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Aristotelian realist philosophy of mathematics holds that mathematics studies properties such as symmetry, quantity, continuity and order that can be realized in the physical world (or in any other world there might be). It contrasts with Platonist realism in holding that the objects of mathematics, such as numbers, do not exist in an abstract world but can be physically realized. It contrasts with nominalism, fictionalism and logicism in holding that mathematics is not about mere names or methods of inference or calculation but about certain real aspects of the world. Aristotelian realists emphasize applied mathematics, especially mathematical modeling, rather than pure mathematics, as the most philosophically central parts of mathematics. The category also includes Aristotle's own philosophy of mathematics and its Thomist developments.

Key works Franklin 2014 is a recent version of Aristotelian realism, arguing that mathematics is the science of quantity and structure. While Aristotelianism was rare in 20th-century philosophy of mathematics, versions of it were revived in Bigelow 1988 and  Maddy 1990. The main Aristotelian view of numbers, as relations between heaps and unit-making universals, is due to Kessler 1980, while Armstrong 1991 gives an Aristotelian account of sets. Frege's influential argument that numbers cannot be properties of physical reality is addressed from an Aristotelian perspective by Irvine 2010 and Katz 2023 .
Introductions Franklin 2022 introduces and surveys the range of Aristotelian options in the philosophy of mathematics. Bostock 2012 introduces Aristotle's philosophy of mathematics. Maurer 1993 introduces Thomist views of mathematics.
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  1. Weyl and Two Kinds of Potential Domains.Laura Crosilla & Øystein Linnebo - forthcoming - Noûs.
    According to Weyl, “‘inexhaustibility’ is essential to the infinite”. However, he distinguishes two kinds of inexhaustible, or merely potential, domains: those that are “extensionally determinate” and those that are not. This article clarifies Weyl's distinction and explains its enduring logical and philosophical significance. The distinction sheds lights on the contemporary debate about potentialism, which in turn affords a deeper understanding of Weyl.
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  2. Thomistic Foundations for Moderate Realism about Mathematical Objects.Ryan Miller - forthcoming - In Proceedings of the Eleventh International Thomistic Congress. Rome: Urbaniana University Press.
    Contemporary philosophers of mathematics are deadlocked between two alternative ontologies for numbers: Platonism and nominalism. According to contemporary mathematical Platonism, numbers are real abstract objects, i.e. particulars which are nonetheless “wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal.” While this view does justice to intuitions about numbers and mathematical semantics, it leaves unclear how we could ever learn anything by mathematical inquiry. Mathematical nominalism, by contrast, holds that numbers do not exist extra-mentally, which raises difficulties about how mathematical statements could be (...)
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  3. How Do I Know That I Know Nothing? The Axiom of Selection and the Arithmetic of Infinity.Matheus Pereira Lobo - 2024 - Open Journal of Mathematics and Physics 6:288.
    We show that the statement "I only know that I know nothing," attributed to the Greek philosopher Socrates, contains, at its core, Zermelo's Axiom of Selection and the arithmetic of the infinite cardinal aleph-0.
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  4. Abstraction and grounding.Louis deRosset & Øystein Linnebo - 2023 - Philosophy and Phenomenological Research 109 (1):357-390.
    The idea that some objects are metaphysically “cheap” has wide appeal. An influential version of the idea builds on abstractionist views in the philosophy of mathematics, on which numbers and other mathematical objects are abstracted from other phenomena. For example, Hume's Principle states that two collections have the same number just in case they are equinumerous, in the sense that they can be correlated one‐to‐one:. The principal aim of this article is to use the notion of grounding to develop this (...)
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  5. Why Aristotle Can’t Do without Intelligible Matter.Emily Katz - 2023 - Ancient Philosophy Today 5 (2):123-155.
    I argue that intelligible matter, for Aristotle, is what makes mathematical objects quantities and divisible in their characteristic way. On this view, the intelligible matter of a magnitude is a sensible object just insofar as it has dimensional continuity, while that of a number is a plurality just insofar as it consists of indivisibles that measure it. This interpretation takes seriously Aristotle's claim that intelligible matter is the matter of mathematicals generally – not just of geometricals. I also show that (...)
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  6. Does Frege Have Aristotle's Number?Emily Katz - 2023 - Journal of the American Philosophical Association 9 (1):135-153.
