This volume offers a selection of the most interesting and important work from recent years in the philosophy of mathematics, which has always been closely linked to, and has exerted a significant influence upon, the main stream of analytical philosophy. The issues discussed are of interest throughout philosophy, and no mathematical expertise is required of the reader. Contributors include W.V. Quine, W.D. Hart, Michael Dummett, Charles Parsons, Paul Benacerraf, Penelope Maddy, W.W. Tait, Hilary Putnam, George Boolos, Daniel Isaacson, Stewart (...) class='Hi'>Shapiro, and Hartry Field. (shrink)

Skeptical of the arguments put forth in Robert Duncan's long-awaited, post-humously published The H. D. Book, this review essay questions the elevation of Pre-Raphaelite, Aestheticist, and Decadent poetry that forms the basis of Duncan's revisionist canon—a revision in which Wallace Stevens and T. S. Eliot are dismissed as “merely rational,” while H. D. and Duncan himself are elevated to the uppermost ranks, just beneath Ezra Pound. The essay focuses on the peculiarity of “Wardour Street” diction returning to poetry in the (...) postmodern era (especially in the verse of Gjertrud Schnackenberg) and connects this development to the nostalgia evidenced in Duncan's occultism. Placing the New Critics with the poets Karl Shapiro and Richard Wilbur, Duncan dismisses them all as “academics” and “descendants of those ministers of the seventeenth and eighteenth century, holding out against the magic of poetry as once they held out against the magic-religion of witch cults.” Emphasizing that it is the witch cults with which Duncan evidently sides, the essay concludes that Duncan's reranking of poets is no more reliable than his attitude toward rationality. (shrink)

One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...) that, if sound, Hale and Wright’s arguments for Frege’s Constraint at most establish a version on which the relevant application of the naturals is transitive counting – roughly, the counting procedure by which numerals are used to answer “how many”-questions. Second, we show that this version of Frege’s Constraint fails to adjudicate in favor of Hume’s Principle. If this is the version of Frege’s Constraint that a foundation for arithmetic must respect, then Hume’s Principle no more – and no less – meets the requirement than the Dedekind-Peano axioms do. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Nevertheless, some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is more “legitmate” in virtue of being “more basic” or “more fundamental”. This paper addresses two related issues. First, (...) we review some of these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege’s foundationalist epistomology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally “legitimate”. (shrink)

There are multiple formal characterizations of the natural numbers available. Despite being inter-derivable, they plausibly codify different possible applications of the naturals – doing basic arithmetic, counting, and ordering – as well as different philosophical conceptions of those numbers: structuralist, cardinal, and ordinal. Some influential philosophers of mathematics have argued for a non-egalitarian attitude according to which one of those characterizations is ‘more basic’ or ‘more fundamental’ than the others. This paper addresses two related issues. First, we review some of (...) these non-egalitarian arguments, lay out a laundry list of different, legitimate, notions of relative priority, and suggest that these arguments plausibly employ different such notions. Secondly, we argue that given a metaphysical-cum-epistemological gloss suggested by Frege's foundationalist epistemology, the ordinals are plausibly more basic than the cardinals. This is just one orientation to relative priority one could take, however. Ultimately, we subscribe to an egalitarian attitude towards these formal characterizations: they are, in some sense, equally ‘legitimate’. (shrink)

The primary argument against mereological singularism—the view that definite plural noun phrases like ‘the students’ refer to “set-like entities”—is that it is ultimately incoherent. The most forceful form of this charge is due to Barry Schein, who argues that singularists must accept a certain comprehension principle which entails the existence of things having the contradictory property of being both atomic and non-atomic. The purpose of this paper is to defuse Schein’s argument, by noting three necessary and independently motivated restrictions on (...) the metalinguistic predicates ‘atom’ and ‘non-atom’: both are sort, property, and context-relative. With these restrictions in place, Schein’s problematic assumption becomes evident: his presumed singularist analysis of ‘non-atom’ conflates the metalanguage with the meta-metalanguage, i.e. the language used to talk about the metalanguage. (shrink)

