathematics, especially an account like constructive empiricism in which mathematical concepts like model and isomorphism play a central role. This thesis is a contribution to the larger project of formulating a constructive empiricist account of mathematics. The philosophy of mathematics developed is fictionalist, with an anti-realist metaphysics. In the thesis, van Fraassen's constructive empiricism is defended and various accounts of mathematics are considered and rejected. (...) Constructive empiricism cannot be realist about abstract objects; it must reject even the realism advocated by otherwise ontologically restrained and epistemologically empiricist indispensability theorists. Indispensability arguments rely on the kind of inference to the best explanation the rejection of which is definitive of constructive empiricism. On the other hand, formalist and logicist anti-realist positions are also shown to be untenable. It is argued that a constructive empiricist philosophy of mathematics must be fictionalist. Borrowing and developing elements from both Philip Kitcher's constructive naturalism and Kendall Walton's theory of fiction, the account of mathematics advanced treats mathematics as a collection of stories told about an ideal agent and mathematical objects as fictions. The account explains what true portions of mathematics are about and why mathematics is useful, even while it is a story about an ideal agent operating in an ideal world; it connects theory and practice in mathematics with human experience of the phenomenal world. At the same time, the make-believe and game-playing aspects of the theory show how we can make sense of mathematics as fiction, as stories, without either undermining that explanation or being forced to accept abstract mathematical objects into our ontology. All of this occurs within the framework that constructive empiricism itself provides the epistemological limitations it mandates, the semantic view of theories, and an emphasis on the pragmatic dimension of our theories, our explanations, and of our relation to the language we use. (shrink)

Two slogans define structuralism: contemporary mathematics studies structures; mathematical objects are places in those structures. Shapiro’s version of structuralism posits abstract objects of three sorts. A system is “a collection of objects with certain relations” between these objects. “An extended family is a system of people with blood and marital relationships.” A baseball defense, e.g., the Yankee’s defense in the first game of the 1999 World Series, is a also a system, “a collection of people with on-field spatial and (...) ‘defensive-role’ relations”. “A structure is the abstract form of a system” ; it consists of “a collection of places [sometimes called positions or offices] and a finite collection of functions and relations on these places”. (shrink)

Beach's Gaelic Symphony is plausibly an abstract object that Beach created. The view that people create some abstract objects is called abstract creationism. There are abstract creationists about many kinds of objects, including musical works, fictional characters, arguments, words, internet memes, installation artworks, bitcoins, and restaurants. Alternative theories include materialism and Platonism. This paper discusses some of the most serious objections against abstract creationism. Arguably, these objections have ramifications for questions in metaphysics pertaining to the (...) abstract/concrete distinction, time, causation, vague existence, vague identity, and inadvertent creation. (shrink)

[EDIT: This book will be published open access. Check back around April 2024 to access the entire book.] One central aim of science is to provide explanations of natural phenomena. What role(s) does mathematics play in achieving this aim? How does mathematics contribute to the explanatory power of science? Rules to Infinity defends the thesis, common though perhaps inchoate among many members of the Vienna Circle, that mathematics contributes to the explanatory power of science by expressing conceptual rules, rules which (...) allow the transformation of empirical descriptions. Mathematics should not be thought of as describing, in any substantive sense, an abstract realm of eternal mathematical objects, as traditional platonists have thought. In Rules to Infinity, this view, which I call mathematical normativism, is updated with contemporary philosophical tools, and it is argued that normativism is compatible with mainstream semantic theory. This allows the normativist to accept that there are mathematical truths, while resisting the platonistic idea that there exist abstract mathematical objects that explain such truths. Furthermore, Rules to Infinity defends a particular account of the distinction between scientific explanations that are in some sense distinctively mathematical – those that explain natural phenomena in some uniquely mathematical way – and those that are only standardly mathematical, and it lays out desiderata for any account of this distinction. Normativism is compared with other prominent views in the philosophy of mathematics such as neo-Fregeanism, fictionalism, conventionalism, and structuralism. Rules to Infinity serves as an entry point into debates at the forefront of philosophy of science and mathematics, and it defends novel positions in these debates. (shrink)

The Polish philosophy of mathematics in the 19th century is not a well-researched topic. For this period, only five philosophers are usually mentioned, namely Jan Śniadecki, Józef Maria Hoene-Wroński, Henryk Struve, Samuel Dickstein, and Edward Stamm. This limited and incomplete perspective does not allow us to develop a well-balanced picture of the Polish philosophy of mathematics and gauge its influence on 19th- and 20th-century Polish philosophy in general. To somewhat complete our picture of the history of the (...) Polish philosophy of mathematics in those times, we here present the profiles of some lesser-known Polish Romantic philosophers of the 19th century, namely Karol Libelt, Bronisław Trentowski, and Józef Kremer. We discuss their contributions to the philosophy of mathematics and their metaphysical perspectives, and we also show how their metaphysical ideas have found some continuity in the studies of some Catholic philosophers. (shrink)

This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; (...) and to the types of intention, when the latter is interpreted as a modal mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic epistemic two-dimensional hyperintensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states, by contrast to availing of modal axiom 4 (i.e. the KK principle). Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional intensions of epistemic two-dimensional semantics solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between topic-sensitive epistemic two-dimensional truthmaker semantics, the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. Chapter \textbf{9} examines the modal profile of $\Omega$-logic in set theory. Chapter \textbf{10} examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter \textbf{11} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of epistemic two-dimensional semantics in order to account for the interaction of the interpretational and objective modalities thereof. Chapter \textbf{12} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal semantics for the different types of intention and the relation of the latter to evidential decision theory. The multi-hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is applied in chapters \textbf{7}, \textbf{8}, \textbf{10}, \textbf{11}, \textbf{12}, and \textbf{14}.}. (shrink)

PUTNAM has made highly regarded contributions to mathematics, to philosophy of logic and to philosophy of science, and in this book he brings his ideas in these three areas to bear on the traditional philosophic problem of materialism versus (objective) idealism. The book assumes that contemporary science (mathematical and physical) is largely correct as far as it goes, or at least that it is rational to believe in it. The main thesis of the book is that consistent acceptance (...) of contemporary science requires the acceptance of some sort of Platonistic idealism affirming the existence of abstract, non-temporal, non-material, non-mental entities (numbers, scientific laws, mathematical formulas, etc.). The author is thus in direct opposition to the extreme materialism which had dominated philosophy of science in the first three quarters of this century. The book can be recommended to the scientifically literate, general reader whose acquaintance with these areas is limited to the literature of the 1950’s and before, when it had been assumed that empiricistic materialism was the only philosophy compatible with a scientific outlook. To this group the book presents an eye-opening challenge fulfilling the author’s intention of “shaking up preconceptions and stimulating further discussion”. QUINE’S book is not easy to read, partly because the level of sophistication fluctuates at high frequency between remote extremes and partly because of convoluted English prose style and devilish terminology. Almost all of the minor but troublesome technical errata in the first printing have been corrected [see reviews, e.g., the reviewer, Philos. Sci. 39 (1972), no. 1, 97–99]. In the opinion of the reviewer the book is not suitable for undergraduate instruction, and without external motivation few mathematicians are likely to have the patience to appreciate it. Nevertheless, a careful study of the book will more than repay the effort and one should expect to find frequent references to this book in coming years. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, undecidable (...) propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. I also develop a novel topic-sensitive truthmaker semantics for dynamic epistemic logic, and develop a novel dynamic two-dimensional semantics. Chapter \textbf{3} provides an abstraction principle for epistemic (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} are availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between my hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest, too, that observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between the interpretational and objective modalities and truthmakers thereof. This yields the first hyperintensional category theory in the literature. I invent a new mathematical trick in which first order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals while bypassing the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

