George Boolos was one of the most prominent and influential logician-philosophers of recent times. This collection, nearly all chosen by Boolos himself shortly before his death, includes thirty papers on set theory, second-order logic, and plural quantifiers; on Frege, Dedekind, Cantor, and Russell; and on miscellaneous topics in logic and proof theory, including three papers on various aspects of the Gödel theorems. Boolos is universally recognized as the leader in the renewed interest in studies of Frege's work on logic and (...) the philosophy of mathematics. John Burgess has provided introductions to each of the three parts of the volume, and also an afterword on Boolos's technical work in provability logic, which is beyond the scope of this volume. (shrink)

This paper is a reply to George Boolos's three papers (Boolos (1987a, 1987b, 1990a)) concerned with the status of Hume's Principle. Five independent worries of Boolos concerning the status of Hume's Principle as an analytic truth are identified and discussed. Firstly, the ontogical concern about the commitments of Hume's Principle. Secondly, whether Hume's Principle is in fact consistent and whether the commitment to the universal number by adopting Hume's Principle might be problematic. Also the so-called `surplus content' worry is discussed, (...) which points out that the conceptual resources to grasp Hume's Principle vastly outstrip the conceptual resources employed in arithmetical reasoning. And lastly whether Hume's Principle is in bad company with other unsuccessful implicit definitions. In the last section, an account towards our entitlement to Hume&'s Principle is sketched. (shrink)

One recent `neologicist' claim is that what has come to be known as "Frege's Theorem"–the result that Hume's Principle, plus second-order logic, suffices for a proof of the Dedekind-Peano postulate–reinstates Frege's contention that arithmetic is analytic. This claim naturally depends upon the analyticity of Hume's Principle itself. The present paper reviews five misgivings that developed in various of George Boolos's writings. It observes that each of them really concerns not `analyticity' but either the truth of Hume's Principle or our entitlement (...) to accept it and reviews possible neologicist replies. A two-part Appendix explores recent developments of the fifth of Boolos's objections–the problem of Bad Company–and outlines a proof of the principle $N^q$, an important part of the defense of the claim that what follows from Hume's Principle is not merely a theory which allows of interpretation as arithmetic but arithmetic itself. (shrink)

Pure mathematics can play an indispensable role explaining empirical phenomena if recent accounts of insect evolution are correct. In particular, the prime life cycles of cicadas and the geometric structure of honeycombs are taken to undergird an inference to the best explanation about mathematical entities. Neither example supports this inference or the mathematical realism it is intended to establish. Both incorrectly assume that facts about mathematical optimality drove selection for the respective traits and explain why they exist. We show how (...) this problem can be avoided, identify limitations of explanatory indispensability arguments, and attempt to clarify the nature of mathematical explanation. (shrink)

Baker (Mind 114:223–238, 2005; Brit J Philos Sci 60:611–633, 2009) has recently defended what he calls the “enhanced” version of the indispensability argument for mathematical Platonism. In this paper I demonstrate that the nominalist can respond to Baker’s argument. First, I outline Baker’s argument in more detail before providing a nominalistically acceptable paraphrase of prime-number talk. Second, I argue that, for the nominalist, mathematical language is used to express physical facts about the world. In endorsing this line I follow moves (...) made by Saatsi (Brit J Philos Sci 62(1):143–154, 2011). But, unlike Saatsi, I go on to argue that the nominalist requires a paraphrase of prime-number talk, for otherwise we lack an account of what that ‘physical fact’ is in the case of mathematics that seemingly makes reference to prime numbers. (shrink)

This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (logicism), (...) the view that the essence of mathematics is the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline. This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy. (shrink)

Baker claims to provide an example of mathematical explanation of an empirical phenomenon which leads to ontological commitment to mathematical objects. This is meant to show that the positing of mathematical entities is necessary for satisfactory scientific explanations and thus that the application of mathematics to science can be used, at least in some cases, to support mathematical realism. In this paper I show that the example of explanation Baker considers can actually be given without postulating mathematical objects and thus (...) cannot be used by the mathematical realist. I also show that, despite this, mathematics keeps playing an important methodological role in the explanation and does not reduce to a merely computational or descriptive framework. (shrink)

