This essay endeavors to define the concept of indefinite extensibility in the setting of category theory. I argue that the generative property of indefinite extensibility for set-theoretic truths in category theory is identifiable with the Grothendieck Universe Axiom and the elementary embeddings in Vopenka's principle. The interaction between the interpretational and objective modalities of indefinite extensibility is defined via the epistemic interpretation of two-dimensional semantics. The semantics can be defined intensionally or hyperintensionally. By characterizing the modal profile of $\Omega$-logical validity, (...) and thus the generic invariance of mathematical truth, modal coalgebras are further capable of capturing the notion of definiteness for set-theoretic truths, in order to yield a non-circular definition of indefinite extensibility. (shrink)

This essay aims to provide a modal logic for rational intuition. Similarly to treatments of the property of knowledge in epistemic logic, I argue that rational intuition can be codified by a modal operator governed by the modal $\mu$-calculus. Via correspondence results between fixed point modal propositional logic and the bisimulation-invariant fragment of monadic second-order logic, a precise translation can then be provided between the notion of 'intuition-of', i.e., the cognitive phenomenal properties of thoughts, and the modal operators regimenting the (...) notion of 'intuition-that'. I argue that intuition-that can further be shown to entrain conceptual elucidation, by way of figuring as a dynamic-interpretational modality which induces the reinterpretation of both domains of quantification and the intensions and hyperintensions of mathematical concepts that are formalizable in monadic first- and second-order formal languages. Hyperintensionality is countenanced via a topic-sensitive epistemic two-dimensional truthmaker semantics. (shrink)

This paper aims to contribute to the analysis of the nature of mathematical modality and hyperintensionality, and to the applications of the latter to absolute decidability. Rather than countenancing the interpretational type of mathematical modality as a primitive, I argue that the interpretational type of mathematical modality is a species of epistemic modality. I argue, then, that the framework of two-dimensional semantics ought to be applied to the mathematical setting. The framework permits of a formally precise account of the priority (...) and relation between epistemic mathematical modality and metaphysical mathematical modality. The discrepancy between the modal systems governing the parameters in the two-dimensional intensional setting provides an explanation of the difference between the metaphysical possibility of absolute decidability and our knowledge thereof. I also advance a topic-sensitive epistemic two-dimensional truthmaker semantics, if hyperintensional approaches are to be preferred to possible worlds semantics. I examine the relation between two-dimensional hyperintensional states and epistemic set theory, providing two-dimensional hyperintensional formalizations of the modal logic of ZFC, large cardinal axioms, $\Omega$-logic, and the Epistemic Church-Turing Thesis. (shrink)

This paper is an exposition and evaluation of the Agustín Rayo's views about the epistemology and metaphysics of mathematics, as they are presented in his book The Construction of Logical Space.

Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ (...) problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof. (shrink)

Essence and Existence: Selected Essays brings together fifteen essays by Bob Hale, mostly written between the publication of his last book, Necessary Beings, in.

In this paper I articulate and defend a view that I call phenomenal dogmatism about intuitive justification. It is dogmatic because it includes the thesis: if it intuitively seems to you that p, then you thereby have some prima facie justification for believing that p. It is phenomenalist because it includes the thesis: intuitions justify us in believing their contents in virtue of their phenomenology—and in particular their presentational phenomenology. I explore the nature of presentational phenomenology as it occurs perception, (...) and I make a case for thinking that it is present in a wide variety of logical, mathematical, and philosophical intuitions. (shrink)

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...) with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)

'Reason, Causation and Compatibility with the Phenomena' strives to give answers to the philosophical problem of the interplay between realism, explanation and experience. This book is a compilation of essays that recollect significant conceptions of rival terms such as determinism and freedom, reason and appearance, power and knowledge. This title discusses the progress made in epistemology and natural philosophy, especially the steps that led from the ancient theory of atomism to the modern quantum theory, and from mathematization to analytic philosophy. (...) Moreover, it provides possible gateways from modern deadlocks of theory either through approaches to consciousness or through historical critique of intellectual authorities. This work will be of interest to those either researching or studying in colleges and universities, especially in the departments of philosophy, history of science, philosophy of science, philosophy of physics and quantum mechanics, history of ideas and culture. Greek and Latin Literature students and instructors may also find this book to be both a fascinating and valuable point of reference. (shrink)