This draft preprint presents a nine step argument for “Connectionist Structuralism” (CS), an account of the ontology of abstract objects that is neither purely nominalist nor purely platonist. CS is a common, often implicit assumption in parts of the artificial intelligence literature, but such discussions have not presented formal accounts of the position or engaged with metaphysical issues that potentially undermine it. By making the position legible and presenting an initial case for it, we hope to support a constructive (...) dialogue between AI researchers and philosophers of metaphysics that helps both sides to refine the position. -/- CS proposes that each abstract object we can draw on in human analysis corresponds to a particular subset of an individual person’s brain structure whose functionality is isomorphic to a subset of the nodes and connections in a suitable connectionist network. In other words, abstract objects are physically realised, but in individual brains, rather than only in the referent objects (pure nominalism) or in metaphysical universals (pure platonism). -/- This paper’s minimum claim is that CS can account for all abstract object predicables regarding sensible properties, such as “is red” or “is a square”. Using evidence from cognitive neuroscience, machine learning, and evolutionary biology, as well as a fully traceable toy example, we describe how CS can support our core cognitive uses of such sensible properties and can account for our core phenomenal experiences of them. In the former, CS provides sufficient albeit imperfect inferential safety, whose limitations are argued to strengthen rather than weaken the case for CS as describing human behaviour. In the latter, four target phenomenal features are accounted for – abstract objects as feeling intangible, non-located, transparent, and unchanging - along with accounts for further phenomena such as semantic refinement, the Stroop effect, synaesthesia, semantic clarity, sensory overload, and satiation. -/- Our minimum claim concerns the day-to-day usage and felt experience of abstract objects, but we suggest also an extended claim in which CS can form the basis of a pragmatic sufficient logic. As such, the initial outline of a response is provided for five common objections to positions that seek to ground abstract objects without reference to metaphysically stand-alone universals: referential opacity; identity of indiscernibles; infinite regression; non-physical concepts; and necessary truths. These outlines lay the foundation for (but do not seek to formally demonstrate) an extended CS account that addresses other abstract objects, including issues relating to their use in mathematics and formal logic. -/- The CS account leads to a four-layer hierarchy of similarity for whether your “red” is the same as mine. Considering both semantic and phenomenal similarity, we conclude that our “reds” are likely non-identical but can be made close enough for practical purposes. Finally, we describe how future work could elaborate CS as a metaphysical project and how confidence in it could be tested through empirical research. (shrink)

