David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
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|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library|
References found in this work BETA
No references found.
Citations of this work BETA
Sean Walsh (2014). Logicism, Interpretability, and Knowledge of Arithmetic. Review of Symbolic Logic 7 (1):84-119.
Jeffrey W. Roland (2009). On Naturalizing the Epistemology of Mathematics. Pacific Philosophical Quarterly 90 (1):63-97.
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.
Jeffrey W. Roland (2010). Concept Grounding and Knowledge of Set Theory. Philosophia 38 (1):179-193.
Similar books and articles
Neil Tennant (2003). Review of B. Hale and C. Wright, The Reason's Proper Study. [REVIEW] Philosophia Mathematica 11 (2):226-241.
Edward N. Zalta (1999). Natural Numbers and Natural Cardinals as Abstract Objects: A Partial Reconstruction of Frege"s Grundgesetze in Object Theory. [REVIEW] Journal of Philosophical Logic 28 (6):619-660.
Richard Heck (1993). The Development of Arithmetic in Frege's Grundgesetze der Arithmetik. Journal of Symbolic Logic 58 (2):579-601.
Roy T. Cook (2002). The State of the Economy: Neo-Logicism and Inflationt. Philosophia Mathematica 10 (1):43-66.
Bob Hale & Crispin Wright (2009). Focus Restored: Comments on John MacFarlane. Synthese 170 (3):457 - 482.
Bob Hale & Crispin Wright, Focus Restored Comment on John MacFarlane's “Double Vision: Two Questions About the Neo-Fregean Programme”.
Bob Hale (2000). Reals by Abstractiont. Philosophia Mathematica 8 (2):100--123.
Matthias Schirn (2003). Fregean Abstraction, Referential Indeterminacy and the Logical Foundations of Arithmetic. Erkenntnis 59 (2):203 - 232.
Added to index2010-01-18
Total downloads24 ( #78,580 of 1,140,270 )
Recent downloads (6 months)2 ( #86,093 of 1,140,270 )
How can I increase my downloads?