Doug (01)
It is not uncommon when characterising mathematical systems to
start with one set of axioms and demonstrate that another set of axioms
is equivalent. From (unreliable) memory, for the integers one might use
either
zero is less than one
or
zero is not equal to one (02)
The concept of "more basic" does not apply here. (03)
With regard to excluding and groups, fields from basic mathematics, that
is equivalent to excluding numbers from basic mathematics, which might
well confuse people. I was taught algebra by starting with sets and
relations, and then gradually adding axioms, which generated more and
more structure in the system. One could generate a lattice of theories
simply by adding different axioms or adding them in different orders.
However, most of the systems resulting are not particularly interesting. (04)
Might I suggest following up on the group of mathematicians that
published under the name of Nicolas Bourbaki. (05)
Sean Barker
Bristol,
> Original Message
> From: ontologforumbounces@xxxxxxxxxxxxxxxx
> [mailto:ontologforumbounces@xxxxxxxxxxxxxxxx] On Behalf Of doug foxvog
> Sent: 03 February 2010 17:12
> To: [ontologforum]
> Subject: Re: [ontologforum] Foundation ontology, CYC, and Mapping
>
>
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> 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.
>
>> Examples of incompatible theories: the axiomatic set theories of
>> mathematics.
>
> Why shouldn't these be able to be described in terms of more basic
> elemets?
>
>> How they are logically incompatible: Search Google on "axiomatic set
>> theories incompatible" for numerous proofs.
>
>> 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.
>
> 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.
>
> 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.
>
>  doug
>
>> Rob
>
>
>
> =============================================================
> doug foxvog doug@xxxxxxxxxx http://ProgressiveAustin.org
>
> "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.
> =============================================================
> (06)
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