For example, if money leaves my account (negative number) it doesn't disappear or go into a universal money vacuum somewhere, it just goes (positive number) to another account. Same for body temperature, photon emission, or anything else in the universe - don't the laws of conservation mean that negative numbers are just measurements for movement between systems but there's never an real "subtraction" of anything anywhere anytime?
And if negative numbers really are contrived to help us deal with systems in isolation, then have we hamstrung our fundamental mathematics by treating them as real numbers instead of imaginary?
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.
In physical applications, relative to a coordinate system negative numbers have perfectly intuitive meanings. A negative spatial coordinate simply indicates a point in one of the quadrants or octants other than the all-positive-coords one. A negative acceleration means something is slowing down.
I see no good purpose served by special-casing "negative" integers, and many needless ensuing headaches.
BTW, read a little more about number sets. There are such sets as "real numbers" and "imaginary numbers". I think you can find some material in mathematics textbooks for ground school and high school (probably not about imaginary numbers, though).
Real and imaginary number sets are the heart of the point (and I think my high school calc - if I can remember that many years ago - did include some work with imaginary numbers, certainly by early college work).
Since an amount can never be negative, then a negative number only makes sense to denote the movement of an amount from one system to another. If we say that negative numbers are real numbers, and especially if we use them as the same class as positive numbers when building blocks to formulate ideas and theories about physical properties and amounts, I'm wondering if there is a potential downside to our fundamental understanding of what our results actually mean.
One of the other replies points out the necessity of units paired with numbers, and I think this is correct. The problem I'm observing, and the heart of my question, is that we have many, many ideas/proofs/theorems in mathematics with unitless numbers (constants and so on, both positive and negative), and then we use the results of this work to jump back into the physical universe to "prove" something. Could there be a problem with our basic use of numbers when we do this?
For a mathematican 42 or -42 makes sense - there are no units necessary. The mathematical intepretation of a negative number is just being element of a set.
For a physicist 42 or -42 are pointless, because a unit is missing. The physical intepretation of a negative number is the quantified absence of a unit.
- by the way: https://xkcd.com/435/
Sometimes a subtraction of a unit is hidden-black-magic and you are switching between two models or contexts but still being in a meta-model or meta-context. (Debits and credits)
It just depends on your model or context how you interprete your value/unit relation, but this is a typical problem in applied physic, engineering or computer science :)
