Re: Mathematical objects and Discernment


LauLuna wrote:
I'm not sure I have quite understood you, but, for all I can gather,
you propose to conceive of sets not really as collections
(extensionally) but in a pure intensional way: a set should perhaps be
identified with the concept its name includes? I assume all names
including the same conceptual content are to be considered the same
name (?).

You discard the first (so to say) axiom of the usual set theory.

If I have correctly understood you, your proposal seems highly
problematic. Sets are extensional from their very definition. If we are
to disregard its extensional nature, what is the use of keeping them;
can we not do with just concepts (or names)?

Regards

It is worth staying with this because I came to the same conclusions
about sets as you did by a different route but from a common source:
The idea of a set as being comprised of intensional objects and the
useful application of this idea to tautologies and paradoxes in current
set theory, was a leap you made in practical reasoning and a
clarification, and I don't doubt that if it is chased up it will be
worth the effort. This leap was made upon my idea that 'collections'
are comprised of objects of our attention. Further, I argued that as a
list of objects of our attention (or intentional objects), a collection
has no other hidden parameter that corales its objects together. What
brings a collection 'together' is our attention; or, as you say,
intentional objects comprise a set. That is, sets are not extensional
but comprised of objects of our attention, or intentional objects.
There are no frameworks or further properties to be considered. Do your
amplifications not suggest that?

(I note that a collection is different from a group. In a group the
properties of the elements are conferred upon the group. So for
example, a group of cows will show the property of a herd, whereas a
set (set/collection) of cows will not. So, in a group we find
relationship, but we find no relationship in a set. I also note what
happens to mathematical objects in a set. Outside of their application
a number becomes a numeral, so we would expect that there can be no
sets of numbers unless the application for making them is presented in
the name of the set. However, in that case, we cannot differentiate
between sets and formulae. Concerning concepts, a concept does not
differentiate between an idea and its representation as particular
objects. So concepts have extension. A set, on the other hand although
it treats all its elements as objects, it is only through the name of
the set that it does this: the elements themselves are empty marks...)

My question for you is this: If sets/collections are intensional
objects, how would you differentiate between these and their extensions?

.

Relevant Pages

  • Re: Mathematical objects and Discernment
    ... I understand that a set is its extension when conceived of as a single ... are comprised of objects of our attention. ... list of objects of our attention (or intentional objects), ... intentional objects comprise a set. ...
    (sci.logic)
  • Re: y^3 -y + 1/8 = 0 Some intellectual muscle sought here
    ... attention by our students:- ... Can anyone help me and by extension him/her throw light on this. ... I mean really help and not to be issuing effortless academic fatwas ... that do not in any way lend themselves to a solution. ...
    (sci.math)
  • Re: Best Match
    ... each addin identifies its own descriptor for the beans so it would ... this is what lead me to the idea of each addin providing its ... > If the UI can differentiate, then you should definitely store these ... >> thinking of each extension of the main app to provide its own piece of ...
    (microsoft.public.sqlserver.datamining)
  • Re: const in place of #define
    ... The lcc-win compilers implements this as an extension, ... your code since (in my opinion) it should work. ... How does it differentiate between an array and a VLA in this case? ...
    (comp.lang.c)