Notes 1 The Aristotelian Realist Point of View 1. The small number of works in that direction are surveyed and compared with the present work in the last section of Chapter 7 below. 2. Introductions to realist views on universals in J.P. Moreland, Universals (Acumen, Chesham, 2001), ch. 1; C. Swoyer, ‘Properties’, Stanford Encyclopedia of Philosophy (1999, revised 2011), http://plato.stanford.edu/entries/ properties/. 3. D.M. Armstrong, A World of States of Affairs (Cambridge University Press, Cambridge, 1997). 4. Moreland, Universals, 28–30; D.M. Armstrong, Universals: An Opinionated Introduction (Westview Press, Boulder, CO, 1989), ch. 1. 5. Also called ‘objects Platonism’ (G. Hellman, Mathematics Without Numbers, Oxford University Press, Oxford, 1989), 3; ‘standard Platonism’ (C. Cheyne and C.R. Pigden, Pythagorean powers, Australasian Journal of Philosophy, 74 (1996), 639–645); ‘full-blooded Platonism’ (M. Balaguer, Platonism and Anti-Platonism in Mathematics, Oxford University Press, Oxford, 1998), 3, 5; G. Restall, Just what is full-blooded Platonism? Philosophia Mathematica, 11 (2003), 82–91); ‘ontological Platonism’ (M. Steiner, Platonism and the causal theory of knowledge, Journal of Philosophy 70 (1973), 57–66); ‘tradi- tional Platonism’ (P. Maddy, Realism in Mathematics, Oxford University Press, Oxford, 1990, 21); it is probably not Plato’s Platonism: discussion in M. Panza and A. Sereni, Plato’s Problem: An Introduction to Mathematical Platonism (Palgrave Macmillan, Basingstoke, 2013), 17–26. 6. D.M. Armstrong, Universals and Scientific Realism (Cambridge University Press, Cambridge, 1978), 66–68; M. Balaguer, Platonism in metaphysics, Stanford Encyclopedia of Philosophy (2004, revised 2009), http://plato.stanford. edu/entries/platonism/. 7. Formulation in F. MacBride, Can ante rem structuralism solve the access problem? Philosophical Quarterly 58 (2008), 155–164; the problem introduced in philosophy of mathematics classically in P. Benacerraf, Mathematical truth, Journal of Philosophy 70 (1973), 661–679; discussion in Balaguer, Platonism in metaphysics, section 5. 8. J.R. Brown, Philosophy of Mathematics: A Contemporary Introduction to the World of Proofs and Pictures (2nd edn, Routledge, New York, 2008), ch. 3. 9. G. Rosen, Abstract objects, Stanford Encyclopedia of Philosophy (2001), http:// plato.stanford.edu/entries/abstract-objects/. 10. Armstrong, Universals and Scientific Realism, vol. 2, ch. 19. 11. J.R. Weinberg, Abstraction, Relation, Induction (University of Wisconsin Press, Madison, 1965), part 2; S.E. Glenn, Relations and Reality: The Metaphysics of 263 264 Notes Parts and Wholes (PhD, Boston College, 2000); D. Odegard, Locke and the unreality of relations, Theoria 35 (1969), 147–152. 12. B. Russell, Mysticism and Logic (Allen & Unwin, London, 1917), 75. 13. B. Russell, Introduction to Mathematical Philosophy (Allen & Unwin, London, 1919), ch. 6. 14. J. Bigelow and R. Pargetter, Science and Necessity (Cambridge University Press, Cambridge, 1990), 82–92; D.M. Armstrong, In defence of structural univer- sals, Australasian Journal of Philosophy 64 (1986), 85–88; J. Pagés, Structural universals and formal relations, Synthese 131 (2002), 215–221. 15. From Armstrong; also called ‘individuative property’ (W.v.O. Quine, Word and Object, MIT Press, Cambridge, Mass, 1960, 90); ‘sortal property’ (in the sense of Strawson, ‘A sortal universal supplies a principle for distinguishing and counting individual particulars which it collects’, Individuals, London, 1959, 168); see R.E. Grandy, Sortals, Stanford Encyclopedia of Philosophy (2006, revised 2007), http://plato.stanford.edu/entries/sortals/. 16. Armstrong, Truth and Truthmakers, 113–115. 17. Armstrong, Truth and Truthmakers, 118–123, building on D. Lewis, Parts of Classes (Blackwell, Oxford, 1991), ch. 1. 18. Armstrong, Truth and Truthmakers, ch. 10; D.M. Armstrong, What is a Law of Nature? (Cambridge, Cambridge University Press, 1983), part II; Bigelow and Pargetter, Science and Necessity, chs 5–6. 19. N. Goodman, Fact, Fiction and Forecast (4th edn Harvard University Press, Cambridge, Mass., 1983), 40–41; discussion in G. Molnar, Powers: A Study in Metaphysics (Oxford University Press, Oxford, 2003), ch. 10. 20. Details in D.M. Armstrong, Belief, Truth and Knowledge (Cambridge University Press, Cambridge, 1973), ch. 12. 21. J.Y. Lettvin, H.R. Maturana, W.S. McCulloch and W.H. Pitts, What the frog’s eye tells the frog’s brain, Proceedings of the Institute of Radio Engineers 47 (1959), 1950–1961; discussion on whether this is belief in F. Dretske, Perception, Knowledge and Belief: Selected Essays (Cambridge University Press, Cambridge, 2000), 67–73. 22. E.g. M.A. Schmuckler, Visual-proprioceptive intermodal perception in infancy, Infant Behavior and Development 19 (1996), 221–232. 