This dissertation 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 dissertation demonstrates how phenomenal consciousness and gradational possible-worlds models in Bayesian perceptual psychology relate to epistemic modal space. The dissertation demonstrates, then, how epistemic modality relates to the computational theory of mind; metaphysical modality; deontic modality; logical modality; the types of mathematical modality; to the (...) epistemic status of undecidable propositions and abstraction principles in the philosophy of mathematics; to the apriori-aposteriori distinction; to the modal profile of rational propositional intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Examining the nature of epistemic logic itself, I develop a novel approach to conditions of self-knowledge in the setting of the modal μ-calculus, as well as novel epistemicist solutions to Curry's, the liar, and the knowability paradoxes. Solutions to previously intransigent issues concerning the first-person concept; the distinction between fundamental and derivative truths; and the unity of intention and its role in decision theory, are developed along the way. (shrink)

In his 1951 Gibbs Lecture Gödel formulates the central implication of the incompleteness theorems as a disjunction: either the human mind infinitely surpasses the powers of any finite machine or there exist absolutely unsolvable diophantine problems (of a certain type). In his later writings in particular Gödel favors the view that the human mind does infinitely surpass the powers of any finite machine and there are no absolutely unsolvable diophantine problems. I consider how one might defend such a view in (...) light of Gödel's remark that one can turn to ideas in Husserlian transcendental phenomenology to show that the human mind ‘contains an element totally different from a finite combinatorial mechanism’. (shrink)

The paper is devoted to phenomenological ideas in conceptions of modern philosophy of mathematics. Views of Husserl, Weyl, Becker andGödel will be discussed and analysed. The aim of the paper is to show the influence of phenomenological ideas on the philosophical conceptions concerning mathematics. We shall start by indicating the attachment of Edmund Husserl to mathematics and by presenting the main points of his philosophy of mathematics. Next, works of two philosophers who attempted to apply Husserl’s phenomenological ideas to the (...) philosophy of mathematics, namely Hermann Weyl and Oskar Becker, will be briefly discussed. Lastly, the connections between Husserl’s ideas and the philosophy of mathematics of Kurt Gödel will be studied. (shrink)

This paper is an exploration of the role of intuition in physics. The ways in which intuition is appealed to in physics are not well understood. To the best of my knowledge, there is no analysis of the different contexts in which we might appeal to intuition in physics, nor is there any analysis of the different potential uses to which intuition might be put. In this paper I look to provide data that goes some way to giving a sense (...) of the different contexts in which intuition is appealed to in physics. As I note in the conclusion, there is still much work to be done but I hope that the work here provides us with a first step in the journey to properly understand the use to which intuitions are put in physics and science more generally. (shrink)

This paper clarifies and discusses Imre Lakatos’ claim that mathematics is quasi-empirical in one of his less-discussed papers A Renaissance of Empiricism in the Recent Philosophy of Mathematics. I argue that Lakatos’ motivation for classifying mathematics as a quasi-empirical theory is epistemological; what can be called the quasi-empirical epistemology of mathematics is not correct; analysing where the quasi-empirical epistemology of mathematics goes wrong will bring to light reasons to endorse a pluralist view of mathematics.

This paper dates from about 1994: I rediscovered it on my hard drive in the spring of 2002. It represents an early attempt to explore the connections between the Julius Caesar problem and Frege's attitude towards Basic Law V. Most of the issues discussed here are ones treated rather differently in my more recent papers "The Julius Caesar Objection" and "Grundgesetze der Arithmetik I 10". But the treatment here is more accessible, in many ways, providing more context and a better (...) sense of how this issue relates to broader issues in Frege's philosophy. (shrink)

Church's thesis asserts that a number-theoretic function is intuitively computable if and only if it is recursive. A related thesis asserts that Turing's work yields a conceptual analysis of the intuitive notion of numerical computability. I endorse Church's thesis, but I argue against the related thesis. I argue that purported conceptual analyses based upon Turing's work involve a subtle but persistent circularity. Turing machines manipulate syntactic entities. To specify which number-theoretic function a Turing machine computes, we must correlate these syntactic (...) entities with numbers. I argue that, in providing this correlation, we must demand that the correlation itself be computable. Otherwise, the Turing machine will compute uncomputable functions. But if we presuppose the intuitive notion of a computable relation between syntactic entities and numbers, then our analysis of computability is circular. (shrink)

