This paper compares the statement ‘Mathematics is the study of structure’ with the actual practice of mathematics. We present two examples from contemporary mathematical practice where the notion of structure plays different roles. In the first case a structure is defined over a certain set. It is argued firstly that this set may not be regarded as a structure and secondly that what is important to mathematical practice is the relation that exists between the structure and the set. In the (...) second case, from algebraic topology, one point is that an object can be a place in different structures. Which structure one chooses to place the object in depends on what one wishes to do with it. Overall the paper argues that mathematics certainly deals with structures, but that structures may not be all there is to mathematics. (shrink)

A number of examples of studies from the field ‘The Philosophy of Mathematical Practice’ (PMP) are given. To characterise this new field, three different strands are identified: an agent-based, a historical, and an epistemological PMP. These differ in how they understand ‘practice’ and which assumptions lie at the core of their investigations. In the last part a general framework, capturing some overall structure of the field, is proposed.

Some forms of analytic reconstructivism take natural language (and common sense at large) to be ontologically opaque: ordinary sentences must be suitably rewritten or paraphrased before questions of ontological commitment may be raised. Other forms of reconstructivism take the commitment of ordinary language at face value, but regard it as metaphysically misleading: common-sense objects exist, but they are not what we normally think they are. This paper is an attempt to clarify and critically assess some common limits of these two (...) reconstructivist strategies. (shrink)

. This paper identifies two aspects of the structuralist position of S. Shapiro which are in conflict with the actual practice of mathematics. The first problem follows from Shapiros identification of isomorphic structures. Here I consider the so called K-group, as defined by A. Grothendieck in algebraic geometry, and a group which is isomorphic to the K-group, and I argue that these are not equal. The second problem concerns Shapiros claim that it is not possible to identify objects in a (...) structure except through the relations and functions that are defined on the structure in which the object has a place. I argue that, in the case of the definition of the so called direct image of a function, it is possible to individuate objects in structures. (shrink)

In an addendum to Naming and Necessity, Saul Kripke argues against his earlier view that Sherlock Holmes is a possible person. In this paper, I suggest a nonstandard interpretation of the addendum. A key feature of this non-standard interpretation is that it attempts to make sense of why Kripke would be rejecting the view that Sherlock Holmes is a possible person without asserting that it is not the case that Sherlock Holmes is a possible person.

In a series of recent publications, Jeffrey King (The nature and structure of content, 2007; Proc Aristot Soc 109(3):257–277, 2009; Philos Stud, 2012) argues for a view on which propositions are facts. He also argues against views on which propositions are set-theoretical objects, in part because such views face Benacerraf problems. In this paper, we argue that, when it comes to Benacerraf problems, King’s view doesn’t fare any better than its set-theoretical rivals do. Finally, we argue that his view faces (...) a further Benacerraf problem, one that threatens to undercut his explanation of why propositions have truth-conditions. If correct, our arguments undercut King’s main motivation for accepting his view over its rivals. (shrink)

I attempt to accommodate the phenomenon of vagueness with classical logic and bivalence. I hold that for any vague predicate there is a sharp cut-off between the things that satisfy it and the things that do not; I claim that this is due to the greater naturalness of one of the candidate meanings of that predicate. I extend the thought to the problem of the many and Benacerraf cases. I go on to explore the idea that it is ontically indeterminate (...) what the most natural meanings are, and hence ontically indeterminate where the sharp cut-off in a sorites series is. (shrink)

It is commonly assumed that facts would be complex entities made out of particulars and universals. This thesis, which I call Compositionalism, holds that parthood may be construed broadly enough so that the relation that holds between a fact and the entities it ‘ties’ together counts as a kind of parthood. I argue firstly that Compositionalism is incompatible with the possibility of certain kinds of fact and universal, and, secondly, that such facts and universals are possible. I conclude that Compositionalism (...) is false. What all these kinds of fact and universal have in common is a violation of supplementation principles governing any relation that may be intelligibly regarded as a kind of parthood. Although my arguments apply to Compositionalism generally, I focus on recent work by David Armstrong, who is a prominent and explicit Compositionalist. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

