We have already dwelt upon the formalism of Number Systems in a previous Chapter, where we reminded ourselves that just counting to ten on paper involves a rather sophisticated and elaborate representational scheme that we all learned as children and which is now tacit in our thought processes until we go to the trouble to dismantle it and consider possible alternatives.
Arithmetic is the basic algebra of Numbers and builds upon our tacit understanding of their conventional representation. However, it would be emphatically wrong to claim that, "Arithmetic is made up of Numbers, so there is nothing to Arithmetic but Numbers." Obviously Arithmetic treats a new level of understanding of the properties of (and the relationships between) Numbers - something like the Frank Lloyd Wright house that was not there in the bricks and mortar of which it is built. [One can argue that in fact the conceptual framework of Number Systems implicitly contains intimations of Arithmetic, but this is like arguing that the properties of atoms are implicit in the behaviour of electrons; let's leave that debate for later.]
We learn Arithmetic at two levels: the actual level ("If I have two apples and I get three more apples, then I have five apples, as long as nothing happens to the first two in the meantime.") and the symbolic level ("2+3=5"). The former level is of course both concrete (as in all the examples) and profoundly abstract in the sense that one learns to understand that two of anything added to three of the same sort of thing will make five of them, independent of words or numerical symbols. The latter level is more for communication (remember, we have to adopt and adapt to a notational convention in order to express our ideas to each other) and for technology - i.e. for developing manipulative tricks to use on Numbers.
Skipping over the simple Arithmetic I assume we all know tacitly, I will use long division as an example of the conventional technology of Arithmetic.4.1 We all know (today) how to do long division. But can we explain how it works? Suppose you were Cultural Attaché to Alpha Centauri IV, where the local intelligent life forms were interested in Earth Math and had just mastered our ridiculous decimal notation. They understand addition, subtraction, multiplication and division perfectly and have developed the necessary skills in Earth-style gimmicks (carrying, etc.) for the first three, but they have no idea how we actually go about dividing one multi-digit number by another. Try to imagine how you would explain the long division trick. Probably by example, right? That's how most of us learn it. Our teacher works out beaucoup examples on the blackboard and then gives us beaucoup homework problems to work out ourselves, hopefully arrayed in a sequence that sort of leads us through the process of induction (not a part of Logic, according to Karl Popper, but an important part of human thinking nonetheless) to a bootstrap grasp on the method. Nowhere, in most cases, does anyone give us a full rigourous derivation of the method, yet we all have a deep confidence in its universality and reliability -- which, I hasten to add, I'm sure can be rigourously derived if we take the trouble. Still, we are awfully trusting....
The point is, as Michael Polanyi has said, "We know more than we can tell." The tacit knowledge of Arithmetic that you possess represents an enormous store of