-/- FILOSOFIA: CRÍTICA À METAFÍSICA -/- PHILOSOPHY: CRITICISM TO METAPHYSICS -/- Por: Emanuel Isaque Cordeiro da Silva - UFRPE Alana Thaís Mayza da Silva - CAP-UFPE RESUMO: A Metafísica (do grego: Μεταφυσική) é uma área inerente à Filosofia, dito isto, é uma esfera que compreende o mundo e os seres humanos sob uma fundamentação suprassensível da realidade, bem como goza de fundamentação ontológica e teológica para explicação dos dilemas do nosso mundo. Logo, não goza da experiência e explicação científica (...) com base na matemática, ciências, observação, análise, etc. e sim da explicação apenas teorética sem a análise empírica. Por fim, filósofos de divergentes escolas e eras da história da filosofia fundamentaram suas críticas quanto a Metafísica; e este breve trabalho tem como objetivo analisar as críticas dos filósofos e trazer uma conclusão clara e objetiva para os leitores, sejam leigos ou não. Palavras-chave: Metafísica. Crítica. Teoria. Suprassensível. Empírico. Experiência. ABSTRACT Metaphysics is an area inherent to philosophy, that is, it is a sphere that comprises the world and human beings under a supersensitive foundation of reality, as well as enjoying ontological and theological foundations for explaining the dilemmas of our world. Therefore, it does not enjoy the scientific experience and explanation based on mathematics, science, observation, analysis, etc., but only the theoretical explanation without empirical analysis. Finally, philosophers from divergent schools and eras in the history of philosophy based their criticism of Metaphysics, and this brief work aims to analyze the criticism of philosophers and bring a clear and objective conclusion to readers, whether lay or not. Keywords: Metaphysics. Criticism. Theory. Supersensitive. Empirical. Experience. INTRODUÇÃO Para uma fundamentação alicerçada na observação, análise e conclusões pragmáticas, é imprescindível aclarar o conceito de Metafísica como esfera inerente da Filosofia, bem como disciplina primordial para o currículo do Filósofo formado nas Universidades globais. Para isso, discorro acerca do conceito de Metafísica desde a alcunha aristotélica de 'Filosofia primeira', passando à análise modernista, em especial à kantiana e seu criticismo (Crítica da Metafísica é a alma desse trabalho), sob análise de autores didáticos e, por fim, sob a elucidação da esfera Metafísica mediante os dicionários de Filosofia de Japiassú e Marcondes, e de Nicola Abbagnano. No conjunto de obras aristotélicas entituladas de Metafísica, Aristóteles buscou elucidar 'o ser enquanto ser', isto é, buscar a objetividade das coisas e do mundo mediante a subjetividade e a realidade suprassensível dessas coisas. Logo, a explicação dos fenômenos que cercavam a Grécia clássica eram explicados além do alicerce das Ciências tradicionais da época (Física, Química, Biologia, etc.). Todavia, para Aristóteles, a Metafísica consiste na 'ciência primeira' no que se refere o fornecimento de um fundamento único para todas as demais ciências, ou seja, dar-lhes o objeto ao qual elas se referem e os princípios dos quais todas elas dependem. Por fim, a Metafísica implica ser uma enciclopédia das ciências um inventário completo e exaustivo de todas as ciências, em suas relações de coordenação e subordinação, nas tarefas e nos limites atribuídos a cada uma, de modo definitivo. Na modernidade, a Metafísica perde a centralidade do mundo da Filosofia, quem é o precursor de tal declínio é Immanuel Kant e seu criticismo. Tal descendência é elucidada e aclarada posteriormente no presente trabalho. A Metafísica moderna ganha uma nova interpretação, sendo alcunhada e levada por analogia à Ontologia, expressada, segundo Kant, como conceito de gnosiologia. Para Kant, Metafísica é a fonte inerente do estudo e explicação das formas alicerçadas na razão, bem como fundamento de toda realidade suprassensível que se deve basear para elucidar e aclarar o mundo moderno. Ainda segundo o filósofo, a Metafísica é a fonte de todos os princípios reais para a explicação da realidade. Sendo ela, por fim um 'sistema filosófico' alicerçado sobre uma perspectiva ontológica, teológica e/ou suprassensível da realidade. No nosso finalismo, a inerência da Metafísica à filosofia transcende a explicação de todo o universo e sua totalidade (matéria e forma como alcunha Aristóteles). Logo, na Filosofia Clássica com Platão e, especialmente em Aristóteles, o Mito passou a ser ciência e essa ciência era determinada de Metafísica. Vale salientar que a alcunha 'Metafísica' foi, primeiramente, usada por Andrônico de Rodes . Por fim, a Metafísica primordial para fundamentar as explicações necessárias para os fenômenos que se decorrera na Grécia antiga, foi perdendo centralidade ao passo da história da filosofia e, na modernidade, com Hume e Kant passa por uma grande crise, discorrida agora no trabalho. -/- DA CRÍTICA À METAFÍSICA Platão criou um sistema metafísico de explicação da realidade e do mundo que influenciou muitos pensadores e religiosos. Boa parte da história da filosofia foi composta de sistemas metafísicos. Mas, se, por um lado, o pensamento metafísico tem uma vida longa e profícua, por outro, é alvo de críticas constantes, principalmente por filósofos modernos e contemporâneos, que questionam a validade das teses metafísicas, seja porque tratam de coisas que não podem ser conhecidas, como alma, Deus e formas inteligíveis, seja porque suas conclusões são consideradas enganosas. ARISTÓTELES E A FILOSOFIA PRIMEIRA As ciências teóricas ou teoréticas são aquelas que produzem um saber universal, válido em qualquer situação, e necessário, isto é, que não pode ser de outro modo. Essas ciências estão voltadas para a contemplação da verdade. O sábio que se dedica a elas encontra um fim em si mesmo, pois o resultado de sua investigação não gera nenhum objeto exterior, como uma construção ou uma escultura, mas beneficia sua própria alma. Elas estão organizadas em Metafísica, Matemática e Física, que inclui a Psicologia ou a ciência da alma. As principais obras teoréticas escritas por Aristóteles são Física, De anima e Metafísica. Filosofia Primeira, que chegou até nós com o nome de Metafísica, também chamada muitas vezes por Aristóteles de Teologia .com a ressalva de que Teologia aqui não tem o sentido que é dado a ela contemporaneamente), é a expressão que designava a mais elevada das ciências teoréticas, diferenciando-se da chamada Filosofia Segunda, ou Física. Aristóteles abre o Livro I da Metafísica com a célebre passagem: Todos os homens têm, por natureza, desejo de conhecer: uma prova disso é o prazer das sensações, pois, fora até da sua utilidade, elas nos agradam por si mesmas e, mais que todas as outras, as visuais [...l. (ARISTÓTELES, 1969). A busca pelo conhecimento encontra-se na própria natureza humana, desde o nascimento, mesmo que se manifeste em seu grau mais rudimentar. Prova disso é o conhecimento por meio das sensações ou dos sentidos, entre os quais se destaca o da visão. É o chamado conhecimento por empiria ou saber empírico, que não pode ser ensinado porque é adquirido imediata e concretamente quando percebemos as coisas sensíveis. Também há um saber que vem da técnica. O técnico é aquele que tem o conhecimento dos meios para chegar a um fim. Como ensina o historiador da Filosofia, Julián Marias, apesar de o saber técnico ser superior ao saber empírico, ambos são necessários em nossa vida: 1..l Portanto, a tékhne [técnica) é superior à empeiria; mas esta também é necessária, por exemplo, para curar, porque o médico não tem de curar o homem, e sim Sócrates, e o homem apenas de modo mediato. (MARIAS, J. 2015). O conhecimento metafísico é menos necessário em nossa vida cotidiana, mas nenhum outro lhe é superior. A Metafísica é o conhecimento pelas causas e, como vimos, o conhecimento é científico quando ele nos dá os princípios e as causas das coisas. Para Aristóteles, tudo o que existe é efeito de uma causa e ela é a responsável por esse algo ser necessariamente de um jeito e não de outro. De acordo com Aristóteles, conhecer algo é conhecer pela causa. Dessa forma, a explicação de algo ou o seu conhecimento deve sempre dizer por que algo é necessariamente de determinado modo. Sem esse tipo de explicaçào, não pode haver propriamente conhecimento. Daí se constata o caráter rigoroso da explicação científica: ela não apenas sabe que "algo é isto", mas é capaz de explicar por que algo é necessariamente isto. A definição aristotélica de Metafísica Chauí indaga em seu livro Iniciação à Filosofia a questão central do que Aristóteles entende por sua alcunha ou Metafísica. Chauí explana: Embora Aristóteles admita que para cada tipo de Ser e suas essências existe uma ciência teorética própria (física, biologia, psicologia, matemática, astronomia), ele também defende a necessidade de uma ciência geral, mais ampla, mais universal, anterior a todas essas, cujo objeto não seja esse ou aquele tipo de Ser, essa ou aquela modalidade de essência, mas o Ser em geral, a essência em geral. Essa ciência, para Aristóteles, é a Filosofia Primeira ou metafísica, que investiga o que é a essência e aquilo que faz com que haja essências particulares diferenciadas, que estuda o Ser enquanto Ser. (CHAUÍ, 2010). O objeto investigado pela Metafísica também é muito diferente do objeto conhecido pelo saber empírico ou pelo saber técnico. O saber empírico nos faz conhecer as coisas particulares, como Sócrates, o Oráculo de Delfos e a caneta esferográfica; o saber técnico nos faz conhecer as diversas técnicas, como os procedimentos médicos e a engenharia de construir templos. O que o saber metafísico, então, nos permite conhecer? Segundo Aristóteles, a Metafisica estuda o ser enquanto ser. Não considera o ser de modo particular como fazem as demais ciências — por exemplo, ao investigar Sócrates, os procedimentos médicos etc. mas busca o que é universal, o que envolve tanto Sócrates quanto os procedimentos médicos, tanto o Oráculo de Delfos quanto a arte de construir templos. Segundo Aristóteles, a Metafísica trata das causas primeiras, que são as quatro seguintes: • a causa material, que é a matéria de que é feita alguma coisa. Em uma escultura, por exemplo, a causa material pode ser o bronze ou o mármore; • a causa formal, que é a forma ou a essência das coisas. No exemplo da escultura, é a forma ou a aparência que possibilita que a reconheçamos como uma escultura (e não como um poste, por exemplo); • a causa eficiente, que é o agente que produz a coisa. Uma escultura é produzida por um artista; • a causa final, que é a razão ou a finalidade das coisas, A finalidade da escultura é o prazer estético. A Metafísica