James Ladyman has recently proposed a view according to which all that exists on the level of microphysics are structures "all the way down". By means of a comparative reading of structuralism in philosophy of mathematics as proposed by Stewart Shapiro, I shall present what I believe structures could not be. I shall argue that, if Ladyman is indeed proposing something as strong as suggested here, then he is committed to solving problems that proponents of structuralism in philosophy of mathematics (...) such as Shapiro are trying to solve. Attempting to do so, however, brings out a tacit tension in Ladyman's position. I shall argue that the upshot of this is that the ontological import that Ladyman attributes to structures is rather epistemological import properly understood. (shrink)

Platonists affirm the existence of abstract mathematical objects, and Nominalists deny the existence of abstract mathematical objects. While there are standard arguments in favor of Nominalism, these arguments fail to account for the necessity of Nominalism. Furthermore, these arguments do nothing to explain why Nominalism is true. They only point to certain theoretical vices that might befall the Platonist. The goal of this paper is to formulate and defend a simple, valid argument for the necessity of Nominalism that seeks to (...) precisify the widespread intuition that mathematical objects are somehow ‘spooky’ or ‘mysterious’. (shrink)

(1988). Platonic explanation: Or, what abstract entities can do for you. International Studies in the Philosophy of Science: Vol. 3, No. 1, pp. 51-67. doi: 10.1080/02698598808573324.

Two fundamental rules of reasoning are Universal Generalisation and Existential Instantiation. Applications of these rules involve stipulations such as ‘Let n be an arbitrary number’ or ‘Let John be an arbitrary Frenchman’. Yet the semantics underlying such stipulations are far from clear. What, for example, does ‘n’ refer to following the stipulation that n be an arbitrary number? In this paper, we argue that ‘n’ refers to a number—an ordinary, particular number such as 58 or 2,345,043. Which one? We do (...) not and cannot know, because the reference of ‘n’ is fixed arbitrarily. Underlying this proposal is a more general thesis: Arbitrary Reference : It is possible to fix the reference of an expression arbitrarily. When we do so, the expression receives its ordinary kind of semantic-value, though we do not and cannot know which value in particular it receives. Our aim in this paper is defend AR. In particular, we argue that AR can be used to provide an account of instantial reasoning, and we suggest that AR can also figure in offering new solutions to a range of difficult philosophical puzzles. (shrink)

It is often possible to know what a speaker intends to communicate without knowing what they intend to say. In such cases, speakers need not intend to say anything at all. Stanley and Szabó's influential survey of possible analysis of quantifier domain restriction is, therefore, incomplete and the arguments made by Clapp and Buchanan against Truth Conditional Compositionality and propositional speaker-meaning are flawed. Two theories should not always be viewed as incompatible when they associate the same utterance with different propositions, (...) as there may be many ways to interpret speakers that are compatible with their intentions. (shrink)

This paper summarises the contributions to our Topical Collection on indeterminacy and underdetermination. The collection includes papers in ethics, metaethics, logic, metaphysics, epistemology, philosophy of science, philosophy of language and philosophy of computation.

ABSTRACTA probability model is underdetermined when there is no rational reason to assign a particular infinitesimal value as the probability of single events. Pruss claims that hyperreal probabilities are underdetermined. The claim is based upon external hyperreal-valued measures. We show that internal hyperfinite measures are not underdetermined. The importance of internality stems from the fact that Robinson’s transfer principle only applies to internal entities. We also evaluate the claim that transferless ordered fields may have advantages over hyperreals in probabilistic modeling. (...) We show that probabilities developed over such fields are less expressive than hyperreal probabilities. (shrink)

This paper considers the attempts put forward by A.N. Whitehead and by Bertrand Russell to ‘construct’ points (and temporal instants) from what they regard as the more basic concept of extended ‘regions’. It is shown how what they each say themselves will not do, and how it should be filled out and amended so that the ‘construction’ may be regarded as successful. Finally there is a brief discussion of whether this ‘construction’ is worth pursuing, or whether it is better—as in (...) today’s mathematics—to prefer a ‘construction’ that goes the other way round, i.e. , to view a region as a set of points. (shrink)

