According to realist structuralism, mathematical objects are places in abstract structures. We argue that in spite of its many attractions, realist structuralism must be rejected. For, first, mathematical structures typically contain intra-structurally indiscernible places. Second, any account of place-identity available to the realist structuralist entails that intra-structurally indiscernible places are identical. Since for her mathematical singular terms denote places in structures, she would have to say, for example, that 1 = − 1 in the group (Z, +). We call this (...) the identity problem and conclude that nominalism is presently the safest route for the structuralist. (shrink)

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

Three principal varieties of mathematical structuralism are compared: set-theoretic structuralism (‘STS’) using model theory, Shapiro's ante rem structuralism invoking sui generis universals (‘SGS’), and the author's modal-structuralism (‘MS’) invoking logical possibility. Several problems affecting STS are discussed concerning, e.g., multiplicity of universes. SGS overcomes these; but it faces further problems of its own, concerning, e.g., the very intelligibility of purely structural objects and relations. MS, in contrast, overcomes or avoids both sets of problems. Finally, it is argued that the modality (...) of MS or a close cousin appears at crucial junctures in both STS and SGS, so that the above outcome is not obviously tendentious. (shrink)

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

Excerpt from Substanzbegriff und Funktionsbegriff: Untersuchungen Uber die Grundfragen der Erkenntniskritik Die erste Anregung zu den Untersuchungen, die dieser Band enthalt, ist mir aus Studien zur Philosophie der Mathe matik erwachsen. Indem ich versuchte, von Seiten der Logik aus einen Zugang zu den Grundbegriffen der Mathematik zu gewinnen, erwies es sich vor allem als notwendig, die B e g r i f f s f u n k t i 0 n e'lhsimaher zu zergliedern und auf ihre Voraussetzungen zuruckzufuhren. Hier (...) aber machte sich alsbald eine eigentumliche Schwierigkeit geltend: die her kommliche logische Lehre vom Begriff zeigte sich in ihren bekannten Hauptzugen als unzureichend, die Probleme, zu denen die Prinzipienlehre der Mathematik hinfuhrt, auch nur vollstandig z u b e z e i c h n e n. Die exakte Wissenschaft war hier, wie sich mir immer deutlicher zu ergeben schien, zu Fragen gelangt, fur welche die Formensprache der tradi tionellen Logik kein genaues Correlat besitzt. Der sachliche Gehalt der mathematischen Erkenntnisse wies auf eine Grund form des Begriffs zuruck, die in der Logik selbst nicht zu klarer Bezeichnung und Anerkennung gekommen war. Ins besondere waren es Untersuchungen jiber den Reihenbegriff und den Grenzbegriff (deren spezielles Ergebnis in die allgemeineren Erorterungen dieses Buches nich? 3iffge nommen werden konnte), die diese Uberzeugung in mir be festigten und damit zu einer erneuten Analyse der Prinzipien der Begriffsbildung selbsfhindrangten. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

First published in 1995. Routledge is an imprint of Taylor & Francis, an informa company.

This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

Steve Awodey and Erich H. Reck. Completeness and Categoricity: 19th Century Axiomatics to 21st Century Senatics.