Pantheism claims: (1) there exists an all-inclusive unity; and (2) that unity is divine. I review three current and scientifically viable ontologies to see how pantheism can be developed in each. They are: (1) materialism; (2) Platonism; and (3) class-theoretic Pythagoreanism. I show how each ontology has an all-inclusive unity. I check the degree to which that unity is: eternal, infinite, complex, necessary, plentiful, self-representative, holy. I show how each ontology solves the problem of evil (its theodicy) and provides for (...) salvation (its soteriology). I conclude that Platonism and Pythagoreanism have the most divine all-inclusive unities. They support sophisticated contemporary pantheisms. (Published Online February 17 2004). (shrink)

Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to provide an arena (...) for exploring relations and interactions between mathematical fields, their relative strengths, etc. Given the different goals, there is little point to determining a single foundation for all of mathematics. (shrink)

I argue that it follows from a very plausible principle concerning understanding that the truth of an ascription of understanding is context-relative. I use this to defend an account of lexical meaning according to which full understanding of a natural kind term or name requires knowing informative, uniquely identifying information about its referent. This point undermines Putnam-style 'elm-beech' arguments against the description theory of names and natural kind terms.

Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the (...) potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover. (shrink)

The aim of this essay is a criticism of reductionism ? both in its ?static? interpretation (usually referred to as the layer model or level?picture of science) and in its ?dynamic? interpretation (as a theory of the growth of scientific knowledge), with emphasis on the latter ? from the point of view of Popperian fallibilism and Feyerabendian pluralism, but without being committed to the idiosyncrasies of these standpoints. In both aspects of criticism, the rejection is based on the proposal of (...) a global alternative. Hummell and Opp's research programme for the reduction of sociology to psychology is used as a starting?point and taken as the primary object of criticism. Following the introductory Section I, Section II analyses the three crucial notions of Hummell and Opp's research programme ? their explications of the notions of ?sociology?, ?psychology? and the concept of reduction itself ? and criticizes the authors? deficient ?logic of reduction?. Although the ?local? shortcomings of our authors? ?logic of reduction? do not affect reductionism as such, i.e. logically sound versions of reductionism as devised by Kemeny, Nagel, Oppenheim, Putnam, Woodger et al., it is argued that the logical soundness of sophisticated reductionism cannot compensate for its additional epistemological and methodological deficiencies. Section III analyses the ?dynamic? interpretation of reductionism as a particular developmental pattern of scientific growth. It is argued that even reductionism at its best can produce only cumulative progress, thus ?a priori? excluding scientific revolutions which are inevitably counter?inductive as well as counter?reductive. Section IV discusses the philosophical background of modern reductionism, and examines the effects both of reductionism and of anti?reductionistic pluralism on the autonomy of scientific fields. It is argued that pluralistic anti?reductionism undermines spurious claims for autonomy much more effectively than reductionism. As a ?local? improvement of the reductionistic research programme, the replacement of the predominant one?way reductionism by a less restrictive many?way reductionism is proposed. It is argued that the appropriate treatment for an allegedly backward science (say sociology) is not its reduction to an allegedly more advanced science (say psychology) but its non?reductive replacement by new theories (of the same or of another field) that do not incorporate the older ones. As a ?global? alternative to the reduction of sociology to psychology, the frontier?crossing direct application of psychological theories to sociological phenomena is proposed. A plea is made for a pluralistic science without reduction, based on intra? and interscientific criticism as the proper method for the advancement of knowledge. (shrink)

One possible interpretation of the species concept is that specifies are natural kinds. Another species concept is that species are individuals whose parts are organisms. Philip Kitcher takes seriously both these ideas; he sees a role for the genealogical/historical conception and also for the one that is “purely qualitative”. I criticize his ideas here. I see the genealogical conception at work in biological discussion of species and it is presupposed by an active and inventive research program, but the natural kind (...) concept is not. I also criticize Kitcher’s criticisms against Hull’s position that species are individuals, including Kitcher’s idea that species are sets of organisms, his explanations of why “all swans are white” isn’t law-like, his discussion of the example of multiple origination. Finally, I argue that given the current evolutionary developments, pluralism is the "null hypothesis" that we should attempt to refute. (shrink)

Number words seemingly function both as adjectives attributing cardinality properties to collections, as in Frege’s ‘Jupiter has four moons’, and as names referring to numbers, as in Frege’s ‘The number of Jupiter’s moons is four’. This leads to what Thomas Hofweber calls Frege’s Other Puzzle: How can number words function as modifiers and as singular terms if neither adjectives nor names can serve multiple semantic functions? Whereas most philosophers deny that one of these uses is genuine, we instead argue that (...) number words, like many related expressions, are polymorphic, having multiple uses whose meanings are systematically related via type shifting. (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. 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)