The Nature and Structure of Content is a lucid, stimulating and occasionally frustrating book about the metaphysics of propositions. King is a realist about propositions, and he assumes throughout that a viable theory must individuate them more finely than sets of possible worlds. His aim in the first three chapters is to motivate an account in which propositions have constituent structure, akin to and dependent on the structure of the sentences that express them. The following chapters defend the use of (...) propositions in semantics against a variety of objections, and in the seventh and final chapter King presents a new solution to the old paradox of analysis. With little exception, the arguments in these later chapters are rigorous and compelling, but being largely independent of the account developed in the preceding chapters, the book feels somewhat disjointed. Here I pass over these provocative vignettes to deal with the main action, the account of propositions.According to tradition, propositions are things expressed by declarative sentences, bearers of alethic and modal properties, and objects of belief and other psychological attitudes. The idea that propositions have constituent structure, mirroring somehow the sentences that express them, comes down to us from Frege . More recently, the idea has been adopted by Kaplan and figures prominently in ‘Russellian’ semantic theories of Salmon , Soames and others. The big debate in recent philosophy of language focuses on the identity of propositional constituents: are they senses, as in Frege? Or the things linguistic expressions designate, as in Russellian theories? Though his sympathies plainly lie with the Russellians, King remains officially neutral on this dispute, focusing instead on the question of structure itself.Structured propositions are often represented as …. (shrink)

This paper develops a metaphysically flexible theory of quantification broad enough to incorporate many distinct theories of objects. Quite different, mutually incompatible conceptions of the nature of objects and of reference find representation within it. Some conceptions yield classical first-order logic; some yield weaker logics. Yet others yield notions of validity that are proper extensions of classical logic.

I show that Wittgenstein's critique of G.H. Hardy's mathematical realism naturally extends to Paul Benacerraf's influential paper, ‘Mathematical Truth’. Wittgenstein accuses Hardy of hastily analogizing mathematical and empirical propositions, thus leading to a picture of mathematical reality that is somehow akin to empirical reality despite the many puzzles this creates. Since Benacerraf relies on that very same analogy to raise problems about mathematical ‘truth’ and the alleged ‘reality’ to which it corresponds, his major argument falls prey to the same critique. (...) The problematic pictures of mathematical reality suggested by Hardy and Benacerraf can be avoided, according to Wittgenstein, by disrupting the analogy that gives rise to them. I show why Tarskian updates to our conception of ‘truth’ discussed by Benacerraf do not answer Wittgenstein's concerns. That is, because they merely presuppose what Wittgenstein puts into question, namely, the essential uniformity of ‘truth’ and ‘proposition’ in ordinary discourse. (shrink)

What makes certain definitions fruitful? And how can definitions play an explanatory role? The purpose of this paper is to examine these questions via an investigation of Frege’s treatment of definitions. Specifically, I pursue this issue via an examination of Frege’s views about the scientific unification of logic and arithmetic. In my view, what interpreters have failed to appreciate is that logicism is a project of unification, not reduction. For Frege, unification involves two separate steps: (1) an account of the (...) content expressed by arithmetical claims and (2) the justification of that content. The distinction between these steps allows us to see that there are two notions of definition at play in Frege’s logicist work, viz., one concerned with conceptual analysis, the other concerned with the construction of gap-free proof. I then use this discussion to explain how Frege employs his definitions to defend an epistemological thesis about arithmetic, and to clarify Grundlagen’s fruitfulness condition of definitions, and thereby address two interpretive puzzles from the recent literature. (shrink)

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)

I argue here that a properly Platonic theory of the nature of number is still viable today. By properly Platonic, I mean one consistent with Plato's own theory, with appropriate extensions to take into account subsequent developments in mathematics. At Parmenides 143a-4a the existence of numbers is proven from our capacity to count, whereby I establish as Plato's the theory that numbers are originally ordinal, a sequence of forms differentiated by position. I defend and interpret Aristotle's report of a Platonic (...) distinction between form and mathematical numbers, arguing that mathematical numbers alone are cardinals, by reference to certain non-technical features of a set-theoretical approach and other considerations in philosophy of mathematics. Finally I respond to the objections that such a conception of number was unavailable in antiquity and that this theory is contradicted by Aristotle's report in Metaph . XIII that Platonic numbers are collections of units. I argue that Aristotle reveals his own misinterpretation of the terms in which Plato's theory was expressed. (shrink)

To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on pro- cedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal (...) calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones. (shrink)

After introducing the adoption problem (AP) as the claim that certain basic logical principles cannot be adopted, I offer a characterization of this notion as a two-phase process consisting in (1) the acceptance of a basic logical principle, and (2) the development, in virtue of Phase 1, of a practice of inferring in accordance with that principle. The case of a subject who does not infer in accordance with universal instantiation is considered in detail. I argue that the AP has (...) deep and wide implications for the epistemology of logic, extending well beyond Kripke’s original target, viz. Putnam’s proposal for the empirical revision of logic and its background Quinean epistemology. In particular, the AP questions whether basic logical principles could have a fundamental role in our inferential practices, drawing our attention to the nature of basic inferences and the need to have a clearer conception of them before taking a stand on the matter of the epistemic justification of the logical principles. (shrink)

