What is the source of logical and mathematical truth? This book revitalizes conventionalism as an answer to this question. Conventionalism takes logical and mathematical truth to have their source in linguistic conventions. This was an extremely popular view in the early 20th century, but it was never worked out in detail and is now almost universally rejected in mainstream philosophical circles. Shadows of Syntax is the first book-length treatment and defense of a combined conventionalist theory of logic and mathematics. It (...) argues that our conventions, in the form of syntactic rules of language use, are perfectly suited to explain the truth, necessity, and a priority of logical and mathematical claims, as well as our logical and mathematical knowledge. (shrink)

From 1953 to 1959 Gödel worked on a response to Carnap’s philosophy of mathematics. The drafts display Gödel’s familiarity with Carnap’s position from The Logical Syntax of Language, but they received a dismissive reaction on their eventual, posthumous, publication. Gödel’s two principal points, however, will here be defended. Gödel, though, had wished simply to append a few paragraphs to show that the same arguments apply to Carnap’s later views. Carnap’s position, however, had changed significantly in the intervening years, and to (...) a significant extent, this mature position realizes much of what Gödel himself wanted out of a philosophy of mathematics. (shrink)

What are intuitions? Stereotypical examples may suggest that they are the results of common intellectual reflexes. But some intuitions defy the stereotype: there are hard-won intuitions that take d...

This paper proposes a possible reconstruction and philosophical-logical clarification of Gödel's idea of causality as the philosophical fundamental concept. The results are based on Gödel's published and non-published texts (including Max Phil notebooks), and are established on the ground of interconnections of Gödel's dispersed remarks on causality, as well as on the ground of his general philosophical views. The paper is logically informal but is connected with already achieved results in the formalization of a causal account of Gödel's onto-theological theory. (...) Gödel's main causal concepts are analysed (will, force, enjoyment, God, time and space, life, form, matter). Special attention is paid to a possible causal account of some of Gödel's logical concepts (assertion, privation, affirmation, negation, whole, part, general, particular, subject, predicate, necessary, possible, implication), as well as of logical antinomies. The problem of mechanical and non-mechanical procedures in the work with and on concepts is addressed in terms of Gödel's causal view. (shrink)

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)

How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when one tries to naturalize (...) in this way the traditional view of mathematics, according to which mathematical knowledge is certain and the method of mathematics is the axiomatic method. This paper suggests that, in order to naturalize mathematics through Darwinism, it is better to take the method of mathematics not to be the axiomatic method. (shrink)

Some have argued for a division of epistemic labor in which mathematicians supply truths and philosophers supply their necessity. We argue that this is wrong: mathematics is committed to its own necessity. Counterfactuals play a starring role.

Gödel’s philosophical rationalism includes a program for “developing philosophy as an exact science.” Gödel believes that Husserl’s phenomenology is essential for the realization of this program. In this article, by analyzing Gödel’s philosophy of idealism, conceptual realism, and his concept of “abstract intuition,” based on clues from Gödel’s manuscripts, I try to investigate the reasons why Gödel is strongly interested in Husserl’s phenomenology and why his program for an exact philosophy is unfinished. One of the topics that has attracted much (...) attention recently is the development of Gödel’s philosophical thoughts and its connection with other philosophical ideas. For instance, some scholars are searching for the possible connections between Gödel’s philosophy and Husserl’s phenomenology and examining if there is any solid evidence of Husserl’s influence on Gödel from Gödel’s works :181–203, 1998; Huaser, Bull Symbolic Logic 12:529–588, 2006). Why is Gödel’ s interested in Husserl? How should this turn to Husserl be interpreted? Is it a dismissal of Leibnizian philosophy, or a different way to achieve similar goals? Way did Gödel turn specifically to Husserl’s transcendental idealism? :425–476, 2003) I believe, the reason is that Gödel has a valuable program for “developing philosophy as an exact science” and he believes that Husserl’s phenomenology is relevant to the realization of this program. So far there are no sufficient evidence to show that there is a direct inheritance relation between Gödel’s and Husserl’s thoughts. However, from the clues in Gödel’s idealistic philosophy, conceptual realism, and his concept of “abstract intuition,” we can perhaps explore some similarities between his thoughts and Husserl’s thoughts, and analyze the reason why Gödel is interested in Husserl’s phenomenology and why his program for an exact philosophy is unfinished. (shrink)

