312 ian mueller
unit α,
t w o αα,
t h r e e ααα,
four αααα,
fi v e ααααα,
and so on.
Nicomachus’ “natural” representation of numbers would seem to break
down the customary Greek contrast between the numbers and the unit, but
Nicomachus insists that it does not:
ii .6.3 Since the unit has the place and character of a point, it will be a principle
( arkhê )... of numbers... and not in itself ( oupô )... a number, just as the point is
a principle of line or distance and not in itself a line or distance.
We fi nd a close analog of Nicomachus’ “natural” representation of
numbers in the account of fi nitary number theory in Hilbert and Bernays’
great work Grundlagen der Mathematik , except that in the Grundlagen the
alphas are replaced by strokes. As that work makes clear, this representa-
tion provides a basis for developing all of elementary arithmetic, including
everything known to the Greeks. Much the most important feature of the
representation in this regard is the treatment of the numbers as formed
from an initial object (the unit or one) by an indefi nitely repeatable succes-
sor operation which always produces a new number. Th is treatment vali-
dates defi nition and proof by mathematical induction, the core of modern
number theory. Th e fi nitary arithmetic of Hilbert and Bernays rests essen-
tially on the intuitive manipulation of sequences of strokes (units) together
with elementary inductive reasoning. 4 It is diffi cult for me to see any sub-
stantial diff erence between the manipulation of sequences of strokes or
alphas and the manipulation of lines and fi gures in what is frequently called
cut-and-paste geometry; the objects are diff erent, but the reasoning seems
to me to be in an important sense the same.
I mention this modern form of elementary arithmetic only to provide a
contrast with its ancient forebears. Nicomachus relies heavily on the notion
of numbers as multiplicities of units and the representation of them as col-
lections of alphas, but, aft er he has introduced his natural representation, it
by and large vanishes in favor of a much more clearly geometric or confi gu-
rational representation in which three is a triangular number, four a square
number, and fi ve a pentagonal number ( Figure 9.1 ).4 In this paper I use words like “inductive” and “induction” only in connection with
mathematical induction.