Articles
Frege on Sense Identity, Basic Law V, and Analysis
Philip A. Ebert
Philosophia Mathematica published 8 October 2015, 10.1093/philmat/nkv032
A Generic Russellian Elimination of Abstract Objects
Kevin C. Klement
Philosophia Mathematica published 1 October 2015, 10.1093/philmat/nkv031
Frege's Cardinals and Neo-Logicism
Roy T. Cook
Philosophia Mathematica published 29 September 2015, 10.1093/philmat/nkv029
A Problem with the Dependence of Informal Proofs on Formal Proofs
Tanswell, F., 2015-10-09 11:49:01 AM
Derivationists, those wishing to explain the correctness and rigour of informal proofs in terms of associated formal proofs, are generally held to be supported by the success of the project of translating informal proofs into computer-checkable formal counterparts. I argue, however, that this project is a false friend for the derivationists because there are too many different associated formal proofs for each informal proof, leading to a serious worry of overgeneration. I press this worry primarily against Azzouni's derivation-indicator account, but conclude that overgeneration is a major obstacle to a successful account of informal proofs in this direction.When Do Some Things Form a Set?
Hewitt, S., 2015-10-09 11:49:01 AM
This paper raises the question under what circumstances a plurality forms a set, parallel to the Special Composition Question for mereology. The range of answers that have been proposed in the literature are surveyed and criticised. I argue that there is good reason to reject both the view that pluralities never form sets and the view that pluralities always form sets. Instead, we need to affirm restricted set formation. Casting doubt on the availability of any informative principle which will settle which pluralities form sets, the paper concludes by affirming a naturalistic approach to the philosophy of set theory.The Representational Foundations of Computation
Rescorla, M., 2015-10-09 11:49:01 AM
Turing computation over a non-linguistic domain presupposes a notation for the domain. Accordingly, computability theory studies notations for various non-linguistic domains. It illuminates how different ways of representing a domain support different finite mechanical procedures over that domain. Formal definitions and theorems yield a principled classification of notations based upon their computational properties. To understand computability theory, we must recognize that representation is a key target of mathematical inquiry. We must also recognize that computability theory is an intensional enterprise: it studies entities as represented in certain ways, rather than entities detached from any means of representing them.Mathematical Structuralism, Modal Nominalism, and the Coherence Principle
Schwartz, J. S. J., 2015-10-09 11:49:01 AM
According to Stewart Shapiro's coherence principle, structures exist whenever they can be coherently described. I argue that Shapiro's attempts to justify this principle (along with his position, ante rem structuralism) are circular, as he relies on criticisms of modal nominalism which presuppose the coherence principle. I argue further that when the coherence principle is not presupposed, his reasoning more strongly supports modal nominalism than ante rem structuralism.Identity in Homotopy Type Theory, Part I: The Justification of Path Induction
Ladyman, J., Presnell, S., 2015-10-09 11:49:01 AM
Homotopy Type Theory (HoTT) is a proposed new language and foundation for mathematics, combining algebraic topology with logic. An important rule for the treatment of identity in HoTT is path induction, which is commonly explained by appeal to the homotopy interpretation of the theory's types, tokens, and identities as (respectively) spaces, points, and paths. However, if HoTT is to be an autonomous foundation then such an interpretation cannot play a fundamental role. In this paper we give a derivation of path induction, motivated from pre-mathematical considerations, without recourse to homotopy theory.Logic, Essence, and Modality Bob Hale. Necessary Beings: An Essay on Ontology, Modality, & the Relations Between Them. Oxford University Press, 2013. ISBN: 978-0-19-966262-3 (hbk); 978-0-19-874803-8 (pbk); 978-0-19-164834-2 (e-book). Pp. ix + 298
Menzel, C., 2015-10-09 11:49:01 AM
Bernard Bolzano. Theory of Science. Volumes I-IV. Paul Rusnock and Rolf George, trans. Oxford: Oxford University Press, 2014. ISBN: 978-0-19-968438-0. Pp. 2044.