My dissertation explores the ways in which Rudolf Carnap sought to make philosophy scientific by further developing recent interpretive efforts to explain Carnap’s mature philosophical work as a form of engineering. It does this by looking in detail at his philosophical practice in his most sustained mature project, his work on pure and applied inductive logic. I, first, specify the sort of engineering Carnap is engaged in as involving an engineering design problem and then draw out the complications of design (...) problems from current work in history of engineering and technology studies. I then model Carnap’s practice based on those lessons and uncover ways in which Carnap’s technical work in inductive logic takes some of these lessons on board. This shows ways in which Carnap’s philosophical project subtly changes right through his late work on induction, providing an important corrective to interpretations that ignore the work on inductive logic. Specifically, I show that paying attention to the historical details of Carnap’s attempt to apply his work in inductive logic to decision theory and theoretical statistics in the 1950s and 1960s helps us understand how Carnap develops and rearticulates the philosophical point of the practical/theoretical distinction in his late work, offering thus a new interpretation of Carnap’s technical work within the broader context of philosophy of science and analytical philosophy in general. (shrink)

This paper provides examples in arithmetic of the account of rational intuition and evidence developed in my book After Gödel: Platonism and Rationalism in Mathematics and Logic . The paper supplements the book but can be read independently of it. It starts with some simple examples of problem-solving in arithmetic practice and proceeds to general phenomenological conditions that make such problem-solving possible. In proceeding from elementary ‘authentic’ parts of arithmetic to axiomatic formal arithmetic, the paper exhibits some elements of the (...) genetic analysis of arithmetic knowledge that is called for in Husserl’s philosophy. This issues in an elaboration on a number of Gödel’s remarks about the meaning of his incompleteness theorems for the notion of evidence in mathematics. (shrink)

Conventionalism about mathematics claims that mathematical truths are true by linguistic convention. This is often spelled out by appealing to facts concerning rules of inference and formal systems, but this leads to a problem: since the incompleteness theorems we’ve known that syntactic notions can be expressed using arithmetical sentences. There is serious prima facie tension here: how can mathematics be a matter of convention and syntax a matter of fact given the arithmetization of syntax? This challenge has been pressed in (...) the literature by Hilary Putnam and Peter Koellner. In this paper I sketch a conventionalist theory of mathematics, show that this conventionalist theory can meet the challenge just raised , and clarify the type of mathematical pluralism endorsed by the conventionalist by introducing the notion of a semantic counterpart. The paper’s aim is an improved understanding of conventionalism, pluralism, and the relationship between them. (shrink)

This thesis articulates the resonances between J. M. Coetzee's lifelong engagement with mathematics and his practice as a novelist, critic, and poet. Though the critical discourse surrounding Coetzee's literary work continues to flourish, and though the basic details of his background in mathematics are now widely acknowledged, his inheritance from that background has not yet been the subject of a comprehensive and mathematically- literate account. In providing such an account, I propose that these two strands of his intellectual trajectory not (...) only developed in parallel, but together engendered several of the characteristic qualities of his finest work. The structure of the thesis is essentially thematic, but is also broadly chronological. Chapter 1 focuses on Coetzee's poetry, charting the increasing involvement of mathematical concepts and methods in his practice and poetics between 1958 and 1979. Chapter 2 situates his master's thesis alongside archival materials from the early stages of his academic career, and thus traces the development of his philosophical interest in the migration of quantificatory metaphors into other conceptual domains. Concentrating on his doctoral thesis and a series of contemporaneous reviews, essays, and lecture notes, Chapter 3 details the calculated ambivalence with which he therein articulates, adopts, and challenges various statistical methods designed to disclose objective truth. Chapter 4 explores the thematisation of several mathematical concepts in Dusklands and In the Heart of the Country. Chapter Five considers Waiting for the Barbarians and Foe in the context provided by Coetzee's interest in the attempts of Isaac Newton to bridge the gap between natural language and the supposedly transparent language of mathematics. Finally, Chapter 6 locates in Elizabeth Costello and Diary of a Bad Year a cognitive approach to the use of mathematical concepts in ethics, politics, and aesthetics, and, by analogy, a central aspect of the challenge Coetzee's late fiction poses to the contemporary literary landscape. (shrink)

Rudolf Carnap's 1930s philosophy of logic, including his adherence to the principle of tolerance, is discussed. What theses did Carnap commit himself to, exactly? I argue that while Carnap did commit himself to a certain multitude thesis—there are different logics of different languages, and the choice between these languages is merely a matter of expediency—there is no evidence that he rejected a language-transcendent notion of fact, contrary to what Warren Goldfarb and Thomas Ricketts have prominently argued. (In fact, it is (...) obscure just what Goldfarb and Ricketts claim about Carnap). Toward the end I critically discuss Michael Friedman's suggestion that Carnap believed in a relative a priori. (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 Institute Vienna Circle held a conference in Vienna in 2003, Cambridge and Vienna a?