    Frege argues that number is so unlike the things we accept as properties of external objects that it cannot be such a property. In particular, (1) number is arbitrary in a way that qualities are not, and (2) number is not predicated of its subjects in the way that qualities are. Most Aristotle scholars suppose either that Frege has refuted Aristotle's number theory or that Aristotle avoids Frege's objections by not making numbers properties of external objects. This has led some (...)
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  7. Aristotle on the Objects of Natural and Mathematical Sciences.Joshua Mendelsohn - 2023 - Ancient Philosophy Today 5 (2):98-122.
    In a series of recent papers, Emily Katz has argued that on Aristotle's view mathematical sciences are in an important respect no different from most natural sciences: They study sensible substances, but not qua sensible. In this paper, I argue that this is only half the story. Mathematical sciences are distinctive for Aristotle in that they study things ‘from’, ‘through’ or ‘in’ abstraction, whereas natural sciences study things ‘like the snub’. What this means, I argue, is that natural sciences must (...)
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  8. Thomas Aquinas’ Mathematical Realism.Jean W. Rioux - 2023 - Cham: Springer Verlag.
    In this book, philosopher Jean W. Rioux extends accounts of the Aristotelian philosophy of mathematics to what Thomas Aquinas was able to import from Aristotle’s notions of pure and applied mathematics, accompanied by his own original contributions to them. Rioux sets these accounts side-by-side modern and contemporary ones, comparing their strengths and weaknesses.
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  9. Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics.James Franklin - 2022 - Foundations of Science 27 (2):327-344.
    There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...)
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  10. 'Aristotle's Intermediates and Xenocrates' Mathematicals'.Phillip Sidney Horky - 2022 - Revue de Philosophie Ancienne 40 (1):79-112.
    This paper investigates the identity and function of τὰ μεταξύ in Aristotle and the Early Academy by focussing primarily on Aristotle’s criticisms of Xenocrates of Chalcedon, the third scholarch of Plato’s Academy and Aristotle’s direct competitor. It argues that a number of passages in Aristotle’s Metaphysics (at Β 2, Μ 1-2, and Κ 12) are chiefly directed at Xenocrates as a proponent of theories of mathematical intermediates, despite the fact that Aristotle does not mention Xenocrates there. Aristotle complains that the (...)
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  11. Generality Explained.Øystein Linnebo - 2022 - Journal of Philosophy 119 (7):349-379.
    What explains the truth of a universal generalization? Two types of explanation can be distinguished. While an ‘instance-based explanation’ proceeds via some or all instances of the generalization, a ‘generic explanation’ is independent of the instances, relying instead on completely general facts about the properties or operations involved in the generalization. This intuitive distinction is analyzed by means of a truthmaker semantics, which also sheds light on the correct logic of quantification. On the most natural version of the semantics, this (...)
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  12. Matter and Mathematics: An Essentialist Account of Laws of Nature.Andrew Younan (ed.) - 2022 - Washington, D.C.: The Catholic University of America Press.
    To borrow a phrase from Galileo: What does it mean that the story of the creation is "written in the language of mathematics?" This book is an attempt to understand the natural world, its consistency, and the ontology of what we call laws of nature, with a special focus on their mathematical expression. It does this by arguing in favor of the Essentialist interpretation over that of the Humean and Anti-Humean accounts. It re-examines and critiques Descartes' notion of laws of (...)
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  13. Aristotle on Geometrical Potentialities.Naoya Iwata - 2021 - Journal of the History of Philosophy 59 (3):371-397.
    This paper examines Aristotle's discussion of the priority of actuality to potentiality in geometry at Metaphysics Θ9, 1051a21–33. Many scholars have assumed what I call the "geometrical construction" interpretation, according to which his point here concerns the relation between an inquirer's thinking and a geometrical figure. In contrast, I defend what I call the "geometrical analysis" interpretation, according to which it concerns the asymmetrical relation between geometrical propositions in which one is proved by means of the other. His argument as (...)
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  14. What could mathematics be for it to function in distinctively mathematical scientific explanations?Marc Lange - 2021 - Studies in History and Philosophy of Science Part A 87 (C):44-53.
    Several philosophers have suggested that some scientific explanations work not by virtue of describing aspects of the world’s causal history and relations, but rather by citing mathematical facts. This paper investigates what mathematical facts could be in order for them to figure in such “distinctively mathematical” scientific explanations. For “distinctively mathematical explanations” to be explanations by constraint, mathematical language cannot operate in science as representationalism or platonism describes. It can operate as Aristotelian realism describes. That is because Aristotelian realism enables (...)
