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Number theory and elementary arithmetic
 Philosophia Mathematica
, 2003
"... Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show t ..."
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Elementary arithmetic (also known as “elementary function arithmetic”) is a fragment of firstorder arithmetic so weak that it cannot prove the totality of an iterated exponential function. Surprisingly, however, the theory turns out to be remarkably robust. I will discuss formal results that show that many theorems of number theory and combinatorics are derivable in elementary arithmetic, and try to place these results in a broader philosophical context. 1
In Defense of the Ideal 2nd DRAFT
"... This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. ..."
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This paper lies at the edge of the topic of the workshop. We can write down a Π1 1 axiom whose models are precisely the ∈structures 〈Rα, ∈ ∩R2 α〉 where α> 0 and Rα is the collection of all (pure) sets of rank < α. From this, one can consider the introduction of new axioms concerning the size of α. The question of the grounds for doing so is perhaps the central question of the workshop. But I want to discuss another question which, as I said, arises at the periphery: How do we know that there are structures 〈Rα, ∈ ∩R2 α〉? How do we know that there exist such things as sets and how do we know that, given such things, the axioms we write down are true of them? These seem very primitive questions, but the skepticism implicit in them has deep (and ancient) roots. In particular, they are questions about ideal objects in general, and not just about the actual infinite. I want to explain why I think the questions (as intended) are empty and the skepticism unfounded. 1 I will be expanding the argument of the first part of my paper “Proof and truth: the Platonism of mathematics”[1986a]. 2 The argument in question
Indispensability Arguments and Their Quinean Heritage
"... Indispensability arguments (IA) for mathematical realism are commonly traced back to Quine. We identify two different Quinean strands in the interpretation of IA, what we label the ‘logical point of view ’ and the ‘theorycontribution ’ point of view. Focusing on each of the latter, we offer two min ..."
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Indispensability arguments (IA) for mathematical realism are commonly traced back to Quine. We identify two different Quinean strands in the interpretation of IA, what we label the ‘logical point of view ’ and the ‘theorycontribution ’ point of view. Focusing on each of the latter, we offer two minimal versions of IA. These both dispense with a number of theoretical assumptions commonly thought to be relevant to IA (most notably confirmational holism and naturalism). We then show that the attribution of both minimal arguments to Quine is controversial, and stress the extent to which this is so in both cases, in order to attain a better appreciation of the Quinean heritage of IA.
Naturalising Mathematics: A Critical Look at the QuineMaddy Debate
"... This paper considers Maddy’s strategy for naturalising mathematics in the context of Quine’s scientific naturalism. The aim of this proposal is to account for the acceptability of mathematics on scientific grounds without committing to revisionism about mathematical practice entailed by the QuinePu ..."
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This paper considers Maddy’s strategy for naturalising mathematics in the context of Quine’s scientific naturalism. The aim of this proposal is to account for the acceptability of mathematics on scientific grounds without committing to revisionism about mathematical practice entailed by the QuinePutnam indispensability argument. It has been argued that Maddy’s mathematical naturalism makes inconsistent assumptions on the role of mathematics in scientific explanations to the effect that it cannot distinguish mathematics from pseudoscience. I shall clarify Maddy’s arguments and show that the objection can be overcome. I shall then reformulate a novel version of the objection and consider a possible answer, and I shall conclude that mathematical naturalism does not ultimately provide a viable strategy for accommodating an antirevisionary stance on mathematics within a Quinean naturalist framework.