23. J.J. Gibson, The Senses Considered as Perceptual Systems (Houghton Mifflin, Boston, 1966). 24. E.g. R.L. Gregory, The Intelligent Eye (Weidenfeld & Nicolson, London, 1970). 25. E.g. P. Geach, Mental Acts: Their Content and Their Objects (Routledge & Kegan Paul, London, 1957), 18–44. 26. Introductory survey in P.C. Quinn and J. Oates, Early category representa- tion and concepts, in J. Oates and A. Grayson, eds, Cognitive and Language Development in Children (Blackwell, Malden, MA, 2004), 21–60. 27. Aristotle, Posterior Analytics, bk I, ch. 13. 28. Aristotle, De Anima, bk III, ch. 5; R.D. McKirahan, Principles and Proofs: Aristotle’s Theory of Demonstrative Knowledge (Princeton University Press, Princeton, NJ, 1992), ch. 18. 29. Works such as J. Maritain, Distinguish to Unite: Or, The Degrees of Knowledge (G. Bles, London, 1959) and B. Lonergan, Insight: A Study of Human Understanding (Philosophical Library, New York, 1957) draw various distinctions within Notes 265 intellectual knowledge, but do not deal substantially with sense knowledge and how it gives rise to intellectual knowledge. 2 Uninstantiated Universals and ‘Semi-Platonist’ Aristotelianism 1. M. Balaguer, Fictionalism in the philosophy of mathematics, Stanford Encyclopedia of Philosophy (2008, revised 2011), http://plato.stanford.edu/ entries/fictionalism-mathematics/ 2. S. Shapiro, Philosophy of Mathematics: Structure and Ontology (Oxford University Press, Oxford, 1997), 86. 3. W.E. Johnson, Logic, Part I (Cambridge University Press, Cambridge, 1921), ch. 11; survey in D.H. Sanford, Determinates vs determinables, Stanford Encyclopedia of Philosophy (2002, revised 2011), http://plato.stanford.edu/ entries/determinate-determinables/. 4. The structure of the space of colours surveyed in J. Cohen, On the structural properties of the colours, Australasian Journal of Philosophy 81 (2003), 78–95; further in Chapter 9 below. 5. Hume’s example of the ‘missing shade of blue’ (Treatise of Human Nature, ed. L.A. Selby-Bigge, 2nd rev. edn, Clarendon Press, Oxford, 1975, 6) concerns epistemology (how can our imagination fill in an unexperienced shade of blue which lies between two experienced ones?), but the example is adapted here to ontology. 6. D.M. Armstrong, Universals: An Opinionated Introduction (Westview Press, Boulder, CO, 1989), 75–82. 7. B. Mundy, The metaphysics of quantity, Philosophical Studies 51 (1987), 29–54; Mundy calls his position ‘naturalistic Platonism’, but it is identical to Aristotelian realism with uninstantiated universals. This is not the same position as ‘naturalized Platonism’, which holds there that a naturalized epis- temology can allow for knowledge of abstract objects: M. Balaguer, Against (Maddian) naturalized Platonism, Philosophia Mathematica 2 (1994), 97–108; B. Linsky and E.N. Zalta, Naturalized Platonism versus Platonized naturalism, Journal of Philosophy 92 (1995), 525–555. 8. B. Ellis, The categorical dimensions of the causal powers, in A. Bird, B.D. Ellis and H. Sankey, eds, Properties, Powers and Structures: Issues in the Metaphysics of Realism (Routledge, New York, 2012), 11–26, section 3. 9. D.M. Armstrong, A Combinatorial Theory of Possibility (Cambridge University Press, Cambridge, 1989), 37. 10. Armstrong, Combinatorial Theory, 125. 11. Armstrong, Combinatorial Theory, 56. 12. Armstrong, Combinatorial Theory, 58–60. 13. D.M. Armstrong, Truth and Truthmakers (Cambridge University Press, Cambridge, 2004), 86–89. 14. It could be called ‘eliminative’ Aristotelianism, on the model of the ‘elimi- native structuralism’ of S. Shapiro, Philosophy of Mathematics: Structure and Ontology (Oxford University Press, New York, 1997), 85–86. 15. M. Colyvan, The Indispensability of Mathematics (Oxford University Press, Oxford, 2001), ch. 3; Ø. Linnebo, Platonism in the philosophy of mathematics, 266 Notes Stanford Encyclopedia of Philosophy (2009, revised 2011), http://plato.stanford. edu/entries/platonism-mathematics/; G. Rosen, Abstract objects, Stanford Encyclopedia of Philosophy (2001, revised 2012), http://plato.stanford.edu/ entries/abstract-objects/ 16. Shapiro, Philosophy of Mathematics, 89. 17. The relevant Stanford Encyclopedia of Philosophy articles do not assume or argue for any such view: M. Gómez-Torrente, Logical truth (2010), http:// plato.stanford.edu/entries/logical-truth/; B. Kment, Varieties of modality (2012), http://plato.stanford.edu/entries/modality-varieties/ . 18. But see Chapter 14 below for a fictionalist account of zero. 19. Armstrong, Combinatorial Theory, 126. 20. A. Pais, Subtle Is the Lord: The Science and the Life of Albert Einstein (Oxford University Press, Oxford, 1982), 210–213. 3 Elementary Mathematics: The Science of Quantity 1. References in Chapter 7 below.