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  15. Immanent Structuralism: A Neo-Aristotelian Account of Mathematics.Alfredo Watkins - 2021 - Dissertation, University of North Carolina Chapel Hill
    The aim of this dissertation is to propose and defend a position in the philosophy of mathematics called “immanent structuralism.” This can be contrasted with the standard Platonist view in the philosophy of mathematics, which holds that mathematics studies a unique category of non-physical, abstract entities. Platonism immediately leads to the epistemological problem of how we can know about these entities if they are not part of the physical world. By contrast, immanent structuralism holds that the things mathematics studies are (...)
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  16. The First Principles of Mathematics in the Light of St. Thomas Aquinas.Eduardo Bernot - 2020 - Dissertation, Universitat Abat Oliba Ceu
    Already as far back as 1934, Peter Hoenen complained that the Scholastics, hardly loyal to Aristotle, had almost entirely neglected the philosophy of mathematics, while several modern schools (logicism, intuitionism, and formalism) vigorously cultivated it. What Hoenen did not know is that the moderns, too, would almost completely abandon this important branch of knowledge in the aftermath of the foundational crisis that was brewing due to the introduction of set theory. Alas, the question is yet to be settled: almost a (...)
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  17. Die aristotelische Konzeption der Mathematik.Ulrich Felgner - 2020 - In Philosophie der Mathematik in der Antike und in der Neuzeit. Cham: Birkhäuser. pp. 27-43.
    Aristotelês (Ἀριστοτέλης) wurde 384 v.u.Z. in Stageira (im Grenzgebiet zwischen Thrakien und Makedonien) geboren. Er trat 367 in Platons „Akademie“ ein und blieb ihr Mitglied bis zu Platons Tod im Jahre 348/347. Im Jahre 343 wurde er am makedonischen Königshof Lehrer des damals 13-jährigen Alexander (später ;der Große‘ genannt). 336 kehrte er nach Athen zurück und wurde schon bald darauf Leiter des Lykeions. Zum Gebäude gehörte eine Wandelhalle (Peripatos, περίπατος). Die Mitglieder dieser Schule wurden daher „Peripatetiker“ genannt (περιπατέω = herumgehen).
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  18. Continuity and Mathematical Ontology in Aristotle.Keren Wilson Shatalov - 2020 - Journal of Ancient Philosophy 14 (1):30-61.
    In this paper I argue that Aristotle's understanding of mathematical continuity constrains the mathematical ontology he can consistently hold. On my reading, Aristotle can only be a mathematical abstractionist of a certain sort. To show this, I first present an analysis of Aristotle's notion of continuity by bringing together texts from his Metaphysica and Physica, to show that continuity is, for Aristotle, a certain kind of per se unity, and that upon this rests his distinction between continuity and contiguity. Next (...)
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  19. Thomas Aquinas and Some Thomists on the Nature of Mathematics.David Svoboda & Prokop Sousedik - 2020 - Review of Metaphysics 73 (4):715-740.
    The authors explicate Aquinas's conception of mathematics. They show that in his work the Aristotelian conception is prevalent, according to which this discipline is—together with physics and metaphysics—a theoretical science, whose subject is the study of real quantity and its necessary properties. But, alongside this dominant and prevalent conception, Aquinas's work contains a number of indications that cast doubt. These sparse and rather marginal reflections lead the authors to conclude that Aquinas's texts contain a "constructivist" conception of mathematics in rudimentary (...)
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  20. Euclid’s Kinds and (Their) Attributes.Benjamin Wilck - 2020 - History of Philosophy & Logical Analysis 23 (2):362-397.
    Relying upon a very close reading of all of the definitions given in Euclid’s Elements, I argue that this mathematical treatise contains a philosophical treatment of mathematical objects. Specifically, I show that Euclid draws elaborate metaphysical distinctions between substances and non-substantial attributes of substances, different kinds of substance, and different kinds of non-substance. While the general metaphysical theory adopted in the Elements resembles that of Aristotle in many respects, Euclid does not employ Aristotle’s terminology, or indeed, any philosophical terminology at (...)
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  21. Aristotle’s argument from universal mathematics against the existence of platonic forms.Pieter Sjoerd Hasper - 2019 - Manuscrito 42 (4):544-581.
    In Metaphysics M.2, 1077a9-14, Aristotle appears to argue against the existence of Platonic Forms on the basis of there being certain universal mathematical proofs which are about things that are ‘beyond’ the ordinary objects of mathematics and that cannot be identified with any of these. It is a very effective argument against Platonism, because it provides a counter-example to the core Platonic idea that there are Forms in order to serve as the object of scientific knowledge: the universal of which (...)