A collection of material on Husserl's Logical Investigations, and specifically on Husserl's formal theory of parts, wholes and dependence and its influence in ontology, logic and psychology. Includes translations of classic works by Adolf Reinach and Eugenie Ginsberg, as well as original contributions by Wolfgang Künne, Kevin Mulligan, Gilbert Null, Barry Smith, Peter M. Simons, Roger A. Simons and Dallas Willard. Documents work on Husserl's ontology arising out of early meetings of the Seminar for Austro-German Philosophy.

In this paper I present a new theory of propositions, according to which propositions are abstract mathematical objects: well-formed formulas together with models. I distinguish the theory from a number of existing views and explain some of its advantages  chief amongst which are the following. On this view, propositions are unified and intrinsically truth-bearing. They are mind- and language-independent and they are governed by logic. The theory of propositions is ontologically innocent. It makes room for an appropriate interface with (...) formal semantics and it does not enforce an overly fine or overly coarse level of granularity. (shrink)

'Philosophy arises through misconceptions of grammar', said Wittgenstein. Few people have believed him, and probably none, therefore, working in the area of the philosophy of mathematics. Yet his assertion is most evidently the case in the philosophy of Set Theory, as this paper demonstrates (see also Rodych 2000). The motivation for twentieth century Set Theory has rested on the belief that everything in Mathematics can be defined in terms of sets [Maddy 1994: 4]. But not only are there notable items (...) which cannot be so defined, including numbers and mereological sums, the very notion of a set, as formalized within this tradition, is based on a series of grammatical confusions. (shrink)

Peter Forrest and D.M. Armstrong have given an argument against a theory of naturalness proposed by David Lewis based on the fact that ordered pairs can be constructed from sets in any of a number of different ways. 1. I think the argument is good, but requires a more thorough defense. Moreover, the argument has important consequences that have not been noticed. I introduce a version of Lewis’s proposal in section one, and then in section two I present and defend (...) my version of the argument. After addressing a worry about my argument in section three, in section four I argue that a similar “argument from arbitrariness” jeopardizes Lewis’s solution to the Kripke/Wittgenstein puzzle of the content of thought. (shrink)

Early sections of the paper develop a view of the natural numbers and a view of counting which are suggested by the remarks of several modern philosophers. Further investigation of these views leads to one of the main theses of the paper: a special kind of quantifier, the "numerical quantifier" is essential to counting. The remainder of the paper suggests the rudiments of a new view of the natural numbers, a view which maintains that numerical quantifiers are one kind of (...) natural number. (shrink)

The paper seeks to answer two new questions about truth and scientific change: What lessons does the phenomenon of scientific change teach us about the nature of truth? What light do recent developments in the theory of truth, incorporating these lessons, throw on problems arising from the prevalence of scientific change, specifically, the problem of pessimistic meta-induction?

Around the turn of the century, Poincare and Hilbert each published an account of geometry that took the discipline to be an implicit definition of its concepts. The terms ‘point’, ‘line’, and ‘plane’ can be applied to any system of objects that satisfies the axioms. Each mathematician found spirited opposition from a different logicist—Russell against Poincare' and Frege against Hilbert— who maintained the dying view that geometry essentially concerns space or spatial intuition. The debates illustrate the emerging idea of mathematics (...) as the science of structure. The article closes with some remarks on structuralist and nonstructuralist themes in Frege's own development of arithmetic. (shrink)

The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) survey of the major, traditional philosophies (...) of mathematics indicating how each is prepared to deal with the present problem. It is shown that (the standard formulations of) some views seem to deny outright that there is a relationship between mathematics and any non-mathematical reality; such philosophies are clearly unacceptable. Other views leave the relationship rather mysterious and, thus, are incomplete at best. The final, more speculative section provides the direction of a positive account. A structuralist philosophy of mathematics is outlined and it is proposed that mathematics applies to reality though the discovery of mathematical structures underlying the non-mathematical universe. (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)

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)

Foundational projects disagree on whether pure and applied mathematics should be explained together. Proponents of unified accounts like neologicists defend Frege’s Constraint (FC), a principle demanding that an explanation of applicability be provided by mathematical definitions. I reconsider the philosophical import of FC, arguing that usual conceptions are biased by ontological assumptions. I explore more reasonable weaker variants — Moderate and Modest FC — arguing against common opinion that ante rem structuralism (and other) views can meet them. I dispel doubts (...) that such constraints are ‘toothless’, showing they both assuage Frege’s original concerns and accommodate neo-logicist intents by dismissing ‘arrogant’ definitions. (shrink)