According to the indispensability argument, the fact that we quantify over numbers, sets and functions in our best scientific theories gives us reason for believing that such objects exist. I examine a strategy to dispense with such quantification by simply replacing any given platonistic theory by the set of sentences in the nominalist vocabulary it logically entails. I argue that, as a strategy, this response fails: for there is no guarantee that the nominalist world that go beyond the set of (...) sentences in the nominalist language such theories entail. However, I argue that what such theories show is that mathematics can enable us to express possibilities about the concrete world that may not be expressible in nominalistically acceptable language. While I grant that this may make quantification over abstracta indispensable, I deny that such indispensability is a reason for accepting them into our ontology. I urge that the nominalist should be allowed to quantify over abstracta whilst denying their existence and I explain how this apparently contradictory practice (a practice I call 'weaseling') is in fact coherent, unproblematic and rational. Finally, I examine the view that platonistic theories are simpler or more attractive than their nominalistic reformulations, and thus that abstract ought to be accepted into our ontology for the same sorts of reasons as other theoretical objects. I argue that, at least in the case of numbers, functions and sets, such arguments misunderstand the kind of simplicity and attractiveness we seek. (shrink)

A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional (...) coverage of introductory material such as sets. Increased flexibility of the text, allowing instructors more choice in how they use the textbook in courses. Reduced mathematical rigour to fit the needs of undergraduate students. (shrink)

The question of the analyticity of Hume’s Principle (HP) is central to the neo-logicist project. We take on this question with respect to Frege’s definition of analyticity, which entails that a sentence cannot be analytic if it can be consistently denied within the sphere of a special science. We show that HP can be denied within non-standard analysis and argue that if HP is taken to depend on Frege’s definition of number, it isn’t analytic, and if HP is taken to (...) be primitive there is only a very narrow range of circumstances where it might be taken to be analytic. The latter discussion also sheds some light on the connections between the Bad Company and Caesar objections. (shrink)

An elementary theory of concatenation,QT+, is introduced and used to establish mutual interpretability of Robinson arithmetic, Minimal Predicative Set Theory, quantifier-free part of Kirby’s finitary set theory, and Adjunctive Set Theory, with or without extensionality. The most basic arithmetic and simplest set theory thus turn out to be variants of string theory.

This book surveys the assortment of methods put forth for fixing Frege's system, in an attempt to determine just how much of mathematics can be reconstructed in ...

Neofregeanism and structuralism are among the most promising recent approaches to the philosophy of mathematics. Yet both have serious costs. We develop a view, structuralist neologicism, which retains the central advantages of each while avoiding their more serious costs. The key to our approach is using arbitrary reference to explicate how mathematical terms, introduced by abstraction principles, refer. Focusing on numerical terms, this allows us to treat abstraction principles as implicit definitions determining all properties of the numbers, achieving a key (...) neofregean advantage, while preserving the key structuralist advantage, which objects play the number role does not matter. (shrink)

The Enhanced Indispensability Argument appeals to the existence of Mathematical Explanations of Physical Phenomena to justify mathematical Platonism, following the principle of Inference to the Best Explanation. In this paper, I examine one example of a MEPP—the explanation of the 13-year and 17-year life cycle of magicicadas—and argue that this case cannot be used defend the EIA. I then generalize my analysis of the cicada case to other MEPPs, and show that these explanations rely on what I will call ‘optimal (...) representations’, which are representations that capture all that is relevant to explain a physical phenomenon at a specified level of description. In the end, because the role of mathematics in MEPPs is ultimately representational, they cannot be used to support mathematical Platonism. I finish the paper by addressing the claim, advanced by many EIA defendants, that quantification over mathematical objects results in explanations that have more theoretical virtues, especially that they are more general and modally stronger than alternative explanations. I will show that the EIA cannot be successfully defended by appealing to these notions. (shrink)

Arguing for mathematical realism on the basis of Field’s explanationist version of the Quine–Putnam Indispensability argument, Alan Baker has recently claimed to have found an instance of a genuine mathematical explanation of a physical phenomenon. While I agree that Baker presents a very interesting example in which mathematics plays an essential explanatory role, I show that this example, and the argument built upon it, begs the question against the mathematical nominalist.

Indispensability-based arguments for mathematical platonism are typically motivated by drawing an analogy between abstract mathematical objects and concrete scientific posits. In this paper, I argue that mathematics can sometimes help to reduce our concrete ontological, ideological, and structural commitments. My focus is on optimization explanations, and in particular the case study involving periodical cicadas. I argue that in this case, stronger mathematical apparatus yields explanations that have fewer concrete commitments. The nominalist cannot accept these more parsimonious explanations without embracing the (...) stronger mathematics, and this poses a challenge for the nominalist position. (shrink)