This article discusses the quiddity of the empty set from its epistemological and linguistic aspects. It consists of four parts. The first part compares the concept of _nihil privativum_ and the empty set in terms of representability, arguing the empty set can be treated as a negative and formal concept. It is argued that, unlike Frege’s definition of zero, the quantitative negation with a full scope is what enables us to represent the empty set conceptually without committing to an antinomy. (...) The second part examines the type and scope of the negation in the concept of _nihil privativum_ and the empty set. In the third part the empty set is interpreted as a rigid abstract general term. The uniqueness of the empty set is explained via a widened version of Kripke’s notion of rigidity. The fourth part proposes a construction for the pure singleton, comparing it with Zermelo’s conception of singletons with the Ur-elements. It is argued that the proposed construction does not face the criterion and ontological inflation problems. The first conclusion of the article is that the empty set can be construed as a negative, formal and unique abstract general term, with quantitative negation full in scope. The second conclusion is that the pure singleton constructed out of the empty set construed in this way overcomes the criterion and ontological inflation problems. (shrink)

Since Benacerraf’s ‘What Numbers Could Not Be, ’ there has been a growing interest in mathematical structuralism. An influential form of mathematical structuralism, modal structuralism, uses logical possibility and second order logic to provide paraphrases of mathematical statements which don’t quantify over mathematical objects. These modal structuralist paraphrases are a useful tool for nominalists and realists alike. But their use of second order logic and quantification into the logical possibility operator raises concerns. In this paper, I show that the work (...) of both these elements can be done by a single natural generalization of the logical possibility operator. (shrink)

Benacerraf has presented two problems for the philosophy of mathematics. These are the problem of identification and the problem of representation. This paper aims to reconstruct the latter problem and to unpack its undermining bearing on the version of Ontic Structural Realism that frames scientific representations in terms of abstract structures. I argue that the dichotomy between mathematical structures and physical ones cannot be used to address the Benacerraf problem but strengthens it. I conclude by arguing that versions of OSR (...) that do not rely on mathematical frameworks for representational purposes need not be vulnerable to Benacerraf’s second problem. (shrink)

My first aim in this paper is to use time diagrams in the style of Brentano to analyze constructions in Brouwer's separable mathematics more precisely. I argue that constructions must involve not only pairing and projecting as basic operations guaranteed by the intuition of twoity, as sometimes assumed in the literature, but also a recalling operation. My second aim is to argue that Brouwer's views on the intuition of twoity and arithmetic lead to an ontological explosion. Redeveloping the constructions of (...) natural numbers and systems sketched in an appendix to Brouwer's Cambridge lectures, I observe that the only plausible way he can make some elementary arithmetic in his separable mathematics is by allowing for the same canonical number to be determined by multiple separable entities, resulting in an overabundant mathematical ontology. (shrink)

Abraham Robinson’s framework for modern infinitesimals was developed half a century ago. It enables a re-evaluation of the procedures of the pioneers of mathematical analysis. Their procedures have been often viewed through the lens of the success of the Weierstrassian foundations. We propose a view without passing through the lens, by means of proxies for such procedures in the modern theory of infinitesimals. The real accomplishments of calculus and analysis had been based primarily on the elaboration of novel techniques for (...) solving problems rather than a quest for ultimate foundations. It may be hopeless to interpret historical foundations in terms of a punctiform continuum, but arguably it is possible to interpret historical techniques and procedures in terms of modern ones. Our proposed formalisations do not mean that Fermat, Gregory, Leibniz, Euler, and Cauchy were pre-Robinsonians, but rather indicate that Robinson’s framework is more helpful in understanding their procedures than a Weierstrassian framework. (shrink)

Cauchy's sum theorem is a prototype of what is today a basic result on the convergence of a series of functions in undergraduate analysis. We seek to interpret Cauchy’s proof, and discuss the related epistemological questions involved in comparing distinct interpretive paradigms. Cauchy’s proof is often interpreted in the modern framework of a Weierstrassian paradigm. We analyze Cauchy’s proof closely and show that it finds closer proxies in a different modern framework.