Sebestik, J., 2015-10-09 11:49:01 AM
Pavel Pudlak. Logical Foundations of Mathematics and Computational Complexity: A Gentle Introduction. Springer Monographs in Mathematics. Springer, 2013. ISBN: 978-3-319-00118-0 (hbk); 978-3-319-00119-7 (ebook). Pp. xiv + 695
Urquhart, A., 2015-10-09 11:49:01 AM
Mark van Atten. Essays on Godel's Reception of Leibniz, Husserl, and Brouwer. Logic, Epistemology, and the Unity of Science; 35. Dordrecht: Springer, 2015. ISBN 978-3-319-10030-2 (hbk); 978-3-319-10031-9 (e-book). Pp. xiv + 328
2015-10-09 11:49:01 AM
Gabriele Lolli, Marco Panza, and Giorgio Venturi, eds. From Logic to Practice: Italian Studies in the Philosophy of Mathematics. Boston Studies in the Philosophy and History of Science; 308. Springer, 2015. ISBN 978-3-319-10433-1 (hbk); 978-3-319-10434-8 (e-book). Pp. xix + 336
2015-10-09 11:49:01 AM
Author Index--Volume 23
2015-10-09 11:49:01 AM
A Generic Russellian Elimination of Abstract Objects
Klement, K. C., 2015-10-09 11:49:03 AM
In this paper I explore a position on which it is possible to eliminate the need for postulating abstract objects through abstraction principles by treating terms for abstracta as ‘incomplete symbols’, using Russell's no-classes theory as a template from which to generalize. I defend views of this stripe against objections, most notably Richard Heck's charge that syntactic forms of nominalism cannot correctly deal with non-first-orderizable quantifcation over apparent abstracta. I further discuss how number theory may be developed in a system treating apparent terms for numbers using these definitions.Frege's Cardinals and Neo-Logicism
Cook, R. T., 2015-10-09 11:49:03 AM
Gottlob Frege defined cardinal numbers in terms of value-ranges governed by the inconsistent Basic Law V. Neo-logicists have revived something like Frege's original project by introducing cardinal numbers as primitive objects, governed by Hume's Principle. A neo-logicist foundation for set theory, however, requires a consistent theory of value-ranges of some sort. Thus, it is natural to ask whether we can reconstruct the cardinal numbers by retaining Frege's definition and adopting an alternative consistent principle governing value-ranges. Given some natural assumptions regarding what an acceptable neo-logicistic theory of value-ranges might look like, successfully implementing this alternative approach is impossible.Adam Olszewski, Bartosz Brozek, and Piotr Urbanczyk, eds. Church's Thesis: Logic, Mind and Nature. Krakow, Poland: Copernicus Center Press, 2014. ISBN 978-83-7886-009-9 (hbk). Pp. 431
2015-10-09 11:49:03 AM
Dedekind's Logicism
Klev, A. M., 2015-10-09 11:49:03 AM
A detailed argument is provided for the thesis that Dedekind was a logicist about arithmetic. The rules of inference employed in Dedekind's construction of arithmetic are, by his lights, all purely logical in character, and the definitions are all explicit; even the definition of the natural numbers as the abstract type of simply infinite systems can be seen to be explicit. The primitive concepts of the construction are logical in their being intrinsically tied to the functioning of the understanding.Brouwer's Conception of Truth
Hansen, C. S., 2015-10-09 11:49:03 AM
In this paper it is argued that the understanding of Brouwer as replacing truth conditions with assertability or proof conditions, in particular as codified in the so-called Brouwer-Heyting-Kolmogorov Interpretation, is misleading and conflates a weak and a strong notion of truth that have to be kept apart to understand Brouwer properly: truth-as-anticipation and truth- in-content. These notions are explained, exegetical documentation provided, and semi-formal recursive definitions are given.Aristotelian Continua
Linnebo, O., Shapiro, S., Hellman, G., 2015-10-09 11:49:03 AM
In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems (such as the existence of bisections) that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal language, and we show that the two approaches are equivalent.A Vindication of Logicism
Roeper, P., 2015-10-09 11:49:03 AM
Frege regarded Hume's Principle as insufficient for a logicist account of arithmetic, as it does not identify the numbers; it does not tell us which objects the numbers are. His solution, generally regarded as a failure, was to propose certain sets as the referents of numerical terms. I suggest instead that numbers are properties of pluralities, where these properties are treated as objects. Given this identification, the truth-conditions of the statements of arithmetic can be obtained from logical principles with the help of definitions, just as the logicist thesis maintains.Rota's Philosophy in its Mathematical Context
Gandon, S., 2015-10-09 11:49:03 AM
