I'm interested in generalizing the underlying mechanism/theory/concept behind subtraction, from natural to negative numbers.
Not tacking on a new concept just for handling the new cases, but applying the same concept to both old and new cases (that in some sense already existed for the old cases). Perhaps identifying some property present for the old cases, and preserving it for the new cases.
This is the farthest I've got:
- Natural numbers can be thought of as a physical objects: you can count how many there are, physically add more objects, or remove them (or take them away). But "take away" is not closed over the naturals; you can't physically take away more than you have.
I say that "take away" cannot be generalized to negative numbers, because you cannot have a negative number of physical objects. However, we can look at it in a different way (a different theory, or concept or model), that gives the same answers for the same cases, but that can be generalized to negative numbers. [This is a kind of a cheat, but if the concept is present and active in the original case, I'll allow it.]
- We have a set of distinct elements, that are ordered, starting from one, and going on forever. We can go to the "next" one in this order (+1), and go to the previous one (-1). By repeating these, we can add a positive number, and subtract a positive number, and get the same results as above.
But from the first element, we can't go to a previous one. It seems a very natural and logical extension to allow this - it seems more like removing an artificial barrier than introducing a new concept.
If we allow this, we have negative numbers, and, given a positive or negative number, we can subtract a positive number $n$, by going to the previous element, $n$ times. There's also nice a nice symmetry, with both ends being infinite, instead of only one.
Although an improvement, this still isn't closed on this expanded set of numbers (positive and negative), because we can't subtract a negative.
I say that we cannot generalize or extend this mechanism/theory to subtraction of negative numbers, because doing something a negative number of times doesn't make any sense. However, we can again change the figure, and use a a different theory or model which can be generalized in this way. We do this by observing properties in the previous case, and choosing a new model that selectively preserves those properties when it is generalized.
A different way of thinking about "going back" $n$ times to get to the result, or of "taking away" is the difference between two numbers, how many next or previous steps it takes to move from second to the first. We might get to this idea in two steps:
- the "going back" $n$ steps is the difference between the starting number and result
- it turns out that if we start from the same number, but instead go back by the result number of steps, we get $n$. i.e. in algebraic notation, with $S,R$ for start and result: $S-n=R \iff S-R=n$
This is quite a different concept, but it does generalize smoothly to any pair of numbers, because they all are separated by some number of steps on the ordered elements. It builds on the generalization of positive to negative, and the property that they can both be moved along in steps.
It also changes the role of the operands: instead of one being an instruction (to move left or right), both operands have equal status, and the result is the relationship between them. This, again, has symmetry.
By this model, from any number (positive or negative) can be taken away or subtracted any other number (positive or negative).
To recap, the steps were:
extend numbers to negatives (see as "next" and "previous", and "previous" continues past $0$)
extend subtraction to negatives (see "take away" as "difference", and "difference" applies between any two numbers).
I think this completes the concept of negative numbers.
But there's another step, to algebraic notation, which has the same properties, but which is different altogether, in that there is no theory or model. It is a purely syntactic system - there is no coherent concept. Just a system of rules. What is most striking to me is that the rules for addition seem overly complex, compared to the above simple and logical model. [Though to anyone proficient in arithmetic, who has internalized the rules, this claim of complexity might be unbelievable].
By Occam's Razor, it is troubling that an unnecessarily complex model is true, that a series of specific cases could he correct. But, unlike the famously complex approximation of "epicycles within epicycles", it is precisely correct.