`At 02:45 PM 7/2/2004, Jesse Mazer wrote:`

As for the non-constructivism definition, is it possible to be a non-constructivist but not a mathematical realist? If not then these aren't really separate definitions.

It may be that all non-constructivists are mathematical realists, but some constructivists are mathematical realists as well (by my definition of "mathematical realism"). So "Platonism == mathematical realism" and "Platonism == non-constructivism" are two different statements. I can imagine a non-constructivist asking "Are you a Platonist?" (thinking "Do you accept the law of excluded middle?"), and a constructivist answering "Yes." (thinking, "yes, valid constructive proofs are valid whether or not any human knows them or believes them.") This miscommunication will lead to confusion later in their conversation.

It may be that all non-constructivists are mathematical realists, but some constructivists are mathematical realists as well (by my definition of "mathematical realism"). So "Platonism == mathematical realism" and "Platonism == non-constructivism" are two different statements. I can imagine a non-constructivist asking "Are you a Platonist?" (thinking "Do you accept the law of excluded middle?"), and a constructivist answering "Yes." (thinking, "yes, valid constructive proofs are valid whether or not any human knows them or believes them.") This miscommunication will lead to confusion later in their conversation.

-- Kory