THE SYDNEY SCHOOL: MATHEMATICS, THE SCIENCE OF STRUCTURE
| AN ARISTOTELIAN REALIST PHILOSOPHY OF MATHEMATICS
We are a school of philosophers of mathematics centred in Sydney, Australia. Our line is realist (about structure and quantity), but Aristotelian rather than Platonist: we hold that mathematics studies real properties of things such as symmetry and continuity . . .
intro . . .
our manifesto . . .
Wikipedia article . . .
a tutorial on Aristotelian realism. . .
review of other schools in philosophy of maths. . .
The Australian's Higher Ed Supplement article. . .
introductory talk on infinity ...
Philosopher's Zone interview 2010 ... 2014 ...
video interview ... podcast (Mar 2020) ... video intro (8 mins) ... interview (Feb 2021)
Paul Thagard writes "The current philosophy of mathematics that fits best with what is known about minds and science is James Franklin's Aristotelian realism."
| || Explanation of diagram|
Jeremiah Joven Joaquin
|James Franklin's book, An Aristotelian Realist Philosophy of Mathematics, appeared from Palgrave Macmillan in 2014.|
(Philosophia Mathematica's review: extract ... (full text) ... New Criterion's review)
|Our articles and books
J Franklin, Mathematics as a science of non-abstract reality: Aristotelian realist philosophies of mathematics, Foundations of Science 25 (2021) ... (Ed Feser's comment)
- R Miller, Thomistic foundations for moderate realism about mathematical objects, Proceedings of the Eleventh International Thomistic Congress (2021)
- A Watkins, Immanent Structuralism: A Neo-Aristotelian Account of Mathematics,
(thesis, University of North Carolina, Chapel Hill, 2021)
J Franklin, Discrete and continuous: a fundamental dichotomy in mathematics, Journal of Humanistic Mathematics 7 (2017), 355-378
J Franklin, Early modern mathematical principles and symmetry arguments, in The Idea of Principles in Early Modern Thought, ed. P. Anstey, Routledge, 2017, ch. 1
C Legg and J Franklin, Perceiving necessity, Pacific Philosophical Quarterly 98 (2017), 320-343
J Franklin, Logical probability and the strength of mathematical conjectures, Mathematical Intelligencer 38 (3) (Sept 2016), 14-19, ... full text
J Franklin, Global and local, Mathematical Intelligencer 36 (4) (Dec 2014), 4-9 ... full text.
J Franklin, Quantity and number, in Neo-Aristotelian Perspectives in Metaphysics, ed. D.D. Novotny and L. Novak (Routledge, New York and London, 2014), 221-44
James Franklin, The mathematical world, Aeon 7 Apr 2014.
(short popular introduction)
- J Franklin, Non-deductive logic in mathematics: the probability of conjectures, in A. Aberdein and I. Dove, eds, The Argument of Mathematics (Springer, Dordrecht, 2013), 11-29.
- A Newstead and J Franklin, Indispensability without Platonism, in Properties, Powers and Structures, ed. A. Bird, B. Ellis and H. Sankey (Routledge, New York, 2012), 81-97
- J Michell, "The constantly recurring argument": Inferring quantity from order, Theory and Psychology 22 (2012), 255-271
- J Franklin, Aristotelianism in the philosophy of mathematics, Studia Neoaristotelica 8 (2011), 3-15
(short but comprehensive statement of our view)
- A Newstead & J Franklin, The epistemology of geometry I: the problem of exactness, ASCS09: Proceedings of the 9th Conference of the Australasian Society for Cognitive Science, 2010, pp. 254-260
- J Franklin, Aristotelian realism, chapter in The Philosophy of Mathematics, ed. A. Irvine (Handbook of the Philosophy of Science series, North-Holland Elsevier, 2009), 101-153
(fullest statement of our view before the book listed above)
- J Michell, Measurement in Psychology: A critical history of a methodological concept (Cambridge University Press, 1999), ch. 3
- P Forrest and D M Armstrong, The nature of number, Philosophical Studies 16 (1987), 165-186
(classic paper explaining counting numbers as relations between universals and heaps)
- J Franklin, Mathematical Necessity and Reality, Australasian Journal of Philosophy, 67 (1989), 286-294
(argues that mathematical statements can be both necessary and about reality)
- J Franklin, The formal sciences discover the philosophers' stone, Studies in History and Philosophy of Science, 25 (1994), 513-33
(argues that the mathematical sciences like operations research and computer science have achieved the goal of necessary truths about real objects)
- A Newstead and J Franklin,
On the reality of the continuum Philosophy 83 (2008), 117-27
(defends the reality of the continuum understood as an actual infinity)
- A Newstead,
Review of Oppy's Philosophical Perspectives on Infinity Australasian J. of Philosophy 85 (2007), 679-82
- J. Michell, The logic of measurement: A realist overview, Measurement, 38 (2005), 285-294
- A Irvine, Introduction to A. Irvine, ed, Physicalism in Mathematics (1990)
(survey of nominalism, realism and physicalism in mathematics)
- A Heathcote, Unbounded operators and the incompleteness of quantum mechanics, Philosophy of Science, (1990), 523-34
(a form of incompleteness in QM follows from the use of unbounded operators)
- A Heathcote, Quantum heterodoxy: realism at the Planck length, Science and Education, 12 (2003), 513-29
(it is possible to be realist about QM without adopting a hidden-variables interpretation)
- A G J Newstead, Aristotle and modern mathematical theories of the continuum, in Aristotle and Contemporary Science, ed. D. Sfendoni-Mentzou, Lang, New York, 2001, vol. 2
- J Michell, Bertrand Russell's 1897 critique of the traditional theory of measurement, Synthese 110 (1997), 257-276
- C Cheyne & C R Pigden, Pythagorean powers Australasian Journal of Philosophy, 74 (1996), 639-45
(the indispensability argument for the existence of mathematical objects
requires them to have causal powers)
- L Lehrer Dive,
An Epistemic Structuralist Account of Mathematical Knowledge, PhD Thesis, University of Sydney, 2003
- J Franklin, Non-deductive logic in mathematics, British J. for the Philosophy of Science 38 (1987), 1-18
(on the logical status of evidence for unproved conjectures in mathematics)
- An interview: Philosophy, mathematics and structure Philosopher 1 (2) (Winter, 1995), 31-38
- Philpapers category 'Mathematical Aristotelianism'.
