The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » sci.math.* » sci.math

Topic: Background Theory
Replies: 4   Last Post: Dec 8, 2012 6:13 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 2,665
Registered: 6/29/07
Re: Background Theory
Posted: Dec 8, 2012 3:57 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On Dec 7, 7:21 pm, fom <> wrote:
> On 12/7/2012 8:46 AM, Zuhair wrote:

> > One might wonder if it is easier to see matters in the opposite way
> > round, i.e. interpret the above theory in set theory? the answer is
> > yes it can be done but it is not the easier direction, nor does it
> > have the same natural flavor of the above,
> > it is just a technical formal piece of work having no natural
> > motivation. Thus I can say with confidence that the case is that Set
> > Theory is conceptually reducible to Representation Mereology and not
> > the converse!

> I have no doubt that you are correct.  In another post
> in your thread I summarized the work of Lesniewski which
> uses the part relation to characterize classes.  His
> method was specifically designed to circumvent the
> grammatical form that leads to Russell's paradox.
> Moreover, the historical record of the philosophy sides
> with you.  Aristotle, Leibniz, Kant, and Russell all
> have had positions asserting how parts are prior to
> individuals.
> The issue becomes, however, whether the theory informs
> differently.  I turned to the investigation of parts
> without any knowledge of mereology because I believe
> that the ontological interpretation of the theory of
> identity is inappropriate for foundational purposes.
> Find any distinction in the literature between the
> application of logic to a linguistic analysis and
> the linguistic synthesis of a theory.
> That is why my proper part relation has a the
> character of a self-defining predicate.  There must
> be a first asserted truth, and, it must be an
> assertion that constructs the language.
> The second sentence has parallel syntax to the
> first.  It took me a long time to understand
> why I should accept that situation.  In Liebniz'
> logical papers, one finds the remark that individuals
> should be identified with a mark.  The membership
> relation is precisely that relation which marks an
> individuated context.  It does so in relation to
> a plural context.
> The priority of the part relation as it applies
> to set theory is that the fundamental relation
> between objects of a domain should reflect the
> object type of the domain.  To the extent that
> sets are
> "collections taken as an object"
> the fundamental relation should relate collections
> to collections.  To the extent that a set is
> "determined by its elements"
> requires that the individuated context be situated
> relative to the defining syntax of the fundamental
> relation.
> One of the primary focuses of my investigation
> was to understand the principle of identity of
> indiscernibles.  If you read Leibniz, you will
> find that what is taught about the identity of
> indiscernibles is not entirely representative
> of what Leibniz said.
> He attributes his motivation to a position held
> by Thomas Aquinas and explains that the typical
> formulation is a consequence of the phrase
> "an individual is the lowest species"
> This is best interpreted mathematically by
> Cantor's nested closed set theorem with vanishing
> set diameter.
> The following assumption:
> Assumption of Aquinian individuation:
> AxAy((Az(ycz -> xez) /\ Ez(xez /\ -ycz)) ->
> Az((xez /\ -ycz) -> (Ew(xew /\ wcy) \/ Aw(zcw -> ycw)))))
> can be used to define membership
> AxAy(xey <-> (Az(ycz -> xez) /\
> Az((xez /\ -ycz) -> (Ew(xew /\ wcy) \/ Aw(zcw -> ycw))))))
> relative to a prior proper part relation "c".  In both
> sentences, the complexity arises from capturing the sense
> of a nested sequence of sets.
> To be quite honest, here, I still need to consider
> these expressions further.  These universals are
> difficult because both the assertion and its
> negation must be considered.
> So, let me suggest that while you continue
> to refine your ideas, you also develop some of
> the claims you have made concerning
> the representations of arithmetic and such.
> Do not take these remarks wrong.  I see that you
> are reading other authors and developing your
> ideas.  So, please continue.  I would like to
> see more.
> Inform us.

I just wanted to note that with this approach a set is a singular
entity that
represent a plurality of singular entities, while with Lewis's
a set is a plurality of singular entities that is represented by a
entity. I see this approach reductive while Lewis's diffusive. However
formally speaking they almost mirror each other, but conceptual wise
I think the approach given here is more faithful to the general
in which sets are mentioned.


Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2017. All Rights Reserved.