The paper presents an interpretation of Platonism, the seeds of which can be found in the writings of Gödel and Wittgenstein. Although it is widely accepted that Wittgenstein is an anti-Platonist the author points to some striking affinities between Gödel’s and Wittgenstein’s accounts of mathematical concepts and the role of feeling and intuition in mathematics. A version of Platonism emerging from these considerations combines realism with respect to concepts with a view of concepts as accessible to feeling and able to (...) guide our behavior through feeling. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

The paper undertakes to characterize Hao Wang's style, convictions, and method as a philosopher, centering on his most important philosophical work From Mathematics to Philosophy, 1974. The descriptive character of Wang's characteristic method is emphasized. Some specific achievements are discussed: his analyses of the concept of set, his discussion, in connection with setting forth Gödel's views, of minds and machines, and his concept of ‘analytic empiricism’ used to criticize Carnap and Quine. Wang's work as interpreter of Gödel's thought and the (...) importance of his writings as a source are also discussed. (shrink)

3.2.2. The principle of conceptual containment ........................... 116 3.3.3. The physical human subject or the “I” ............................... 119 3.4. The hyperverse and its EDWs – the antimetaphysical foundation of the EDWs perspective ........................................... 150 Part II. Applications Chapter 4. Applications to some notions from philosophy of mind .. 159 4.1. Levels and reduction vs. emergence ............................................. 160 4.2. Qualia, Kant and the “I” ............................................................... 181 4.3. Mental causation and supervenience ............................................ 190 Chapter 5. Applications to some notions from cognitive science (...) ...... 200 5.1. Computationalism ........................................................................ 200 5.2. Connectionism .............................................................................. 211 5.3. The dynamical system approach .................................................. 223 5.4. Robotics ........................................................................................ 232 5.5. Dichotomies concerning the notion of mental representation and processing .............................................................................. 243 5.6. The EDWs perspective and some key elements in cognitive science .......................................................................................... 249 5.7. The relation between key elements and some philosophical distinctions ................................................................................... 264 5.8. Cognitive neuroscience ................................................................ 267 5.9. The status of any living entity ...................................................... 277 Chapter 6. Applications to some notions from philosophy of science and science ............................................................................ 281 6.1. A glance at logical positivism ...................................................... 285 6.2. Carnap’s linguistic frameworks .................................................... 289 6.3. Carnap vs. Gödel or syntactic vs. semantic .................................. 292 6.4. Carnap vs. Quine or rational reconstruction vs. naturalized epistemology ................................................................................ 295 6.5. Quine’s ontological relativity ....................................................... 296 6.6. Goodman’s relativity .................................................................... 298 6.7. Putnam and the rejection of the “thing-in-itself” .......................... 299 6.8. Friedman’s relative constitutive a priori principles....................... 301 6.9. Some notions from quantum mechanics ....................................... 305 6.10. The status of the external non-living epistemologically different entities ....................................................................................... 343 Conclusion .............................................................................................. 361 References .............................................................................................. 369. (shrink)

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the “unreasonable effectiveness of mathematics in (...) philosophy”—to adapt Wigner’s phrase—is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate. (shrink)

Derivationists, those wishing to explain the correctness and rigour of informal proofs in terms of associated formal proofs, are generally held to be supported by the success of the project of translating informal proofs into computer-checkable formal counterparts. I argue, however, that this project is a false friend for the derivationists because there are too many different associated formal proofs for each informal proof, leading to a serious worry of overgeneration. I press this worry primarily against Azzouni's derivation-indicator account, but (...) conclude that overgeneration is a major obstacle to a successful account of informal proofs in this direction. (shrink)

