{"id":11602,"date":"2010-05-16T17:29:07","date_gmt":"2010-05-16T17:29:07","guid":{"rendered":"https:\/\/maverickphilosopher.blog\/index.php\/2010\/05\/16\/on-the-abstractness-of-mathematical-sets\/"},"modified":"2010-05-16T17:29:07","modified_gmt":"2010-05-16T17:29:07","slug":"on-the-abstractness-of-mathematical-sets","status":"publish","type":"post","link":"https:\/\/maverickphilosopher.blog\/index.php\/2010\/05\/16\/on-the-abstractness-of-mathematical-sets\/","title":{"rendered":"On the Abstractness of Mathematical Sets"},"content":{"rendered":"

Let us agree that x is concrete iff x is causally\/active passive and abstract otherwise.  Many say that mathematical sets ('sets' hereafter: 'mathematical' as opposed to 'commonsense') are abstract objects, abstract entities, abstracta.  Why?<\/font><\/p>\n

Argument One:<\/strong>  In set theory there are singleton sets, e.g. {Quine}.  Obviously, Quine is not identical to {Quine}.  The second is a set, the first is not.  Yet the difference cannot be the difference between two concreta.  Quine is a concretum.  Therefore, {Quine} is an abstractum.  This is of course meant generally: singletons are abstracta.  Now if singletons are abstracta, then all sets are.  <\/font><\/p>\n

Argument Two:<\/strong>  In set theory there is a null set.  It is not nothing, but something despite having no members. Yet it cannot be a concrete something.  Therefore, it is an abstract something.  And if one set is abstract, all are.<\/font><\/p>\n

Contra Argument One:<\/strong>  A statue and the lump of clay that constitute it are numerically distinct.  (For the one has properties the other doesn't have, e.g., the lump, but not the statue, can exist without having the form of a statue.)  And yet both are concrete, i.e., both are causally active\/passive.  If this is possible, why should it not also be possible that Quine and {Quine} both be concrete?  One could say that Quine and {Quine} occupy the same 'plime' to borrow a term form D. C. Williams, the same place-time, in the way statue and lump do.<\/font><\/p>\n

Contra Argument Two:<\/strong>  Possibly, there is a concrete atomic entity. Being atomic, it has no parts.  So why should a set's having no members rule out its being concrete?<\/font><\/p>\n

Are any of these arguments compelling?<\/font><\/p>\n","protected":false},"excerpt":{"rendered":"

Let us agree that x is concrete iff x is causally\/active passive and abstract otherwise.  Many say that mathematical sets ('sets' hereafter: 'mathematical' as opposed to 'commonsense') are abstract objects, abstract entities, abstracta.  Why? Argument One:  In set theory there are singleton sets, e.g. {Quine}.  Obviously, Quine is not identical to {Quine}.  The second is … Continue reading “On the Abstractness of Mathematical Sets”<\/span><\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[481],"tags":[],"class_list":["post-11602","post","type-post","status-publish","format-standard","hentry","category-set-theory"],"_links":{"self":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/11602","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=11602"}],"version-history":[{"count":0,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/posts\/11602\/revisions"}],"wp:attachment":[{"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/media?parent=11602"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/categories?post=11602"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/maverickphilosopher.blog\/index.php\/wp-json\/wp\/v2\/tags?post=11602"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}