This equation, and an unanswered question about it, has fascinated me most of my life: a * b = exp(log a + log b) Equivalently: a + b = log(exp a * exp b)

This equation, and an unanswered question about it, has fascinated me most of my life: a * b = exp(log a + log b) Equivalently: a + b = log(exp a * exp b) The reason I like this is that it is the numerical equivalent to De Morgan's Law of Boolean logic: a and b = not(not a or not b) Equivalently: a or b = not(not a and not b) They are both dualities, where AND and PRODUCT can be thought of as intersections, with OR and PLUS being unions. However their is one crucial difference, in the Boolean case the duality is based on a symmetric operation, in the numerical case they are not. a = not b, b = not a a = log b, b = exp a It is very interesting that if you: 1) take a set of symbols, then define various adds and subtractions of those symbols you will get some result, which will be a sum of positive/negative versions of those symbols, 2) then replace all your additions and subtractions with multiplications and divisions of the same symbols, ...you will get an answer with exactly the same form, and with the same symbols, except with products/divisions instead of additions/subtractions. In isolation, addition and product behave exactly the same way! We could call the addition/multiplier operator the combiner operator and the subtraction/divisor operator the remover operator, and forget about whether we were doing addition or multiplication and would still get the right answer either way. It is only when you combine BOTH additions with multiplications in equations, that they operate different RELATIVE to each other, with the LOG/EXP relation "adjusting" values between their additive to multiplicative forms and back. INSIGHT So add and multiply are the same operations, but on values that are either the log or exp version of themselves respectively. INSIGHT TAKES US TO AN ODD PLACE Looked at that way, numbers live on an "Addition Ladder" where numbers start at the bottom rung where addition happens and to move up to multiplication we apply EXP before continuing to combine them with addition and then LOG them to get back to our home rung again. Now that we have two rungs, why not operate at more levels up (EXP EXP) or below (LOG). The math is easy to do this, but for some reason nature doesn't seem to give us much use for "super-multiplication" or "sub-addition". MY UNANSWERED QUESTION Why doesn't nature use super-multiplication or sub-addition? Or does it somewhere and I am just not aware of it? * If anyone has any insight on this, please pass it on! nevermark.mail at marks.house