John Edser wrote:
>>JE:-
>>I would like to thank AW for his detailed
>>reply. However, we disagree on basics.
>>I cannot reply fully to AW since I
>>require an answer to the basic question below.
>>After AW has replied to this question, I
>>will fully respond to his post.
> 
> 
>>>JE:-
>>>Set intersection is the most basic process of pure mathematics.
>>>It represents a formalisation of how logic is applied within
>>
> mathematics.
> 
>>>Any two sets can be intersected. However, for any set elements
>>>to exist within the intersection the set elements must be equivalent.
>>>Equivalent means mathematically the same e.g. a = b means a
>>>is equivalent to b. Equivalence is simply, defined i.e. you are
>>>free to define set elements to be equivalent or not. The definition
>>>of set element equivalence lies outside of mathematics. As Godel
>>>noted, some propositions used within mathematics are not propositions
>>>of mathematics. The definition of set element equivalence is one
>>>of them.
>>
> 
>>AW:
>>This is, in a word, incorrect. I do not see that I can add anything to
>>my original comment above, but let's try. Intersection is NOT an
>>operation. Read that again. Intersection is a property.
> 
> 
>>JE:-
>>________________________________________________
>>If set intersection is just a _property_,
>>exactly what _process_ is it just a property
>>of?
>>________________________________________________
>>
> 
> 
> AW:
> I want to thank JE for the question, as it provided me with some points
> to ponder that I found interesting. In any event, I thought that this
> assertion (intersection a property rather than an operation) might come
> back to bite me. Though my argument may end in being hoist by its own
> petard, I'll try to explicate my thinking on the matter.
> I made the observation in response to JE's statement, in several posts,
> that sets of parental fitnesses are allowed to 100% intersect, from
> which I inferred that it would be possible to 50% (or some other
> percentage) intersect sets. This is clearly not possible, hence my
> statement that once sets are defined, their intersection is also
> defined - one cannot choose to 50 or 75 or 100% intersect them. In the
> quote above, JE says that any two sets can be intersected; my point is
> that once two sets are defined, so is the intersection - it is not a
> question of intersecting them, the intersection already exists.
> 
> JE:-
> propositions _outside_ of mathematics control the level of set
> intersection. Mathematically, only the following intersections are
> possible between a set A = 3 and set B = 5. Two equal, partly
> overlapping circles should be drawn out on a sheet of paper
> providing three separates areas. These areas from left
> to right are:
> 
> Area 1 represents  all the set elements in circle A.
> 
> Area 2 represents all the set elements in the intersection
> of both circles (in the overlap).
> 
> Area 3 represents all the set elements within circle B.
> 
> An empty intersection is represented by the numbers 3,0,5.
> Here, 3 is in area 1, zero in area 2 (the overlap)
> and 5 in area 3, respectively. All the possible intersections
> between sets A and B are represented by 3,0,5 or 1,2,3,
> or 2,1,4, or 0,3,2. Four possible self exclusive levels of set
> intersection  exist ranging from nothing in the intersection to
> a maximum of 3.
> 
> Maximum intersection is 0,3,2. Here all of set A is in set B,
> i.e. A is just a subset of B. The level of intersection between
> set A and set B can only be determined by the problem mathematics
> is being applied to and NOT just the mathematics.
>
This is the nub of the problem - you seem to be the only person who 
thinks this is true.

 From http://whatis.techtarget.com/definition/0,,sid9_gci838379,00.html>:
"Given two sets A and B, the intersection of A and B, written A INT B, 
is the set C of all elements that are in both A and B."

i.e. the mathematics tells you which elements are in the intersection 
(and hence you can count them).

 From http://www.shu.edu/projects/reals/logic/defs/sets.html>:
"A INT B: A intersection B is the set of all elements that are in both 
sets A and B."
Again, which elements are in the intersection is defined.

 From 
http://www.quantnotes.com/fundamentals/backgroundmaths/settheory.htm>:
"Intersection - Denotes the set of elements that are members of all the 
sets under consideration. "
And again, the set of elements in the intersection is explicitly 
defined.  There is also an example following the definition.


> 
> AW:
> Here we have two sets. Set A consists of the 3 offspring of parent A,
> and set B consists of the 4 offspring of parent B. Now Mother Nature
> looks at the set intersection. I say the intersection is the empty set,
> as the offspring of A are not members of set B and the offspring of B
> are not members of set A. JE gets around this by defining the offspring
> as "fitness elements", asserting they are equivalent, and then looking
> at the intersection. This makes no sense.
> 
> JE:-
> Weather this "makes no sense" was not the
> point under discussion. What was under discussion
> is: does mathematics ALLOW this intersection? The answer
> is YES. Why? 

Oddly enough, I would guess that you would both agree on his.  The 
difference is that AW then goes on to ask the question "given that an 
intersection is possible, does one actually exist?".  And his answer is 
that it does not (or, to be precise, it exists but is empty).  John has 
spent many months avoiding this question, apparently because he doesn't 
understand what an intersection is.  Or at least what the rest of the 
world thinks an intersection is.

Bob

-- 
Bob O'Hara

Rolf Nevanlinna Institute
P.O. Box 4 (Yliopistonkatu 5)
FIN-00014 University of Helsinki
Finland
Telephone: +358-9-191 23743
Mobile: +358 50 599 0540
Fax:  +358-9-191 22 779
WWW:  http://www.RNI.Helsinki.FI/~boh/
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