GS Chandy
Posts:
4,348
From:
Hyderabad, Mumbai/Bangalore, India
Registered:
9/29/05
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Re: Some important demonstrations on negative numbers
Posted:
Nov 28, 2012 6:49 AM
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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
> <niederberger@comcast.net> 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
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