Numeration, Number Systems

Part Three: Our numeration blind spot

We live in (collective) ignorance of what numbers are. While the lay person would object with indignation to that statement, it would come as no surprise to the mathematician. This harks back to the italicized phrase ‘purely logical consequences’ in the passage we quoted from Nagel/Newman in the Prologue. But even if Queen Mathematics has summarily swept ‘What numbers are’ off the table, there is no law that forbids others from expressing curiosity about the crumbs of that question that remain on the floor. But before confronting the topic directly (with help from Figure 5), let’s get a feel for how it seems almost to exist, somewhere in the vastness of the conventional math universe.

At the very moment of its birth (say 1889), number theory simultaneously gives a perfunctory nod to the numeration question, and promptly washes its hands of it. (Cf. Joseph p. 35. See also the fleeting mention of ‘symbols of number’ in Jourdain p. 21.) Consider the following reflection on how/why Peano axiomized numbers, taking the ‘natural numbers’ (aka ‘counting numbers’) as his foundation:

It might seem strange that Peano should need to [develop our ‘counting numbers’ from a set of propositions] but those familiar numbers we use all the time [...] have to come from somewhere [...] it’s easy to think of them as real things [...] But in reality these numbers [...] are just symbols we use to represent the cardinality of a set. I can’t hold 15 in my hand. [Peano in 1889] lets us build those numbers in terms of a series of sets that are almost hauled up by their own bootstraps. — Clegg 2003:152, italics added.

Early on, Peano would have been well aware of (indeed driven by) the fact that our numbers are not ‘real things,’ but once he entered the realm of axioms, all attention was focused there and the philosophical question would have been spirited away to the far horizon, where it became ‘not my job’ for the mathematician. Let’s look at some representative examples to see what is typically covered, just ‘inches away’ from what would be numeration theory, if such existed.