“Structuralism, Fictionalism, and the Applicability of Mathematics in Science”. This article has two objectives. The first one is to review some of the most important questions in the contemporary philosophy of mathematics: What is the nature of mathematical objects? How do we acquire knowledge about these objects? Should mathematical statements be interpreted differently than ordinary ones? And, finally, how can we explain the applicability of mathematics in science? The debate that guides these reflections is the one between mathematical realism and (...) anti-realism. On the other hand, the second objective is to discuss the arguments that use the applicability of mathematics in science to justify mathematical realism, and show that none of them reaches its aim. To this end, we will distinguish three aspects of the problem of the applicability of mathematics: the utility of mathematics in science, the unexpected utility of some mathematical theories, and the apparent indispensability of mathematics in our best scientific theories - in particular, in our best scientific explanations. Finally, I argue that none of these aspects constitutes a reason to adopt mathematical realism. (shrink)

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Carey has argued that there is a system of core numerical cognition – the analog magnitude system – in which cardinal numbers are explicitly represented in iconic format. While the existence of this system is beyond doubt, this paper aims to show that its representations cannot have the combination of features attributed to them by Carey. According to the argument from abstractness, the representation of the cardinal number of a collection of individuals as such requires the representation of individuals as (...) such, and this in turn requires non-iconic format, from which it is concluded that the explicit representation of the cardinal number of some individuals requires non-iconic representational format. In support of the first premise, an account is given of what approximate cardinal numbers might be, and in support of the second, a direct argument is articulated and defended. Finally, in response to an objection, a second argument for the central thesis is provided. While the discussion is couched in the terms of Carey’s work, the considerations it adduces are perfectly general, and the conclusion should therefore be taken into consideration by all those aiming to characterize the AM system. (shrink)

A response is given here to Benacerraf's (1965) non-uniqueness (or multiple-reductions) objection to mathematical platonism. It is argued that non-uniqueness is simply not a problem for platonism; more specifically, it is argued that platonists can simply embrace non-uniqueness—i.e., that one can endorse the thesis that our mathematical theories truly describe collections of abstract mathematical objects while rejecting the thesis that such theories truly describe unique collections of such objects. I also argue that part of the motivation for this stance is (...) that it dovetails with the correct response to Benacerraf's other objection to platonism, i.e., his (1973) epistemological objection. (shrink)

? Mark BALAGUER Philosophical forum 34:3-43-4, 459-475, Blackwell, 2003.

One recent trend in the philosophy of mathematics has been to approach the central epistemological and metaphysical issues concerning mathematics from the perspective of the applications of mathematics to describing the world, especially within the context of empirical science. A second area of activity is where philosophy of mathematics intersects with foundational issues in mathematics, including debates over the choice of set-theoretic axioms, and over whether category theory, for example, may provide an alternative foundation for mathematics. My central claim is (...) that these latter issues are of direct relevance to philosophical arguments connected to the applicability of mathematics. In particular, the possibility of there being distinct alternative foundations for mathematics blocks the standard argument from the indispensable role of mathematics in science to the existence of specific mathematical objects. (shrink)

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

While developing his theory of world 3, Popper rejects two claims made by Plato: first, that the inhabitants of world 3, ideas, are a source of ultimate explanation, a divine revelation of truth, and second, that these ideas are unchanging. I will rehabilitate the second claim. Man does not construct world 3 by creating his theories, nor is it a source of ultimate truth. Instead, world 3 is discovered by man, and it destroys some of his theories: destructive Platonism. I (...) discuss the impact of the modified position on the Grundlagenstreit in mathematics, taking into account Gillies’s constructive Aristotelianism. I then turn to the philosophy of computers and discuss how destructive Platonism relates to computers and the Internet. In particular, I criticize Popper’s claim that computers are nothing but elaborate pens as too pessimistic and incautious, and the fashionable “singularity” theory as too optimistic and at the same time too fatalistic. In my closing remarks, as an antidote against both views, I provide a more modest and reasonable speculation about the ultimate goal and meaning of life. (shrink)

Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of (...) arithmetic. In doing so, we provide set-theoretical facts which, we believe, are crucial for informed assessment of reductionism. (shrink)

A version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante- rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the ante-rem structuralist (...) fails to explain reference in a way that makes her account different to, and privileged over, that of her eliminativist rivals. Both problems undercut the motivation behind ante-rem structuralism. (shrink)