Fictionalism and the attitudes

This paper distinguishes revolutionary fictionalism from other forms of fictionalism and also from other philosophical views. The paper takes fictionalism about mathematical objects and fictionalism about scientific unobservables as illustrations. The paper evaluates arguments that purport to show that this form of fictionalism is incoherent on the grounds that there is no tenable distinction between believing a sentence and taking the fictionalist's distinctive attitude to that sentence. The argument that fictionalism about mathematics is ‘comically immodest’ is also evaluated. In place of those arguments, an argument against fictionalism about abstract objects of any kind is presented in the last section. This argument takes the form of a trilemma against the fictionalist.

Mathematizing phenomenology

Husserl is well known for his critique of the “mathematizing tendencies” of modern science, and is particularly emphatic that mathematics and phenomenology are distinct and in some sense incompatible. But Husserl himself uses mathematical methods in phenomenology. In the first half of the paper I give a detailed analysis of this tension, showing how those Husserlian doctrines which seem to speak against application of mathematics to phenomenology do not in fact do so. In the second half of the paper I focus on a particular example of Husserl’s “mathematized phenomenology”: his use of concepts from what is today called dynamical systems theory.

Towards a New Epistemology of Mathematics

In this introduction we discuss the motivation behind the workshop “Towards a New Epistemology of Mathematics” of which this special issue constitutes the proceedings. We elaborate on historical and empirical aspects of the desired new epistemology, connect it to the public image of mathematics, and give a summary and an introduction to the contributions to this issue.

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The Role of Axioms in Mathematics

To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide variety of philosophical positions. Once the axioms are generally accepted, mathematicians can expend their energies on proving theorems instead of arguing philosophy. Given this account of the role of axioms, I give four criteria that axioms must meet in order to be accepted. Penelope Maddy has proposed a similar view in Naturalism in Mathematics, but she suggests that the philosophical questions bracketed by adopting the axioms can in fact be ignored forever. I contend that these philosophical arguments are in fact important, and should ideally be resolved at some point, but I concede that their resolution is unlikely to affect the ordinary practice of mathematics. However, they may have effects in the margins of mathematics, including with regards to the controversial “large cardinal axioms” Maddy would like to support.

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Quine, Putnam, and the ‘Quine–Putnam’ Indispensability Argument

Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine’s argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam’s writings reveals some neglected arguments for platonism which may be superior to the indispensability argument.

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Explanation and Understanding: Action as “Historical Structure”

The first part of this essay is basically historical. It introduces the explanation–understanding divide, focusing in particular on the general–unique distinction. The second part is more philosophical and it presents two different claims on action. In the first place, I will try to say what it means to understand an action. Secondly, we will focus on the explanation of action as it is seen in some explanatory sciences. I will try to argue that in some cases these sciences commit what I call an “external contradiction”.

Visualizations in Mathematics

In this paper we discuss visualizations in mathematics from a historical and didactical perspective. We consider historical debates from the 17th and 19th centuries regarding the role of intuition and visualizations in mathematics. We also consider the problem of what a visualization in mathematical learning can achieve. In an empirical study we investigate what mathematical conclusions university students made on the basis of a visualization. We emphasize that a visualization in mathematics should always be considered in its proper context.

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Intuition and Philosophical Methodology

Intuition serves a variety of roles in contemporary philosophy. This paper provides a historical discussion of the revival of intuition in the 1970s, untangling some of the ways that intuition has been used and offering some suggestions concerning its proper place in philosophical investigation. Contrary to some interpretations of the results of experimental philosophy, it is argued that generalized skepticism with respect to intuition is unwarranted. Intuition can continue to play an important role as part of a methodologically conservative stance towards philosophical investigation. I argue that methodological conservatism should be sharply distinguished from the process of evaluating individual propositions. Nevertheless, intuition is not always a reliable guide to truth and experimental philosophy can serve a vital ameliorative role in determining the scope and limits of our intuitive competence with respect to various areas of inquiry.

Remarks on Grandi’s Comments

This note is a reply to some of Giovanni Grandi’s comments on my paper “Berkeley’s Contingent Necessities.”

Husserl’s philosophy of mathematics: its origin and relevance

This paper offers an exposition of Husserl's mature philosophy of mathematics, expounded for the first time in Logische Untersuchungen and maintained without any essential change throughout the rest of his life. It is shown that Husserl's views on mathematics were strongly influenced by Riemann, and had clear affinities with the much later Bourbaki school.