também se ocupa da substância. A substância responde pelos significados do ser. Mas o que é a substância? Trata-se de uma questão complexa no pensamento aristotélico. Aristóteles recusa-se a entender a substância como sendo a forma platónica. Ela nào é o suprassensível. Seria entao a substância o sensível individual? As coisas são matéria e forma; amatéria é o substrato da coisa (por exemplo, o mármore é a matéria da estátua). Sem sua forma, ela é apenas potencialidade, mas sem a matéria qualquer realidade sensível se desvaneceria. Assim, a matéria pode ser dita substância em sentido impróprio. A forma, por seu turno, é aquilo que determina a matéria, é a sua essência; ela é substancia em sentido próprio. Mas a matéria é também matéria enformada. As coisas sensíveis são, a um só tempo, matéria e forma. E esse composto também pode ser legitimamente chamado de substância. Aristóteles apresenta três gêneros de substâncias: • as substâncias sensíveis, que nascem, morrem e, por isso, passam por todo tipo de mudança; • as substancias sensíveis e incorruptíveis, que não passam por nenhum tipo de mudança, como os planetas e as estrelas; • a substância suprassensível, que é superior às outras duas: o Primeiro Motor Imóvel, o deus aristotélico, causa de todo movimento, mas sendo ele mesmo imóvel. (Ele precisa ser imóvel porque se ele também se movesse precisaria haver uma causa para esse movimento, o que levaria a uma espécie de regressão ao infinito). Assim como o demiurgo de Platão, o deus aristotélico nao é um deus criador. Não criou o Universo nem gerou o movimento dos corpos. Ao contrário, o deus de Aristóteles exerce uma atraçáo sobre o Universo, motivando sua existência, e sobre os corpos, gerando seu movimento, ou seja, atrai tudo para si como objeto de amor. Não é, portanto, causa eficiente do mundo e do movimento do mundo, mas causa final. O deus aristotélico nao cria o mundo do nada por um ato de vontade. Ele nao é causa eficiente do mundo, como indicamos ser o artista causa eficiente de sua escultura. A METAFÍSICA CLÁSSICA OU MODERNA Grandes revoluções no pensamento do século XVII deram estopim à uma grande crise na esfera metafísica. De modo resumido a Metafísica sofreu uma grande transformação e uma nova elaboração, sem a fundamentação embasada no pensamento platônico, aristotélico ou neoplatônico. Sendo assim a nova metafísica é caracterizada por: [...] afirmação da incompatibilidade entre fé e razão, acarretando a separação de ambas, de sorte que a religião e a filosofia possam seguir caminhos próprios, mesmo que a segunda não esteja publicamente autorizada a expor ideias que contradigam as verdades ou dogmas da fé [...] [...] redefinição do conceito de Ser ou substância. Os modernos conservam a definição tradicional da substância como o Ser que existe em si e por si mesmo, que subsiste em si e por si mesmo. Porém, em lugar de considerar que a substância se define por gênero e espécie, havendo tantos tipos de substâncias quantos gêneros e espécies houver, passa-se a definir a substância levando em consideração seus predicados essenciais ou seus atributos essenciais, isto é, aquelas propriedades ou atributos sem os quais uma substância não é o que ela é. Após o supracitado, os cartesianos e Descartes dirão que há somente três substâncias essenciais, divergindo totalmente do pensamento aristotélico, as substâncias cartesianas são a Alma, a matéria dos corpos e Deus. Para empiristas, só se é possível conhecer a substância corpórea, logo não se é possível elaborar uma metafísica, e sim uma geometria ou uma física que se ocupa do estudo dessas matérias corpóreas. Baruch Spinoza (1632-1677) explanou a primordialidade de se elaborar uma definição genérica e universal, comumente aceita por toda a comunidade de filósofos de Aristóteles a modernidade. Podemos reduzir o conceito proposto da seguinte maneira: “substância é aquilo que existe em si e por si e não depende de outros para existir”.Logo, diz Espinosa, que há apenas uma substância no Universo que não depende de outra para existir, tal substância é o próprio Deus ou a natureza. Nessa revolução do pensamento filosófico, em peculiar à metafísica, houve também a redefinição do conceito de causa e causalidade. Em Aristóteles haviam quatro tipos de causas, agora, na modernidade, se propunha apenas dois tipos: a eficiente e a final. Chauí explana que a Causa eficiente é aquela na qual uma ação anterior determina como consequência necessária a produção de um efeito, e que seu alcance é universal na natureza. Causa final é aquela que determina, para os seres pensantes, a escolha da realização ou não realização de uma ação, e que só opera na ação de Deus e nas ações humanas. Houve também, nessa revolução, uma quebra do paradigma metafísico, isto é, a metafísica não se dividia mais entre a teologia, a psicologia racional e a cosmologia racional. Agora, a metafísica ganha um novo rumo, um rumo próprio de estudo embasado em suas próprias fundamentações. Logo, a metafísica se apropria de três e apenas três ideias de substância. A substância infinita, ou o próprio Deus. A substância pensante que é a consciência como faculdade de reflexão e de representação da realidade alicerçada na razão. E, por fim, a substância extensa que é a natureza embasada nos princípios e leis regidas pela matemática e a mecânica. DAVID HUME E A CRISE DA METAFÍSICA O filósofo escocês David Hume (1711-1776) nos ajuda a entender em parte o teor das críticas ao pensamento metafísico. "Se tomarmos em nossas mãos um volume qualquer, de teologia ou metafísica escolástica, por exemplo, façamos a pergunta: contém ele qualquer raciocínio abstrato referente a números e quantidades? Não. Contém qualquer raciocínio experimental referente a questões de fato e de existência? Não. Às chamas com ele, então, pois não pode conter senão sofismas e ilusão." (HUME, 2004) Hume critica a teologia e a metafísica escolástica, que foi a metafísica praticada principalmente nas escolas cristãs durante a Idade Média, porque suas concepções não tratam nem de conceitos matemáticos, que são formais e podem levar a conclusões seguras - por exemplo, "2 + 2 = 4" -, nem de afirmações sobre fatos, que podem ser verificados empiricamente, isto é, pelos órgãos dos sentidos —como "a aceleração de um corpo em queda livre é de 9,8 metros por segundo", Assim, Hume aconselha o leitor a jogar as afirmações metafísicas na fogueira, pois são consideradas ilusões, fantasias ou produto de erros de raciocínio e nada têm a acrescentar ao conhecimento humano. A partir de Hume, a metafísica, tal como existira desde os gregos, tornara-se impossível. O CRITICISMO KANTIANO E O FIM DA METAFÍSICA CLÁSSICA Entre a Dissertação de 1770 e a Crítica da razão pura, mais de dez anos se passaram. Apesar de a obra ser volumosa, a redaçào da Crítica da razão pura não chegou a cinco meses de trabalho. De início, Kant pretendia apenas revisar sua dissertação, mas o trabalho acabou por levá-lo a figurar entre os grandes nomes do pensamento filosófico, como Aristóteles e Descartes. Kant fez um diagnóstico bastante negativo da situação da Filosofia de sua época, em particular da Metafísica, que, embora fosse considerada a Filosofia Primeira, patinava em questões impossíveis de serem resolvidas. Ao mesmo tempo, buscou inserir a Filosofia no rumo seguido pelas Ciências Naturais, pela Lógica e pela Matemática, que já tinham encontrado o caminho da Ciência. Em alguns pontos, o projeto kantiano se aproximou de algumas características do projeto cartesiano. Descartes duvidou de todo o saber conhecido, pois queria encontrar um fundamento seguro que sustentasse a edificação da Filosofia e, mais particularmente, da Ciência. Diferentemente de Descartes, contudo, Kant nào duvidou de todo o conhecimento que havia aprendido. Para ele, bastava estabelecer os limites do conhecimento racional, ou seja, o que a razão podia conhecer e o que era impossível de ser conhecido. No fim do século XVIII, de acordo com a proposta de Wolff, a Metafísica era dividida em duas partes: Metafísica geral e Metafísica especial. A primeira tratava da Ontologia (ciência do ser enquanto ser, como definido por Aristóteles), e a segunda se ocupava do ser humano (Psicologia), do mundo (Cosmologia) e de Deus (Teologia racional), Os principais temas in-vestigados eram a imortalidade da alma, a liberdade a finitude ou infinitude do mundo e a existência de Deus. No entanto, de acordo com Kant, as questões metafísicas clássicas extrapolavam as capacidades de conhecimento do ser humano, pois estavam tão afastadas da experiência que a razão só podia pensá-las, sem conhecê-las realmente. As questões postas pela Metafísica clássica extrapolavam o terreno de qualquer experiência possível e, no entanto, a razão, por sua própria natureza, apresenta questões que não pode evitar, mas que também náo pode responder por não terem nenhuma pedra de toque na experiência. Disso decorre a pergunta fundamental sobre a possibilidade de haver um conhecimento metafísico. Publicada em 1781, a Crítica da razão pura foi uma tentativa de responder a essa pergunta. A obra instituiu um "tribunal da razao" para examinar a própria razão, e não para tomar partido no conflito entre racionalistas, como Descartes e Leibniz, que elaboraram sistemas metafísicos, e empiristas, como Locke e Hume, que basearam o conhecimento unicamente nos sentidos. Criticar a razão é determinar as possibilidades, os limites e o alcance do próprio conhecimento. A transformação foi tao grande para a Filosofia que essa obra nao pôde ser ignorada. A situação da Metafísica No texto a seguir, retirado do célebre prefácio à primeira edição de Crítica da razão pura, podemos ver o diagnóstico de Kant sobre a situaçao da Metafísica no período em que vivia. A razão humana, num determinado dominio dos seus conhecimentos, possui o singular destino de se ver atormentada por questões, que náo pode evitar, pois lhe são impostas pela sua natureza, mas às quais também não pode dar resposta por ultrapassarem completamente as suas possibilidades. Não é por culpa sua que cai nessa perplexidade. Parte de princípios, cujo uso é inevitável no decorrer da experiência e, ao mesmo tempo, suficientemente garantido por esta. Ajudada por estes princípios eleva-se cada vez mais alto (como de resto lho consente a natureza) para condições mais remotas. Porém, logo se apercebe de que, desta maneira, a sua tarefa há de ficar sempre inacabada, porque as questões nunca se esgotam; vê-se obrigada, por conseguinte, a refugiar-se em princípios, que ultrapassam todo o uso possível da experiência e, não obstante, estão ao abrigo de qualquer suspeita, pois o senso comum está