Responding to Robert Hansen's (RH's) post of Nov 28, 2012 4:03 PM (pasted for reference below my signature):
I agree whole-heartedly. When you, RH, do anything whatsoever, you are NOT using 'common sense'.
GSC ("Still Shoveling Away!")
Robert Hansen posted Nov 28, 2012 4:03 PM: > > On Nov 27, 2012, at 11:21 PM, Joe Niederberger > <firstname.lastname@example.org> 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. > > Bob Hansen