Date: Nov 28, 2012 5:33 AM
Author: Robert Hansen
Subject: Re: Some important demonstrations on negative numbers
On Nov 27, 2012, at 11:21 PM, Joe Niederberger <email@example.com> wrote:
> Your "reasoned thought" verbiage yet again. Mere repetition is neither reason nor thought.
I don't understand why you are having such a problem with this. When I work out a complicated problem or proof, entirely symbolically, never once referring to or thinking of something "concrete", then how can I be using common sense? Or how about realizing that dividing a fraction by a fraction is the same as multiplying it by its reciprocal simply because algebraic reasoning says it should work that way. That after I think about it for awhile, no other interpretation would fit. No references to pizzas or pictographs. Just algebra. Or how about the examples in this thread, multiplying two negative numbers or the use of fractional exponents. Where is the common sense in all of that? Clyde's theory that people don't understand this stuff because we don't concoct enough common sense examples is just another way of saying "People don't understand this stuff!" By the time you get to these oddities, if you don't already have a theory of algebra that can stand on its own, separate from your "common sense", then you are way outside your zone of understanding. Or how about those metaphors of yours regarding imaginary numbers? Or do you still protest me calling them "metaphors"? They are not even analogies, like what many concrete examples are, they are essentially mnemonics. Aids invented by instructors to help newbies (and oldies) keep the terms straight, hopefully long enough for the real reasons to sink in. You can either work with a god awful collection of trigonometric (cyclic) expressions or you can use complex arithmetic. You decide. What if I gave you some arbitrary mathematical situation involving trigonometric relationships, void of any concrete physical examples, and asked you to simplify the whole thing by refactoring it using complex numbers? Are you saying that you would fail because you wouldn't have anything concrete with which to guide you? What about when we employ coordinate transformations or switch to parametric representations? Is that just more of that common sense?
I am not suggesting that it is mere coincidence that the concrete world is held accountable to the same mathematics and logic we strive to understand so deeply. If the world operated according to some other set of principles then I am sure that those would be the principles that we would strive to understand so deeply. But our understanding of those principles is not grounded in our common sense perception of the world. That is where we all start (where else would we start?) but the destination (mathematics) and all of its elements are entirely imagined (abstract) and must therefore be grounded in an ability to work with imagined things. This ability to not only work with imagined things but to build a whole world of imagined things, consistent unto itself, is what I call reasoned thought. Very distinct from common sense, which is merely perception, not thought.