The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such as (...) Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. (shrink)

This paper addresses three problems: the problem of formulating a coherent relativism, the Sorites paradox and a seldom noticed difficulty in the best intuitionistic case for the revision of classical logic. A response to the latter is proposed which, generalised, contributes towards the solution of the other two. The key to this response is a generalised conception of indeterminacy as a specific kind of intellectual bafflement-Quandary. Intuitionistic revisions of classical logic are merited wherever a subject matter is conceived both as (...) liable to generate Quandary and as subject to a broad form of evidential constraint. So motivated, the distinctions enshrined in intuitionistic logic provide both for a satisfying resolution of the Sorites paradox and a coherent outlet for relativistic views about, e.g., matters of taste and morals. An important corollary of the discussion is that an epistemic conception of vagueness can be prised apart from the strong metaphysical realism with which its principal supporters have associated it, and acknowledged to harbour an independent insight. (shrink)

"The central task of phenomenology is to investigate the nature of consciousness and its relations to objects of various types. The present book introduces students and other readers to several foundational topics of phenomenological inquiry, and illustrates phenomenology's contemporary relevance. The main topics include consciousness, intentionality, perception, meaning, and knowledge. The book also contains critical assessments of Edmund Husserl's phenomenological method. It argues that knowledge is the most fundamental mode of consciousness, and that the central theses constitutive of Husserl's "transcendental (...) idealism" are compatible with metaphysical realism regarding the objects of thought, perception, and knowledge. Helpful tools include introductions that help the reader segue from the previous chapter to the new one, chapter conclusions, and suggested reading lists of primary and some key secondary sources. Key Features: Elucidates and engages with contemporary work in analytic epistemology and philosophy of mind. Provides clear prose explanations of the necessary distinctions and arguments required for understanding the subject. Places knowledge at the center of phenomenological inquiry"--. (shrink)

It is often claimed that anti-realism is a form of transcendental idealism or that Kant is an anti-realist. It is also often claimed that anti-realists are committed to some form of knowability principle and that such principles have problematic consequences. It is therefore natural to ask whether Kant is so committed, and if he is, whether this leads him into difficulties. I argue that a standard reading of Kant does indeed have him committed to the claim that all empirical truths (...) are knowable and that this claim entails that there is no empirical truth that is never known. I extend the result to a priori truths and draw some general philosophical lessons from this extension. However, I then propose a re-examination of Kant’s notion of experience according to which he carefully eschews any commitment to empirical knowability. Finally I respond to a remaining problem that stems from a weaker, justified believability principle. (shrink)

Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl's third claim is wrong, by his own standards.

Phenomenology: The Basics is a concise and engaging introduction to one of the important philosophical movements of the twentieth century and to a subject that continues to grow and diversify. Yet it is also a challenging subject, the elements of which can be hard to grasp. This lucid book provides an introduction to the core ideas of phenomenology and to the arguments of its principal thinkers, including Husserl, Heidegger and Merleau-Ponty. Written by a leading expert in the field, Dan Zahavi (...) examines and explains key questions such as: What is a phenomenological analysis? What are the methodological foundations of phenomenology? What does phenomenology have to say about intentionality, embodiment, intersubjectivity and the lifeworld? How do ideas from classic phenomenology relate to ongoing debates in the cognitive sciences and qualitative research? This second edition has been thoroughly revised and expanded. It contains a new chapter on critical phenomenology and updated discussions of the application of phenomenology in psychiatry, psychology, and qualitative research. Including a glossary of key terms and suggestions for further reading, Phenomenology: The Basics is a superb starting point for anyone seeking a concise and accessible introduction to this rich and fascinating subject. (shrink)

Perennial philosophers' hopes are unlikely victims of swift, natural deduction. Yet anti-realism has been thought one. Not hoping for anti-realism myself I here show it, lest it be underestimated, to survive the following argument, adapted from W. D.Hart pp. 156, 164-5; he credits first publication to Fitch).

