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Switch to: References
Citations of:
Introduction
In Philosophy of Mathematics. Princeton, NJ: Princeton University Press. pp. 1-3 (2017)
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I show that Wittgenstein's critique of G.H. Hardy's mathematical realism naturally extends to Paul Benacerraf's influential paper, ‘Mathematical Truth’. Wittgenstein accuses Hardy of hastily analogizing mathematical and empirical propositions, thus leading to a picture of mathematical reality that is somehow akin to empirical reality despite the many puzzles this creates. Since Benacerraf relies on that very same analogy to raise problems about mathematical ‘truth’ and the alleged ‘reality’ to which it corresponds, his major argument falls prey to the same critique. (...) |
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The neo-Fregean project of basing mathematics on abstraction principles faces “the bad company problem,” namely that a great variety of unacceptable abstraction principles are mixed in among the acceptable ones. In this paper I propose a new solution to the problem, based on the idea that individuation must take the form of a well-founded process. A surprising aspect of this solution is that every form of abstraction on concepts is permissible and that paradox is instead avoided by restricting what concepts (...) |
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Thin Objects has two overarching ambitions. The first is to clarify and defend the idea that some objects are ‘thin’, in the sense that their existence does not make a substantive demand on reality. The second is to develop a systematic and well-motivated account of permissible abstraction, thereby solving the so-called ‘bad company problem’. Here I synthesise the book by briefly commenting on what I regard as its central themes. |
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In this short paper, we analyse whether assuming that mathematical objects are “thin” in Linnebo's sense simplifies the epistemology of mathematics. Towards this end, we introduce the notion of transparency and show that not all thin objects are transparent. We end by arguing that, far from being a weakness of thin objects, the lack of transparency of some thin objects is a fruitful characteristic mark of abstract mathematics. |
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According to the iterative conception of set, sets are formed in stages. According to the purely iterative conception of set, sets are formed by iterated application of a set-of operation. The cumulative hierarchy is a mathematical realization of the iterative conception of set. A mathematical realization of the purely iterative conception can be found in Peter Aczel’s type-theoretic model of constructive set theory. I will explain Aczel’s model construction in a way that presupposes no previous familiarity with the theories on (...) |
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Alter (The Matter of Consciousness: From the Knowledge Argument to Russellian Monism, GB: Oxford University Pres, 2023) argues for something surprising: despite being widely rejected by philosophers, including Frank Jackson himself, Jackson’s knowledge argument succeeds. Alter’s defense of Jackson’s argument is not only surprising; it’s also exciting: the knowledge argument, if it’s sound, underscores the power of armchair philosophy, the power of pure thought to arrive at substantial conclusions about the world. In contrast, I aim to make a case for (...) |
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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. |
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We contribute to the ongoing discussion on mathematical structuralism by focusing on a question that has so far been neglected: when is a structure part of another structure? This paper is a first step towards answering the question. We will show that a certain conception of structures, abstractionism about structures, yields a natural definition of the parthood relation between structures. This answer has many interesting consequences; however, it conflicts with some standard mereological principles. We argue that the tension between abstractionism (...) |
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Kant's claim that arithmetical truths are synthetic is famously contradicted by Frege, who considers them to be analytical. It may seem that this is a mere dispute about linguistic labels, since both Kant and Frege agree that arithmetical truths are a priori and informative, and, therefore, it is only a matter of how one chooses to call them. I argue that the choice between calling arithmetic “synthetic” or “analytic” has a deeper significance. I claim that the dispute is not a (...) |
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Aristotle ends Metaphysics books M–N with an account of how one can get the impression that Platonic Form-numbers can be causes. Though these passages are all admittedly polemic against the Platonic understanding, there is an undercurrent wherein Aristotle seems to want to explain in his own terms the evidence the Platonist might perceive as supporting his view, and give any possible credit where credit is due. Indeed, underlying this explanation of how the Platonist may have formed his impression, we discover (...) |
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