Re: justifying fuzzy sets

[Goran Topic <topic@STUDENT.MATH.HR>]

> If someone wants to use numbers to describe an experience, that's
> fine, but the numbers are not being used as *numbers*.  They are being
> used as symbolic representatives for adjectives.  I bet that any of your
> patients could give a description, in words, of what 4-pain is, and what
> 8-pain is, but to conclude that 8-pain is exactly twice as "bad" as
> 4-pain is absurd... pain is a complex phenomenon, and not numerical in a
> simple, scalar fashion.
(li'o)
> However, you don't treat these numbers
> as numbers.  You can't do math with them.  You can't do statistics with
> them.  Everyone realizes this, except the fuzzy logic folks who think that
> a person who is a "10" is twice as good looking as a person who is a "5".

Well, numbers as treated in mathematics are not absolute. 2+2 are not
necessarily 4, just usually. I know of perfectly nice examples where for
all x, x squared is x. We usually work with the field of real numbers;
but ignore the higher properties and take just the set Z(n) (or I(n), I
don't know what is the usual English notation for the group of first n
integers, starting from 0, and the isomorphic groups), not as a group
but as a set. A function with such a co-domain serves only as a classifier.
You can do statistics with it, but not much of it... About the only
thing you can do is to calculate the distribution. Nothing else makes
sense. For each individual such function, you can define an arbitrary
operator that models the reality the way you want it to... Mathematically
all very correct. The thing you cannot do is to take the operation of
addition intuitively. For example, the set {0,1,2,3} is isomorphic with the
set {no, little, much, all}, but, although you see that three times as much
hair as "little" is not neccessarily "all", you argue that fuzzy
logicians say that 3 times 1-hair is 3-hair. The error is not in numbers,
but in operation, "times". If you say 3 times x, you most probably mean
x+x+x. If you don't have addition defined, this yields nonsense.

I've gone ba'e so out of topic...

co'o mi'e. goran.

--
GAT/CS/O d?@ H s:-@ !g p1(2)@ !au(0?) a- w+(+++) (!)v-@(+) C++(++++)
UU/H(+) P++>++++ L(>+) !3 E>++ N+ K(+) W--(---) M-- !V(--) -po+ Y(+)
t+@(+++) !5 !j R+@ G-@(J++) tv+(++) b++@ D++ B? e+* u@ h!$ f?(+) r--
!n(+@) y+. GeekCode v2.1, modifications left to reader to puzzle out