Arguably no concept is more fundamental to science than that of causality, for investigations into cases of existence, persistence, and change in the natural world are largely investigations into the causes of these phenomena. Yet the metaphysics and epistemology of causality remain unclear. For example, the ontological categories of the causal relata have been taken to be objects (Hume 1739), events (Davidson 1967), properties (Armstrong 1978), processes (Salmon 1984), variables (Hitchcock 1993), and facts (Mellor 1995). (For convenience, causes and effects (...) will usually be understood as events in what follows.) Complicating matters, causal relations may be singular (Socrates’s drinking hemlock caused Socrates’s death) or general (Drinking hemlock causes death); hence the relata might be tokens (e.g., instances of properties) or types (e.g., types of events) of the category in question. Other questions up for grabs are: Are singular causes metaphysically and/or epistemologically prior to general causes or vice versa (or neither)? What grounds the intuitive asymmetry of the causal relation? Are macro-causal relations reducible to micro-causal relations? And perhaps most importantly: Are causal facts (e.g., the holding of causal relations) reducible to non-causal facts (e.g., the holding of certain spatiotemporal relations)? (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)

Erratum to: Axiomathes DOI 10.1007/s10516-014-9234-yIn the original publication of the article, some of the references were published incorrectly. Please find below the corrected version of these references.

The view that mathematics deals with ideal objects to which we have epistemic access by a kind of perception has troubled many thinkers. Using ideas from Husserl’s phenomenology, I will take a different look at these matters. The upshot of this approach is that there are non-material objects and that they can be recognized in a process very closely related to sense perception. In fact, the perception of physical objects may be regarded as a special case of this more universal (...) way of recognizing objects of any kind. (shrink)

Michael Friedman maintains that Carnap did not fully appreciate the impact of Gödel's first incompleteness theorem on the prospect for a purely syntactic definition of analyticity that would render arithmetic analytically true. This paper argues against this claim. It also challenges a common presumption on the part of defenders of Carnap, in their diagnosis of the force of Gödel's own critique of Carnap in his Gibbs Lecture. The author is grateful to Michael Friedman for valuable comments. Part of the research (...) towards this paper was carried out while the author was a Visiting Fellow at the Center for Philosophy of Science at the University of Pittsburgh. The paper was presented to the Center's Fellowship Reunion Conference in Athens in 1992. It was committed for publication in the Proceedings of that conference, but those Proceedings never appeared. By the time it became evident that they would never appear, both the hard copy and the source file had been mislaid. The hard copy re-surfaced in 2007. The literature on this topic since 1992 appears to leave some space for the ideas and arguments presented here. Although the paper has been updated in light of the more recent literature, its basic thesis, presented in 1992, remains the same. Only 3 is new, questioning a basic presumption made by more recent commentators in their presentation of Gödel's criticism of Carnap in his Gibbs Lecture. For helpful comments on the current version, the author is indebted to Robert Kraut, Stewart Shapiro, and Adam Podlaskowski. CiteULike Connotea Del.icio.us What's this? (shrink)

, Rudolf Carnap became a chief proponent of the doctrine that the statements of intuitionism carry nonstandard intuitionistic meanings. This doctrine is linked to Carnap's ‘Principle of Tolerance’ and claims he made on behalf of his notion of pure syntax. From premises independent of intuitionism, we argue that the doctrine, the Principle, and the attendant claims are mistaken, especially Carnap's repeated insistence that, in defining languages, logicians are free of commitment to mathematical statements intuitionists would reject. I am grateful to (...) Nathan Carter, Gary Ebbs, Janet Folina, Luise Prior McCarty, Stewart Shapiro, Neil Tennant, Christopher Tillman, Beth Tropman, Wen-fang Wang, and two anonymous referees for their comments and suggestions. CiteULike Connotea Del.icio.us What's this? (shrink)

According to the dialetheist argument from the inconsistency of informal mathematics, the informal version of the Gödelian argument leads us to a true contradiction. On one hand, the dialetheist argues, we can prove that there is a mathematical claim that is neither provable nor refutable in informal mathematics. On the other, the proof of its unprovability is given in informal mathematics and proves that very sentence. We argue that the argument fails, because it relies on the unjustified and unlikely assumption (...) that the informal Gödel sentence is informally provable. (shrink)

This article presents three extracts from the introductory course in mathematical logic that Gödel gave at the University of Notre Dame in 1939. The lectures include a few digressions, which give insight into Gödel's views on logic prior to his philosophical papers of the 1940s. The first extract is Gödel's first lecture. It gives the flavour of Gödel's leisurely style in this course. It also includes a curious definition of logic and a discussion of implication in logic and natural language. (...) The second extract is a discussion on undecidability and on Leibniz. The third extract concerns the paradoxes and Russell's theory of types. (shrink)

Hans Hahn's long-neglected philosophy of mathematics is reconstructed here with an eye to his anticipation of the doctrine of logical pluralism. After establishing that Hahn pioneered a post-Tractarian conception of tautologies and attempted to overcome the traditional foundational dispute in mathematics, Hahn's and Carnap's work is briefly compared with Karl Menger's, and several significant agreements or differences between Hahn's and Carnap's work are specified and discussed.

