On Sat, 7 Feb 2004 05:52:54 +0000 (UTC), "John Edser"
 wrote:

>
>
>>JM:-
>>...C' contains only two things, not four.  Those two things are both
>>sets, and each of them contains two of the original objects.  The set
>>"container" does not "dissolve" when a set
"moves inside" another set.
>
>>JE:-
>>The set container inside, does dissolve when sets are
>>joined by set union. This is a critical difference
>>between union and intersection. AW severely criticised me when
>>I suggested that sets merged by set union lose their integrity
>>because of this fact. Set union is a non reversible logic
>>but set intersection is a reversible logic. Understanding
>>this difference is critical for any application of set theory
>>to evolutionary theory. If independent sets of fitness are merged
>>by set union they lose their fitness independence but this
>>is 100% preserved when fitness sets intersect.
>>Can I make a suggestion?
>>Could AW and/or JM please supply a valid example
>>of any applied set intersection where all of one set
>>is a subset of the other, i.e. provide a simple problem
>>and then show how set theory correctly illustrates this problem
>>where  the illustration shows one set as a subset of the other.
>>Could you please point out why you think the intersection
>>was mathematically valid in the example provided.
>
>RN:-
>I find it extremely difficult to understand just why John Edser has so
>much trouble with elementary set theory.  There is no big conceptual
>difference between set union and set intersection as he claims.  There
>is no such thing as a "set container".
>Consider the sets  S1 = { a, b, c } and S2 = { a, b, d }.
>
>
>JE:-
>We were attempting to represent a _biological_ problem
>using set theory, specifically: illustrate one selective
>event. I represent this event as the total intersection
>between a minimum of 2 independent parental fitness
>sets, say: A = 3 B = 4.  Two overlapping Venn circles
>are drawn where the 3 areas from left to right contain:
>
>		0,3,1
>
>Set A has all of its fitness elements in the
>intersection. Set B has only 3 of them.
>
>Thus:
>Set A is a subset of set B so nature
>only has just one set to select: set B.
>Thus no cognition is needed to make
>a selection so “automatic selection”
>becomes logically explicable.
>When set B is selected set A is also
>selected because it is a subset of B.
>But set A is only selected _after_ and
>not _simultaneous_ to, set B. Thus
>B is only sub selected after A is
>selected.
>
>_______________________________________
>The main dispute here is about the
>validity of my proposed set intersection
>between independent parental fitnesses
>set totals within the same Darwinian
>population.
>________________________________________
>
>The “set container” refers to the edge of _separate_ sets,
>that is all. Unless sets can be represented as
>separate then set theory cannot even exist. When representing
>independent set of _fitness_ it is of upmost importance to
>represent a barrier between independent sets because
>they are all competing against each other within the defined
>universal set: one Darwinian population.  Clearly,
>if all fitness sets are competing but you only represent
>them as the one universal set of fitness by merging all of
>them using set union (this is the normal Neo Darwinian way
>of representing  absolute fitness) then only zero competition
>can now exist because only one set exists and this set cannot
>compete (be compared) against itself. However if you represent
>them as intersections they can all be compared to each other as
>independent sets of fitness because were NOT merged.
>
>RN:-
>Then the intersection of S1 and S2 is simply the set { a, b } and the
>union of S1 and S2 is the set {a, b, c, d }.  Taking the union or
>taking the intersection do absolutely nothing to the original sets.
>They still exist unchanged. The intersection and the union are brand
>new sets, not modifications of the original.
>
>JE:-
>Please explain the logical _difference_ between set
>union and set intersection. If they are logically the
>same then it must be possible for one set to become a
>subset of the other using set union, but it isn’t.
>If I compare sets of fitness I do nothing to each set yet.
>However a comparison can represented using set intersection because
>the  sets are _not_ merged, they _remain_ separate. I can’t compare
>independent  sets of fitness using set union because the sets
>are lost when they are merged and merging them provides
>no information re: their relative size difference. Set union is not
>logically the same as set intersection. Adding to a total is set
>union. Comparing totals is set intersection. Making and comparing
>totals are not logically the same thing.
>
>RN:-
>Now consider the case requested where one set is a proper subset of
>the other:  S1 = {a, b, c } and S3 = {a, b}.  Now the intersection of
>S1 and S3 is {a, b} which is equal to S3 while the union of S1 and S3
>is {a, b, c} which is equal to S1.  The fact that the union and the
>intersection happen to be equal to (contain the same elements) as one
>of the originals has absolutely no signficance on whether the original
>sets are somehow modified -- they are not.
>What is the big deal?  Why has there been a thread about set
>intersection for all these many months?
>
>JE:-
>Nobody can agree re: exactly HOW independent
>sets of fitness can be represented within set
>theory to illustrate one selective event. Since
>one selective event is just a default comparison
>of two independent fitness totals, then this must
>be able to be illustrated using set theory.
>
>Please provide what was requested:
>an application of set theory, OUTSIDE OF
>MATHEMATICS, that allows one set to
>become a subset of the other as a valid set
>theory illustration of that specific problem,
>using set intersection.
>
Frankly, I have no interest in getting involved in your dispute over
how you represent "independent sets of fitness".  It is simply that
you are either misinterpreting or misusing the notions of set theory.
Not everything in the work must necessarily be able to  be illustrated
using set theory.  Especially, if you define the problem in a totally
inappropriate way, set theory is inappropriate or improperly used.

Of course I didn't say that set intersection and set union are
identical.  I said they shared many features including the fact that
neither of them has anything at all to do with the notion of a "set
container", whatever that may be.  On both cases, you act on the
elements of two separate sets to produce a new set containing elements
selected from the two original sets.  The difference is in which
elements you select.  The similarity is that an operation on two sets
yields a new set  Nowhere does the notion of boundaries arise.

When you compute  set union, the sets are NOT merged.  The sets are
NOT lost. 

Set theory is part of mathematics.  You are simply abusing mathematics
by improperly defining sets to try to model your notion of fitness.
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