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  22. Geometrical Objects as Properties of Sensibles: Aristotle’s Philosophy of Geometry.Emily Katz - 2019 - Phronesis 64 (4):465-513.
    There is little agreement about Aristotle’s philosophy of geometry, partly due to the textual evidence and partly part to disagreement over what constitutes a plausible view. I keep separate the questions ‘What is Aristotle’s philosophy of geometry?’ and ‘Is Aristotle right?’, and consider the textual evidence in the context of Greek geometrical practice, and show that, for Aristotle, plane geometry is about properties of certain sensible objects—specifically, dimensional continuity—and certain properties possessed by actual and potential compass-and-straightedge drawings qua quantitative and (...)
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  23. Aristotle's Mathematicals in Metaphysics M.3 and N.6.Andrew Younan - 2019 - Journal of Speculative Philosophy 33 (4):644-663.
    Aristotle ends Metaphysics books M–N with an account of how one can get the impression that Platonic Form-numbers can be causes. Though these passages are all admittedly polemic against the Platonic understanding, there is an undercurrent wherein Aristotle seems to want to explain in his own terms the evidence the Platonist might perceive as supporting his view, and give any possible credit where credit is due. Indeed, underlying this explanation of how the Platonist may have formed his impression, we discover (...)
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  24. The Mixed Mathematical Intermediates.Emily Katz - 2018 - Plato Journal 18:83-96.
    In Metaphysics B.2 and M.2, Aristotle gives a series of arguments against Platonic mathematical objects. On the view he targets, mathematicals are substances somehow intermediate between Platonic forms and sensible substances. I consider two closely related passages in B2 and M.2 in which he argues that Platonists will need intermediates not only for geometry and arithmetic, but also for the so-called mixed mathematical sciences, and ultimately for all sciences of sensibles. While this has been dismissed as mere polemics, I show (...)
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  25. Mathematical Substances in Aristotle’s Metaphysics B.5: Aporia 12 Revisited.Emily Katz - 2018 - Archiv für Geschichte der Philosophie 100 (2):113-145.
    : Metaphysics B considers two sets of views that hypostatize mathematicals. Aristotle discusses the first in his B.2 treatment of aporia 5, and the second in his B.5 treatment of aporia 12. The former has attracted considerable attention; the latter has not. I show that aporia 12 is more significant than the literature suggests, and specifically that it is directly addressed in M.2 – an indication of its importance. There is an immediate problem: Aristotle spends most of M.2 refuting the (...)
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  26. Philosophy’s Loss of Logic to Mathematics: An Inadequately Understood Take-Over.Woosuk Park - 2018 - Cham, Switzerland: Springer Verlag.
    This book offers a historical explanation of important philosophical problems in logic and mathematics, which have been neglected by the official history of modern logic. It offers extensive information on Gottlob Frege’s logic, discussing which aspects of his logic can be considered truly innovative in its revolution against the Aristotelian logic. It presents the work of Hilbert and his associates and followers with the aim of understanding the revolutionary change in the axiomatic method. Moreover, it offers useful tools to understand (...)
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  27. Early Modern Mathematical Principles and Symmetry Arguments.James Franklin - 2017 - In Franklin J. W. (ed.), The Idea of Principles in Early Modern Thought Interdisciplinary Perspectives. Routledge. pp. 16-44.
    The leaders of the Scientific Revolution were not Baconian in temperament, in trying to build up theories from data. Their project was that same as in Aristotle's Posterior Analytics: they hoped to find necessary principles that would show why the observations must be as they are. Their use of mathematics to do so expanded the Aristotelian project beyond the qualitative methods used by Aristotle and the scholastics. In many cases they succeeded.
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  28. Abstraction and Diagrammatic Reasoning in Aristotle’s Philosophy of Geometry.Justin Humphreys - 2017 - Apeiron 50 (2):197-224.
    Aristotle’s philosophy of geometry is widely interpreted as a reaction against a Platonic realist conception of mathematics. Here I argue to the contrary that Aristotle is concerned primarily with the methodological question of how universal inferences are warranted by particular geometrical constructions. His answer hinges on the concept of abstraction, an operation of “taking away” certain features of material particulars that makes perspicuous universal relations among magnitudes. On my reading, abstraction is a diagrammatic procedure for Aristotle, and it is through (...)