Øystein Linnebo*. Philosophy of Mathematics. Princeton University Press, 2017. ISBN: 978-0-691-16140-2 ; 978-1-40088524-4. Pp. xviii + 203.

Knowledge requires both freedom and friction . Freedom to set up our epistemic goals, choose the subject matter of our investigations, espouse cognitive norms, design research programs, etc., and friction (constraint) coming from two directions: the object or target of our investigation, i.e., the world in a broad sense, and our mind as the sum total of constraints involving the knower. My goal is to investigate the problem of epistemic friction, the relation between epistemic friction and freedom, the viability of (...) foundationalism as a solution to the problem of friction, an alternative solution in the form of a neo-Quinean model, and the possibility of solving the problem of friction as it applies to logic and the philosophy of logic within that model. (shrink)

Nosso objetivo neste artigo é o de lançar luz sobre alguns aspectos das concepções de Gödel acerca da percepção de conceitos. Começamos investigando a natureza e o papel da analogia entre percepção sensível e percepção de conceitos. A seguir, examinamos as conexões entre percepção de conceitos, razão e compreensão, tentando mostrar que a percepção de conceitos é compreensão de conceitos. Por fim, examinamos aqueles aspectos da concepção de Gödel em que a percepção de conceitos de fato se aproxima perigosamente da (...) ideia de um "olho da mente". Contudo, argumentamos que esses aspectos problemáticos não são intrínsecos à noção de percepção de conceitos, mas sim resultam de uma má compreensão da distinção entre conceito e objeto. Our goal in this article is to illuminate some aspects of Gödel's conception of the perception of concepts. We begin by investigating the nature and function of Gödel's analogy between sense perception and perception of concepts. After that, we examine the connections between perception of concepts, reason, and understanding, trying to show that the perception of concepts consists in the understanding of concepts. Finally, we examine those aspects of Gödel's conception where the perception of concepts actually comes dangerously close to the idea of a "mind's eye". However, we argue that these problematic aspects are not intrinsic to the notion of perception of concepts, but result from a misunderstanding of the distinction between concept and object. (shrink)

The paper examines the interrelationship between mathematics and logic, arguing that a central characteristic of each has an essential role within the other. The first part is a reconstruction of and elaboration on Paul Bernays’ argument, that mathematics and logic are based on different directions of abstraction from content, and that mathematics, at its core it is a study of formal structures. The notion of a study of structure is clarified by the examples of Hilbert’s work on the axiomatization of (...) geometry and Hilbert et al.’s formalist proof theory. It is further argued that the structural aspect of logic puts it under the purview of the mathematical, analogously to how the deductive nature of mathematics puts it under the purview of logic. This is then linked, in the second part, to certain aspects of Gödel’s critique of Carnap’s conventionalism, that ‘mere syntax’ cannot capture the full content of mathematics, which is revealed to be closely related to the characteristic of mathematics argued for by Bernays. Finally, this is connected with Gödel’s latter-day views about two kinds of formality, intensional and extensional, and the relationship between them. (shrink)

This is a companion to a paper by the authors entitled “Gödel on deduction”, which examined the links between some philosophical views ascribed to Gödel and general proof theory. When writing that other paper, the authors were not acquainted with a system of natural deduction that Gödel presented with the help of Gentzen’s sequents, which amounts to Jaśkowski’s natural deduction system of 1934, and which may be found in Gödel’s unpublished notes for the elementary logic course he gave in 1939 (...) at the University of Notre Dame. Here one finds a presentation of this system of Gödel accompanied by a brief reexamination in the light of the notes of some points concerning his interest in sequents made in the preceding paper. This is preceded by a brief summary of Gödel’s Notre Dame course, and is followed by comments concerning Gödel’s natural deduction system. (shrink)

