https://riviste.fupress.net/index.php/jpm/issue/feedJournal for the Philosophy of Mathematics2024-09-10T00:00:00+00:00Leon Horstenleon.horsten@uni-konstanz.deOpen Journal Systems<p>The Journal for the Philosophy of Mathematics's (JPM) core mission is to publish high-quality research in the philosophy of mathematics and in the application of formal methods to philosophical problems concerning mathematics. JPM will be happy to consider contributions ranging from technical articles (having, however, clear philosophical motivations and aims) to purely philosophical, formalism-free articles. JPM also welcomes research that departs from traditional philosophical methodology and is interested in submissions employing methods from historical and social sciences, cognitive science, and neuroscience.</p>https://riviste.fupress.net/index.php/jpm/article/view/2930Editorial2024-09-02T13:33:35+00:00Leon Horstenleon.horsten@uni-konstanz.de2024-09-10T00:00:00+00:00Copyright (c) 2024 Leon Horstenhttps://riviste.fupress.net/index.php/jpm/article/view/2931Enumerative Induction in Mathematics2024-09-02T13:35:07+00:00Alan Bakerabaker1@swarthmore.edu<div class="page" title="Page 5"> <div class="layoutArea"> <div class="column"> <p>In my 2007 paper, “Is There a Problem of Induction for Mathematics?” I rejected the idea that enumerative induction has force for mathematical claims. My core argument was based on the fact that we are restricted to examining relatively small numbers, so our samples are always biased, and hence they carry no inductive weight. In recent years, I have come to believe that this argument is flawed. In particular, while arithmetical samples are indeed biased, my new view is that this bias actually strengthens the inductive support that accrues from them. The reason is that small numbers typically provide a more severe test of general arithmetical claims due to the greater frequency of significant properties and boundary cases among such numbers. In this paper, I describe and defend this new view, which a call Positive Bias Pro-Inductivism.</p> </div> </div> </div>2024-09-10T00:00:00+00:00Copyright (c) 2024 Alan Bakerhttps://riviste.fupress.net/index.php/jpm/article/view/2932Construction Theorems and Constructive Proofs in Geometry2024-09-02T13:36:39+00:00John B. Burgessnull@null.com<div class="page" title="Page 29"> <div class="layoutArea"> <div class="column"> <p>Given Tarski’s version of Euclidean straightedge and compass geometry, it is shown how to express construction theorems, and shown that for any purely existential theorem there is a construction theorem implying it. Some related results and open questions are then briefly described.</p> </div> </div> </div>2024-09-10T00:00:00+00:00Copyright (c) 2024 John B. Burgesshttps://riviste.fupress.net/index.php/jpm/article/view/2933Definiteness in Early Set Theory2024-09-02T13:38:15+00:00Laura Crosillanull@null.comØystein Linnebooystein.linnebo@ifikk.uio.no<div class="page" title="Page 53"> <div class="layoutArea"> <div class="column"> <p>The notion of definiteness has played a fundamental role in the early developments of set theory. We consider its role in work of Cantor, Zermelo and Weyl. We distinguish two very different forms of definiteness. First, a condition can be definite in the sense that, given any object, either the condition applies to that object or it does not. We call this intensional definiteness. Second, a condition or collection can be definite in the sense that, loosely speaking, a totality of its instances or members has been circumscribed. We call this extensional definiteness. Whereas intensional definiteness concerns whether an intension applies to objects considered one by one, extensional definiteness concerns the totality of objects to which the intension applies. We discuss how these two forms of definiteness admit of precise mathematical analyses. We argue that two main types of explication of extensional definiteness are available. One is in terms of completability and coexistence (Cantor), the other is based on a novel idea due to Hermann Weyl and can be roughly expressed in terms of proper demarcation. We submit that the two notions of extensional definiteness that emerges from our investigation enable us to identify and understand some of the most important fault lines in the philosophy and foundations of mathematics.</p> </div> </div> </div>2024-09-10T00:00:00+00:00Copyright (c) 2024 Laura Crosilla, Øystein Linnebohttps://riviste.fupress.net/index.php/jpm/article/view/2934Categoricity-like Properties in the First Order Realm2024-09-02T13:40:38+00:00Ali Enayatali.enayat@gu.seMateusz Łełykmlelyk@uw.edu.pl<div class="page" title="Page 79"> <div class="layoutArea"> <div class="column"> <p>By classical results of Dedekind and Zermelo, second order logic imposes categoricity features on Peano Arithmetic and Zermelo-Fraenkel set theory. However, we have known since Skolem’s anti-categoricity theorems that the first order formulations of Peano Arithmetic and Zermelo- Fraenkel set theory (i.e., PA and ZF) are not categorical. Here we investigate various categoricity-like properties (including tightness, solidity, and internal categoricity) that are exhibited by a distinguished class of first order theories that include PA and ZF, with the aim of understanding what is special about canonical foundational first order theories.