Perhaps the most pressing challenge for singularism—the predominant view that definite plurals like ‘the students’ singularly refer to a collective entity, such as a mereological sum or set—is that it threatens paradox. Indeed, this serves as a primary motivation for pluralism—the opposing view that definite plurals refer to multiple individuals simultaneously through the primitive relation of plural reference. Groups represent one domain in which this threat is immediate. After all, groups resemble sets in having a kind of membership-relation and iterating: (...) we can have groups of groups, groups of groups of groups, etc. Yet there cannot be a group of all non-self-membered groups. In response, we develop a potentialist theory of groups according to which we always can, but do not have to, form a group from any sum. Modalizing group-formation makes it a species of potential, as opposed to actual or completed, infinity. This allows for a consistent, plausible, and empirically adequate treatment of natural language plurals, one which is motivated by the iterative nature of syntactic and semantic processes more generally. (shrink)

According to commitment accounts of assertion, asserting is committing oneself to something’s being the case, where such commitment is understood in terms of norms governing a social practice. I elaborate and compare two version of such accounts, liability accounts (associated with C.S. Peirce) and dialectical norm accounts (associated with Robert Brandom), concluding that the latter are more defensible. I argue that both versions of commitment account possess a potential advantage over rival normative accounts of assertion in that they needn’t presuppose (...) any notion of an assertion’s correctness. Additionally, I show how dialectical norm accounts can explain relations between assertion and truth. After setting forth objections that have been raised against commitment accounts, I argue that responses are available on behalf of dialectical norm accounts. Finally, I propose that a liberalized dialectical norm account can illuminate phenomena sometimes seen as supporting truth relativism. (shrink)

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Thought: A Journal of Philosophy, EarlyView.

There is a parallel between the debate between Gottlob Frege and David Hilbert at the turn of the twentieth century and at least some aspects of the current controversy over whether category theory provides the proper framework for structuralism in the philosophy of mathematics. The main issue, I think, concerns the place and interpretation of meta-mathematics in an algebraic or structuralist approach to mathematics. Can meta-mathematics itself be understood in algebraic or structural terms? Or is it an exception to the (...) slogan that mathematics is the science of structure? (shrink)

There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants. Taking a cue from some work in linguistics and (...) the philosophy of language, I suggest that i functions like a parameter in natural deduction systems. This may require some rethinking of the role of singular terms, at least in mathematical languages. (shrink)

Brandom's "inferentialism"—his theory that contentfulness consists in being governed by inferential norms—proves dubiously compatible with his own deflationary approach to intentional objectivity. This is because a deflationist argument, adapted from the case of truth to that of correct inference, undermines the criterion of adequacy Brandom employs in motivating inferentialism. Once that constraint is abandoned, moreover, the very constitutive-explanatory availability of Brandom's inferential norms becomes suspect. Yet Brandom intertwines inferentialism with a separate explanatory project, one that in explaining the pragmatic significance (...) of meaning-attributions does yield a convincing construal of the claim that the concept of meaning is normative. (shrink)

A number of authors have recently weighed in on the issue of whether it is coherent to have bound variables that range over absolutely everything. Prima facie, it is difficult, and perhaps impossible, to coherently state the “relativist” position without violating it. For example, the relativist might say, or try to say, that for any quantifier used in a proposition of English, there is something outside of its range. What is the range of this quantifier? Or suppose we ask the (...) relativist if there are some things that cannot appear in the range of any bound variable. The likely response would be along these lines: “No. For each object o, it possible to include o in the range of quantifiers, but one cannot quantify over everything at once.” This sentence contains unrestricted quantifiers, or so it seems, pending some clever move from a relativist. On the other hand, in the context of set theory, the reasoning behind the Burali-Forti paradox strongly suggests that there are well-orderings strictly longer than the collection of all ordinals. And set theorists regularly do transfinite recursions and transfinite reductions along such well-orderings. The relativist simply points out that one can always define new ordinals, and thus expand the range of one’s bound variables. The purpose of this paper is to explore the iterative framework, proposed in Zermelo’s 1930 paper, “Über Grenzzahlen und Mengenbereiche” (“On boundary numbers and domains of sets”), in order to shed light on these issues, and see what is involved in resolving them. (shrink)