The article deals with the question of in which sense the notion of explanation can be applied to Kurt Gödel’s philosophy of mathematics. Gödel, as a mathematical realist, claims that in mathematics we are dealing with facts that have an objective character. One of these facts is the solvability of all well-formulated mathematical problems—and this fact requires a clarification. The assumptions on which Gödel’s position is based are: metaphysical realism: there is a mathematical universe, it is objective and independent of (...) us; epistemological optimism: we are equipped with sufficient cognitive power to gain insight into the universe. Gödel’s concept of a solution to a mathematical problem is much broader than of a mathematical proof—it is rather about finding reliable axioms that lead to a solution of the problem. I analyse the problem presented in the article, taking as an example the continuum hypothesis. (shrink)

After a brief discussion of Kreisel’s notion of informal rigour and Myhill’s notion of absolute proof, Gödel’s analysis of the subject is presented. It is shown how Gödel avoids the notion of informal proof because such a use would contradict one of the senses of “formal” that Gödel wants to preserve. This Gödelian notion of “formal” is directly tied to his notion of absolute proof and to the question of the general applicability of concepts, in a way that overcomes both (...) Kreisel and Myhill’s conceptions. This paper aims to contribute to the present-day debate on informal and epistemic mathematics, focusing on what appears necessary for a better understanding of the issues at stake. (shrink)

This paper examines Quine’s web of belief metaphor and its role in his various responses to conventionalism. Distinguishing between two versions of conventionalism, one based on the under-determination of theory, the other associated with a linguistic account of necessary truth, I show how Quine plays the two versions of conventionalism against each other. Some of Quine’s reservations about conventionalism are traced back to his 1934 lectures on Carnap. Although these lectures appear to endorse Carnap’s conventionalism, in exposing Carnap’s failure to (...) provide an explanatory account of analytic truth, they in fact anticipate Quine’s later critique of conventionalism. I further argue that Quine eventually deconstructs both his own metaphor and the thesis of under-determination it serves to illustrate. This enables him to hold onto under-determination, but at the cost of depleting it of any real epistemic significance. Lastly, I explore the implications of this deconstruction for Quine’s indeterminacy of translation thesis. (shrink)

Gödel's philosophical views were to a significant extent influenced by the study not only of Leibniz or Husserl, but also of Kant. Both Gödel and Kant aimed at the secure foundation of philosophy, the certainty of knowledge and the solvability of all meaningful problems in philosophy. In this paper, parallelisms between the foundational crisis of metaphysics in Kant's view and the foundational crisis of mathematics in Gödel's view are elaborated, especially regarding the problem of finding the “secure path of a (...) science” for both mathematics and philosophy. Gödel's temporal subjectivism and metaphysical conceptual objectivism are presented as positively or negatively motivated by Kant's viewpoints. A remark on Gödel's collapse of modalities (in accordance with the collapse of objective time) is added. (shrink)

This paper is concerned with the use of logic to solve philosophical problems. Such use of logic goes counter to the prevailing empiricist tradition in analytic circles. Specifically, model-theoretic tools are applied to three fundamental issues in the philosophy of logic and mathematics, namely, to the issue of the existence of mathematical entities, to the dispute between first- and second-order logic and to the definition of analyticity.

Artykuł dotyczy zagadnienia, w jakim sensie można stosować kategorię wyjaśnienia do interpretacji filozofii matematyki Kurta Gödla. Gödel – jako realista matematyczny – twierdzi bowiem, że w wypadku matematyki mamy do czynienia z niezależnymi od nas faktami. Jednym z owych faktów jest właśnie rozwiązywalność wszystkich dobrze postawionych problemów matematycznych – i ten fakt domaga się wyjaśnienia. Kluczem do zrozumienia stanowiska Gödla jest identyfikacja założeń, na których się opiera: metafizyczny realizm: istnieje uniwersum matematyczne, ma ono charakter obiektywny, niezależny od nas; optymizm epistemologiczny: (...) jesteśmy wyposażeni w wystarczająco dobre środki poznawcze, aby uzyskać wgląd w owo uniwersum. Pojęcie rozwiązania problemu matematycznego Gödel rozumie znacznie szerzej niż jako podanie matematycznego dowodu – chodzi raczej o znalezienie wiarogodnych aksjomatów, prowadzących do rozwiązania. Stawiany w artykule problem analizuję na przykładzie hipotezy kontinuum. (shrink)