<!DOCTYPE html><HTML lang="en"> <head><meta charset="utf-8"> <title>Shapiro (Stewart) - Structure and Identity (Theo Todman's Book Collection - Paper Abstracts) </title> <link href="../../TheosStyle.css" rel="stylesheet" type="text/css"><link rel="shortcut icon" href="../../TT_ICO.png" /></head> <BODY> <CENTER> <div id="header"><HR><h1>Theo Todman's Web Page - Paper Abstracts</h1><HR></div><A name="Top"></A> <TABLE class = "Bridge" WIDTH=950> <tr><th><A HREF = "../../PaperSummaries/PaperSummary_07/PaperSummary_7165.htm">Structure and Identity</A></th></tr> <tr><th><A HREF = "../../Authors/S/Author_Shapiro (Stewart).htm">Shapiro (Stewart)</a></th></tr> <tr><th>Source: MacBride - Identity and Modality, 2006, Chapter 5</th></tr> <tr><th>Paper - Abstract</th></tr> </TABLE> </CENTER> <P><CENTER><TABLE class = "Bridge" WIDTH=600><tr><td><A HREF = "../../PaperSummaries/PaperSummary_07/PaperSummary_7165.htm">Paper Summary</A></td><td><A HREF = "../../PaperSummaries/PaperSummary_07/PaperCitings_7165.htm">Books / Papers Citing this Paper</A></td><td><A HREF="#ColourConventions">Text Colour-Conventions</a></td></tr></TABLE></CENTER></P> <hr><P><FONT COLOR = "0000FF"><u>Editor s <U><A HREF="#On-Page_Link_P7165_1">Introduction</A></U><SUB>1</SUB><a name="On-Page_Return_P7165_1"></A></u><FONT COLOR = "800080"><ol type="1"><li>In his 'Structure and Identity' Stewart Shapiro reflects upon the doctrine (advanced in his <em>Philosophy of Mathematics: Structure and Ontology</em> (Oxford: OUP, 1997)) that mathematical objects are places in structures where the latter are conceived as <em>ante rem</em> <a name="1"></a><A HREF="../../Notes/Notes_10/Notes_1008.htm">universals</A><SUP>2</SUP>. </li><li>This doctrine  that Shapiro dubs '<em>ante rem</em> structuralism'  suggests that there is no more to a mathematical object than the (structural) relations it bears to the other objects within the structure to which it belongs. </li><li>However, as Shapiro recognizes, when conceived in this way <em>ante rem</em> structuralism is open to a variety of criticisms. This is because there appears to be more to a mathematical object than the relations it bears to other objects within its parent structure. </li><li>Mathematical objects enjoy relations to <ul type="disc"><li>(i) items outside the mathematical realm (e.g. the concrete objects they are used to measure or count) and </li><li>(ii) objects that belong to other structures inside the mathematical realm. </li><li>Moreover, (iii) there are mathematical objects (e.g. points in a Euclidean plane) that are indiscernible with respect to their (structural) relations but nevertheless distinct. </li></ul>This makes it appear that <em>ante rem</em> structuralism is committed to the absurdity of identifying these objects. </li><li>Shapiro seeks to overcome these difficulties by a series of interlocking manoeuvres. <ul type="disc"><li>First, he seeks to overturn the metaphysical tradition about numbers, suggesting that it may be contingent whether a given mathematical object is abstract or concrete. </li><li>Second, Shapiro questions whether mathematical discourse is semantically determinate. </li><li>Finally, Shapiro rejects the requirement that ante rem structuralism provide for the non-trivial <a name="2"></a><A HREF="../../Notes/Notes_0/Notes_77.htm">individuation</A><SUP>3</SUP> of mathematical objects. </li></ul></li></ol></FONT> <BR><u>Author s Abstract</u><FONT COLOR = "800080"><ol type="1"><li>The purpose of this paper is to further articulate my preferred version of mathematical structuralism & <em>ante rem</em> structuralism, the thesis that mathematical structures exist prior to, and independent of, any exemplifications they may have in the non-mathematical world. </li><li>The contrast is with an approach that either adopts an Aristotelian <em>in re</em> view that a given structure exists only in the systems that exemplify it or the more common eliminative thesis that structures do not exist at all  talk of structures is to be paraphrased away. </li></ol></FONT><BR><u>Sections</u><FONT COLOR = "800080"><ol type="1"><li>What is (Ante Rem) Structuralism?</li><li>Cross-Structural Identity</li><li>Identity and Indiscernibility</li></ol></FONT><BR><HR><BR><U><B>In-Page Footnotes</U></B><a name="On-Page_Link_P7165_1"></A><BR><BR><U><A HREF="#On-Page_Return_P7165_1"><B>Footnote 1</B></A></U>: <ul type="disc"><li>From <a name="3"></a>"<A HREF = "../../Abstracts/Abstract_07/Abstract_7161.htm">MacBride (Fraser) - Identity and Modality: Introduction</A>", </li><li>Bullet numbering is mine. </li></ul><FONT COLOR = "0000FF"><HR></P><a name="ColourConventions"></a><p><b>Text Colour Conventions (see <A HREF="../../Notes/Notes_10/Notes_1025.htm">disclaimer</a>)</b></p><OL TYPE="1"><LI><FONT COLOR = "0000FF">Blue</FONT>: Text by me; &copy; Theo Todman, 2018</li><LI><FONT COLOR = "800080">Mauve</FONT>: Text by correspondent(s) or other author(s); &copy; the author(s)</li></OL> <BR><HR><BR><CENTER> <TABLE class = "Bridge" WIDTH=950> <TR><TD WIDTH="30%">&copy; Theo Todman, June 2007 - August 2018.</TD> <TD WIDTH="40%">Please address any comments on this page to <A HREF="mailto:theo@theotodman.com">theo@theotodman.com</A>.</TD> <TD WIDTH="30%">File output: <time datetime="2018-08-02T06:55" pubdate>02/08/2018 06:55:33</time> <br><A HREF="../../Notes/Notes_10/Notes_1010.htm">Website Maintenance Dashboard</A></TD></TR> <TD WIDTH="30%"><A HREF="#Top">Return to Top of this Page</A></TD> <TD WIDTH="40%"><A HREF="../../Notes/Notes_11/Notes_1140.htm">Return to Theo Todman's Philosophy Page</A></TD> <TD WIDTH="30%"><A HREF="../../index.htm">Return to Theo Todman's Home Page</A></TD> </TR></TABLE></CENTER><HR> </BODY> </HTML>