David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Ezio Di Nucci
Jack Alan Reynolds
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Notre Dame Journal of Formal Logic 41 (4):335--364 (2000)
This paper uses neo-Fregean-style abstraction principles to develop the integers from the natural numbers (assuming Hume’s principle), the rational numbers from the integers, and the real numbers from the rationals. The first two are first-order abstractions that treat pairs of numbers: (DIF) INT(a,b)=INT(c,d) ≡ (a+d)=(b+c). (QUOT) Q(m,n)=Q(p,q) ≡ (n=0 & q=0) ∨ (n≠0 & q≠0 & m⋅q=n⋅p). The development of the real numbers is an adaption of the Dedekind program involving “cuts” of rational numbers. Let P be a property (of rational numbers) and r a rational number. Say that r is an upper bound of P, written P≤r, if for any rational number s, if Ps then either s<r or s=r. In other words, P≤r if r is greater than or equal to any rational number that P applies to. Consider the Cut Abstraction Principle: (CP) ∀P∀Q(C(P)=C(Q) ≡ ∀r(P≤r ≡ Q≤r)). In other words, the cut of P is identical to the cut of Q if and only if P and Q share all of their upper bounds. The axioms of second-order real analysis can be derived from (CP), just as the axioms of second-order Peano arithmetic can be derived from Hume’s principle. The paper raises some of the philosophical issues connected with the neo-Fregean program, using the above abstraction principles as case studies.
|Keywords||neologicism Frege real numbers|
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Citations of this work BETA
Sean Walsh (2014). Logicism, Interpretability, and Knowledge of Arithmetic. Review of Symbolic Logic 7 (1):84-119.
Bob Hale (forthcoming). Second-Order Logic: Properties, Semantics, and Existential Commitments. Synthese:1-27.
Stewart Shapiro & Geoffrey Hellman (forthcoming). Frege Meets Aristotle: Points as Abstracts. Philosophia Mathematica:nkv021.
Øystein Linnebo (2009). Bad Company Tamed. Synthese 170 (3):371 - 391.
Øystein Linnebo & Gabriel Uzquiano (2009). Which Abstraction Principles Are Acceptable? Some Limitative Results. British Journal for the Philosophy of Science 60 (2):239-252.
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