Hi Brian Tenneson Could be, but I'll stick with Kronecer.
Roger Clough, rclo...@verizon.net 9/6/2012 Leibniz would say, "If there's no God, we'd have to invent him so that everything could function." ----- Receiving the following content ----- From: Brian Tenneson Receiver: everything-list Time: 2012-09-06, 11:09:25 Subject: Re: Where do numbers and geometry come from ? Correct me if I'm wrong but my understanding is that sets and membership cannot be defined in terms of a more primary mathematical concept.? Functions can be defined in terms of this primitive called sets.? Numbers are sets; natural numbers are defined directly in terms of sets (via the Von Neumann approach) and every more complicated number set can be defined in terms of the previous type of number set all the way up to real numbers, complex numbers, and nonstandard number sets.? The only type of number I am not sure how they can be seen as sets is that of surreal numbers described by Conway I believe.? I don't know much about surreal numbers. Yes, this approach necessitates the existence of sets and membership. There probably are other ways to define numbers but all properties that we want numbers to have can come from how they are defined in terms of sets.? In other words, the set theoretical description of various number sets is sufficient. Kronecker said "God made the integers; all else is the work of man."? I would amend that to say God made sets (and membership); all else is the work of man. On Thu, Sep 6, 2012 at 7:45 AM, Stephen P. King <stephe...@charter.net> wrote: Dear Brian, ?? "can be defined ..." implies the necessary existence of something or process or whatever that does the act of defining the set. Truth values do not do this, btw. Sets are collections defined in terms of functions, but numbers in-themselves are not those functions.. Unless you are considering some other ideas of what sets are... If we are going to think of set as having ontological primacy we have to have a notion of a set that does not need a membership function. On 9/6/2012 10:28 AM, Brian Tenneson wrote: All numbers can be defined in terms of sets.? The question becomes this: do sets have ontological primacy relative to mankind or are sets invented or created by mankind? On Thu, Sep 6, 2012 at 5:11 AM, Roger Clough <rclo...@verizon.net> wrote: Hi Stephen P. King ? ? Yes, of course, but I wanted a more obvious, dramatic?xample. The philosophy of mathematics says something like the numbers belong to a static or eternal world, change?tself ?s a property of geometry. Numbers and geometry thus belong to the platonic world, which is forbidden or at least not consistent with the philosophy of materialism, IMHO. ? If numbers are platonic,? wonder what the? presumably materialist Steven Hawkings has to say about their origin in his recent book on numbers. ? ? ? Roger Clough, rclo...@verizon.net 9/6/2012 Leibniz would say, "If there's no God, we'd have to invent him so that everything could function." ----- Receiving the following content ----- From: Stephen P. King Receiver: everything-list Time: 2012-09-06, 07:53:18 Subject: Re: Could we have invented the prime numbers ? Dear Roger, ?? Could the mere possibility of being a number (without the specificity of which one) be considered to be "there from the beginning"? -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- 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. -- 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. -- 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.
