Journal for the Philosophy of Mathematics
https://riviste.fupress.net/index.php/jpm
<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>en-US<ul> <li class="show">Copyright on any open access article in JPM published by FUP is retained by the author(s).</li> <li class="show">Authors grant FUP a license to publish the article and identify itself as the original publisher.</li> <li class="show">Authors also grant any third party the right to use the article freely as long as its integrity is maintained and its original authors, citation details and publisher are identified.</li> <li class="show">The <a class="is-external" href="http://creativecommons.org/licenses/by/4.0" target="_blank" rel="noopener">Creative Commons Attribution License 4.0</a> formalizes these and other terms and conditions of publishing articles.</li> <li class="show">In accordance with our Open Data policy, the <a class="is-external" href="http://creativecommons.org/publicdomain/zero/1.0/" target="_blank" rel="noopener">Creative Commons CC0 1.0 Public Domain Dedication waiver</a> applies to all published data in JPM open access articles.</li> </ul>leon.horsten@uni-konstanz.de (Leon Horsten)redazione.riviste@fup.unifi.it (Redazione Riviste FUP)Tue, 10 Sep 2024 00:00:00 +0000OJS 3.3.0.14http://blogs.law.harvard.edu/tech/rss60Enumerative Induction in Mathematics
https://riviste.fupress.net/index.php/jpm/article/view/2931
<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>Alan Baker
Copyright (c) 2024 Alan Baker
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https://riviste.fupress.net/index.php/jpm/article/view/2931Tue, 10 Sep 2024 00:00:00 +0000Construction Theorems and Constructive Proofs in Geometry
https://riviste.fupress.net/index.php/jpm/article/view/2932
<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>John B. Burgess
Copyright (c) 2024 John B. Burgess
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https://riviste.fupress.net/index.php/jpm/article/view/2932Tue, 10 Sep 2024 00:00:00 +0000Definiteness in Early Set Theory
https://riviste.fupress.net/index.php/jpm/article/view/2933
<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>Laura Crosilla, Øystein Linnebo
Copyright (c) 2024 Laura Crosilla, Øystein Linnebo
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https://riviste.fupress.net/index.php/jpm/article/view/2933Tue, 10 Sep 2024 00:00:00 +0000Categoricity-like Properties in the First Order Realm
https://riviste.fupress.net/index.php/jpm/article/view/2934
<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>Ali Enayat, Mateusz Łełyk
Copyright (c) 2024 Ali Enayat, Mateusz Łełyk
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https://riviste.fupress.net/index.php/jpm/article/view/2934Tue, 10 Sep 2024 00:00:00 +0000Logical Constants and Unrestricted Quantification
https://riviste.fupress.net/index.php/jpm/article/view/2935
<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>Volker Halbach
Copyright (c) 2024 Volker Halbach
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https://riviste.fupress.net/index.php/jpm/article/view/2935Tue, 10 Sep 2024 00:00:00 +0000How the Continuum Hypothesis could have been a Fundamental Axiom
https://riviste.fupress.net/index.php/jpm/article/view/2936
<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>Joel David Hamkins
Copyright (c) 2024 Joel David Hamkins
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https://riviste.fupress.net/index.php/jpm/article/view/2936Tue, 10 Sep 2024 00:00:00 +0000Aristotle meets Frege: from Potentialism to Frege Arithmetic
https://riviste.fupress.net/index.php/jpm/article/view/2937
<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>Stewart Shapiro
Copyright (c) 2024 Stewart Shapiro
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https://riviste.fupress.net/index.php/jpm/article/view/2937Tue, 10 Sep 2024 00:00:00 +0000Why be a Height Potentialist?
https://riviste.fupress.net/index.php/jpm/article/view/2938
<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>Zeynep Soysal
Copyright (c) 2024 Zeynep Soysal
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https://riviste.fupress.net/index.php/jpm/article/view/2938Tue, 10 Sep 2024 00:00:00 +0000On the Limits of Comparing Subset Sizes within N
https://riviste.fupress.net/index.php/jpm/article/view/2939
<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>Sylvia Wenmackers
Copyright (c) 2024 Sylvia Wenmackers
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https://riviste.fupress.net/index.php/jpm/article/view/2939Tue, 10 Sep 2024 00:00:00 +0000Editorial
https://riviste.fupress.net/index.php/jpm/article/view/2930
Leon Horsten
Copyright (c) 2024 Leon Horsten
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https://riviste.fupress.net/index.php/jpm/article/view/2930Tue, 10 Sep 2024 00:00:00 +0000