The aim of this paper is to open a new front in the debate between platonism and nominalism by arguing that the degree of explanatory entanglement of mathematics in science is much more extensive than has been hitherto acknowledged. Even standard examples, such as the prime life cycles of periodical cicadas, involve a penumbra of mathematical features whose presence can only be explained using relatively sophisticated mathematics. I introduce the term ‘mathematical spandrel’ to describe these penumbral properties, and focus on (...) the property that cicada period lengths are expressible as the sum of two perfect squares. I argue that mathematical spandrels pose a particular problem for nominalism because of the way in which they are entangled with scientific explanations. (shrink)

Does mathematics ever play an explanatory role in science? If so then this opens the way for scientific realists to argue for the existence of mathematical entities using inference to the best explanation. Elsewhere I have argued, using a case study involving the prime-numbered life cycles of periodical cicadas, that there are examples of indispensable mathematical explanations of purely physical phenomena. In this paper I respond to objections to this claim that have been made by various philosophers, and I discuss (...) potential future directions of research for each side in the debate over the existence of abstract mathematical objects. (shrink)

According to one popular nominalist picture, even when mathematics features indispensably in scientific explanations, this mathematics plays only a purely representational role: physical facts are represented, and these exclusively carry the explanatory load. I think that this view is mistaken, and that there are cases where mathematics itself plays an explanatory role. I distinguish two kinds of explanatory generality: scope generality and topic generality. Using the well-known periodical-cicada example, and also a new case study involving bicycle gears, I argue that (...) what is picked out by the mathematics are structural facts that go beyond any specific physical facts. (shrink)

Many explanations in science make use of mathematics. But are there cases where the mathematical component of a scientific explanation is explanatory in its own right? This issue of mathematical explanations in science has been for the most part neglected. I argue that there are genuine mathematical explanations in science, and present in some detail an example of such an explanation, taken from evolutionary biology, involving periodical cicadas. I also indicate how the answer to my title question impacts on broader (...) issues in the philosophy of mathematics; in particular it may help platonists respond to a recent challenge by Joseph Melia concerning the force of the Indispensability Argument. (shrink)

First published in 1971, Professor Putnam's essay concerns itself with the ontological problem in the philosophy of logic and mathematics - that is, the issue of whether the abstract entities spoken of in logic and mathematics really exist. He also deals with the question of whether or not reference to these abstract entities is really indispensible in logic and whether it is necessary in physical science in general.

What in our theoretical pronouncements commits us to objects? The Quinean standard for ontological commitment involves (nearly enough) commitments when we utter “there is” or “there are” statements without hope of eliminating these by paraphrase. Coupled with the indispensability of the truth of applied mathematical doctrine, the result is that the ontologically hard-nosed scientist is a Platonist—haplessly commited to abstracta. In this book Azzouni offers a way around the Quinean straitjacket: ontological commitment turns on how theories are (nearly enough) nailed (...) to the world. The specifics of how theories are applied indicates which among the posits of a theory are mere mathematical garb and which are genuine connections to items out there. In the first part of the book Azzouni undercuts the arguments, both actual and possible, in support of Quine’s criterion. An alternative criterion for what exists—ontological independence—is offered, one in sturdy accord with ordinary folk views on the matter. In the second part of the book, a beginning is made of bringing this alternative to bear upon scientific theories with a rich mathematical component. Along the way, old philosophical issues about absolute space and time versus relative space and time, the status of mathematical posits, such as spatial and temporal points, and so on, are illuminated. (shrink)

In this paper, I develop a new defense of logicism: one that combines logicism and nominalism. First, I defend the logicist approach from recent criticisms; in particular from the charge that a cruciai principie in the logicist reconstruction of arithmetic, Hume's Principle, is not analytic. In order to do that, I argue, it is crucial to understand the overall logicist approach as a nominalist view. I then indicate a way of extending the nominalist logicist approach beyond arithmetic. Finally, I argue (...) that a nominalist can use the resulting approach to provide a nominalization strategy for mathematics. In this way, mathematical structures can be introduced without ontological costs. And so, if this proposal is correct, we can say that ultimately all the nominalist needs is logic (and, rather loosely, ali the logicist needs is nominalism). (shrink)

Edward Nelson published in 1986 a book defending an extreme formalist view of mathematics according to which there is an impassable barrier in the totality of exponentiation. On the positive side, Nelson embarks on a program of investigating how much mathematics can be interpreted in Raphael Robinson's theory of arithmetic Q. In the shadow of this program, some very nice logical investigations and results were produced by a number of people, not only regarding what can be interpreted in Q but (...) also what cannot be so interpreted. We explain some of these results and rely on them to discuss Nelson's position. (shrink)