ABSTRACT Recruitment into the scientific community is one oft-stated goal of science education—in the post-Sputnik United States, for example—but this obscures the fact that science textbooks are often read by people who will never be scientists. It cannot be presupposed that science textbooks for younger audiences, students in primary and secondary schools, function in this way. For this reason, precollegiate-level science textbooks are sometimes discussed as a subset of literature popularizing science. The high school science classroom and the textbook are (...) forums for exposing the public to science. The role of governments and educational institutions in regulating the consumption of these texts not only determines which books are used; it influences how they are written, read, and deemed authoritative. Therefore such science textbooks should not be seen as (at best) the disjunction of texts-for-training and books-for-popularization. A changing sense of what “textbooks” are compels a different understanding of their use in the history of science. (shrink)

Saul Kripke once noted that there is a tight connection between computation and de re knowledge of whatever the computation acts upon. For example, the Euclidean algorithm can produce knowledge of _which number_ is the greatest common divisor of two numbers. Arguably, algorithms operate directly on syntactic items, such as strings, and on numbers and the like only via how the numbers are represented. So we broach matters of _notation_. The purpose of this article is to explore the relationship between (...) the notations acceptable for computation, the usual idealizations involved in theories of computability, flowing from Alan Turing’s monumental work, and de re propositional attitudes toward numbers and other mathematical objects. (shrink)

Three essays discuss aspects of Nietzsche's Thus Spake Zarathustra: the place of giftgiving in the portrayed economy, the meaning of feasting and parasitism, and references to the classical myth of Alcyone.

According to relativist accounts of discourse about, e.g., epistemic possibility and matters of taste, the truth of propositions must be relativized to nonstandard parameters. This paper argues that the central thrust of such accounts should be understood independently of relative truth, in terms of a perspectival account of assertoric force. My point of departure is a stripped-down version of Brandom’s analysis of the normative structure of discursive practice. By generalizing that structure, I make room for an analogue of the “assessment (...) sensitivity” MacFarlane characterizes in terms of relative truth. I argue that my reformulation supplies a stronger rationale for the most distinctive feature of MacFarlane’s brand of relativism, its account of when speakers ought to retract assertions. Furthermore, I show that the view usually regarded as a “moderate” alternative to MacFarlane’s “radical” relativism requires the more radical deviation from an absolutist account of assertoric force. (shrink)

The primary argument against mereological singularism—the view that definite plural noun phrases like ‘the students’ refer to “set-like entities”—is that it is ultimately incoherent. The most forceful form of this charge is due to Barry Schein, who argues that singularists must accept a certain comprehension principle which entails the existence of things having the contradictory property of being both atomic and non-atomic. The purpose of this paper is to defuse Schein’s argument, by noting three necessary and independently motivated restrictions on (...) the metalinguistic predicates ‘atom’ and ‘non-atom’: both are sort, property, and context-relative. With these restrictions in place, Schein’s problematic assumption becomes evident: his presumed singularist analysis of ‘non-atom’ conflates the metalanguage with the meta-metalanguage, i.e. the language used to talk about the metalanguage. (shrink)

Most philosophers are familiar with the metaphysical puzzle of the statue and the clay. A sculptor begins with some clay, eventually sculpting a statue from it. Are the clay and the statue one and the same thing? Apparently not, since they have different properties. For example, the clay could survive being squashed, but the statue could not. The statue is recently formed, though the clay is not, etc. Godehart Link 1983’s highly influential analysis of the count/mass distinction recommends that English (...) draws a distinction between uncountable “stuff” and countable “things”. There are two mereological relations, related in specific ways. Our primary question here is whether an empirically adequate account of the mass/count distinction really does require distinguishing “things” from “stuff”, and thus postulating two corresponding mereological relations, or if instead positing only one sort of entity and corresponding mereological relation is sufficient, as other semantic theories would have it. This question is meant to be one of what we call natural language mereology. We are asking about the mereological commitments of English, or perhaps competent speakers of English, and not about ultimate reality as such. There is no pretense that we will definitively solve the metaphysical puzzle of the statue and clay. (shrink)

Computationalism, the notion that cognition is computation, is a working hypothesis of many AI researchers and Cognitive Scientists. Although it has not been proved, neither has it been disproved. In this paper, I give some refutations to some well-known alleged refutations of computationalism. My arguments have two themes: people are more limited than is often recognized in these debates; computer systems are more complicated than is often recognized in these debates. To underline the latter point, I sketch the design and (...) abilities of a possible embodied computer system. (shrink)