- J Franklin, ‘Let no-one ignorant of geometry…’: Mathematical parallels for understanding the objectivity of ethics, Journal of Value Inquiry, 2021
- J Franklin, On the parallel between mathematics and morals, Philosophy 79 (2004), 97-119
(defends the absolute objectivity of both mathematical and ethical truths; winner of the 2005 Eureka Prize for Research in Ethics)
- J Franklin, Last bastion of reason, New Criterion 18 (9) (May 2000), 74-8
(polemical review on Lakatos's philosophy of mathematics)
- A Heathcote, a forthcoming review/essay on complex numbers
- P Forrest, Sets as mereological tropes, Metaphysica 2 (2002), 5-10
- J Franklin & A Daoud, Proof in Mathematics: An Introduction
(textbook on proof from an Aristotelian perspective)
- J Franklin, Two caricatures II: Leibniz's best world, International Journal for Philosophy of Religion 52 (2002), 45-56
(defence of Leibniz's solution to the problem of evil in terms of mathematical restrictions on local and global structure)
- J Franklin, Chapter on 'Artifice and the natural world: Mathematics, logic, technology' in
Cambridge History of Eighteenth Century Philosophy
(full account of eighteenth century philosophy of mathematics and logic)
- J Franklin, Achievements and fallacies in Hume's account of infinite divisibility,
20 (1994), 85-101.
(Hume was right to say we can't prove space is infinitely divisible, but wrong
to claim it isn't)
- J Franklin, Corrupting the Youth: a history of philosophy in Australia
(contains a chapter on Australian philosophy of science and mathematics)
- J Franklin, Resurrecting logical probability, Erkenntnis 55 (2001), 277-305
(philosophy of probability considered as logic)
Philosophy of mathematics resources we recommend
- David Armstrong, Truth and Truthmakers
(a defence of Aristotelian realism in general, including applications to numbers and sets)
- John Bigelow, The Reality of Numbers
(an Australian book defending an Aristotelian realism about numbers)
John Baez's pages
(a must for anyone interested in the role of mathematics in physics)
- David Corfield's site
"The n-Category Cafe"
- Stanford Encyclopedia of Philosophy article on
indispensability arguments in mathematics
- Doron Zeilberger's
defence of finitism; Norman Wildberger's attack on infinities.
(all mathematics is finite: crazy but interesting)
- Philosophia Mathematica
(the journal of philosophy of mathematics)
- R W Hamming, The unreasonable effectiveness of mathematics
(a leading applied mathematician reflects on the applicability of mathematics)
- Dale Jacquette, Toward a Neoaristotelian inherence philosophy of mathematical entities, Studia Neoaristotelica 11 (2) (2014)
(another Aristotelian approach to the philosophy of mathematics)
- Donald Gillies, An Aristotelian approach to mathematical ontology, in Mathematics, Substance and Surmise, 2015.
- Keith Hossack, Knowledge and the Philosophy of Number
(largely Aristotelian philosophy of numbers - cardinal, ratio and ordinal) (review)
- Bob Knapp, Mathematics is About the World (2014)
(book based on Ayn Rand's version of Aristotelianism)
- Vladislav Shaposhnikov, The applicability problem and a naturalistic perspective on mathematics (2014)
(article reflecting on prospects for an Aristotelian applied-mathematics-centred philosophy of mathematics)
- Marc Lange, What could mathematics be for it to function in distinctively mathematical scientific explanations?
(shows the advantages of Aristotelian realism in mathematical explanation)
- Eduardo Bernot, The First Principles of Mathematics in the Light of St. Thomas Aquinas
(thesis, Universitat Abat Oliba CEU, 2020)
- Wikipedia's list of philosophers of mathematics
- Institute for the Study of Nature, devoted to Aristotelian philosophy of science
- Macquarie University's Centre for Research in Mathematics and Science Education has a project on Pattern & structure in early mathematics learning
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