Jesse Mazer wrote: > > > > Date: Tue, 9 Jun 2009 12:54:16 -0700 > > From: meeke...@dslextreme.com > > To: everything-list@googlegroups.com > > Subject: Re: The seven step-Mathematical preliminaries > > > > > You don't justify definitions. How would you justify Peano's axioms > as being > > the "right" ones? You are just confirming my point that you are > begging the > > question by assuming there is a set called "the natural numbers" > that exists > > independently of it's definition and it satisfies Peano's axioms. > > What do you mean by "exists" in this context? What would it mean to > have a well-defined, non-contradictory definition of some mathematical > objects, and yet for those mathematical objects not to "exist"?
A good question. But if one talks about some mathematical object, like the natural numbers, having properties that are unprovable from their defining set of axioms then it seems that one has assumed some kind of existence apart from the particular definition. Everybody believes arithmetic, per Peano's axioms, is consistent, but we know that can't be proved from Peano's axioms. So it seems we are assigning (or betting on, as Bruno might say) more existence than is implied by the definition. When Quentin insists that Peano's axioms are the right ones for the natural numbers, he is either just making a statement about language conventions, or he has an idea of the natural numbers that is independent of the axioms and is saying the axioms pick out the right set of natural numbers. Brent --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---
