Rob Freeman wrote:
>
> Yes, my key doubt is that "incompatible theories can be described in
> terms of common, more basic elements".
>
> On Tue, Feb 2, 2010 at 12:02 PM, Patrick Cassidy <pat@xxxxxxxxx> wrote:
>> ...
>> ... If you doubt that incompatible
>> theories can be described in terms of common, more basic elements, try
>> presenting some incompatible theories (and show how they are logically
>> incompatible) and I will show how it can be done. (01)
> Examples of incompatible theories: the axiomatic set theories of
> mathematics. (02)
Why shouldn't these be able to be described in terms of more basic
elemets? (03)
> How they are logically incompatible: Search Google on "axiomatic set
> theories incompatible" for numerous proofs. (04)
> If you want to build me a FO, can you build me a FO of maths first.
> Show me how all theories of maths can be derived from a single theory. (05)
The theories include more than the terms which are used to define them.
Although the theories are not DERIVED from a single theory, they should
be able to be DEFINED using terms less rigorously defined. Parallel
lines can be defined as two straight lines in the same plane that never
meet. One subtheory may state that there is exactly one parallel line to a
given line through any given point outside that line. A second may state
that there are none, while a third may axiomatize that there are an
infinite number. None of these three subtheories can be DERIVED from the
more basic one, but they can all be DEFINED using its terminology. (06)
There will be terms defined in subtheories which are not included in
the more general ones. A basic theory of mathematics would not include
groups, fields, transfinites, or complex numbers. The appropriate concepts
would be defined in the appropriate contexts, but defined using the
terminology of the more general context or using locally defined terms
so defined. (07)
-- doug (08)
> -Rob (09)
=============================================================
doug foxvog doug@xxxxxxxxxx http://ProgressiveAustin.org (010)
"I speak as an American to the leaders of my own nation. The great
initiative in this war is ours. The initiative to stop it must be ours."
- Dr. Martin Luther King Jr.
============================================================= (011)
_________________________________________________________________
Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/
Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/
Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx
Shared Files: http://ontolog.cim3.net/file/
Community Wiki: http://ontolog.cim3.net/wiki/
To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J
To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx (012)
|