A typical interpreted formal language has (first‐order) variables that range over a collection of objects, sometimes called a domain‐of‐discourse. The domain is what the formal language is about. A language may also contain second‐order variables that range over properties, sets, or relations on the items in the domain‐of‐discourse, or over functions from the domain to itself. For example, the sentence ‘Alexander has all the qualities of a great leader’ would naturally be rendered with a second‐order variable ranging over qualities. Similarly, (...) the sentence ‘there is a property that holds of all and only the prime numbers’ has a variable ranging over properties of natural numbers. Third‐order variables range over properties of properties, sets of sets, functions from properties to sets, etc. For example, according to some logicist accounts, the number 4 is the property shared by all properties that apply to exactly four objects in the domain. Accordingly, the number 4 is a third‐order item. Fourth‐order variables, and beyond, are characterized similarly. The phrase ‘higher‐order variable’ refers to the variables beyond first‐order. (shrink)

Jaegwon Kim has argued that if psychological kinds are multiply realizable then no single psychological theory can describe regularities ranging over psychological states. Instead, psychology must be fractured, with human psychology covering states realized in the human way, martian psychology covering states realized in the martian way, and so on. I show that even if one accepts the principles that motivate Kim.

The paradox of the Unexpected Hanging, related prediction paradoxes, and the Sorites paradoxes all involve reasoning about ordered collections of entities: days ordered by date in the case of the Unexpected Hanging; men ordered by the number of hairs on their heads the case of the bald man version of the Sorites. The reasoning then assigns each entity a value that depends on the previously assigned value of one of the neighboring entities. The final result is paradoxical because it conflicts (...) with the obviously correct, commonsensical value. The paradox is due to the serial procedure of assigning a value based on the newly assigned value of the neighbor. An alternative procedure is to assign each value based only on the original values of neighbors - a parallel procedure. That procedure does not give paradoxical answers. (shrink)

Minds are obscure things. This is especially obvious and especially onerous to those interested in understanding the mind. One way to begin an investigation of mind, given its abstruseness, is to explore the implications of something we believe must be true of minds. This is the approach I take in this paper. Whatever uncertainties we have about the mind, it’s a safe bet that the mind is an adaptation. So, I begin with this truth about minds: minds are the product (...) of evolution by natural selection. In what follows, I trace some of the consequences of this fact. Doing so will take us some distance toward answering a variety of questions about the mind. (shrink)

On this, the 97th anniversary of the year of his birth, thoughts turn naturally to Willard Van Orman Quine. Quine, known as ‘Van’ to his friends but ‘That putz with the beret’ to everyone else, was one of the great systematists of the last century. The range of topics he addressed is awesome: epistemology, confirmation, philosophical logic, set theory, analyticity, modality, and, perhaps most familiarly, the indeterminacy of translation. My focus in this, my final and most challenging address as Chair (...) of the Department of Philosophy, will be on this last topic. (shrink)

This paper suggests that young people can explore moral philosophy in ways that will help them both think and act in ways that are consistent with good moral reasoning. It describes several games and exercises that allow children to explore various moral principles in their behavior toward others. Participating in activities that give children practice in making moral decisions helps them to appreciate the role of principles in moral reasoning. The author contends that it is important for young people to (...) examine ethical dilemmas from the “inside out”; that is, not by listening to the wisdom of philosophers telling them how to approach these issues, but by facing them head on themselves. (shrink)

High‐capacity sequencing technologies have dramatically reduced both the cost and time required to generate complete human genome sequences. Besides expanding our knowledge about existing diversity, the nature of these technologies makes it possible to extend knowledge in yet another dimension: time. Recently, the complete genome sequence of a 4,000‐year‐old human from the Saqqaq culture of Greenland was determined to 20‐fold coverage. These data make it possible to investigate the population affinities of this enigmatic culture and, by identifying several phenotypic traits (...) of this individual, provide a limited glimpse into how these people may have looked. While undoubtedly a milestone in ancient DNA research, the cost to generate an ancient genome, even from such an exceptionally preserved specimen, remains out of reach for most. Nonetheless, recently developed DNA capture methods, already applied to Neanderthal and fossil human mitochondrial DNA, may soon make large‐scale genome‐wide analysis of ancient human diversity a reality, providing a fresh look at human population history. (shrink)