The natural numbers (the non-negative integers) are abstractions of the additive combinations of units. The number 10 is an abstraction of the set of all sets of that number of units. (This definition is similar to that given by Bertrand Russell).
In the consideration of the rational foundations of Mathematics, one of the first concepts which is a cause of concern is that of the “negative numbers”. As conventionally presented, negative numbers have no grounding in Reality. One can not have -3 cows in a field. There is either some natural number of cows present, or no cows at all. (I am deferring the concept of zero for a future discussion).
However, there are valid interpretations for the negative number notation. Consider the concept of electric charge, and assume, for the moment, the existence of unit quantities of “positive” and “negative” charge. To measure the “total charge” of a body, we count the number of positive charge units and subtract the number of negative charge units. Invoking the concept of negative numbers, we then obtain the total as a positive or negative integer. The proper interpretation of this total is the positive number of non-cancelling units of positive or negative charge. Note that the number of units in the total is always positive, and it is the nature of the unit which changes as the sign of the integral total is changed.
For clarity, lets try a second example. Determining the value of financial assets, we count the number of units of wealth, and subtract the number of units of debt. Both the number of wealth units and the number of debt units are positive values. These units cancel one another, and the total value is then the (positive) number of non-cancelling units. The sign of the total is set based upon whether the excess units are of wealth or debt.
The use of signed integers represents the combination of dissimilar units in arithmetic. The units represented must be of equivalent magnitude and exactly opposite meaning. This implies that any of the classes being combined in an arithmetic summation or subtraction, whose result will be a signed integer, must be subclasses of a class of entities measurable with a unit whose magnitude is common to the subclass units. The sign of the result is set based upon which of the opposing units is in excess within the total.
That was tough to write and tougher to read. In our first example, “charge” is the parent class, “positive charge” and “negative charge” the subclasses. A unit of “charge” is of the same magnitude as that of a unit of “positive charge” or “negative charge”. When I write C=P-N, with C=Total Units of Charge, P=Units of Positive Charge and N=Units of Negative Charge, I am combining equal magnitude but cancelling units of Positive and Negative Charge to obtain a total in units of the parent class “Charge”. The sign of this value is determined by which of the subclasses is in excess.
It is essential that the distinction between arithmetic involving natural numbers, and arithmetic involving signed integers be maintained in the use of mathematics. The underlying arithmetic meaning of:
Cows to Bring Home = Cows Brought to Market – Cows Sold
is not at all the same as the underlying arithmetic meaning of:
Total Asset Value = Wealth – Debt