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>> But how do you derive the count from the weight if you have never
weighed a known number of cans? How do you know the number of cans
without counting any? How do you count cans--even one can--if cans
are unidentifiable. <<
I can also work with just the weight of the bag without any reference to an indivdual element ("I bought 20 pounds of canned cat food on sale!" or simply "I bought a big bag of canned cat food on sale!").
All I need is to distinquish -- rather than count -- one can, then make the assumption that this separate entity has "the weight of one can" and that all cans in the bag have the same weight -- basically, the inductive hypothesis, but for unordered collections. Distinction is a more primitive operator than counting.
>> Yes, I recall. You argued that a sum over zero elements is
undefined, and I argued that it is defined as zero. <<
For the lurkers, my original article is archived at this url and I will not do a monster "cut & paste" on it.
http://www.dbmsmag.com/9611d06.html
>> You cited a book to defend your position and that book said in
black and white that it is defined as zero. <<
I also gave a few thousand words, some short programs and illustrations to support that convention.
That was exercise one of chapter 2, page 62, of Concrete Mathematics, by Graham, Knuth, and Patashnik (Addison-Wesley, 1994) where they stated that "the generally accepted convention that the sum of n values = 0 when n = 0" then the authors also make a point that empty index sets are not defined and that this is strictly a convention, used because it works with original the summation of a series -- the procedural stuff we did before we knew about sets.
They then have an exercise at the end of that chapter that asks the student to give several interpretations of some "Big Sigma" expressions with index start and stop values that do not define a proper series of cardinal numbers. They then announce which ones they prefer.
Going back to section 1.2.3, "Sums and Products," exercise 23 of Knuth's The Art of Computer Programming, Vol. 1, "Fundamental Algorithms" (Addison-Wesley, 1973), the author asks the student to explain why it is a good idea to define the summation and product of an operator with an empty set as zero and one respectively. The argument was that you can simplify procedural code. Without a "missing value" token or to handle sets as a whole, your other choice is to raise an exception.
>> You were an idiot then, and I have seen nothing to indicate you
have changed. <<
Thank you for that carefully reasoned argument <g>. Received on Sat Feb 22 2003 - 13:06:58 CST