On 9/6/2012 11:09 AM, Brian Tenneson wrote:
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.
Hi Brian,
Surreals and hyperreals and non-standard numbers and so on, the
list is long! My point is that there really is no such thing as an
absolutely irreducible entity.
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.
Depending on what intends to try to explain, but sure.
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.
Balderdash! We can use the God concept as a way to capture the sum
of what exists and its evolution and so forth, but it is just another
word that may not refer to anything that really exists.
On Thu, Sep 6, 2012 at 7:45 AM, Stephen P. King <[email protected]
<mailto:[email protected]>> 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 <[email protected]
<mailto:[email protected]>> wrote:
Hi Stephen P. King
Yes, of course, but I wanted a more obvious, dramatic example.
The philosophy of mathematics says something like the numbers
belong to a static or eternal world, change itself is 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, I wonder what the presumably
materialist
Steven Hawkings has to say about their origin in his recent
book on numbers.
Roger Clough, [email protected] <mailto:[email protected]>
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 <mailto:[email protected]>
*Receiver:* everything-list
<mailto:[email protected]>
*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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