de acordo com eles. Assim, a razão humana cai em obscuridades e contradições, que a autorizam a concluir dever ter-se apoiado em erros, ocultos algures, sem contudo os poder descobrir. Na verdade, os princípios de que se serve, uma vez que ultrapassam os limites de toda experiência, já não reconhecem nesta qualquer pedra de toque. O teatro destas disputas infindáveis chama-se Metafísica. (KANT, I. 2010). O fim da Metafísica clássica Podemos agora retomar o questionamento sobre os motivos pelos quais a física newtoniana é possível como ciência e entender também por que a metafísica racionalista tradicional não é conhecimento científico, segundo a teoria kantiana. A física trata de regras e leis dos fenômenos, das representações conformadas pela sensibilidade e pelo entendimento, ou seja, refere-se a objetos do conhecimento humano. A metafísica clássica, por sua vez, trata de coisas que não fazem parte da experiência sensível. Deus e a alma, por exemplo, não são apreendidos pela sensibilidade, não são impressões conformadas ou fenômenos e, portanto, não podem ser conhecidos pelo ser humano. Ideias ou conceitos dessa espécie podem ser pensados pela razão, mas é impossível conhecê-los, pois estão além da capacidade humana de conhecimento. Por esse motivo, a física é possível como ciência e a metafísica tradicional não. Com base nisso, Kant provocou uma revolução na teoria do conhecimento, que ele mesmo designou como "virada copernicana na filosofia", Se Copémico mudou a forma de entender o mundo ao defender a ideia de que o Sol está no centro do sistema solar, Kant mudou a forma de o ser humano compreender o mundo e a si próprio ao estabelecer a ideia de que o ser humano, e não os objetos externos, é o centro do conhecimento, O ser humano não é um agente passivo, que só recebe informações, mas atua decisivamente na constituição do conhecimento e do que se compreende como realidade. Os elementos a priori que determinam essa constituição do conhecimento são o escopo da filosofia kantiana, também conhecida como transcendental. Logo, o pensamento kantiano alterou profundamente a Metafísica. A existência de Deus ou a imortalidade da alma, por exemplo, passaram a ser questões que já não podiam ser respondidas. Se percebemos apenas as coisas que ocorrem em um tempo específico (antes, agora, depois), como seremos capazes de perceber algo que está na eternidade (Deus)? Como afirmar que a alma é imortal se não conseguimos percebê-la com os nossos sentidos? O conhecimento a partir de Kant, então, é sempre conhecimento de objetos, isto é, de conteúdos elaborados e modificados pelo sujeito que conhece. O conhecimento passou a se ocupar, assim, de fenómenos, e não mais da própria coisa, como pretendiam os metafísicos clássicos. Com a crítica kantiana, tornou-se impossível conhecer efetivamente os temas da Metafísica clássica e emitir qualquer opinião segura sobre assuntos que não podem ser assentados na experiência. Podemos pensar sobre Deus, podemos querer que nossa a seja imortal, mas jamais teremos a possibilidade de verificar esses assuntos em nossa experiência ou mesmo ter sobre eles verdadeiro conhecimento. Nada nos impede de acreditar em Deus ou na imortalidade da alma, porém não podemos exigir que a Filosofia se pronuncie a respeito desses temas. Na verdade, a crítica kantiana não diz que Deus existe nem que não existe, mas apenas que não é capaz de saben Deus, não obstante sua importância para a Filosofia do próprio Kant, é uma questão de fé. AUGUSTE COMTE E A METAFÍSICA NO SÉCULO XIX O filósofo francês Auguste Comte (1798-1857), considerado o pai do positivismo, fundamenta sua filosofia também em oposição às concepções metafísicas. Em seu entendimento, a busca por princípios gerais ou essências, entidades que estão além do que podemos observar ou perceber pelos órgãos dos sentidos, é algo infrutífero. Comte explana: "[...] não é supérfluo assinalar agora, de modo direto, a preponderância contínua da observação sobre a imaginação, como o principal caráter lógico da sã filosofia moderna, dirigindo nossas pesquisas, não para causas essenciais, mas para leis efetivas, dos diversos fenômenos naturais. Sem ser doravante imediatamente contestado, permanece este princípio fundamental muitas vezes desconhecido nos trabalhos especiais. Embora as diferentes ordens de especulações concedam, sem dúvida, à imaginação uma alta participação, isto é, constan temente empregada para criar ou aperfeiçoar os meios da vinculação entre os fatos constatados, mas o ponto de partida e a sua direçào não lhe poderiam pertencer em nenhum caso. Ainda quando procedemos verdadeiramente a priori, é claro que as considerações gerais que nos guiam foram inicialmente fundadas, quer na ciência correspondente, quer em outra, na simples observação, única fonte de sua realidade e também de sua fecundidade. Ver para prever: tal é o caráter permanente da verdadeira ciência. Tudo prever sem ter nada visto constitui somente uma absurda utopia metafísica, ainda muito seguida." (COMTE, 1978). A observação é considerada a única fonte de realidade. Abandonando a observação e apoiando-se única e exclusivamente na imaginação, como seria característico das especulaçôes metafísicas, o pensamento pode levar não para o conhecimento das coisas, mas para a fantasia. A ciência, então, deve ter como base as coisas que podem ser observadas. Todas as especulações e as teorias científicas, em última instância, devem ter como fundamento a realidade observável. LUDWIG WITTGENSTEIN E A METAFÍSICA CONTEMPORÂNEA O filósofo austríaco Ludwig Wittgenstein (1889-1951) também se opõe às afirmações metafísicas por considerar que estas tratam de problemas (ou pseudoproblemas) que não podem ser formulados claramente e, portanto, não podem ser respondidos. “O método correto da filosofia seria o seguinte: só dizer o que pode ser dito, ou seja, as proposições das ciências naturais [...] e depois, quando alguém quisesse dizer algo metafísico, mostrar-lhe que nas suas proposições existem sinais aos quais não foi dada uma denotação.” (WITTGENSTEIN, 1995). Quer dizer, as sentenças ou os termos metafísicos se referem a coisas, objetos ou seres enigmáticos ou misteriosos, que não podem ser tratados com clareza, isto é, não se sabe exatamente o que são, o que impossibilitaria a formulação de problemas e soluções reais. "Deus", "alma" ou "vida após a morte", por exemplo, são termos que se referem especificamente a quê? Os "problemas" metafísicos estariam no campo do mistério e não no da investigação racional. Por isso, como afirma Wittgenstein, "acerca daquilo de que se não pode falar, tem que se ficar em silêncio". CONCLUSÃO (ÕES) Ora, a Metafísica na Grécia antiga foi imprescindível para o entendimento dos mais variados dilemas do mundo antigo, sejam perguntas simples, ou complexas. Iniciada como um sistema filosófico platônico, passando a ser filosofia primeira em Aristóteles, a metafísica incorporou elementos teológicos, psicológicos e cosmológicos durante o mundo antigo, e ainda mais durante o período medieval e a hegemônica Igreja Católica. No mundo moderno e embasado pelo “Luz da razão”, a metafísica começa a ser refutada pelos pensadores da época em questão, dentre eles o célebre cético Hume, e o emblemático Kant, além de Descartes e seus cartesianistas. Kant e seu criticismo incorporaram à Metafísica uma nova interpretação, esta que fez a mesma entrar em declínio e sair da centralidade do mundo filosófico. Kant, graças às indagações e refutações de Hume, pôde acordar do célebre “sono dogmático”. O que fez Kant se importa com a Metafísica. Kant trouxe à luz o fim da Metafísica clássica e deu legado ao início da metafísica contemporânea alcunhada de Ontologia. No decorrer da história da filosofia, a Metafísica teve intensa ascenção e vertiginosos declives, sendo seu maior declive o crítico Kant. Por fim, muitos filósofos se apropriaram de ser heterodoxos e enfrentaram de frente o mundo metafísica e a base do catolicismo sobre a realidade. Posta realidade que hoje, com o legado de Kant, da fenomenologia de Husserl, etc., originaram a nova escola filosófica chamada de Ontologia. -/- REFERÊNCIAS BIBLIOGRÁFICAS ABBAGNANO, N. Dicionário de filosofia. 5ª. ed. Trad. Ivone Castillo Benedetti. São Paulo: Martins Fontes, 2007. ARANHA, M. L. de A.; MARTINS, M. H. P. Filosofando: introdução à filosofia. 5ª. ed. São Paulo: Moderna, 2013. ARISTÓTELES. Metafísica. Trad. L. Vallandro. Porto Alegre: Globo, 1969. BELO, R. dos S. Filosofia. Vol. Único. 2ª. ed. São Paulo: FTD, 2016. (Coleção Filosofia). BENOIT, L. O. Augusto Comte: fundador da física social. 1ª. ed. São Paulo: Moderna, 2002. CHAUÍ, M. Iniciação à Filosofia. 1ª. ed. São Paulo: Ática, 2010. COMTE, A. Curso de filosofia positiva. In: FERNANDES, F.; FILHO, E. de M. Comte: sociologia. São Paulo: Ática, 1978. (Coleção Grandes Cientistas Sociais). COTRIM, G.; FERNANDES, M. Fundamentos de filosofia. 2ª. ed. São Paulo: Saraiva, 2013. FERREIRO, H. Siete ensayos sobre la muerte de la Metafísica: Una introducción al idealismo absoluto a partir de la ontología. 1ª. ed. Porto Alegre: Editora Fi, 2016. FIGUEIREDO, V. de. (Org.); et. al. Filosofia: Temas e Percursos. Vol. Único. 2ª. ed. São Paulo: Berlendis & Vertecchia, 2016. FILHO, J. S. Filosofia e filosofias: existência e sentidos. 1ª. ed. Belo Horizonte: Autêntica, 2016. HUME, D. Investigações sobre o entendimento humano e sobre os principios da moral. São Paulo: Unesp, 2004. ________. Tratado da natureza humana: uma tentativa de introduzir o método experimental de raciocínio nos assuntos morais. São Paulo: Unesp, 2001. JAPIASSÚ, H.; MARCONDES, D. Dicionário básico de filosofia. 3ª. ed. Rio de Janeiro: Zahar, 2001. KANT, I. Crítica da razão prática. Lisboa: Edições 70, 1989. [Textos filosóficos]. ______. Crítica da razão pura. São Paulo: Abril Cultural, 1979. (Col. Os Pensadores). MARIAS, J. História da filosofia. 2ª. ed. Trad. Claudia Berliner. São Paulo: Martins Fontes, 2015. MELANI, R. Diálogo: primeiros estudos em filosofia. Vol. Único. 2ª. ed. São Paulo: Moderna, 2016. OTERO, J. G. Compendio de filosofía: para uso de los jóvenes estudiosos. 1ª. ed. Bogotá: Imprensa de San Bernardo, 1919. SILVA, F. L. e. Descartes, a metafísica da modernidade. 1ª. ed. São Paulo: Moderna, 2006. WITTGENSTEIN, L. Tratado lógico-filosófico. 2ª. ed. Lisboa: Fundação Calouste Gulbenkian, 1995. -/- Emanuel Isaque Cordeiro da Silva – estudante, pesquisador, professor, agropecuarista, altruísta e defensor dos direitos humanos e dos animais. -/- Alana Thaís Mayza da Silva – estudante, potterhead, pesquisadora, projetista, musicista, filantropa, defensora dos direitos humanos e dos animais, LGBT. (shrink)