Not Always Enslaved,Yet Not Quite Free: Philosophical Challenges from the Underside of the New World

This article is the keynote address of the University of the West Indies at Cave Hill, Barbados, philosophy symposium in celebration of the 200th Anniversary of the British outlawing the Atlantic Slave Trade. The paper explores questions of enslavement and freedom through challenges of philosophical anthropology, philosophy of social change, and metacritical reflections posed by African Diasporic or Africana philosophy. Such challenges include the relevance and legitimacy of philosophical reflection to the lives of racialized slaves and concludes with a discussion of the implications of the analysis for an understanding of the “face” of political life and the importance of the concept of “home” for a cogent theory of freedom.

Comparing Biographical Backgrounds of Religious Founders and Converts to Those Religions: An Exploratory Study

This paper is an exploratory, preliminary investigation of the possible links between the biographical backgrounds and developmental trajectories of major religious figures such as Jesus Christ, Muhammad, Buddha, and Baha’u’llah, and the backgrounds of those who convert to these religions (or certain groups within these religions) in the West. This article ends with the hypothesis that in terms of biographical backgrounds and motivations for conversion, followers’ narratives resemble those of their religious leaders in some areas.

On Promising to Supererogate: A Response to Heyd

In my “Promising and Supererogation” I argue that one cannot fulfill promises to perform supererogatory actions (such as “I hereby promise to perform one supererogatory action every month”). In a response to my paper, David Heyd argues that there is an alternative solution to the problem I raise. While I agree with much that Heyd says about the examples he discusses, his proposed solution involves a crucial alteration of the problem; his proposed solution does not solve the problem I present.

Acceptance,Belief, and Descartes’s Provisional Morality

This paper explores Descartes’s work with an eye towards abiding issues in moral epistemology. In so doing, I focus on the role played by the so-called provisional morality that surfaces in “Discourse on the Method”. What I argue is that despite the tenuousness with which it seems to be held, Descartes remained committed to the truth of this morality even in the midst of his most strenuous philosophical reflections. Put in the contemporary epistemological terms which provide the context of my discussion, I argue that Descartes believed in the goodness of his provisional morality as opposed to merely accepting its maxims.

Union and Difference: A Dialectical Structuring of St. John of the Cross’ Mysticism

This paper intends to append the frame of dialectic upon St. John of the Cross’ delineation of mysticism. Its underlying hypothesis is that the dialectical structuring of St. John’s mystical theology promises to unravel the web of relational concepts embedded within his immense writings on this unique phenomenon. It is hoped that as a consequence of this undertaking, relevant pairs of correlative opposites that figure prominently in mysticism can be elucidated and perhaps come to some form of resolution.

What can the Philosophy of Mathematics Learn from the History of Mathematics?

This article canvasses five senses in which one might introduce an historical element into the philosophy of mathematics: 1. The temporal dimension of logic; 2. Explanatory Appeal to Context rather than to General Principles; 3. Heraclitean Flux; 4. All history is the History of Thought; and 5. History is Non-Judgmental. It concludes by adapting Bernard Williams’ distinction between ‘history of philosophy’ and ‘history of ideas’ to argue that the philosophy of mathematics is unavoidably historical, but need not and must not merge with historiography.

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Who is the Addressee of Philosophical Argumentation?

Chaim Perelman invokes the idea of “universal audience” for explaining the nature of philosophical argumentation as rational rhetoric. As opposed to this view, centuries before Perelman, Socrates argues that philosophy should be conducted as a dialogue between concrete individuals with very specific qualities. The paper presents these different views in order to claim that the philosopher addresses neither a universal audience nor a particular other, but mainly and essentially the philosopher herself/himself. This brings to light the problem of self-deception as a central problem of philosophical thinking. In posing this view the paper uses Nietzsche’s definition of “the will to truth” as the will not to deceive, not even myself.

The Chrysippus intuition and contextual theories of truth

Contextual theories of truth are motivated primarily by the resolution they provide to paradoxical reasoning about truth. The principal argument for contextual theories of truth relies on a key intuition about the truth value of the proposition expressed by a particular utterance made during paradoxical reasoning, which Anil Gupta calls “the Chrysippus intuition.” In this paper, I argue that the principal argument for contextual theories of truth is circular, and that the Chrysippus intuition is false. I conclude that the philosophical motivation for contextual theories of truth fails.

God and The Possibility of Random Creation

In this paper I discuss a number of problems associated with the suggestion that it is possible for God to randomly select a possible world for actualization.

A consistent way with paradox

Consideration of a paradox originally discovered by John Buridan provides a springboard for a general solution to paradoxes within the Liar family. The solution rests on a philosophical defence of truth-value-gaps and is consistent (non-dialetheist), avoids ‘revenge’ problems, imports no ad hoc assumptions, is not applicable to only a proper subset of the semantic paradoxes and implies no restriction of the expressive capacities of language.