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  1. 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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  2. 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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  3. 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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  4. 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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  5. Early Modern Mathematical Principles and Symmetry Arguments.James Franklin - 2017 - In The Idea of Principles in Early Modern Thought Interdisciplinary Perspectives. New York, USA: 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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  6. 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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  7. An Aristotelian Realist Philosophy of Mathematics by James Franklin. [REVIEW]Alex Koo - 2016 - Mathematical Intelligencer 38:81-84.
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  8. 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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  9. 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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  10. Quantity and number.James Franklin - 2014 - In Daniel D. Novotný & Lukáš Novák (eds.), Neo-Aristotelian Perspectives in Metaphysics. New York, USA: 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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  11. 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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  12. Indispensability Without Platonism.Anne Newstead & James Franklin - 2012 - In Alexander Bird, Brian Ellis & Howard Sankey (eds.), Properties, Powers, and Structures: Issues in the Metaphysics of Realism. New York, USA: 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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  13. 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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  14. Aristotle on Mathematical and Eidetic Number.Daniel P. Maher - 2011 - Hermathena 190:29-51.
    The article examines Greek philosopher Aristotle's understanding of mathematical numbers as pluralities of discreet units and the relations of unity and multiplicity. Topics discussed include Aristotle's view that a mathematical number has determinate properties, a contrast between Aristotle and French philosopher René Descartes in terms of their understanding of number and Aristotle's description of ways to understand eidetic numbers.
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  15. Aristotle’s prohibition rule on kind-crossing and the definition of mathematics as a science of quantities.Paola Cantù - 2010 - Synthese 174 (2):225-235.
    The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior (...)
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  16. Extreme Science: Mathematics as the Science of Relations as such.R. S. D. Thomas - 2008 - In Bonnie Gold & Roger Simons (eds.), Proof and Other Dilemmas: Mathematics and Philosophy. Mathematical Association of America. pp. 245.
    This paper sets mathematics among the sciences, despite not being empirical, because it studies relations of various sorts, like the sciences. Each empirical science studies the relations among objects, which relations determining which science. The mathematical science studies relations as such, regardless of what those relations may be or be among, how relations themselves are related. This places it at the extreme among the sciences with no objects of its own (A Subject with no Object, by J.P. Burgess and G. (...)
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  17. Aristotelian Infinity.John Bowin - 2007 - Oxford Studies in Ancient Philosophy 32:233-250.
    Bowin begins with an apparent paradox about Aristotelian infinity: Aristotle clearly says that infinity exists only potentially and not actually. However, Aristotle appears to say two different things about the nature of that potential existence. On the one hand, he seems to say that the potentiality is like that of a process that might occur but isn't right now. Aristotle uses the Olympics as an example: they might be occurring, but they aren't just now. On the other hand, Aristotle says (...)
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  18. Aristotle and modern mathematical theories of the continuum.Anne Newstead - 2001 - In Demetra Sfendoni-Mentzou & James Brown (eds.), Aristotle and Contemporary Philosophy of Science. Peter Lang.
    This paper is on Aristotle's conception of the continuum. It is argued that although Aristotle did not have the modern conception of real numbers, his account of the continuum does mirror the topology of the real number continuum in modern mathematics especially as seen in the work of Georg Cantor. Some differences are noted, particularly as regards Aristotle's conception of number and the modern conception of real numbers. The issue of whether Aristotle had the notion of open versus closed intervals (...)
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  19. Pythagorean powers or a challenge to platonism.Colin Cheyne & Charles R. Pigden - 1996 - Australasian Journal of Philosophy 74 (4):639 – 645.
    The Quine/Putnam indispensability argument is regarded by many as the chief argument for the existence of platonic objects. We argue that this argument cannot establish what its proponents intend. The form of our argument is simple. Suppose indispensability to science is the only good reason for believing in the existence of platonic objects. Either the dispensability of mathematical objects to science can be demonstrated and, hence, there is no good reason for believing in the existence of platonic objects, or their (...)
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  20. The formal sciences discover the philosophers' stone.James Franklin - 1994 - Studies in History and Philosophy of Science Part A 25 (4):513-533.
    The formal sciences - mathematical as opposed to natural sciences, such as operations research, statistics, theoretical computer science, systems engineering - appear to have achieved mathematically provable knowledge directly about the real world. It is argued that this appearance is correct.
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  21. Mathematical necessity and reality.James Franklin - 1989 - Australasian Journal of Philosophy 67 (3):286 – 294.
    Einstein, like most philosophers, thought that there cannot be mathematical truths which are both necessary and about reality. The article argues against this, starting with prima facie examples such as "It is impossible to tile my bathroom floor with regular pentagonal tiles." Replies are given to objections based on the supposedly purely logical or hypothetical nature of mathematics.
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  22. Mathematics, The Computer Revolution and the Real World.James Franklin - 1988 - Philosophica 42:79-92.
    The philosophy of mathematics has largely abandoned foundational studies, but is still fixated on theorem proving, logic and number theory, and on whether mathematical knowledge is certain. That is not what mathematics looks like to, say, a knot theorist or an industrial mathematical modeller. The "computer revolution" shows that mathematics is a much more direct study of the world, especially its structural aspects.
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