In this article, I present a novel account of a priori warrant, which I then use to examine the relationship between a priori and a posteriori warrant in mathematics. According to this account of a priori warrant, the reason that a posteriori warrant is subordinate to a priori warrant in mathematics is because processes that produce a priori warrant are reliable independent of the contexts in which they are used, whereas this is not true for processes that produce a posteriori (...) warrant. Following this, I show how this difference explains why a priori warranting processes, such as proof (and mathematical intuition) can, and a posteriori warranting processes cannot, definitively establish mathematical results. I then show why, in the event of a conflict between a priori and a posteriori warranting processes, the former undermine, and cannot be undermined by, the latter. Finally, I explain the connection between apriority and mathematical necessity, but do so in a way that allows for the existence of contingent a priori knowledge. (shrink)

Though many working mathematicians embrace a rough and ready form of Platonism, that venerable position has suffered a checkered philosophical career. Indeed the three schools of thought with which most of us began our official philosophizing about mathematics—Intuitionism, Formalism, and Logicism—all stand in fundamental disagreement with Platonism. Nevertheless, various versions of Platonistic thinking survive in contemporary philosophical circles. The aim of this paper is to describe these views, and, as my title suggests, to trace their roots.I'll begin with some preliminary (...) remarks about the big three schools. This seems a reasonable approach to the issues both because most observers are familiar, at least in a general way, with the tenets of Intuitionism, Formalism, and Logicism, and because it is in reaction to these that contemporary Platonism has taken shape. (shrink)

In a series of papers published in the sixties and seventies, Jaakko Hintikka, drawing upon Kant’s conception, defines an argument to be analytic whenever it does not introduce new individuals into the discussion and argues that there exists a class of arguments in polyadic first-order logic that are to be synthetic according to this sense. His work has been utterly overlooked in the literature. In this paper, I claim that the value of Hintikka’s contribution has been obscured by his formalisation (...) of the original definition. Therefore, I provide (i) a brief reconstruction of the historical framework of the problem and the revolutionary import of Hintikka’s contribution, (ii) a clarification of the most complicated steps of Hintikka’s elaboration of his insight, (iii) a criticism of several features that play a fundamental role in Hintikka’s formalisation and (iv) a selection from Hintikka’s own material of some valuable suggestions towards a clear and workable formalisation. As for the pars construens, I isolate in the approach of depth-bounded first-order logics (D'Agostino et al. 2021) an alternative formalisation of the notion of syntheticity as the introduction of new individuals in the reasoning, and I show that it is not affected by the same difficulties as Hintikka’s proposal. In so doing, I hope to have contributed to the realisation of the project of rehabilitating Kant’s analytic–synthetic distinction in the context of modern first-order logic with the purpose of showing, against the logical empiricist movement, that logic is not analytic. (shrink)

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of mathematical theorems can cover at most (...) one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

This paper offers a defense of Charles Parsons’ appeal to mathematical intuition as a fundamental factor in solving Benacerraf’s problem for a non-eliminative structuralist version of Platonism. The literature is replete with challenges to his well-known argument that mathematical intuition justifies our knowledge of the infinitude of the natural numbers, in particular his demonstration that any member of a Hilbertian stroke string ω-sequence has a successor. On Parsons’ Kantian approach, this amounts to demonstrating that for an “arbitrary” or “vaguely represented” (...) string of strokes, we can always “add” one more stroke. Critics have contested the cogency of a notion of an arbitrary object, our capacity to vaguely represent a definite object, and the role of spatial and temporal representation in the demonstration that we can “add” one more. The bulk of this paper is devoted to demonstrating how to meet all extant criticisms of his key argument. Critics have also suggested that Parsons’ whole approach is misbegotten because the appeal to mathematical intuition inevitably falls short of providing a complete solution to Benacerraf’s problem. Since the natural numbers are essentially and exclusively characterized by their structural properties, they cannot be identified with any particular model of arithmetic, and thus a notion of intuition will fail to capture the universality of arithmetic, its applicability to all entities. This paper also explains why we should not reject appeal to mathematical intuition even though it is not itself sufficient to fully “close the gap” on Benacerraf’s challenge. (shrink)

