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The ordered pair: how its history and philosophy has pedagogical importance in teaching mathematics

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Abstract:

The concept of an ordered pair and the problem of providing a formal explication of it has a rich history in late 19th century and early 20th century mathematics—Peano, Peirce, Frege, Schröder, Wiener, Russell, Whitehead, Hausdorff, Kuratowski, VonNeumann, Bernays, Gödel, Tarski, Quine are key figures. Teaching the history of the ordered pair in the mathematics classroom provides several opportunities for enhancing a mathematics education. First, the distinction between intensional and extensional concepts is nicely illustrated by the task of providing a formal explication of an ordered pair (as is the idea of philosophical explication of a concept). Second, the reduction of the intensional notions of a function and of relations via the ordered pair to extensional ones—in terms of sets or classes—reveals one of the ways in which modern mathematics differs from classical mathematics. Third, that there are several, incompatible formal explications of the ordered pair each of which satisfies the formal criterion of an ordered pair (Peano’s criterion: = iff x = v and y = w) provides an object lesson in how definitions of mathematical objects are accepted in mathematical practice.
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Association:
Name: The Mathematical Association of America MathFest
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http://www.maa.org


Citation:
URL: http://citation.allacademic.com/meta/p435137_index.html
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MLA Citation:

Buechner, Jeff. "The ordered pair: how its history and philosophy has pedagogical importance in teaching mathematics" Paper presented at the annual meeting of the The Mathematical Association of America MathFest, Omni William Penn, Pittsburgh, PA, Aug 05, 2010 <Not Available>. 2014-11-27 <http://citation.allacademic.com/meta/p435137_index.html>

APA Citation:

Buechner, J. , 2010-08-05 "The ordered pair: how its history and philosophy has pedagogical importance in teaching mathematics" Paper presented at the annual meeting of the The Mathematical Association of America MathFest, Omni William Penn, Pittsburgh, PA <Not Available>. 2014-11-27 from http://citation.allacademic.com/meta/p435137_index.html

Publication Type: Conference Paper/Unpublished Manuscript
Review Method: Peer Reviewed
Abstract: The concept of an ordered pair and the problem of providing a formal explication of it has a rich history in late 19th century and early 20th century mathematics—Peano, Peirce, Frege, Schröder, Wiener, Russell, Whitehead, Hausdorff, Kuratowski, VonNeumann, Bernays, Gödel, Tarski, Quine are key figures. Teaching the history of the ordered pair in the mathematics classroom provides several opportunities for enhancing a mathematics education. First, the distinction between intensional and extensional concepts is nicely illustrated by the task of providing a formal explication of an ordered pair (as is the idea of philosophical explication of a concept). Second, the reduction of the intensional notions of a function and of relations via the ordered pair to extensional ones—in terms of sets or classes—reveals one of the ways in which modern mathematics differs from classical mathematics. Third, that there are several, incompatible formal explications of the ordered pair each of which satisfies the formal criterion of an ordered pair (Peano’s criterion: = iff x = v and y = w) provides an object lesson in how definitions of mathematical objects are accepted in mathematical practice.


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