Abstract
The story here involves F. Ramsey’s realization that the nineteenth century mathematical debate about functions had implications for the expression of statements of arithmetic in Russell and Whitehead’s Principia. We believe that it is the same flaw, – expressive inadequacy – that lies at the heart of what is wrong with D. Armstrong’s account of scientific laws.
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Notes
- 1.
It’s the exclusive used of non-extensional functions that gives rise to severe problems. Many sciences and even classical logic are not so resrtricted. For example classical probability is a clear case because the probability function is non-extensional. In the case of classical first-order logic, it has been argued that the quantifiers are non-extension al. Cf. A Koslow, “The modality and non-extensionality of the quantifiers” Synthese, 2014.
- 2.
F.P. Ramsey, Philosophical papers. D.H.Mellor (ed.).
- 3.
Principia Mathematica to ∗56.Cambridge Mathematical Library Pb, 1997.
- 4.
This is an unpublished typescript which was Braithwaite’s Fellowship Dissertation. It is on deposit in the King’s College Archives, Cambridge University, England.
- 5.
F. Hausdorff, Mengenlehre, First German ed., 1914. Potter says that unfortunately there are no underlined or marked passages in it. The quotation above is from the preface to the third English edition, Set Theory, Chelsia Publishing Company, Tr. J.R. Aumann et al.(15–17)
- 6.
That is R(x, x), If Rx, y) then R(y, x), and If R(x, y) and R(y, z), then R(x, z), for all x, y, and z.
- 7.
Mellor 1990, (215).
- 8.
[Mellor 1990, (215)].
- 9.
There is a letter of Wittgenstein to Ramsey critical of Ramsey’ paper “Foundations of Mathematics (reprinted in Mellor, [1990]., and Wittgenstein asks Ramsey to convey his response to M. Schlick. Part of Ramsey’s response is conveyed by Schlick to Wittgenstein. These letters and responses are reprinted in Wittgenstein and the Vienna Circle, Blackwell, Oxford 1984. Ramsey then drafted two responses to Wittgenstein directly. It’s not clear whether they were ever sent. They are reprinted in M.C. Galavotti, Notes on Philosophy, Probability and Mathematics, Bibliopolis, 1991, pp. 337–346). Also cf. the noteworthy paper of P.M. Sullivan, “Wittgenstein on “The Foundations of Mathematics”, June 1927, in Theoria 61(2), 1995, pp. 105–42.
- 10.
Admirably discussed in I. Grattan-Guinness, The Development Of the Foundations of Mathematical Analysis From Euler to Riemann, MIT Press (1970), pp. 2–12), and I. Kleiner, “Evolution of the Function Concept: A Brief Survey, College Mathematics Journal, 20 (1989), pp. 282–399.
- 11.
That is, aRb if and only if A[a] & A[b].
- 12.
The Development of the Foundations of Mathematical Analysis from Euler to Riemann, MIT, Cambridge, 1970, p. 50.
- 13.
I. Grattan-Guinness, p. 51.
- 14.
I. Grattan-Guinness, p. 10.
- 15.
I. Grattan-Guinness, pp. 50–51.
- 16.
I. Grattan-Guinness, p. 6.
- 17.
Here and above, we follow the usual definition of “algebraic function” according to which u = f(x,y,…,z) is an algebraic function if and only if there is a polynomial F, such that F(x,y,…,z,u) = 0. Cf for example, R, Courant, Differential and Integral Calculus, vol.I, Interscience, N.Y.1937, tr.by E.J. McShane, p. 485.
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Koslow, A. (2019). Prelude to Armstrong: A Mathematical Revolution That Inspired F. Ramsey, and Left Russell and Armstrong Unmoved. In: Laws and Explanations; Theories and Modal Possibilities. Synthese Library, vol 410. Springer, Cham. https://doi.org/10.1007/978-3-030-18846-7_5
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