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Infinitesimals, Nations, and Persons

In memoriam D.A.P. 11 December 1942 − 1 January 2017

Published online by Cambridge University Press:  09 October 2019

Ian Rumfitt
Ian Rumfitt*
Affiliation:
All Souls College, Oxford
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Abstract

I compare three sorts of case in which philosophers have argued that we cannot assert the Law of Excluded Middle for statements of identity. Adherents of Smooth Infinitesimal Analysis deny that Excluded Middle holds for statements saying that an infinitesimal is identical with zero. Derek Parfit contended that, in certain sci-fi scenarios, the Law does not hold for some statements of personal identity. He also claimed that it fails for the statement ‘England in 1065 was the same nation as England in 1067’. I argue that none of these cases poses a serious threat to Excluded Middle. My analysis of the last example casts doubt on the principle of the Determinacy of Distinctness. While David Wiggins's ‘conceptualist realism’ provides a metaphysics which can dispense with that principle, it leaves no house-room for infinitesimals.

Type
Research Article
Information
Philosophy , Volume 94 , Issue 4 , October 2019 , pp. 513 - 528
Copyright
Copyright © The Royal Institute of Philosophy 2019 

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Footnotes

This paper is a record of a talk I gave at a number of schools in the years 2014-18 with the aim of showing how a joint honours degree in mathematics and philosophy could offer students more than the sum of its parts. Friends have suggested that a wider readership might enjoy it. Each of the topics treated is the subject of a vast literature. I have not, though, come across another paper which explores the relations between them.

References

1 Alexander, Amir, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (New York: Farrar, Straus and Giroud, 2014)Google Scholar.

2 ‘And what are these same evanescent Increments?  They are not finite Quantities nor Quantities infinitely small, nor yet nothing.  May we not call them the Ghosts of departed Quantities?’  Berkeley, George, The Analyst, or A Discourse Addressed to an Infidel Mathematician (London: Jacob Tonson, 1734), §35Google Scholar.

3 Russell, Bertrand, The Principles of Mathematics, 2nd ed. (London: George Allen and Unwin, 1937) §324, 345Google Scholar.

4 Robinson, Abraham, Non-standard Analysis, revised ed. (Princeton, NJ: Princeton University Press, 1996)CrossRefGoogle Scholar.

5 See Kock, Anders, Synthetic Differential Geometry (Cambridge: Cambridge University Press, 1981)Google Scholar; Lawvere, F. William, ‘Categorical Dynamics’, in Kock, ed., Topos-Theoretic Methods in Geometry (Aarhus: Aarhus Mathematical Institute Publications Series, 30, 1979), 1−28Google Scholar.  For an accessible introduction to SIA, see also Bell, John, A Primer of Infinitesimal Analysis, 2nd ed. (Cambridge: Cambridge University Press, 2008)CrossRefGoogle Scholar.

6 For further discussion, see Rumfitt, Ian, The Boundary Stones of Thought: An Essay in the Philosophy of Logic (Oxford: Clarendon Press, 2015), 213−16CrossRefGoogle Scholar.

7 Brouwer, L.E.J., Cambridge Lectures on Intuitionism, ed. van Dalen, D. (Cambridge: Cambridge University Press, 1981), 8Google Scholar.

8 For a discussion of various intuitionistic continuity principles, and proofs that they contradict classical logical laws, see Dummett, Michael, Elements of Intuitionism, 2nd ed. (Oxford: Clarendon Press, 2000), 56−62Google Scholar.

9 Parfit, Derek, ‘Personal Identity’, The Philosophical Review 80 (1971), 3−27CrossRefGoogle Scholar, at 3−4.

10 Wiggins, David, Identity and Spatio-Temporal Continuity (Oxford: Blackwell, 1967)Google Scholar, 53f.

11 A and B ‘will be in different places and have separate experiences from now on.  And they will communicate interpersonally’.  Wiggins op. cit., 53.

12 Cf. his example of the candidate intentionally dividing his mind as he comes towards the end of a mathematics exam: Parfit op. cit., 6−7.

13 Frege, Gottlob, Grundgesetze der Arithmetik I (Jena: Hermann Pohle, 1893), xviGoogle Scholar.

14 Parfit op.cit., 8.

15 Barcan, Ruth, ‘The Identity of Individuals in a Strict Functional Calculus of Second Order’, The Journal of Symbolic Logic 12 (1947), 12−15CrossRefGoogle Scholar; Evans, Gareth, ‘Can There Be Vague Objects?’ Analysis 38 (1978), 208CrossRefGoogle Scholar.

16 See Parsons, Terence, Indeterminate Identity: Metaphysics and Semantics (Oxford: Clarendon Press, 2000)CrossRefGoogle Scholar.

17 Williamson, Timothy, ‘The Necessity and Determinacy of Distinctness’, in Lovibond, Sabina and Williams, Stephen, eds., Essays for David Wiggins: Identity, Truth, and Value (Oxford: Blackwell, 1996), 1−17Google Scholar; see esp. 10−13.

18 Wiggins, David, ‘On Singling Out an Object Determinately’, in Pettit, Philip and McDowell, John, eds., Subject, Thought, and Context (Oxford: Clarendon Press, 1986), 169−80Google Scholar.

19 For his recantations, see Wiggins, ‘Reply to Timothy Williamson’, in Lovibond and Williams, eds., op. cit., 231−38; and Wiggins, Continuants: Their Activity, Their Being and Their Identity (Oxford: Oxford University Press, 2016), 24−5.

20 Wiggins, ‘On Singling Out an Object Determinately’, 171.

21 Wiggins op. cit., 177.

22 Wiggins op. cit., 179, replacing ‘clubs’ (his example) with ‘nations’.