A thriving literature has developed over logical and mathematical pluralism – i.e. the views that several rival logical and mathematical theories can be equally correct. These have unfortunately grown separate; instead, they both could gain a great deal by a closer interaction. Our aim is thus to present some novel forms of abstractionist mathematical pluralism which can be modeled on parallel ways of substantiating logical pluralism (also in connection with logical anti-exceptionalism). To do this, we start by discussing the Good (...) Company Problem for neo-logicists recently raised by Paolo Mancosu (2016), concerning the existence of rival abstractive definitions of cardinal number which are nonetheless equally able to reconstruct Peano Arithmetic. We survey Mancosu’s envisaged possible replies to this predicament, and suggest as a further path the adoption of some form of mathematical pluralism concerning abstraction principles. We then explore three possible ways of substantiating such pluralism—Conceptual Pluralism, Domain Pluralism, Pluralism about Criteria—showing how each of them can be related to analogous proposals in the philosophy of logic. We conclude by considering advantages, concerns, and theoretical ramifications for these varieties of mathematical pluralism. (shrink)

HellmanGeoffrey ** and ShapiroStewart. **** Mathematical Structuralism. Cambridge Elements in the Philosophy of Mathematics, RushPenelope and ShapiroStewart, eds. Cambridge University Press, 2019. Pp. iv + 94. ISBN 978-1-108-45643-2, 978-1-108-69728-6. doi: 10.1017/9781108582933.

The author of “Evidence, Explanation, Enhanced Indispensability” advances a criticism to the Enhanced Indispensability Argument and the use of Inference to the Best Explanation in order to draw ontological conclusions from mathematical explanations in science. His argument relies on the availability of equivalent though competing explanations, and a pluralist stance on explanation. I discuss whether pluralism emerges as a stable position, and focus here on two main points: whether cases of equivalent explanations have been actually offered, and which ontological consequences (...) should follow from these. (shrink)

Rudolf Carnap’s mature work on the logical reconstruction of scientific theories consists of two components. The first is the elimination of the theoretical vocabulary of a theory in terms of its Ramsification. The second is the reintroduction of the theoretical terms through explicit definitions in a language containing an epsilon operator. This paper investigates Carnap’s epsilon-reconstruction of theories in the context of pure mathematics. The main objective here is twofold: first, to specify the epsilon logic underlying his suggested definition of (...) theoretical terms and a suitable choice semantics for it. Second, to analyze whether Carnap’s approach is compatible with a structuralist conception of mathematics. (shrink)

One way of identifying the beginning of the Investigations is by deciding to regard remark 1, and hence neither the motto nor the Preface but the famous quotation from Augustine, as the real starting point of Wittgenstein’s reflections as developed in this book. One point implicit in this decision is that the notion of a language-game is placed in the foreground of Wittgenstein’s discussion. In a way, the language-game of the builders is Wittgenstein’s paradigm of a language-game – but why (...) is it treated differently from the shopkeeper scene in the sense that the latter is not given a separate number? This question appears particularly urgent in view of the fact that in earlier manuscript versions of the Investigations Wittgenstein did allot a separate number to the shopkeeper scene. Towards the end of this paper I make an attempt to answer that question. But in order to get down to this a number of additional questions are raised focussing on Wittgenstein’s use of central terms like “game”, “operate”, “sample” and drawing on distinctions elaborated in the literature, such as Benacerraf’s distinction between “transitive” and “intransitive” kinds of counting. This reading of the beginning of the Investigations is impregnated with the conviction that later parts of the book can fruitfully be seen as growing out of the remarks immediately following the quotation from Augustine. (shrink)

The paper outlines a novel version of mathematical structuralism related to invariants. The main objective here is twofold: first, to present a formal theory of structures based on the structuralist methodology underlying work with invariants. Second, to show that the resulting framework allows one to model several typical operations in modern mathematical practice: the comparison of invariants in terms of their distinctive power, the bundling of incomparable invariants to increase their collective strength, as well as a heuristic principle related to (...) the search for new invariants. (shrink)

