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by twangist
3776 days ago
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They're quite as real as nonnegative numbers, which are also abstract things. It's ironic that in your last line you use the terms "real" and "imaginary", in their colloquial meanings, as these have long since been adopted and coopted to denote... the real and complex numbers respectively :) which, some of us contend, are no less real than the integers. Face it: groups abound in physics. A group has a binary operation such that for every element has a (unique) inverse element w.r.t. that operation. When the group is Abelian we write the operation as "+", and then the "additive" inverse of an element x is -x, also an element of the group. It's silly to claim that, for example, rotation through an angle $\theta > 0$ should be considered more "real" than rotation through $-\theta$. One is counterclockwise, the other, clockwise, by the same amount. |
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It's true that nonnegative numbers are just abstract ideas that we come up with to represent quantity, but that's just it: quantity is a real thing. Is there such a thing as negative quantity? That's my question.
I like the reference to the contention of what we currently denote as imaginary numbers and how they should be compared as just as real as other numbers. This is the same issue I'm questioning but from the other side. I'm questioning if we should count negative numbers as real as opposed to understanding them as a set that is at least imaginary, if not complex or irrational.
I also like the point about a counterclockwise rotation being just as real as a clockwise rotation. This is where the non/negative sign is used to indicate a vector. As long as you have a point of reference, this is always ok because it's simply referring to what type of amount of whatever quantity/magnitude/etc we're dealing with.
But I believe this is different than what we often do with negative numbers when working through many of our problems that involve unitless numbers/constants or take an abstraction and apply it as a physical theory - in these cases the negative numbers were treated as separate and real when in fact they should have been directions, unit movement, or other information about actual physical quantities. This is where I'm wondering if we should treat negative numbers as a different set altogether.