The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

19
Structure and Abstraction

PETER SIMONS

Ave Caesar! Morituri te salutant.


1. INTRODUCTION

My ontological inclinations are nominalistic, and nominalism traditionally faces a number of stern challenges. One is from the theory of predication and universals. I think this challenge is, for a philosophical one, fairly easy for a nominalist to face. A far more daunting prospect is to reconcile the nominalist's denial of the existence of abstract objects with the acceptance of a vast body of mathematical propositions by a large number of highly intelligent people. Taken at face value, the mathematical propositions these people hold true entail the existence of untold infinities of abstract mathematical objects.

However, rather than presuppose or work within a nominalist framework I shall be exploring aspects of the kind of Platonistic realism of mathematical objects which I consider the most promising and plausible, namely structuralism. I shall not attempt to argue for structuralism against other kinds of Platonistic mathematical ontology. Rather I wish to canvass the possibility of a union--some might consider it a shotgun wedding-- between the philosophies of mathematics of Dedekind and Frege. This is not in order to rescue the logicism they held in common, but because each emphasizes in his philosophy of mathematics something the other can profitably use, namely structure ( Dedekind) and abstraction ( Frege).


2. ABSTRACTION

It may seem anachronous to take abstraction as the Fregean component of the proposed synthesis because after all Frege vociferously rejected and indeed poured scorn on what other mathematicians such as Dedekind and Cantor and various empiricists called 'abstraction'. By 'abstraction' I shall intend the procedure of contextual introduction of singular terms and their

-485-

Notes for this page

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this book

This book has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this book

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this page

Cited page

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited page

Bookmark this page
The Philosophy of Mathematics Today
Table of contents

Table of contents

Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this book

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Full screen
/ 646

matching results for page

Cited passage

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

"Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited passage

Thanks for trying Questia!

Please continue trying out our research tools, but please note, full functionality is available only to our active members.

Your work will be lost once you leave this Web page.

For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

Already a member? Log in now.