here is an old puzzle about ontology, one that has been puzzling enough to cast a shadow of doubt over the legitimacy of ontology as a philosophical project. The puzzle concerns in particular ontological questions about natural numbers, properties, and propositions, but also some other things as well. It arises as follows: ontological questions about numbers, properties, or propositions are questions about whether reality contains such entities, whether they are part of the stuff that the world is made of. The ontological questions about numbers, properties, or propositions thus seem to be substantive metaphysical questions about what is part of reality. Complicated as these questions may be, they can nonetheless be stated simply in ordinary English with the words ‘Are there numbers/properties/propositions?’ However, it seems that such a question can be answered quite immediately in the affirmative. It seems that there are trivial arguments that have the conclusion that there are numbers/properties/

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This paper is divided into four sections. The first two identify different logicist theses, and show that their truth-values can be conclusively established on minimal assumptions. Section 3 sets forth a notion of ‘content-recarving ’ as a possible constraint on logicist theses. Section 4—which is largely independent from the rest of the paper—is a discussion of ‘Neo-Logicism’. 1 Logicism 1.1 What is Logicism? Briefly, logicism is the view that mathematics is a part of logic. But this formulation is imprecise because it fails to distinguish between the following three claims: 1. Language-Logicism The language of mathematics consists of purely logical expressions.

Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege’s version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influenced other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within a broadly Dummettian framework. The conclusions are mostly negative: Dummett’s views on analyticity and the logical/non-logical boundary leave little room for logicism. Dummett’s considerations concerning manifestation and separability lead to a conservative extension requirement: if a sentence S is logically true, then there is a proof of S which uses only the introduction and elimination rules of the logical terms that occur in S. If basic arithmetic propositions are logically true—as the logicist contends—then there is tension between this conservation requirement and the ontological commitments

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Abstract (Short version): Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe these abilities set the stage for later mathematics: the natural numbers and arithmetic. But the connection between these early and later skills is far from obvious. We sketch what we think is the most likely model for infant abilities and argue that children could not extrapolate mature math concepts from these beginnings. We suggest instead that children may arrive at natural numbers by constructing mathematical schemas on the basis of innate abilities and math principles. Abstract (Long version): Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe these abilities set the stage for later mathematics: the natural numbers and arithmetic. But the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (a) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children’s understanding of number terms do not necessarily tap these concepts. (b) True

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