what is an operator?
you've seen them before: +*-/^ are common examples.
operatos map sets unto other sets. for example a + b the addition operator is mapping the addition of a and b unto a single quantity. but what is addition? we all know how to do it but that's not the same thing as knowing what it is. we can define a lot of concepts in arithmetic by startng at the beginning: 0.
then we can define another number/ quantity as its succesor. we invent the operator k. so that 0k is 1 and 0kk is 2 then we can define addition as m+n as n succesions to m and multiplication as j*k as k additions of j. the reader can prove commutativity laws for all these by noticing that the fifth succesor to one and the first succesr to 5 are the same quantity. we can define powers as g^h as h multiplications of g. this however is not commutative.
what's the next step? what does y powers of x. what's before the succesor operator. can we generalize all these orderings to any order? (valga la redundancia).
I think about this sometimes thanks to m. shachar.
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2 comments:
good stuff abraham,
is this from like a class, or do you just like to read about math?
the second one i guess. i didn't read about it though me and the guy mentioned in the end were talking about that the other day
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