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.

Repr. as Essay 14 in Davidson, Donald, _Inquiries into Truth and Interpretation_, 2nd ed. Oxford, UK (Clarendon, 2001). 215-226.

Machine generated contents note: Essay 1.

In this brief book one of the most distinguished living American philosophers takes up the question of whether ethical judgments can properly be considered objective--a question that has vexed philosophers over the past century. Reviewing what he deems the disastrous consequences of ontology's influence on analytic philosophy--in particular, the contortions it imposes upon debates about the objective of ethical judgments--Putnam proposes abandoning the very idea of ontology.

In the course of the discussion, Professor Quine pinpoints the difficulties involved in translation, brings to light the anomalies and conflicts implicit in our ...

Professor Hilary Putnam has been one of the most influential and sharply original of recent American philosophers in a whole range of fields. His most important published work is collected here, together with several new and substantial studies, in two volumes. The first deals with the philosophy of mathematics and of science and the nature of philosophical and scientific enquiry; the second deals with the philosophy of language and mind. Volume one is now issued in a new edition, including an (...) essay on the philosophy of logic first published in 1971. (shrink)

Quine and Putnam argued for mathematical realism on the basis of the indispensability of mathematics to science. They claimed that the mathematics that is used in physical theories is confirmed along with those theories and that scientific realism entails mathematical realism. I argue here that current theories of confirmation suggest that mathematics does not receive empirical support simply in virtue of being a part of well confirmed scientific theories and that the reasons for adopting a realist view of scientific theories (...) do not support realism about mathematical entities, despite the use of mathematics in formulating scientific theory. (shrink)

Quine and Putnam argued for mathematical realism on the basis of the indispensability of mathematics to science. They claimed that the mathematics that is used in physical theories is confirmed along with those theories and that scientific realism entails mathematical realism. I argue here that current theories of confirmation suggest that mathematics does not receive empirical support simply in virtue of being a part of well confirmed scientific theories and that the reasons for adopting a realist view of scientific theories (...) do not support realism about mathematical entities, despite the use of mathematics in formulating scientific theory. (shrink)

Realists persuaded by indispensability arguments af- firm the existence of numbers, genes, and quarks. Van Fraassen's empiricism remains agnostic with respect to all three. The point of agreement is that the posits of mathematics and the posits of biology and physics stand orfall together. The mathematical Platonist can take heart from this consensus; even if the existence of num- bers is still problematic, it seems no more problematic than the existence of genes or quarks. If the two positions just described (...) were the only ones possible, there could be no objection to this melding of numbers with genes and quarks. However, the position I call contrastive empiricism (Sober 1990a) stands opposed to both realism and to Van Fraassen's em- piricism. As it turns out, contrastive empiricism entails that coalesc- ing mathematics with empirical science is highly problematic. I believe that there is an important kernel of truth in abductive ar- guments for genes and quarks. But no counterpart argument exists for the case of numbers. Of course, the existence of this third way would be uninteresting, if contrastive empiricism were wholly implausible. However, I will argue that contrastive empiricism captures what makes sense in standard versions of realism and empiricism, while avoiding the excesses of each. This third way is a middle way; it cannot be dismissed out of hand. (shrink)

Precis of my book by this title, for a symposium.

Persistence through time is like extension through space. A road has spatial parts in the subregions of the region of space it occupies; likewise, an object that exists in time has temporal parts in the various subregions of the total region of time it occupies. This view — known variously as four dimensionalism, the doctrine of temporal parts, and the theory that objects “perdure” — is opposed to “three dimensionalism”, the doctrine that things “endure”, or are “wholly present”.1 I will (...) attempt to resolve this dispute in favor of four dimensionalism by means of a novel argument based on considerations of vagueness. But before argument in this area can be productive, I believe we must become much clearer than is customary about exactly what the dispute is, for the usual ways of formulating the dispute are flawed, especially where three dimensionalism is concerned. (shrink)

Penelope Maddy and Elliott Sober recently attacked the confirmational indispensability argument for mathematical realism. We cannot count on science to provide evidence for the truth of mathematics, they say, because either scientific testing fails to confirm mathematics (Sober) or too much mathematics occurs in false scientific theories (Maddy). I present a pragmatic indispensability argument immune to these objections, and show that this argument supports mathematical realism independently of scientific realism. Mathematical realism, it turns out, may be even more firmly established (...) than scientific realism. (shrink)

This book expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defense of realism about the metaphysics of mathematics--the view that mathematics (...) is about things that really exist. (shrink)

Things and Their Place in Theories Our talk of external things, our very notion of things, is just a conceptual apparatus that helps us to foresee and ...

