# Re: Re: Where do numbers and geometry come from ?

```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
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

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
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
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