The goal of the research programme I describe in this article is a realist epistemology for arithmetic which respects arithmetic's special epistemic status (the status usually described as a prioricity) yet accommodates naturalistic concerns by remaining fundamentally empiricist. I argue that the central claims which would allow us to develop such an epistemology are (i) that arithmetical truths are known through an examination of our arithmetical concepts; (ii) that (at least our basic) arithmetical concepts are accurate mental representations of elements (...) of the arithmetical structure of the independent world; (iii) that (ii) obtains in virtue of the normal functioning of our sensory apparatus. The first of these claims protects arithmetic's special epistemic status relative, for example, to the laws of physics, the second preserves the independence of arithmetical truth, and the third ensures that we remain empiricists. Preliminaries Justifying and grounding concepts Cameras and filters An epistemology for arithmetic Concluding remarks. (shrink)

At the most general level, the concept of finitism is typically characterized by saying that finitistic mathematics is that part of mathematics which does not appeal to completed infinite totalities and is endowed with some epistemological property that makes it secure or privileged. This paper argues that this characterization can in fact be sharpened in various ways, giving rise to different conceptions of finitism. The paper investigates these conceptions and shows that they sanction different portions of mathematics as finitistic.

The question of how we can know anything about ideal entities to which we do not have access through our senses has been a major concern in the philosophical tradition since Plato's Phaedo. This article focuses on the paradigmatic case of mathematical knowledge. Following a suggestion by Gödel, we employ concepts and ideas from Husserlian phenomenology to argue that mathematical objects – and ideal entities in general – are recognized in a process very closely related to ordinary perception. Our analysis (...) combines Husserl's insights into the nature of perception and the phenomena of evidence and justification with case studies of the role of intuition in axiomatic set theory. (shrink)

There is a pervasive view that inference—as opposed, notably, to a grasp of objects—is an intralinguistic process that does not draw on extralinguistic resources. The present paper aims to show that this dichotomy between inferring and grasping objects can be resisted. Specifically, I offer an alternative view: a phenomenological account according to which our most basic inferences draw on our grasp of objects. I motivate this account on the grounds that, although it is restricted to such basic inferences, it has (...) significant implications for the general question of whether inference involves taking one's premises to support one's conclusion. The proposed account implies that basic inferences need not involve corresponding “takings,” (even though such inferences involve conceptual content), but it leaves open the possibility that non-basic inferences do involve corresponding “takings,” while relying on basic inferences. It will turn out that such a disjunctive view, according to which non-basic reasoning is somewhat parasitic on basic reasoning, manages to avoid many of the problems with which the literature on the “taking condition” deals. (shrink)

Finitism is given an interpretation based on two ideas about strings (sequences of symbols): a replacement principle extracted from Hilberts class 2 can be justified by means of an additional finitistic choice principle, thus obtaining a second equational theory . It is unknown whether is strictly stronger than since 2 may coincide with the class of lower elementary functions.

The debate on structuralism in the philosophy of mathematics has brought into focus a question about the status of meta-mathematics. It has been raised by Shapiro (2005), where he compares the ongoing discussion on structuralism in category theory to the Frege-Hilbert controversy on axiomatic systems. Shapiro outlines an answer according to which meta-mathematics is understood in structural terms and one according to which it is not. He finds both options viable and does not seem to prefer one over the other. (...) The present paper reconsiders the nature of the formulae and symbols meta-mathematics is about and finds that, contrary to Charles Parsons' influential view, meta-mathematical objects are not "quasi-concrete". It is argued that, consequently, structuralists should extend their account of mathematics to meta-mathematics. (shrink)

Geometrical and physical intuition, both untutored andcultivated, is ubiquitous in the research, teaching,and development of mathematics. A number ofmathematical ``monsters'', or pathological objects, havebeen produced which – according to somemathematicians – seriously challenge the reliability ofintuition. We examine several famous geometrical,topological and set-theoretical examples of suchmonsters in order to see to what extent, if at all,intuition is undermined in its everyday roles.

The term ‘continuous’ in real analysis wasn’t given an adequate formal definition until 1817. However, important theorems about continuity were proven long before that. How was this possible? In this paper, I introduce and refine a proposed answer to this question, derived from the work of Frank Jackson, David Lewis and other proponents of the ‘Canberra plan’. In brief, the proposal is that before 1817 the meaning of the term ‘continuous’ was determined by a number of ‘platitudes’ which had some (...) special epistemic status. (shrink)

The Quine-Putnam indispensability argument urges us to place mathematical entities on the same ontological footing as (other) theoretical entities of empirical science. Recently this argument has attracted much criticism, and in this paper I address one criticism due to Elliott Sober. Sober argues that mathematical theories cannot share the empirical support accrued by our best scientific theories, since mathematical propositions are not being tested in the same way as the clearly empirical propositions of science. In this paper I defend the (...) Quine-Putnam argument against Sober's objections. (shrink)