This volume consists of the first of the John Dewey Lectures delivered under the auspices of Columbia University's Philosophy Department as well as other essays by the author. Intended to clarify the meaning of the philosophical doctrines propounded by Professor Quine in 'Word and Objects', the essays included herein both support and expand those doctrines.

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.

In this brief book one of the most distinguished living American philosophers takes up the question of whether ethical judgments can properly be considered ...

Mark Colyvan uses applications of mathematics to argue that mathematical entities exist. I claim that his argument is invalid based on the assumption that a certain way of thinking about applications, called `the mapping account,' is correct. My main contention is that successful applications depend only on there being appropriate structural relations between physical situations and the mathematical domain. As a variety of non-realist interpretations of mathematics deliver these structural relations, indispensability arguments are invalid.

This precis explains that _Philosophical naturalism contains three parts. Part I examines arguments for physicalism and maintains I) that all causally relevant special science properties must be realized by physical ones, and II) that all special science laws must reduce to physical ones, apart from the significant category of special laws that result from selection processes. Part II defends a teleological theory of representation and an identity theory of consciousness. Part III defends reliabilism and applies it to inductive scepticism and (...) also defends a fictionalist account of mathematics. (shrink)

Alex Oliver; XI*—A Few More Remarks on Logical Form, Proceedings of the Aristotelian Society, Volume 99, Issue 1, 1 June 1999, Pages 247–272, https://doi.org/10.

Yah boo sucks to the grammer wot we lernt in skool! Grammar (and the bad old traditional logic) says that quantifier phrases such as 'nobody', 'everyone', 'all women', 'some men' and 'a man' are in the same category as names such as 'Milly', 'Molly' and 'Mandy'. So, prior to their first corrective lessons, students are awfully muddled, the first and fundamental problem being the Woozle hunt for somebody called 'nobody'. Hoorah for modern logic and logic teachers! The story used to (...) justify our current logics is entirely fictional. The claims about names and quantifier phrases in English are wildly false. Two of the heroes of modern logic, Russell and Hilbert, make the very mistakes which are falsely blamed on traditional logic. The villain, Meinong, turns out to have been working a different patch. Ideas ascribed to traditional grammar are modern inventions. Neither logicians nor grammarians can be trusted to tell the history of either grammar or logic. (shrink)

This engaging and accessible introduction to the philosophy of language provides an important guide to one of the liveliest and most challenging areas of study in philosophy. Interweaving the historical development of the subject with a thematic overview of the different approaches to meaning, the book provides students with the tools necessary to understand contemporary analytical philosophy. The second edition includes new material on: Chomsky, Wittgenstein and Davidson as well as new chapters on the causal theory of reference, possible worlds (...) semantics and semantic externalism. (shrink)

Starting with Gottlob Frege's foundational theories of sense and reference, Miller provides a useful introduction to the formal logic used in all subsequent philosophy of language. He communicates a sense of active philosophical debate by confronting the views of the early theorists concerned with building systematic theories - such as Frege, Bertrand Russell, and the logical positivists - with the attacks mounted by sceptics - such as W.O. Quine, Saul Kripke, and Ludwig Wittgenstein. This leads to important excursions into related (...) areas of metaphysics, philosophy of mind, and cognitive science that present the more recent attempts to save the notions of sense and meaning by philosophers such as Paul Grice, John Searle, Jerry Fodor, Colin McGinn, and Crispin Wright. Miller then returns to the systematic program by examining the formal theories of Donald Davidson, concluding with a chapter surveying the relevance of philosophy of language to the broader metaphysical debates between realists and anti-realists. Miller's clear, engaged, and coherently structured approach makes Philosophy of Language an ideal text for undergraduate courses. The guides to further reading provided in each chapter help the reader pursue interesting topics further and facilitate using the book in conjunction with primary sources. (shrink)

This engaging and accessible introduction to the philosophy of language provides an important guide to one of the liveliest and most challenging areas of study in philosophy. Interweaving the historical development of the subject with a thematic overview of the different approaches to meaning, the book provides students with the tools necessary to understand contemporary analytical philosophy.