Neo-logicism is, not least in the light of Frege’s logicist programme, an important topic in the current philosophy of mathematics. In this essay, I critically discuss a number of issues that I consider to be relevant for both Frege’s logicism and neo-logicism. I begin with a brief introduction into Wright’s neo-Fregean project and mention the main objections that he faces. In Sect. 2, I discuss the Julius Caesar problem and its possible Fregean and neo-Fregean solution. In Sect. 3, I raise (...) what I take to be a central objection to the position of neo-logicism. In Sect. 4, I attempt to clarify how we should understand Frege’s stipulation that the two sides of an abstraction principle qua contextual definition of a term-forming operator shall be “gleichbedeutend”. In Sect. 5, I consider the options that Frege might have had to establish the analyticity of Hume’s Principle: The number that belongs to the concept F is equal to the number that belongs to the concept G if and only if F and G are equinumerous. Section 6 is devoted to Frege’s two criteria of thought identity. In Sects. 7 and 8, I defend the position of the neo-logicist against an alleged “knock-down argument”. In Sect. 9, I comment on Frege’s description of abstraction in Grundlagen, §64 and the use of the terms “recarving” and “reconceptualization” in the relevant literature on Fregean abstraction and neo-logicism. I argue that Fregean abstraction has nothing to do with the recarving of a sentence content or its decomposition in different ways. I conclude with remarks on global logicism versus local logicisms. (shrink)

The paper argues that no second order theory is ontologically commited to anything beyond what its individual variables range over.

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, namely, ZFC set theory, all mathematical objects are represented by sets, while ordinary, namely, non–set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of (...) basic theorems named after Tietze, Heine, and Weierstrass changes significantly upon the replacement of “second-order representations” by “third-order functions.” We discuss the implications and connections to the reverse mathematics program and its foundational claims regarding predicativist mathematics and Hilbert’s program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations. (shrink)

In 1909, Peirce recorded in a few pages of his logic notebook some experiments with matrices for three-valued propositional logic. These notes are today recognized as one of the first attempts to create non-classical formal systems. However, besides the articles published by Turquette in the 1970s and 1980s, very little progress has been made toward a comprehensive understanding of the formal aspects of Peirce's triadic logic (as he called it). This paper aims to propose a new approach to Peirce's matrices (...) for three-valued propositional logic. We suggested that his logical matrices give rise to three different systems, one of them - which we called P3 - is an original and non-explosive logic. Besides that, we will show that the P3 system can easily be transformed into paraconsistent and paracomplete calculi, adding to it, respectively, unary operators of consistency and intuitionistic negation. We conclude with a discussion about philosophical motivations. (shrink)

I present an account of mental representation based upon the ‘SINBAD’ theory of the cerebral cortex. If the SINBAD theory is correct, then networks of pyramidal cells in the cerebral cortex are appropriately described as representing, or more specifically, as modelling the world. I propose that SINBAD representation reveals the nature of the kind of mental representation found in human and animal minds, since the cortex is heavily implicated in these kinds of minds. Finally, I show how SINBAD neurosemantics can (...) provide accounts of misrepresentation, equivocal representation, twin cases, and Frege cases. (shrink)

Dans cet article je présente une analyse critique de l’approche habituelle de l’identité mathématique qui a son origine dans les travaux de Frege et Russell, en faisant un contraste avec les approches alternatives de Platon et Geach. Je pose ensuite ce problème dans un cadre de la théorie des catégories et montre que la notion d’identité ne peut pas être « internalisée » par les moyens catégoriques standards. Enfin, je présente deux approches de l’identité mathématique plus spécifiques: une avec la (...) fibration catégorielle et l’autre avec des catégories supérieures faibles. (shrink)

Since the linguistic turn, many have taken semantics to guide ontology. Here, I argue that semantics can, at best, serve as a partial guide to ontological commitment. If semantics were to be our guide, semantic data and semantic treatments would need to be taken seriously. Through an examination of plurals and their treatments, I argue that there can be multiple, equally semantically adequate, treatments of a natural language theory. Further, such treatments can attribute different ontological commitments to a theory. Given (...) this, I argue that semantics can fail to deliver determinate ontological commitments and determinate answers to ontological questions, more generally. (shrink)