In this paper I assume that we have some intuitive knowledge—i.e. beliefs that amount to knowledge because they are based on intuitions. The question I take up is this: given that some intuition makes a belief based on it amount to knowledge, in virtue of what does it do so? We can ask a similar question about perception. That is: given that some perception makes a belief based on it amount to knowledge, in virtue of what does it do so? (...) A natural idea about perception is that a perception makes a belief amount to knowledge in part by making you sensorily aware of the concrete objects it is about. The analogous idea about intuition is that an intuition makes a belief amount to knowledge in part by making you intellectually aware of the abstract objects it is about. I expand both ideas into fuller accounts of perceptual and intuitive knowledge, explain the main challenge to this sort of account of intuitive knowledge (i.e. the challenge of making sense of intellectual awareness), and develop a response to it. (shrink)

Awareness is a two-place determinable relation some determinates of which are seeing, hearing, etc. Abstract objects are items such as universals and functions, which contrast with concrete objects such as solids and liquids. It is uncontroversial that we are sometimes aware of concrete objects. In this paper I explore the more controversial topic of awareness of abstract objects. I distinguish two questions. First, the Existence Question: are there any experiences that make their subjects aware of abstract objects? Second, the Grounding (...) Question: if an experience makes its subject aware of an abstract object, in virtue of what does it do so? I defend the view that intuitions, specifically mathematical intuitions, sometimes make their subjects aware of abstract objects. In defending this view, I develop an account of the ground of intuitive awareness. (shrink)

This paper defends the Quine-Putnam mathematical indispensability argument against two objections raised by Penelope Maddy. The objections concern scientific practices regarding the development of the atomic theory and the role of applied mathematics in the continuum and infinity. I present two alternative accounts by Stephen Brush and Alan Chalmers on the atomic theory. I argue that these two theories are consistent with Quine’s theory of scientific confirmation. I advance some novel versions of the indispensability argument. I argue that these new (...) versions accommodate Maddy’s history of the atomic theory. Counter-examples are provided regarding the role of the mathematical continuum and mathematical infinity in science. (shrink)

The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.

I dub a certain central tradition in philosophy of language (and mind) the de re tradition. Compelling thought experiments show that in certain common cases the truth conditions for thoughts and public-language expressions categorically turn on external objects referred to, rather than on linguistic meanings and/or belief assumptions. However, de re phenomena in language and thought occur even when the objects in question don't exist. Call these empty de re phenomena. Empty de re thought with respect to numeration is explored (...) in this paper, and such thought with respect to hallucinations is touched on. (shrink)

This book is both an introduction to and a research work on Meinongianism. “Meinongianism” is taken here, in accordance with the common philosophical jargon, as a general label for a set of theories of existence – probably the most basic notion of ontology. As an introduction, the book provides the first comprehensive survey and guide to Meinongianism and non-standard theories of existence in all their main forms. As a research work, the book exposes and develops the most up-to-date Meinongian theory (...) (called modal Meinongianism), applies it to specific fields, and discusses its open problems. Part I of the book provides a historical introduction to, and critical discussion of, the dominant philosophical view of existence: the “Kantian-Fregean-Quinean” perspective. Part II is the full-fledged introduction to the Meinongian theories of existence as a real property of individuals: after starting with the so-called naïve Meinongian conception and its problems, it provides a self-contained presentation of the main neo-Meinongian proposals, and a detailed discussion of their strengths and weaknesses. Part III develops a specific neo-Meinongian theory of existence employing a model-theoretic semantic framework. It discusses its application to the ontology and semantics of fictional objects, and its open problems. The methodology of the book follows the most recent trends in analytic ontology. In particular, the meta-ontological point of view is largely privileged. (shrink)

Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are mathematical objects.

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. (...) Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics. (shrink)

Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and nonmental. Platonism in this sense is a contemporary view. It is obviously related to the views of Plato in important ways, but it is not entirely clear that Plato endorsed this view, as it is defined here. In order to remain neutral on this question, the (...) term ‘platonism’ is spelled with a lower-case ‘p’. (See entry on Plato.) The most important figure in the development of modern platonism is Gottlob Frege (1884, 1892, 1893-1903, 1919). The view has also been endorsed by many others, including Kurt Gödel (1964), Bertrand Russell (1912), and W.V.O. Quine (1948, 1951). (shrink)