Husserl’s first work formulated what proved to be an algorithmically complete arithmetic, lending mathematical clarity to Kronecker’s reduction of analysis to finite calculations with integers. Husserl’s critique of his nominalism led him to seek a philosophical justification of successful applications of symbolic arithmetic to nature, providing insight into the “wonderful affinity” between our mathematical thoughts and things without invoking a pre-established harmony. For this, Husserl develops a purely descriptive phenomenology for which he found inspiration in Mach’s proposal of a “universal (...) physical phenomenology.” To account for applications to any domain, Husserl envisages a theory of all possible deductive systems, which he develops extensively in his Göttingen lectures wherein he engages with Hilbert’s work on deductive systems for geometry, real arithmetic, and physics. This leads Husserl to formulate claims of decidability and proofs of completeness for various arithmetics that result from his analysis of Kronecker’s general arithmetic. Careful attention to these proofs seem to show that Husserl was not oblivious to problems that underlie our incompleteness theorems, namely that of showing that some inversions of his algorithmic arithmetic are undefined. His growing preoccupation with the issue of a pre-established harmony between mathematical thought and reality motivate his pursuit of a “supramathematics” of all possible complete theory forms to demystify such harmony, by having such a form on hand for describing any empirical domain. He soon decides that a transcendental idealism of nature will reveal the wonderful affinity of thoughts and things comprising such harmony, to be a wonderful “parallelism of objective unities and constituted manifolds of consciousness.” But the paradoxes of logic and set theory cloud the clarity of mathematics, which Weyl would restore with Brouwer’s intuitionism and Hilbert with his metamathematics. Husserl informed Weyl that his student Becker had formulated a phenomenological foundation not only for Weyl’s generalization of relativity theory but also for the Brouwer-Weyl continuum. But Weyl eventually rejected much of Becker’s work, especially when it became clear that his phenomenological intuitionism could not account for the success of Hilbert’s transfinite mathematics in quantum physics. Becker responded to this “crisis of phenomenological method” with his mantic phenomenology celebrating the magic of mathematical mysticism, which Husserl finally rejects in favor of a pluralistic phenomenology of mathematics and nature. (shrink)

First, some reminiscences. In the years 1973-80, when I was an undergraduate and then graduate student at Oxford, Michael Dummett’s formidable and creative philosophical presence made his arguments impossible to ignore. In consequence, one pole of discussion was always a form of anti-realism. It endorsed something like the replacement of truth-conditional semantics by verification-conditional semantics and of classical logic by intuitionistic logic, and the principle that all truths are knowable. It did not endorse the principle that all truths are known. (...) Nor did it mention the now celebrated argument, first published by Frederic Fitch (1963), that if all truths are knowable then all truths are known. (shrink)

The knowability paradox derives from a proof by Frederic Fitch in 1963. The proof purportedly shows that if all truths are knowable, it follows that all truths are known. Antirealists, wed as they are to the idea that truth is epistemic, feel threatened by the proof. For what better way to express the epistemic character of truth than to insist that all truths are knowable? Yet, if that insistence logically compels similar assent to some omniscience claim, antirealism is in jeopardy. (...) Response to the paradox has drifted toward a common theme, a theme I will argue is a non-starter in resolving the paradox. Seeing this point will also make clear the philosophical inadequacy of simply viewing the paradox as a refutation of a wide range of antirealisms. (shrink)

The naive anti-realist holds the following principle: (◊K) All truths are knowable. This unrestricted generalization (◊K), as is now well known, falls prey to Fitch’s Paradox (Fitch 1963: 38, Theorem 1). It can be used as the only suspect principle, alongside others that cannot be impugned, to prove quite generally, and constructively, that the set {p, ¬Kp} is inconsistent (Tennant 1997: 261). From this it would follow, intuitionistically, that any proposition that is never actually known to be true (by anyone, (...) at any time) is false; and it would follow, classically, that every truth is known (by someone, at some time). (shrink)

The knowability paradox threatens metaphysical or semantical antirealism, the view that truth is epistemic, by revealing an awful consequence of the claim [i] that all truths are knowable. Various attempts have been made to find a way out of the paradox.

I argue that Brouwer’s notion of the construction of purely mathematical objects and Husserl’s notion of their constitution by the transcendental subject coincide. Various objections to Brouwer’s intuitionism that have been raised in recent phenomenological literature are addressed. Then I present objections to Gödel’s project of founding classical mathematics on transcendental phenomenology. The problem for that project lies not so much in Husserl’s insistence on the spontaneous character of the constitution of mathematical objects, or in his refusal to allow an (...) appeal to higher minds, as in the combination of these two attitudes. (shrink)