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  29. Aristoteles’in Matematik Felsefesi ve Matematik Soyut­lama.Murat Kelikli - 2017 - Beytulhikme An International Journal of Philosophy 7 (2):33-49.
    Although there are many questions to be asked about philosophy of mathematics, the fundamental questions to be asked will be questions about what the mathematical object is in view of being and what the mathematical reasoning is in view of knowledge. It is clear that other problems will develop in parallel within the framework of the answers to these questions. For this rea­ son, when we approach Aristotle's philosophy of mathematics over these two basic problems, we come up with the (...)
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  30. An Aristotelian Realist Philosophy of Mathematics by James Franklin. [REVIEW]Alex Koo - 2016 - Mathematical Intelligencer 38:81-84.
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  31. Beyond Platonism and Nominalism?: James Franklin: An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure[REVIEW]Vassilis Livanios - 2016 - Axiomathes 26 (1):63-69.
    Review of James Franklin: An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure, Palgrave Macmillan, 2014, x + 308 pp.
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  32. Review of An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure[REVIEW]William Lane Craig - 2015 - Philosophia Christi 17 (1):225-230.
    James Franklin aspires to a realist view of mathematical objects as concrete, rather than abstract, objects. It is shown that he fails to carry out his program but is forced to revert to Platonism.
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  33. Do we see numbers?: James Franklin: An Aristotelian realist philosophy of mathematics[REVIEW]James Davies - 2015 - Metascience 24 (3):483-486.
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  34. An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure by James Franklin. [REVIEW]Jude P. Dougherty - 2015 - Review of Metaphysics 68 (3):658-660.
  35. James Franklin: An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure. [REVIEW]Peter Forrest - 2015 - Studia Neoaristotelica 12 (1):105-109.
    This paper is a book review of "An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure" by James Franklin.
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  36. An Aristotelian approach to mathematical ontology.Donald Gillies - 2015 - In E. Davis & P. Davis (eds.), Mathematics, Substance and Surmise. Springer. pp. 147–176.
    The paper begins with an exposition of Aristotle’s own philosophy of mathematics. It is claimed that this is based on two postulates. The first is the embodiment postulate, which states that mathematical objects exist not in a separate world, but embodied in the material world. The second is that infinity is always potential and never actual. It is argued that Aristotle’s philosophy gave an adequate account of ancient Greek mathematics; but that his second postulate does not apply to modern mathematics, (...)
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  37. Aristotle on mathematical objects.Janine Gühler - 2015 - Dissertation, University of St Andrews
    My thesis is an exposition and defence of Aristotle’s philosophy of mathematics. The first part of my thesis is an exposition of Aristotle’s cryptic and challenging view on mathematics and is based on remarks scattered all over the corpus aristotelicum. The thesis’ central focus is on Aristotle’s view on numbers rather than on geometrical figures. In particular, number is understood as a countable plurality and is always a number of something. I show that as a consequence the related concept of (...)
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  38. Review of An Aristotelian Realist Philosophy of Mathematics[REVIEW]Max Jones - 2015 - Philosophia Mathematica 23 (2):281-288.
    In An Aristotelian Realist Philosophy of Mathematics Franklin develops a tantalizing alternative to Platonist and nominalist approaches by arguing that at least some mathematical universals exist in the physical realm and are knowable through ordinary methods of access to physical reality. By offering a third option that lies between these extreme all-or-nothing approaches and by rejecting the ‘dichotomy of objects into abstract and concrete’, Franklin provides potential solutions to many of these traditional problems and opens up a whole new terrain (...)
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  39. Semi-Platonist Aristotelianism: Review of James Franklin, An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure[REVIEW]Catherine Legg - 2015 - Australasian Journal of Philosophy 93 (4):837-837.
    This rich book differs from much contemporary philosophy of mathematics in the author’s witty, down to earth style, and his extensive experience as a working mathematician. It accords with the field in focusing on whether mathematical entities are real. Franklin holds that recent discussion of this has oscillated between various forms of Platonism, and various forms of nominalism. He denies nominalism by holding that universals exist and denies Platonism by holding that they are concrete, not abstract - looking to Aristotle (...)
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  40. Aristotelian finitism.Tamer Nawar - 2015 - Synthese 192 (8):2345-2360.
    It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has (...)
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  41. An Aristotelian Realist Philosophy of Mathematics: Mathematics as the science of quantity and structure.James Franklin - 2014 - London and New York: Palgrave MacMillan.
    An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...)
