## Notre Dame Journal of Formal Logic

### Frege Meets Dedekind: A Neologicist Treatment of Real Analysis

Stewart Shapiro

#### Abstract

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:

$(\mathrm{DIF})\qquad \mathrm{INT}(a,b) = \mathrm{INT}(c,d)\equiv(a+d)=(b+c).$

$(\mathrm{QUOT}) \qquad \begin{eqnarray}\mathrm{Q}(m,n)=\mathrm{Q}(p,q)\equiv(n=0 \: \& \: q=0) \\ \qquad \vee\: (n\neq 0 \: \& \: q\neq 0 \: \& \: m \cdot q=n \cdot p).\end{eqnarray}$

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 $P\leq r$, if for any rational number $s$, if $Ps$ then either $s<r$ or $s=r$. In other words, $P\leq r$ if $r$ is greater than or equal to any rational number that $P$ applies to. Consider the Cut Abstraction Principle:

$(\mathrm{CP}) \qquad \forall P \forall Q(C(P)=C(Q) \equiv \forall r(P\leq r \equiv Q\leq r)).$

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.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 41, Number 4 (2000), 335-364.

Dates
First available in Project Euclid: 26 November 2002

https://projecteuclid.org/euclid.ndjfl/1038336880

Digital Object Identifier
doi:10.1305/ndjfl/1038336880

Mathematical Reviews number (MathSciNet)
MR1963486

Zentralblatt MATH identifier
1014.03013

Subjects
Primary: 00A25
Secondary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}

#### Citation

Shapiro, Stewart. Frege Meets Dedekind: A Neologicist Treatment of Real Analysis. Notre Dame J. Formal Logic 41 (2000), no. 4, 335--364. doi:10.1305/ndjfl/1038336880. https://projecteuclid.org/euclid.ndjfl/1038336880

#### References

• Boolos, G., "The consistency of F"rege's Foundations of Arithmetic, pp. 3–20 in On Being and Saying: Essays in Honor of Richard Cartwright, The MIT Press, Cambridge, 1987.
• Boolos, G., "Iteration again", Philosophical Topics, vol. 17 (1989), pp. 5–21.
• Boolos, G., “Is Hume's Principle analytic?” pp. 245–61 in Language, Thought, and Logic, edited by Jr. Heck, R. G., Oxford University Press, Oxford, 1997.
• Clark, P., "Indefinite extensibility and set theory", Talk to Arché Workshop on Abstraction, University of St. Andrews, 2000.
• Cook, R. T., "The state of the economy: N"eo-logicism and inflation, Philosophia Mathematica. Series 3, vol. 10 (2002), pp. 43–66.
• Dedekind, R., "Continuity and irrational numbers", pp. 1–27 in Essays on the Theory of Numbers, edited by W. W. Beman, Dover Publications Inc., New York, 1963.
• Dedekind, R., "The nature and meaning of numbers", pp. 31–115 in Essays on the Theory of Numbers, edited by W. W. Beman, Dover Publications Inc., New York, 1963.
• Dummett, M., The Seas of Language, The Clarendon Press, Oxford, 1993.
• Field, H., Science Without Numbers. A Defence of Nominalism, Princeton University Press, Princeton, 1980.
• Fine, K., "The limits of abstraction", pp. 503–629 in The Philosophy of Mathematics Today, Oxford University Press, New York, 1998.
• Frege, G., Begriffsschrift, Eine der arithmetischen Nachgebildete Formelsprache des Reinen Denkens, Louis Nebert, Halle, 1879.
• Frege, G., "Über die G"rundlagen der Geometrie, Jahresbericht der Mathematiker-Vereinigung, vol. 12 (1903), pp. 319–24, 368–75.
• Frege, G., "Über die G"rundlagen der Geometrie, Jahresbericht der Mathematiker-Vereinigung, vol. 15 (1906), pp. 293–309, 377–403, 423–30.
• Frege, G., Die Grundlagen der Arithmetik, Philosophical Library, New York, 1950.
• Frege, G., Grundgesetze der Arithmetik I, Georg Olms Verlagsbuchhandlung, Hildesheim, 1962.
• Frege, G., On the Foundations of Geometry and Formal Theories of Arithmetic, Yale University Press, New Haven, 1971.
• Goldfarb, W. D., "Logic in the twenties: T"he nature of the quantifier, The Journal of Symbolic Logic, vol. 44 (1979), pp. 351–68.
• Hale, B., "Reals by abstraction", Philosophia Mathematica, Series III, vol. 8 (2000), pp. 100–123.
• Hale, B., and C. Wright, "To bury C"aesar..., pp. 335–96 in The Reason's Proper Study, Oxford University Press, Oxford, 2001.
• Heck, R. G., Jr., "Finitude and H"ume's Principle, Journal of Philosophical Logic, vol. 26 (1997), pp. 589–617.
• Ramsey, F., "The foundations of mathematics", Proceedings of the London Mathematical Society, vol. 25 (1925), pp. 338–84.
• Russell, B., Introduction to Mathematical Philosophy, 2d edition, Dover Publications Inc., New York, 1993.
• Shapiro, S., Foundations Without Foundationalism. A Case for Second-Order Logic, The Clarendon Press, Oxford, 1991.
• Shapiro, S., Philosophy of Mathematics: Structure and Ontology, Oxford University Press, New York, 1997.
• Shapiro, S., and A. Weir, "New V", ZF, and abstraction, Philosophia Mathematica, Series III, vol. 7 (1999), pp. 293–321.
• Shapiro, S., Thinking About Mathematics: The Philosophy of Mathematics, Oxford University Press, Oxford, 2000.
• Shapiro, S., "Prolegomenon to any future neo-logicist set theory: E"xtensionality and indefinite extensibility, forthcoming in British Journal for the Philosophy of Science.
• van Heijenoort, J., "Logic as calculus and logic as language", Synthese, vol. 17 (1967), pp. 324–30.
• Weir, A., “Neo-Fregeanism: An embarrassment of riches?” Talk to Arché workshop on abstraction, University of St. Andrews, 2000.
• Wright, C., Frege's Conception of Numbers as Objects, Aberdeen University Press, Aberdeen, 1983.
• Wright, C., "On the philosophical significance of F"rege's theorem, pp. 201–44 in Language, Thought, and Logic, edited by Jr.,R. G. Heck, Oxford University Press, Oxford, 1997.
• Wright, C., “Is Hume's Principle analytic?” Notre Dame Journal of Formal Logic, vol. 40 (1999), pp. 6–30.
• Wright, C., "Neo-F"regean foundations for real analysis: Some reflections on Frege's Constraint, Notre Dame Journal of Formal Logic, vol. 41 (2000), pp. 317–34.
• Wright, C., and B. Hale, "Implicit definition and the a priori", pp. 286–319 in New Essays on the a Priori, edited by P. Boghossian and C. Peacocke, Oxford University Press, Oxford, 2000.