THE JOURNAL OF PHILOSOPHY - ONTOLOGICAL REDUCTION AND THE WORLD OF NUMBERS

Sep 21, 2026

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Discussion of how reduction works in analytic and scientific philosophy, focusing on number theories. Compares Frege’s definition of natural numbers as classes of n-member classes with von Neumann’s construction as classes of all numbers less than n, and contrasts them with Carnap’s treatment of measure and impure numbers. Argues that many reductions are ontological rather than coextensive in meaning, and that what matters for the success of these approaches is structure preservation, especially preserving arithmetic, even when translations of mixed contexts are required.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages30

Reading Time1 h 16 min

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Book Reviews

HERMAN-J.DE VLEESCHAUWER

MARY J.GREGOR

THE JOURNAL OF PHILOSOPHY

ONTOLOGICAL REDUCTION AND THE WORLDOF NUMBERS

VOL.LXI,No.7:MARCH 26,1964 Ontological Reduction and the World of Numbers:W.V. ....................209 QUINE .............. Comments and Criticism A Note on Some Epistemic Principles of Chisholm and Martin:R.C.SLEIGH ......................216 # Book Reviews HERMAN-J.DE VLEESCHAUWER:The Development of Kantian Thought:The History of a Doctrine:BELLA K.MILMED ....218 MARY J.GREGOR:Laws of Freedom:A Study of Kant's Method of Applying the Categorical Imperative in the Metaphysik der Sitten:ROBERT PAUL WoLFF ..…….......... …..226 Program,MEETINGS OF THE AMERICAN PHILOSOPHICAL AssoCI-ATION,WESTERN DIVISION ...........................232 BY THE JOURNAL OF PHILOSOPHY,INC. # THE JOURNAL OF PHILOSOPHY FoUNDED BY FBEDERICK J.E.WoODBRIDGE AND WENDELL T.BUSH EDITORS:John H.Randall,Jr.,Robert D.Cumming,George L.Kline,Sidney Morgenbesser,Arthur Danto,and James j.Walsh. CoNSULTING EDITORS:Herbert W.Schneider,Alexandre Koyré,andJosé Ferrater Mora. MANAGING EDITOR:Leigh S.Cauman. Communieations to the editors should be addressed to the Managing Edi-tor,720 Philosophy Hall,Columbia University,New York,N.Y,10027.Subscriptions and advertisements should be sent to THE JOURNALOF PHILOSOPHY,720 Philosophy Hall,Columbia University,NewYork,N.Y.,10027.The annual subseription price(24 numbers)is $7.50;to students,$6.00.The price of single numbers is $0.50;double numbers,$1.00;complete volumes(from 1960 on),$12.00. THE JOURNAL OF PHILOSOPHY is owned by the Journal ofPhilosophy,Ine.,and published bi-weekly(every four weeks in Julyand August)by Lancaster Press,Ine.,Prince and Lemon Sts.,Lan-caster,Pa.,17604.Editorial Offce:720 Philosophy Hall,ColumbiaUniversity,New York,N.Y,10027.President,Albert G.Redpath;Vice President,John H.Randall,Jr.;Treasurer,Frederick A.Dewey;Secretary,Corliss Lamont. Second-class postage paid at Lancaster,Pa. | PHILOSOPHY OF SCIENCE
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PUBLISHED BY THE ST.CATHERINE PRESS,BRUGES,BELGIUM | | --- | MARCH 26,1964 THE JOURNAL OF PHILOSOPHY # ONTOLOGICAL REDUCTION AND THE WORLDOF NUMBERS NE conspicuous concern of analytieal or scientific philosophyhae hoon towdan como notione to othonc nnofonahlv to laccnas been to reduce some notions to otners,prererabiy to lessputative ones.A familiar case of such reduction is Frege's defini-tion of number.Each natural number n became,if I may speakin circles,the class of all n-member classes.As is also well known,Frege's was not the only good way.Another was von Neumann's.Under it,if I may again speak in circles,each natural number nbecame the class of all numbers less than n. In my judgment we have satisfactorily reduced one predicateto others,certainly,if in terms of these others we have fashionedan open sentence that is coextensive with the predicate in questionas originally interpreted;i.e.,that is satisfied by the same valuesof the variables.But this standard does not suit the Frege andvon Neumann reductions of number;for these reductions are bothgood,yet not coextensive with each other. Again consider Carnap's clarification of measure,or impurenumber,where he cons

FAQs

What does ontological reduction mean in the discussion of numbers?

Reduction is treated as ontological when it replaces one kind of entity or predicate with others in a way that is not coextensive in meaning, such as Carnap’s banishment of impure numbers while using only pure ones.

How do Frege and von Neumann reduce natural numbers?

Frege reduces each natural number n to the class of all n-member classes. Von Neumann reduces n to the class of all numbers less than n.

What common feature makes these reductions successful?

They are viewed as structure-preserving models of natural numbers: each preserves arithmetic, which is considered sufficient for the reduction’s adequacy even though further translation issues for mixed contexts may arise.

Sep 21, 2026

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