When young children attempt to locate numbers along a number line, they show logarithmic (or other compressive) placement. For example, the distance between “5” and “10” is larger than the distance between “75” and “80.” This has often been explained by assuming that children have a logarithmically scaled mental representation of number (e.g., Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, 2010; Siegler & Opfer, 2003). However, several investigators have questioned this argument (e.g., Barth & Paladino, 2011; Cantlon, Cordes, Libertus, & Brannon, (...) 2009; Cohen & Blanc-Goldhammer, 2011). We show here that children prefer linear number lines over logarithmic lines when they do not have to deal with the meanings of individual numerals (i.e., number symbols, such as “5” or “80”). In Experiments 1 and 2, when 5- and 6- year-olds choose between number lines in a forced-choice task, they prefer linear to logarithmic and exponential displays. However, this preference does not persist when Experiment 3 presents the same lines without reference to numbers, and children simply choose which line they like best. In Experiments 4 and 5, children position beads on a number line to indicate how the integers 1 100 are arranged. The bead placement of 4- and 5-year-olds is better fit by a linear than by a logarithmic model. We argue that previous results from the number line task may depend on strategies specific to the task. (shrink)

ABSTRACT Historical structuralist views have been ontological. They either deny that there are any mathematical objects or they maintain that mathematical objects are structures or positions in them. Non-ontological structuralism offers no account of the nature of mathematical objects. My own structuralism has evolved from an early sui generis version to a non-ontological version that embraces Quine’s doctrine of ontological relativity. In this paper I further develop and explain this view.

This paper is concerned with the genesis of mathematical knowledge. While some philosophers might argue that mathematics has no real subject matter and thus is not a body of knowledge, I will not try to dissuade them directly. I shall not attempt such a refutation because it seems clear to me that mathematicians do know such things as the Mean Value Theorem, The Fundamental Theorem of Arithmetic, Godel's Theorems, etc. Moreover, this is much more evident to me than any philosophical (...) view of mathematics I know of — including my own. So I am going to take mathematics as my starting point. (shrink)

This is a survey of the concept of continuity. Efforts to explicate continuity have produced a plurality of philosophical conceptions of continuity that have provably distinct expressions within contemporary mathematics. I claim that there is a divide between the conceptions that treat the whole continuum as prior to its parts, and those conceptions that treat the parts of the continuum as prior to the whole. Along this divide, a tension emerges between those conceptions that favor philosophical idealizations of continuity and (...) those that are more readily available for mathematico-scientific use. (shrink)

In recent philosophy of mathematics avariety of writers have presented ``structuralist''views and arguments. There are, however, a number ofsubstantive differences in what their proponents take``structuralism'' to be. In this paper we make explicitthese differences, as well as some underlyingsimilarities and common roots. We thus identifysystematically and in detail, several main variants ofstructuralism, including some not often recognized assuch. As a result the relations between thesevariants, and between the respective problems theyface, become manifest. Throughout our focus is onsemantic and metaphysical issues, (...) including what is orcould be meant by ``structure'' in this connection. (shrink)

Various contributors to recent philosophy of mathematics havetaken Richard Dedekind to be the founder of structuralismin mathematics. In this paper I examine whether Dedekind did, in fact, hold structuralist views and, insofar as that is the case, how they relate to the main contemporary variants. In addition, I argue that his writings contain philosophical insights that are worth reexamining and reviving. The discussion focusses on Dedekind''s classic essay Was sind und was sollen die Zahlen?, supplemented by evidence from Stetigkeit und (...) irrationale Zahlen, his scientific correspondence, and his Nachlaß. (shrink)

I show that any sentence of nth-order (pure or applied) arithmetic can be expressed with no loss of compositionality as a second-order sentence containing no arithmetical vocabulary, and use this result to prove a completeness theorem for applied arithmetic. More specifically, I set forth an enriched second-order language L, a sentence A of L (which is true on the intended interpretation of L), and a compositionally recursive transformation Tr defined on formulas of L, and show that they have the following (...) two properties: (a) in a universe with at least [HEBREW LETTER BET] $_{n-2}$ objects, any formula of nth-order (pure or applied) arithmetic can be expressed as a formula of L, and (b) for any sentence $\ulcorner \phi \urcorner$ of L, $\ulcorner \phi^{Tr} \urcorner$ is a second-order sentence containing no arithmetical vocabulary, and nth $\mathcal{A} \vdash \ulcorner \phi \longleftrightarrow \phi^{Tr} \urcorner$. (shrink)

Simon Evnine’s Making Objects and Events: A Hylomorphic Theory of Artifacts develops amorphic hylomorphism. I critically discuss three of its main themes. One theme is its attempt to do the work of form without forms. A second theme is the requirement that hylomorphs have ‘metabolisms at work’. A third theme is the use of artifacts as the paradigms for hylomorphs. I will raise some criticisms of each of these themes. Although the themes might at first appear disconnected, I believe the (...) third underwrites the first two. So the criticisms of the third theme also bear on the rest. (shrink)