Objective The relations between thought and reality are studied in many fields of philosophy and science. Examples include ontology and metaphysics in general, linguistics, neuroscience and even mathematics. Each one has its postulates, its language, its methods and its own constraints. It would be unreasonable, however, for them to ignore each other. In the pages that follow we will try to identify areas of proximity between the ideas of contemporary philosophers of language and those issued mainly by Ontology of (...) Knowledge but also by mathematics and neuroscience. We will try to take advantage of the clarity and the perfect structuring of the lecture « La philosophie contemporaine du langage » (the contemporary philosophy of language) given by Professor Denis Vernant . We will make use of this lecture, both for the ideas presented and as a reference process. The goal of this article is to bring out, through a benevolent confrontation, new ideas for the benefit of knowledge in general. (shrink)

The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior (...) analytics is used to distinguish between conceptions that share the same name but are substantively different: for example the search for a broader genus including all mathematical objects; the search for a common character of different species of mathematical objects; and the effort to treat magnitudes as numbers. (shrink)

We propose a way to explain the diversification of branches of mathematics, distinguishing the different approaches by which mathematical objects can be studied. In our philosophy of mathematics, there is a base object, which is the abstract multiplicity that comes from our empirical experience. However, due to our human condition, the analysis of such multiplicity is covered by other empirical cognitive attitudes (approaches), diversifying the ways in which it can be conceived, and consequently giving rise to different mathematical (...) disciplines. This diversity of approaches is founded on the manifold categories that we find in physical reality. We also propose, grounded on this idea, the use of Aristotelian categories as a first model for this division, generating from it a classification of mathematical branches. Finally we make a history review to show that there is consistency between our classification, and the historical appearance of the different branches of mathematics. (shrink)

I defend a new position in philosophy of mathematics that I call mathematical inferentialism. It holds that a mathematical sentence can perform the function of facilitating deductive inferences from some concrete sentences to other concrete sentences, that a mathematical sentence is true if and only if all of its concrete consequences are true, that the abstract world does not exist, and that we acquire mathematical knowledge by confirming concrete sentences. Mathematical inferentialism has several advantages over mathematical realism and (...) fictionalism. (shrink)

the Mona Lisa, the Mondscheinsonate, the Chanson d’automne are works of art, the salt shaker on your table, the car in your garage, or the pijamas on your bed are not. the basic question of the metaphysics of works of art is this: what makes a thing a work of art? that is: what sort of property do works of art have in virtue of which they are works of art? or more simply: what sort of property being a work (...) of art is? In this paper we argue that things are works of art in virtue of what they are like, their intrinsic features, that is, in virtue of the fact that they have the perceptual (auditory, visual, etc.) properties they have. In other words: being a work of art supervenes on perceptual-intrinsic features. Currently, this metaphysical view is extremely unpopular within the philosophy of art. It is unpopular because there allegedly exists a knock-down objection to it, the well-known argument from indiscernible counterparts. our thesis implies, among other things, that every perceptual duplicate of a work of art is also a work of art. according to the argument from indiscernible counterparts, however, there could be (or even: there are) indiscernible counterparts such that one of them is a work of art while the other is not. hence things cannot be works of art solely in virtue of what they are like. Our paper divides into three parts. In the first part we state our views. In the second part we defend it against various versions of the argument from indiscernible counterparts. (In doing so our position will become more plausible, we hope). In the final part we provide some meta-reflections on the matter. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Newton's Metaphysics: Essays by Eric SchliesserMarius StanEric Schliesser. Newton's Metaphysics: Essays. Oxford: Oxford University Press, 2021. Pp. 328. Hardback, $99.90.Newton owes his high regard to the quantitative science he left us, but his overall picture of the world had some robustly metaphysical threads woven in as well. Posthumous judgment about the value of these threads has varied wildly. Christian Wolff thought him a metaphysical rustic, as did (...) Hans Reichenbach some two centuries later ("Die Bewegungslehre bei Newton, Leibniz und Huygens," Kant-Studien 29 [1924]: 416–38). In the 1960s, the tide would turn, as Howard Stein and James Edward McGuire separately began to show that Newton's metaphysics was not just sophisticated, but often more compelling than its early modern alternatives (see respectively "Newtonian Spacetime," Texas Quarterly 10 [1967]: 174–200; and Tradition and Innovation: Newton's Metaphysics of Nature [Dordrecht: Kluwer Academic Publishers, 1995]). Eric Schliesser's book unfolds in that same register of appreciative, high scholarship, and it attests to the enduring attraction of its subject.The book grew out of previous discrete papers, to which he added a few more for this occasion. Accordingly, it does not come with a master argument. Rather, it is an in-depth exploration of some key metaphysical themes in Newton. As such, it befits its subject figure, who reflected on themes and concepts while stopping short of working out systems.The first major theme is Newton and Spinozism. The latter does not denote Spinoza's metaphysics. In fact, Schliesser explains, Newton shared with Spinoza some weighty commitments, for example, to space being actually infinite, and to substance monism: "for Newton, there is strictly speaking only one genuine substance," namely, God (33). Rather, "Spinozism" is an interpreter's category for a bundle of three theses: identifying God and nature; denying final causation in the physical world; and the assumption of "blind" metaphysical necessity. Some counted Hobbes and John Toland as Spinozist in this sense, and even Epicurus, avant la lettre. Newton and his followers argued vehemently against this package, as Schliesser shows in chapters 4, 5, and 8. Their chief complaint was that Spinozism is unable to account for the "origin of motion," for a certain type of order, and for the stability of cosmological structure—and to do so in a way that "meets the standards of Newtonian mechanics" (122). This interpretive lens allows Schliesser to draw Kant in, by reading his youthful Theory of Heaven as a possible Spinozist reply to their objections (chapter 3). [End Page 157]A second theme is the metaphysics of mechanics, where Schliesser weaves together several strands. One is the ontology of gravity, for which he offers a novel construal: for Newton, gravity counts as real, but in a qualified sense. Namely, it is not essential and not intrinsic: "even after creation, a lonely partless particle of matter in the universe would not be said to gravitate." And gravity is relational: a "shared quality" of two or more bits of matter, which obtains in virtue of their sharing a nature (19). Another strand is Newton's ontology of time—really, a subtle, difficult topic long neglected. Famously, Newton in Principia defends absolute, true, and mathematical time. The question is what these three qualifiers denote, and whether Newton thought they were synonymous. Schliesser in chapter 7 argues for the provocative view that "absolute" and "true" time are not identical concepts; rather, they pick out different entities. These entities share a common structure, namely, metric and topological. That makes them species of "mathematical" time (179). But, Schliesser contends, Newton has unequal warrant for his two concepts of time.Yet another theme is formal causation. It comes to the fore in chapter 5, which grapples with Newton's opaque idea that space is an "emanative effect." Schliesser explains that here, "emanation picks out a species of formal causation" (147), but in a novel, early modern sense that comes from Bacon, not Aristotle. If space has a formal cause, then so has its twin brother, time, the topic of chapter 7. God is their common formal cause. Then in chapter 6 (coauthored with Zvi Biener) Schliesser adds that, for Newton, laws... (shrink)

