{"id":1524,"date":"2023-04-07T11:24:10","date_gmt":"2023-04-07T11:24:10","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2023\/04\/07\/spherical-trianges-and-incongruent-counterparts\/"},"modified":"2023-04-07T11:24:10","modified_gmt":"2023-04-07T11:24:10","slug":"spherical-trianges-and-incongruent-counterparts","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2023\/04\/07\/spherical-trianges-and-incongruent-counterparts\/","title":{"rendered":"Kant, Spherical Triangles, and Incongruent Counterparts"},"content":{"rendered":"

Buckner demands an argument from incongruent counterparts to the ideality of space. But before we get to that, I am having trouble understanding how the 'spherical triangles' Kant mentions in the Prolegomena to Any Future Metaphysics<\/em>, sec. 13,  are incongruent counterparts. Perhaps my powers of visualization are weak. Maybe someone can help me<\/a>.<\/span><\/p>\n

I understand how a hand and its mirror image are incongruent counterparts. If I hold up my right hand before a mirror what I see is a left hand.  As Kant says, "I cannot put such a hand as is seen in the glass in the place of its original; for if this is a right hand, that in the glass is a left one . . . ." (p. 13)  That is clear to me.<\/span><\/p>\n

Now visualize a sphere and two non-plane 'spherical triangles' the common base of which is an arc of the sphere's equator. The remaining two sides of the one triangle meet at the north pole; the remaining two sides of the other at the south pole.  The two triangles are exact counterparts, equal in all such internal respects as lengths of sides, angles, etc.  They are supposed to be incongruent in that "the one cannot be put in place of the other (that is, upon the opposite hemisphere)." (ibid.)  That is not clear to me.<\/span><\/p>\n

Imagine the southern triangle detached from the sphere and rotated through 180 degrees so that the south vertex is pointing north and the base is directly south. Now imagine the southern triangle place on top of the northern triangle.  To my geometrical intuition they are congruent!<\/span><\/p>\n

So, as I see it, hands and gloves are chiral<\/em> but Kant's spherical triangles are not.<\/span><\/p>\n

Wikipedia<\/a>:<\/span><\/p>\n

\n

In geometry<\/a>, a figure is chiral<\/strong> (and said to have chirality<\/strong>) if it is not identical to its mirror image<\/a>, or, more precisely, if it cannot be mapped to its mirror image by rotations<\/a> and translations<\/a> alone. An object that is not chiral is said to be achiral<\/em>.<\/span><\/p>\n

A chiral object and its mirror image are said to be enantiomorphs<\/strong>. The word chirality<\/em> is derived from the Greek \u03c7\u03b5\u03af\u03c1<\/span><\/span> (cheir), the hand, the most familiar chiral object; the word enantiomorph<\/em> stems from the Greek \u1f10\u03bd\u03b1\u03bd\u03c4\u03af\u03bf\u03c2<\/span><\/span> (enantios) 'opposite' + \u03bc\u03bf\u03c1\u03c6\u03ae<\/span><\/span> (morphe) 'form'.<\/span><\/p>\n<\/blockquote>\n","protected":false},"excerpt":{"rendered":"

Buckner demands an argument from incongruent counterparts to the ideality of space. But before we get to that, I am having trouble understanding how the 'spherical triangles' Kant mentions in the Prolegomena to Any Future Metaphysics, sec. 13,  are incongruent counterparts. Perhaps my powers of visualization are weak. Maybe someone can help me. I understand … Continue reading “Kant, Spherical Triangles, and Incongruent Counterparts”<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[270,476],"tags":[],"class_list":["post-1524","post","type-post","status-publish","format-standard","hentry","category-kant","category-mathematics"],"_links":{"self":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/1524","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/comments?post=1524"}],"version-history":[{"count":0,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/1524\/revisions"}],"wp:attachment":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/media?parent=1524"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/categories?post=1524"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/tags?post=1524"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}