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)

This chapter compares and contrasts Quine’s naturalism with the versions of two post-Quineans on the nature of science, logic, and mathematics. The role of indispensability in the philosophy of mathematics is treated in detail.

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...) She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

For many philosophers not automatically inclined to Platonism, the indispensability argument for the existence of mathematical objectshas provided the best (and perhaps only) evidence for mathematicalrealism. Recently, however, this argument has been subject to attack, most notably by Penelope Maddy (1992, 1997),on the grounds that its conclusions do not sit well with mathematical practice. I offer a diagnosis of what has gone wrong with the indispensability argument (I claim that mathematics is indispensable in the wrong way), and, taking my cue (...) from Mark Colyvan''s (1998) attempt to provide a Quinean account of unapplied mathematics as `recreational'', suggest that, if one approaches the problem from a Quinean naturalist starting point, one must conclude that all mathematics is recreational in this way. (shrink)

I propose a way of formulating scientific laws and magnitude attributions which eliminates ontological commitment to mathematical entities. I argue that science only requires quantitative sentences as thus formulated, and hence that we ought to deny the existence of sets and numbers. I argue that my approach cannot plausibly be extended to the concrete "theoretical" entities of science.

Is conceptual relativity a genuine phenomenon? If so, how is it properly understood? And if it does occur, does it undermine metaphysical realism? These are the questions we propose to address. We will argue that conceptual relativity is indeed a genuine phenomenon, albeit an extremely puzzling one. We will offer an account of it. And we will argue that it is entirely compatible with metaphysical realism. Metaphysical realism is the view that there is a world of objects and properties that (...) is independent of our thought and discourse (including our schemes of concepts) about such a world. Hilary Putnam, a former proponent of metaphysical realism, later gave it up largely because of the alleged phenomenon that he himself has given the label ‘conceptual relativity’. One of the key ideas of conceptual relativity is that certain concepts—including such fundamental concepts as object, entity, and existence—have a multiplicity of different and incompatible uses (Putnam 1987, p. 19; 1988, pp. 110 14). According to Putnam, once we recognize the phenomenon of conceptual relativity we must reject metaphysical realism: The suggestion . . . is that what is (by commonsense standards) the same situation can be described in many different ways, depending on how we use the words. The situation does not itself legislate how words like “object,” “entity,” and “exist” must be used. What is wrong with the notion of objects existing “independently” of conceptual schemes is that there are no standards for the use of even the logical notions apart from conceptual choices.” (Putnam 1988, p. 114) Putnam’s intriguing reasoning in this passage is difficult to evaluate directly, because conceptual [1] relativity is philosophically perplexing and in general is not well understood. In this paper we propose a construal of conceptual relativity that clarifies it considerably and explains how it is possible despite its initial air of paradox. We then draw upon this construal to explain why, contrary to Putnam and others, conceptual relativity does not conflict with metaphysical realism, but in fact comports well with it.. (shrink)

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

When a debate seems intractable, with little agreement as to how one might proceed towards a resolution, it is understandable that philosophers should consider whether something might be amiss with the debate itself. Famously in the last century, philosophers of various stripes explored in various ways the possibility that at least certain philosophical debates are in some manner deficient in sense. Such moves are no longer so much in vogue. For one thing, the particular ways they have been made have (...) themselves undergone much critical scrutiny, so that many philosophers now feel that there is, for example, a Quinean response to Carnap, a Gricean reply to Austin, and a diluting proliferation of Wittgenstein interpretations.2 Be that as it may,3 there do of.. (shrink)