Art plays a significant role in Iris Murdoch’s moral philosophy, a major part of which may be interpreted as a proposal for the revision of religious belief. In this paper, I identify within Murdoch’s philosophical writings five distinct but related ways in which great art can assist moral/religious belief and practice: art can reveal to us “the world as we were never able so clearly to see it before”; this revelatory capacity provides us with evidence for the existence of (...) the Good, a metaphor for a transcendent reality of which God was also a symbol; art is a “hall of reflection” in which “everything under the sun can be examined and considered”; art provides us with an analogue for the way in which we should try to perceive our world; and art enables us to transcend our selfish concerns. I consider three possible objections: that Murdoch’s theory is not applicable to all forms of art; that the meaning of works of art is often ambiguous; and that there is disagreement about what constitutes a great work of art. I argue that none of these objections are decisive, and that all forms of art have at least the potential to furnish us with important tools for developing the insight required to live a moral/religious life. (shrink)

ABSTRACT. May scientists rely on substantive, a priori presuppositions? Quinean naturalists say "no," but Michael Friedman and others claim that such a view cannot be squared with the actual history of science. To make his case, Friedman offers Newton's universal law of gravitation and Einstein's theory of relativity as examples of admired theories that both employ presuppositions (usually of a mathematical nature), presuppositions that do not face empirical evidence directly. In fact, Friedman claims that the use of such presuppositions (...) is a hallmark of "science as we know it." But what should we say about the special sciences, which typically do not rely on the abstruse formalisms one finds in the exact sciences? I identify a type of a priori presupposition that plays an especially striking role in the development of empirical psychology. These are ontological presuppositions about the type of object a given science purports to study. I show how such presuppositions can be both a priori and rational by investigating their role in an early flap over psychology's contested status as a natural science. The flap focused on one of the field's earliest textbooks, William James's Principles of Psychology. The work was attacked precisely for its reliance on a priori presuppositions about what James had called the "mental state," psychology's (alleged) proper object. I argue that the specific presuppositions James packed into his definition of the "mental state" were not directly responsible to empirical evidence, and so in that sense were a priori; but the presuppositions were rational in that they were crafted to help overcome philosophical objections (championed by neo-Hegelians) to the very idea that there can be a genuine science of mind. Thus, my case study gives an example of substantive, a priori presuppositions being put to use—to rational use—in the special sciences. In addition to evaluating James's use of presuppositions, my paper also offers historical reflections on two different strands of pragmatist philosophy of science. One strand, tracing back through Quine to C. S. Peirce, is more naturalistic, eschewing the use of a priori elements in science. The other strand, tracing back through Kuhn and C. I. Lewis to James, is more friendly to such presuppositions, and to that extent bears affinity with the positivist tradition Friedman occupies. (shrink)

Taking mathematics as a language based on empirical experience, I argue for an account of mathematics in which its objects are abstracta that describe and communicate the structure of reality based on some of our ancestral interactions with their environment. I argue that mathematics as a language is mostly invented. Nonetheless, in being a general description of reality it cannot be said that it is fictional; and as an intersubjective reality, mathematical objects can exist independent of any one person’s mind.

The question of how people change artworks is important for the metaphysics of art. It’s relatively easy for anyone to change a painting or sculpture, but who may change a literary or musical work is restricted and varies with context. Authors of novels and composers of symphonies often have a special power to change their artworks. Mary Shelley revised Frankenstein, and Tchaikovsky revised his Second Symphony. I cannot change these artworks. In other cases, such as those involving jazz standards and (...) folk songs, performers and ordinary folks have more power to change artworks. My preferred explanation of these facts is the created-abstract-simples view, according to which literary and musical works, unlike paintings and sculptures, are created abstract objects that have no parts. On this view, the way to change a literary or musical work is for an individual, empowered by social practices, to change rules about how a literary work should be published or a musical work should be performed. A. R. J. Fisher and Caterina Moruzzi object that the created-abstract-simples view doesn’t allow for literary and musical works to genuinely change, and Nemesio Garcia-Carríl Puy objects that the view doesn’t allow for these artworks to be repeatable. This paper clarifies the created-abstract-simples view and defends the view against these objections. -/- . (shrink)

Gottlob Frege’s conception of works of art has received scant notice in the literature. This is a pity since, as this paper undertakes to reveal, his innovative philosophy of language motivated a theoretically and historically consequential, yet unaccountably marginalized Wittgenstinian line of inquiry in the domain of aesthetics. The element of Frege’s approach that most clearly inspired this development is the idea that only complete sentences articulate thoughts and that what sentences in works of drama and literary art express (...) are ‘mock thoughts’ (Schiengedanken). The early Wittgenstein closely followed Frege’s lead on this theme. One sees this, for example, in the Tractatus, where Wittgenstein announces that only sentential propositions model (‘picture’) states of affairs whereas works of art are objects we perceive sub specie aeternitatis (1961, 83). By the 1930s, however, Wittgenstein began to revise his view beyond his initial Frege-inspired standpoint. He came to insist that works of art can convey thoughts as well, but that thoughts do not model (picture) the world of facts and hence do not convey information about measurable objects and events. Rather, his contention was that aesthetically configured thoughts that artworks communicate impart information about our perspective on reality and reconfigure our view of life. To be more explicit, successful (gelungene) or ‘good’ works of art (ibid.), in Wittgenstein’s view, can supply aesthetically instructive ‘gestures’ that open to living experience promising new ways of being. It is in this respect, he held, that artists ‘have something to teach’ (1980, 36). (shrink)

My goal in this paper is, to tentatively sketch and try defend some observations regarding the ontological dignity of object references, as they may be used from within in a formalized language. -/- Hence I try to explore, what properties objects are presupposed to have, in order to enter the universe of discourse of an interpreted formalized language. -/- First I review Frege′s analysis of the logical structure of truth value definite sentences of scientific colloquial language, to draw suggestions from (...) his saturated vs. unsaturated sentence components paradigm. -/- Next try investigate, in how far reference to non pure math objects might allow for a role as argument of a truth value function. Object kinds to be considered are: common sense objects, technical objects, humanities object kinds (social, psychical, ...), objects of art, ... , be they abstract, concrete, or in this respect mixed objects. -/- Then have a comment on the just referenced label abstract objects. -/- Next try to get an idea wrt the ontological significance of the fact, that pure math objects and functions in some important classical cases can be uniquely defined by means of categorical theories. Here, in the course of my argument, I have a little corollary wrt the standard model of first order Peano arithmetic, reducing the epistemological significance of the existence of the non standard models. -/- Next, wrt a special concept of a formalized empirical theory, I care for whether the impure math objects and mixed objects described here, and complying best with truth value function mapping, are also reliable candidates for the ontological commitment of such theories: and discuss an alternative, which reduces the ontological significance of the universe of discourse of the theories intended models. -/- In the end I sum up what is - from my point of view - the status quo, according to my respective findings. (shrink)

We develop a series of theses on the philosophical aesthetics of design art. A sketch of an outline of a theory of objects is drawn from within a naturalistic worldview, that of abstract materialism and the general, still ongoing, quest to build a comprehensive philosophy of nature encompassing not only the physical world, but also culture, art, and politics.

This article investigates a tension that arises in Hegel’s aesthetic theory between theoretical and practical forms of reason. This tension, I argue, stems from Hegel’s appropriation of an Aristotelian framework for a historically unfolding social teleology which puts practical reason to work for the aims of theoretical reason. Recognizing that this aspect of Hegel’s dialectic is essential in overcoming problems left in Kant’s transcendental idealism, the appearance of incongruence does not lessen. Grouped together with absolute spirit, Hegel positions art as (...) a transitory mode of mind, a vehicle, which aims to raise spirit to the higher cognition of philosophy. When the unfolding absolute concept becomes too complex for articulation in the material, art must end, as spirit’s message can be expressed only through the non-material form of philosophy. This study focuses on the ambivalence found in Hegel’s writings regarding his account of historical completion. Though Hegel sees in the Absolute a metaphysical solution to the unity of subject and object, the practical aspects of the unity appear to falter when philosophy becomes the dominant mode of expression at the close of a historical cycle. In Lectures on the History of Philosophy, Hegel links his notion of the Absolute, albeit with modification, to Aristotle’s nous. As described in De Anima, this entails a progression in which active and possible intellect rise to the level of the eternal, while passive intellect, the imaginative element, passes on with the body. Because the architecture of Aristotle’s nous, which is not in line with his defense of poetry, is integrated into the blueprint of Hegel’s absolute, an unresolved tension emerges in the spirit of art. A divergence of aims is forced to the surface through Hegel’s application of a template for achievement of theoretical knowledge, with an end in the universal, to a form of practical knowing which has an end in the particular. (shrink)

In this paper a parallel is drawn between the problem of epistemic access to abstract objects in mathematics and the problem of epistemic access to idealized systems in the physical sciences. On this basis it is argued that some recent and more traditional approaches to solving these problems are problematic.