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  42. Toward a Neoaristotelian Inherence Philosophy of Mathematical Entities.Dale Jacquette - 2014 - Studia Neoaristotelica 11 (2):159-204.
    The fundamental idea of a Neoaristotelian inherence ontology of mathematical entities parallels that of an Aristotelian approach to the ontology of universals. It is proposed that mathematical objects are nominalizations especially of dimensional and related structural properties that inhere as formal species and hence as secondary substances of Aristotelian primary substances in the actual world of existent physical spatiotemporal entities. The approach makes it straightforward to understand the distinction between pure and applied mathematics, and the otherwise enigmatic success of applied (...)
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  43. Mathematical One and Many: Aquinas on Number.David Svoboda & Prokop Sousedik - 2014 - The Thomist 78 (3):401-418.
  44. Aristotle and euclid's postulates.Fabio Acerbi - 2013 - Classical Quarterly 63 (2):680-685.
    Book 1 of Euclid's Elements opens with a set of unproved assumptions: definitions, postulates, and ‘common notions’. The common notions are general rules validating deductions that involve the relations of equality and congruence. The attested postulates are five in number, even if a part of the manuscript tradition adds a sixth, almost surely spurious, that in some manuscripts features as the ninth, and last, common notion. The postulates are called αἰτήματα both in the manuscripts of the Elements and in the (...)
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  45. Quantity and number.James Franklin - 2013 - In Daniel Novotný & Lukáš Novák (eds.), Neo-Aristotelian Perspectives in Metaphysics. London: Routledge. pp. 221-244.
    Quantity is the first category that Aristotle lists after substance. It has extraordinary epistemological clarity: "2+2=4" is the model of a self-evident and universally known truth. Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity.
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  46. Aristotle’s Critique of Platonist Mathematical Objects: Two Test Cases from Metaphysics M 2.Emily Katz - 2013 - Apeiron 46 (1):26-47.
    Books M and N of Aristotle's Metaphysics receive relatively little careful attention. Even scholars who give detailed analyses of the arguments in M-N dismiss many of them as hopelessly flawed and biased, and find Aristotle's critique to be riddled with mistakes and question-begging. This assessment of the quality of Aristotle's critique of his predecessors (and of the Platonists in particular), is widespread. The series of arguments in M 2 (1077a14-b11) that targets separate mathematical objects is the subject of particularly strong (...)
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  47. Indispensability Without Platonism.Anne Newstead & James Franklin - 2013 - In Alexander Bird, Brian David Ellis & Howard Sankey (eds.), Properties, Powers and Structures: Issues in the Metaphysics of Realism. New York: Routledge. pp. 81-97.
    According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of metaphysics, 'Aristotelian (...)
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  48. Aristotle's philosophy of mathematics.David Bostock - 2012 - In Christopher John Shields (ed.), The Oxford Handbook of Aristotle. Oxford University Press USA. pp. 465.
    Much of Aristotle's thought developed in reaction to Plato's views, and this is certainly true of his philosophy of mathematics. To judge from his dialogue, the Meno, the first thing that struck Plato as an interesting and important feature of mathematics was its epistemology: in this subject we can apparently just “draw knowledge out of ourselves.” Aristotle certainly thinks that Plato was wrong to “separate” the objects of mathematics from the familiar objects that we experience in this world. His main (...)
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  49. Aristotle on Mathematical Truth.Phil Corkum - 2012 - British Journal for the History of Philosophy 20 (6):1057-1076.
    Both literalism, the view that mathematical objects simply exist in the empirical world, and fictionalism, the view that mathematical objects do not exist but are rather harmless fictions, have been both ascribed to Aristotle. The ascription of literalism to Aristotle, however, commits Aristotle to the unattractive view that mathematics studies but a small fragment of the physical world; and there is evidence that Aristotle would deny the literalist position that mathematical objects are perceivable. The ascription of fictionalism also faces a (...)
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  50. Aristotelianism in the Philosophy of Mathematics.James Franklin - 2011 - Studia Neoaristotelica 8 (1):3-15.
    Modern philosophy of mathematics has been dominated by Platonism and nominalism, to the neglect of the Aristotelian realist option. Aristotelianism holds that mathematics studies certain real properties of the world – mathematics is neither about a disembodied world of “abstract objects”, as Platonism holds, nor it is merely a language of science, as nominalism holds. Aristotle’s theory that mathematics is the “science of quantity” is a good account of at least elementary mathematics: the ratio of two heights, for example, is (...)
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