Of course, this is a result of our interpreting the definition of magnitude in an "operational" sense, as opposed to a "quantitative" sense. The difference is enlightening. Hestenes traces the history of mathematical development with respect to the Clifford algebras and the geometric product, which Clifford developed by combining the algebraic ideas of Grassmann and Hamilton, and he explains the meaning of the two interpretations that was uncovered by Clifford:
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Clifford may have been the first person to find significance in the fact that two different interpretations of number can be distinguished, the quantitative and the operational. On the first interpretation, number is a measure of 'how much' or 'how many' of something. On the second, number describes a relation between different quantities.
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It is the operational relation between the units of the space/time progression that constitutes the magnitude of scalar motion, and it is only because of this that the possibility exists that it can have two "directions." The two "directions" may be designated positive and negative, up and down, left and right, or whatever. The important thing to understand, however, is that the special meaning of "direction," with respect to this magnitude, has a scalar meaning of "direction" that is not geometrical. For this reason, the distinction will be made between the two by placing quotation marks around the word to indicate its scalar meaning, as opposed to its geometric meaning.
There is a point of deep significance here. The story is told of a young school girl in a poor Welsh village who, when she was introduced to negative numbers, "got into a crying jag:" [5]
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Hanmer school left its mark on my mental life, though. For instance, one day in a grammar school maths lesson I got into a crying jag over the notion of minus numbers. Minus one threw out my universe, it couldn’t exist, I couldn’t understand it. This, I realised tearfully, under coaxing from an amused (and mildly amazed) teacher, was because I thought numbers were things. In fact, cabbages. We’d been taught in Miss Myra’s class to do addition and subtraction by imagining more cabbages and fewer cabbages. Every time I did mental arithmetic I was juggling ghostly vegetables in my head. And when I tried to think of minus one I was trying to imagine an anti-cabbage, an anti-matter cabbage, which was as hard as conceiving of an alternative universe.
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I think this anecdote ought to included in every textbook for elementary math teachers. I had a similar experience when I was introduced to imaginary numbers by an inept teacher who couldn’t, or didn’t bother, to adequately explain the idea behind the “number” ‘i’. What I have learned since is that, just because we can grasp an idea abstractly, by divorcing it from physical concepts, it doesn’t necessarily mean that we understand the meaning underlying the concept, only that we can understand how to use it as a method.
For instance, the idea of negative space is absurd, but, nevertheless, we can use it to great advantage. However, it took mankind centuries to take this giant leap for the first time, and it was not done without a lot of hand wringing and pain. Fortunately, I’ve finally found that we can understand the meaning of the idea as well as how to use it abstractly, but only if we are willing to ferret it out by thinking on our own, definitely not by reading textbooks. The key is to understand that space doesn’t exist as stuff that has properties, like cabbages, or fabric, and that the early Greeks were wise in keeping the ideas of magnitude and numbers separate.
It turns out that the ideas of operationally defined magnitudes as opposed to quantitatively defined magnitudes, and the proper use of real numbers in these respective definitions, can help us avoid a lot of grief, not only for naïve school children, trying to learn mathematics, but also for the sophisticated adults they later turn into, who then try to formulate physical theories.
Under the current definition of space, as a set of points satisfying the postulates of geometry, negative space doesn't exist. But under the new definition of space, as the reciprocal aspect of time, in the equation of motion, a negative magnitude of motion does exist, and just as positive space can be generated by positive motion, so inverse space, or time, can be generated by negative motion. But this is getting ahead of ourselves. The important thing to understand now is that two "directions" of scalar motion exist in the universe of motion, under the new definition.
Not only is it important to understand the two possible "directions" of scalar motion, but it's also very important to understand that the datum for these possible magnitudes, the "zero" reference from which they are measured, is 1/1, not zero. The easiest way to keep this in mind is to think of the motion as an old fashioned pan balance, where equal weights on either side balance out, so that 1:1 is actually zero, and 1:2 or 2:1 is a magnitude of one, in two different "directions."
Larson calls this unit motion, where the space/time progression ratio is balanced, the natural reference system. It is an absolute reference system from which magnitudes of scalar motion may be reckoned, providing, at long last, the widely sought basis for a background free definition of motion.
References:
5) Sean Carroll, "Minus Numbers," preposterousuniverse blog,
http://cosmicvariance.com/2005/08/03...bers/#comments