Putnam, Hilary FPhilosophy of logic. Harper Essays in Philosophy. Harper Torchbooks, No. TB 1544. Harper & Row, Publishers, New York-London, 1971. v+76 pp. The author of this book has made highly regarded contributions to mathematics, to philosophy of logic and to philosophy of science, and in this book he brings his ideas in these three areas to bear on the traditional philosophic problem of materialism versus (objective) idealism. The book assumes that contemporary science (mathematical and physical) is (...) largely correct as far as it goes, or at least that it is rational to believe in it. The main thesis of the book is that consistent acceptance of contemporary science requires the acceptance of some sort of Platonistic idealism affirming the existence of abstract, non-temporal, non-material, non-mental entities (numbers,scientific laws, mathematical formulas, etc.). The author is thus in direct opposition to the extreme materialism which had dominated philosophy of science in the first three quarters of this century. the book can be especially recommended to the scientifically literate, general reader whose acquaintance with these areas is limited to the earlier literature of when it had been assumed that empiricistic materialism was the only philosophy compatible with a scientific outlook. To this group the book presents an eye-opening challenge fulfilling the author’s intention of “shaking up preconceptions and stimulating further discussion”. (shrink)

Abstract In the new millennium there have been important empirical developments in the philosophy of mathematics. One of these is the so-called “Empirical Philosophy of Mathematics”(EPM) of Buldt, Löwe, Müller and Müller-Hill, which aims to complement the methodology of the philosophy of mathematics with empirical work. Among other things, this includes surveys of mathematicians, which EPM believes to give philosophically important results. In this paper I take a critical look at the sociological part of EPM as (...) a case study of ... (shrink)

This paper argues that Carnap both did not view and should not have viewed Frege's project in the foundations of mathematics as misguided metaphysics. The reason for this is that Frege's project was to give an explication of number in a very Carnapian sense — something that was not lost on Carnap. Furthermore, Frege gives pragmatic justification for the basic features of his system, especially where there are ontological considerations. It will be argued that even on the question of the (...) independent existence of abstract objects, Frege and Carnap held remarkably similar views. I close with a discussion of why, despite all this, Frege would not accept the principle of tolerance. (shrink)

Ordinary language and scientific discourse are filled with linguistic expressions for dispositional properties such as “soluble,” “elastic,” “reliable,” and “humorous.” We characterize objects in all domains – physical objects as well as human persons – with the help of dispositional expressions. Hence, the concept of a disposition has historically and systematically played a central role in different areas of philosophy ranging from metaphysics to ethics. The contributions of this volume analyze the ancient foundations of the discussion about disposition, examine (...) the problem of disposition within the context of the foundation of modern science, and analyze this dispute up to the 20th century. Furthermore, articles explore the contemporary theories of dispositions. (shrink)

The descriptions and theoretical laws scientists write down when they model a system are often false of any real system. And yet we commonly talk as if there were objects that satisfy the scientists’ assumptions and as if we may learn about their properties. Many attempt to make sense of this by taking the scientists’ descriptions and theoretical laws to define abstract or fictional entities. In this paper, I propose an alternative account of theoretical modelling that draws upon Kendall (...) Walton’s ‘make-believe’ theory of representation in art. I argue that this account allows us to understand theoretical modelling without positing any object of which scientists’ modelling assumptions are true. (shrink)

This paper aims to provide hyperintensional foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by (...) Hale and Wright and examined in Hale (2013), and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. I countenance a hyperintensional semantics for novel epistemic abstractionist modalities. I suggest that observational type theory can be applied to first-order abstraction principles in order to make abstraction principles recursively enumerable. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics. (shrink)

Abstract artifacts such as musical works and fictional entities are human creations; they are intentional products of our actions and activities. One line of argument against abstract artifacts is that abstract objects are not the kind of objects that can be created. This is so, it is argued, because abstract objects are causally inert. Since creation requires being caused to exist, abstract objects cannot be created. One common way to refute this argument is to reject (...) the causal inefficacy of abstracta. I argue that creationists should rather reject the principle that creation requires causation. Creation, in my view, is a non-causal relation that can be explained using an appropriate notion of ontological dependence. The existence and the creation of abstract artifacts depend on certain individuals with appropriate intentions, along with events of a certain kind that include but are not limited to creations of certain concrete objects. (shrink)

Diagrams by Connor Camburn -/- The relationship between axiomatization, mechanization, creative individuation, and virtual/physical individuation presents a fascinating interplay of concepts that have significantly influenced various fields, including mathematics, physics, philosophy, and art. This essay explores these relationships by drawing insights from André Weil's "From Metaphysics to Mathematics," Gilles Châtelet's works, and Schmid's discussion on Gnostic Futurism. -/- Axiomatization: Weil and Grothendieck -/- Axiomatization, as discussed in André Weil's "From Metaphysics to Mathematics," represents the transformation of metaphysical concepts into (...) formal mathematical structures. Weil's document highlights the progression from vague metaphysical ideas to concrete mathematical theories. This process is evident in the works of mathematicians like Lagrange and Galois, where initial metaphysical notions eventually crystallized into the formal structures of modern algebra. Similarly, Alexander Grothendieck's work in algebraic geometry involved the creation of an abstract framework that could unify various mathematical concepts, demonstrating the power of axiomatization in providing a foundational structure for complex ideas. -/- Mechanization: Oppenheimer, Watson-Crick, and Von Neumann -/- Mechanization, in the context of scientific development, refers to the application of mechanical principles to solve problems and the automation of processes. Figures like Oppenheimer in physics, Watson and Crick in biology, and Von Neumann in computing and mathematics, represent the pinnacle of mechanization in their respective fields. Their work exemplifies how mechanization has enabled the simplification and automation of complex processes, leading to alienating discoveries such as the atomic bomb, the structure of DNA, and the development of modern computers. -/- Creative Individuation: Gnostic Futurism -/- Eric Schmid's discussion on Gnostic Futurism, as found in his document, offers a perspective on creative individuation. Gnostic Futurism, rooted in Kantian-Hegelian transcendental universalism, represents a departure from conventional realism, emphasizing a transcendental nature and integration of mythopoeia. This movement in art and philosophy underscores the role of creativity and individual expression in transcending traditional forms and norms. It embodies the idea of creative individuation, where individual creativity leads to the formation of new, often revolutionary, artistic and philosophical expressions. -/- Virtual/Physical Individuation: Châtelet's Perspective -/- Gilles Châtelet's works, particularly "Figuring Space: Philosophy, Mathematics, and Physics," delve into the concept of virtual/physical individuation. Châtelet discusses the configuration of space in mathematics and physics, emphasizing the role of diagrams in scientific discovery. This exploration of space, both virtual (as in mathematical abstractions) and physical (as in tangible representations), reflects the process of individuation in both realms. Châtelet's perspective highlights how mathematical and physical representations are not just tools for understanding the world but are also expressions of individual and collective intellectual endeavors. -/- Interconnections and Synthesis -/- The synthesis of these concepts reveals a complex tapestry of intellectual development across disciplines. Axiomatization provides a foundational structure that enables the mechanization of processes, as seen in the works of Oppenheimer, Watson-Crick, and Von Neumann. This mechanization, in turn, paves the way for creative individuation, where individuals can transcend established norms and frameworks, as exemplified in Gnostic Futurism. Finally, the concept of virtual/physical individuation, as explored by Châtelet, represents the culmination of these ideas, where the abstract and the tangible intersect, leading to new forms of understanding and expression. -/- In conclusion, the relationship between axiomatization, mechanization, creative individuation, and virtual/physical individuation is a testament to the dynamic and interconnected nature of intellectual progress. From the formal structures of mathematics to the revolutionary changes in art and philosophy, these concepts demonstrate how human thought and creativity continue to evolve, challenging existing paradigms and forging new paths of understanding and expression. (shrink)