The goal of this paper is to connect Rota's discussion of the Husserlian notion of Fundierung with Rota's project of giving combinatorics a foundation in his 1964 paper ‘On the foundations of combinatorial theory I’. Section 2 gives the basic tenets of this seminal paper. Sections 3 and 4 spell out the connections made there between Rota's philosophical writings and his mathematical achievements. Section 5 shows how these two developments fit into Rota's analysis of the place of combinatorics in mathematics.Using Mathematics to Explain a Scientific Theory
Friend, M., Molinini, D., 2015-10-09 11:49:03 AM
We answer three questions: 1. Can we give a wholly mathematical explanation of a physical phenomenon? 2. Can we give a wholly mathematical explanation for a whole physical theory? 3. What is gained or lost in giving a wholly, or partially, mathematical explanation of a phenomenon or a scientific theory? To answer these questions we look at a project developed by Hajnal Andréka, Judit Madarász, István Németi and Gergely Székely. They, together with collaborators, present special relativity theory in a three-sorted first-order formal language.Frege Meets Aristotle: Points as Abstracts
Shapiro, S., Hellman, G., 2015-10-09 11:49:03 AM
There are a number of regions-based accounts of space/time, due to Whitehead, Roeper, Menger, Tarski, the present authors, and others. They all follow the Aristotelian theme that continua are not composed of points: each region has a proper part. The purpose of this note is to show how to recapture ‘points’ in such frameworks via Scottish neo-logicist abstraction principles (instead of Whiteheadian ‘extensive abstraction’). The results recapitulate some Aristotelian themes. A second agenda is to provide a new arena to help decide what is at stake when adjudicating issues concerning the identity of neo-logicist abstracts — so-called ‘Caesar questions’.Is Frege's Definition of the Ancestral Adequate
Heck, R. G., 2015-10-09 11:49:03 AM
Why should one think Frege's definition of the ancestral correct? It can be proven to be extensionally correct, but the argument uses arithmetical induction, and that seems to undermine Frege's claim to have justified induction in purely logical terms. I discuss such circularity objections and then offer a new definition of the ancestral intended to be intensionally correct; its extensional correctness then follows without proof. This new definition can be proven equivalent to Frege's without any use of arithmetical induction. This proves, without any use of arithmetical induction, that Frege's definition is extensionally correct and so answers the circularity objections.Bernulf Kanitscheider. Natur und Zahl: Die Mathematisierbarkeit der Welt [Nature and Number: The Mathematizability of the World]. Berlin: Springer Verlag, 2013. ISBN: 978-3-642-37707-5 (hbk); 978-3-642-37708-2 (e-book). Pp. vii + 385
Lane Craig, W., 2015-10-09 11:49:03 AM
Ernesto Damiani, Ottavio D'Antona, Vincenzo Marra, Fabrizio Palombi, eds. From Combinatorics to Philosophy: The Legacy of G.-C. Rota. Springer, 2009. ISBN 978-0-387-88752-4 (hbk); 978-1-4899-8298-8 (pbk); 978-0-387-88753-1 (e-book). Pp. xviii + 260
2015-10-09 11:49:03 AM
Dedekind and Cassirer on Mathematical Concept Formation
Yap, A., 2015-10-09 11:49:03 AM
Dedekind's major work on the foundations of arithmetic employs several techniques that have left him open to charges of psychologism, and through this, to worries about the objectivity of the natural-number concept he defines. While I accept that Dedekind takes the foundation for arithmetic to lie in certain mental powers, I will also argue that, given an appropriate philosophical background, this need not make numbers into subjective mental objects. Even though Dedekind himself did not provide that background, one can nevertheless be found in the work of the neo-Kantian Ernst Cassirer on mathematical concept formation.Dedekind's Abstract Concepts: Models and Mappings
Sieg, W., Schlimm, D., 2015-10-09 11:49:03 AM
Dedekind's mathematical work is integral to the transformation of mathematics in the nineteenth century and crucial for the emergence of structuralist mathematics in the twentieth century. We investigate the essential components of what Emmy Noether called, his ‘axiomatic standpoint’: abstract concepts (for systems of mathematical objects), models (systems satisfying such concepts), and mappings (connecting models in a structure-preserving way).Mathematical Objects arising from Equivalence Relations and their Implementation in Quine's NF
Forster, T., 2015-10-09 11:49:03 AM
Many mathematical objects arise from equivalence classes and invite implementation as those classes. Set-existence principles that would enable this are incompatible with ZFC's unrestricted aussonderung but there are set theories (e.g., NF and Church's CUS) which admit more instances than does ZF. NF provides equivalence classes for stratified relations only. Church's construction provides equivalence classes for "low" sets, and thus, for example, a set of all (low) ordinals. However, that set has an ordinal in turn which is not a member of the set constructed; so no set of all ordinals is obtained thereby. This "recurrence problem" is discussed.
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