</p> </div> </div> </div>2024-09-10T00:00:00+00:00Copyright (c) 2024 Ali Enayat, Mateusz Łełykhttps://riviste.fupress.net/index.php/jpm/article/view/2935Logical Constants and Unrestricted Quantification2024-09-02T13:42:38+00:00Volker Halbachvolker.halbach@new.ox.ac.uk<div class="page" title="Page 123"> <div class="layoutArea"> <div class="column"> <p>Variants of the so-called permutation criterion have been used for distinguishing between logical and non-logical operations or expressions. Roughly, an operation is defined as logical if, and only if, it is invariant under arbitrary permutations on every domain. Thus a logical operation behaves on all objects in the same way. An expression is logical if, and only if, the operation expressed by it is logical. I consider a variant of the permutation criterion that eliminates domains: An operation is permutation-invariant if, and only if, it is invariant under arbitrary permutations of the universe. An expression is logical if, and only if, it expresses an operation that is permutation-invariant in this sense. This domain-free definition of the invariance criterion matches definitions of logical consequence without domains where first-order quantifiers are taken to range over all (first-order) objects in all interpretations. Without domains some problems of the invariance criterion disappear. In particular, an operation can behave on all objects of any domain in the same way, while still behaving very differently in each domain. On the criterion without domains, a logical operation always behaves on all objects in the same way, not only on all objects of any given domain.</p> </div> </div> </div>2024-09-10T00:00:00+00:00Copyright (c) 2024 Volker Halbachhttps://riviste.fupress.net/index.php/jpm/article/view/2936How the Continuum Hypothesis could have been a Fundamental Axiom2024-09-02T13:44:13+00:00Joel David Hamkinsjdhamkins@nd.edu<div class="page" title="Page 141"> <div class="layoutArea"> <div class="column"> <p>I describe a simple historical thought experiment showing how we might have come to view the continuum hypothesis as a fundamental axiom, one necessary for mathematics, indispensable even for calculus.</p> </div> </div> </div>2024-09-10T00:00:00+00:00Copyright (c) 2024 Joel David Hamkinshttps://riviste.fupress.net/index.php/jpm/article/view/2937Aristotle meets Frege: from Potentialism to Frege Arithmetic2024-09-02T13:45:49+00:00Stewart Shapiroshapiro.4@osu.edu<div class="page" title="Page 161"> <div class="layoutArea"> <div class="column"> <p>The purpose of this paper is to present a genuinely potentialist account of Frege arithmetic. The (cardinal) numbers are not generated from Hume’s Principle, but rather from more or less standard principles of potentialism. The relevant version of Hume’s Principle is a principle stating a condition for numbers to be identical with each other. Essentially, (HP) tells us what we are generating— cardinal numbers—but the generation does not go through (HP) itself. We also develop an Aristotelian, potentialist set theory—in effect, a theory of hereditarily finite sets—a theory that is definitionally equivalent to Dedekind-Peano arithmetic.</p> </div> </div> </div>2024-09-10T00:00:00+00:00Copyright (c) 2024 Stewart Shapirohttps://riviste.fupress.net/index.php/jpm/article/view/2938Why be a Height Potentialist?2024-09-02T13:47:33+00:00Zeynep Soysalzeynep.soysal@rochester.edu<div class="page" title="Page 195"> <div class="layoutArea"> <div class="column"> <p>According to height potentialism, the height of the universe of sets is “potential” or “indefinitely extensible,” and this is something that a (formal) theory of sets should capture. Height actualism is the rejection of height potentialism: the height of the universe of sets isn’t potential or indefinitely extensible, and our standard non-modal theories of sets don’t need to be supplemented with or reinterpreted in a modal language. In this paper, I examine and (mostly) criticize arguments for height potentialism. I first argue that arguments for height potentialism that appeal to its explanatory powers are unsuccessful. I then argue that the most promising argument for height potentialism involves</p> <p>the claim that height potentialism follows from our intuitive conception of sets. But, as I explain, on the most plausible way of developing this argument from an intuitive conception of sets, it turns out that whether height potentialism or height actualism is true is a verbal dispute, i.e., a matter of what meanings we choose to assign to our set-theoretic expressions. I explain that only pragmatic considerations can settle such a dispute and that these weigh in favor of actualism over potentialism. My discussion is also intended to serve two broader aims: to develop what I take to be the most promising line of argument for height potentialism, and to elaborate the height actualist position in greater detail than is standardly done.</p> </div> </div> </div>2024-09-10T00:00:00+00:00Copyright (c) 2024 Zeynep Soysalhttps://riviste.fupress.net/index.php/jpm/article/view/2939On the Limits of Comparing Subset Sizes within N2024-09-02T13:49:18+00:00Sylvia Wenmackerssylvia.wenmackers@kuleuven.be<div class="page" title="Page 223"> <div class="layoutArea"> <div class="column"> <p>Wereviewandcomparefivewaysofassigningtotallyorderedsizestosubsetsofthenatural numbers: cardinality, infinite lottery logic with mirror cardinalities, natural density, generalised density, and α-numerosity. Generalised densities and α-numerosities lack uniqueness, which can be traced to intangibles: objects that can be proven to exist in ZFC while no explicit example of them can be given. As a sixth and final formalism, we consider a recent proposal by Trlifajová (2024), which we call c-numerosity. It is fully constructive and uniquely determined, but assigns merely partially ordered numerosity values. By relating all six formalisms to each other in terms of the underlying limit operations, we get a better sense of the intrinsic limitations in determining the sizes of subsets of N.</p> </div> </div> </div>2024-09-10T00:00:00+00:00Copyright (c) 2024 Sylvia Wenmackers