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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...
In the section of the Antinomy of pure Reason Kant presents three notions of infinity. By investigating these concepts of infinity, this paper highlights important ‘building blocks’ of the structure of the mathematical antinomies, such as the ability of reason of producing ascending and descending series, as well as the notions of given and givable series. These structural features are discussed in...
In this paper I will examine what Blaise Pascal means by “infinite distance”, both in his works on projective geometry and in the apologetics of the Pensées’s. I suggest that there is a difference of meaning in these two uses of “infinite distance”, and that the Pensées’s use of it also bears relations to the mathematical concept of heterogeneity. I also consider the relation between the finite and...
Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles coming...
It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some...
We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review...
How are we to account for the epistemic contribution of our perceptual experiences to the reasonableness of our perceptual beliefs? It is well known that a conception heavily influenced by Cartesian thinking has it that experiences do not enable the experiencing subject to have direct epistemic contact with the external world; rather, they are regarded as openness to a kind of private inner realm...
Many philosophers think that requirements of rationality are “wide-scope”. That is to say: they are requirements to make true some material conditional, such that one counts as satisfying the requirement iff one either makes the conditional’s antecedent false or makes its consequent true. These contrast with narrow-scope requirements, where the ‘requires’ operator takes scope only over the consequent...
A triviality result for what Lewis (Mind 105: 303–313, 1996) called “the Desire by Necessity Thesis” and Broome (Mind 100(2): 265–267, 1991) called “the Desire as Expectation Thesis” is presented. The result shows that this thesis and three other reasonable conditions can be jointly satisfied only in trivial cases. Some meta-ethical implications of the result are discussed. The discussion also highlights...
Cognitive psychologists have uncovered a number of natural tendencies to systematic errors in thinking. This paper proposes some ways that intellectual character virtues might help correct these sources of epistemic unreliability. We begin with an overview of some insights from recent work in dual-process cognitive psychology regarding ‘biases and heuristics’, and argue that the dozens of hazards...
The medieval distinction between categorematic and syncategorematic words is usually given as the distinction between words which have signification or meaning in isolation from other words (such as nouns, pronouns, verbs) and those which have signification only when combined with other words (such as conjunctions, quantifiers, and articles). Some words, however, are classified as both categorematic...
This essay will examine some rather serious trouble confronting claims that mathematicalia might be social constructs. Because of the clarity with which he makes the case and the philosophical rigor he applies to his analysis, our exemplar of a social constructivist in this sense is Julian Cole, especially the work in his 2009 and 2013 papers on the topic. In a 2010 paper, Jill Dieterle criticized...
In this paper I develop a philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions. The account is based on a simple metaphor in which we think of indefinitely continuing processes as defining objects. It is shown that such a metaphor is valid in terms of mathematical practice, as well as in line with empirical data on arithmetical...
Stewart Cohen argues that basic knowledge is problematic, as it implies that subjects can acquire knowledge or justified beliefs about certain matters in ways that are supposedly too easy. Cohen raises two versions of the problem of easy knowledge, one involving the principle of closure and the other track-record style bootstrapping reasoning. In this paper I confront the problem of easy knowledge...
GRW theory offers precise laws for the collapse of the wave function. These collapses are characterized by two new constants, $$\lambda $$ λ and $$\sigma $$ σ . Recent work has put experimental upper bounds on the collapse rate, $$\lambda $$ λ . Lower bounds on $$\lambda $$ λ have been more controversial since GRW begins to take on a many-worlds character for small...
This study provides a critical appraisal of Duncan Pritchard’s (Synthese 189:255–272, 2012) argument to the effect that ability to preserve certain eminently plausible transmission and/or closure principles for knowledge serves as a powerful adequacy test on alternative accounts of so-called Wittgensteinian certainties or hinge commitments. I argue that Pritchard fails to establish this claim—the...
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