Euclid's Elements, Book VII, Definitions 1 and 2

Definitions 1 and 2

Def. 1. A unit is that by virtue of which each of the things that exist is called one.

Def. 2. A number is a multitude composed of units.

These 23 definitions at the beginning of Book VII are the definitions for all three books VII through IX on number theory. Some won't be used until Books VIII or IX.

These first two definitions are not very constructive towards a theory of numbers. The numbers in definition 2 are meant to be whole positive numbers greater than 1, and definition 1 is meant to define the unit as 1. The word "monad," derived directly from the Greek, is sometimes used instead of "unit."

Euclid treats the unit, 1, separately from numbers, 2, 3, and so forth. This makes his proofs awkward in some cases. Chrysippus (280–207), a Stoic philosopher, claimed that 1 is a number, but his pronouncement was not accepted for some time.

Throughout these three books on number theory Euclid exhibits numbers as lines. In the diagram above, if A is the unit, then BE is the number 3. But, just because he draws them as lines does not mean they are lines, and he never calls them lines.

It is not clear what the nature of these numbers is supposed to be. But their nature is irrelevant. Euclid could illustrate the unit as a line or as any other magnitude, and numbers would then be illustrated as multiples of that unit.

There is a major distinction between lines and numbers. Lines are infinitely divisible, but numbers are not, in particular, the unit is not divisible into smaller numbers.

Euclid has no postulates to elaborate the concept of number (other than the Common Notions which apparently are meant to apply to numbers as well as magnitudes of various kinds).

Mathematical induction

Mathematicians did not develop foundations for number theory until the late nineteenth century. The Dedekind/Peano axioms for numbers are the best known. The most important of these axioms encodes the principle of mathematical induction which states that

if a property of numbers holds for 1,
and whenever property holds for a number n then it also holds for n + 1,
then the property holds for all numbers.

Euclid does not use the principle of mathematical induction, but he does implicitly use a similar property of numbers, namely, that any decreasing sequence of numbers is finite. That property is known as the "descending chain condition." It's closely realated to the "well-ordering principle" which says that any nonempty set of numbers contains a least number.

The Dedekind/Peano axioms for numbers

Dedekind defined the natural numbers as a set N whose elements are called numbers, along with a specific number 1 and a successor function which associates with each number another number satisfying the following conditions

1 is not the successor of any number.
No two distinct numbers have the same successor.
The smallest subset of N that contains 1 and is closed under the successor function is N itself.

The last condition assures that the principle of mathematical induction holds on N.

Next definitions: VII.3-5

Book VII introduction