## Friday, September 16, 2011

### Best words in the best order

Writers of mathematics strive for clear and careful wording, especially in the formulation of definitions. Well-specified definitions can enable theorems to be proved succinctly. For example, the relation "less than" (denoted <) for the positive integers {1,2,3,...} may be defined as follows:

If  a  and  are integers, then
a < b  if  b - a  is a positive integer.

Although the simple definition of "less than" as "to the left of" in the list {1,2,3,...} is intuitively clear, the formal definition above is better suited for mathematical arguments. It defines "less than" in terms of the known term, "positive." This sort of sequencing of definitions is common in mathematics -- one may go on to define "greater than" in terms of "less than," and so on.
Saying things in the best way is also a goal of poetry. Well known to many are these words of poet Samuel Taylor Coleridge (1772–1834):
I wish our clever young poets would remember
my homely definitions of prose and poetry;
that is, prose, — words in their best order;
poetry, — the best words in their best order.

On March 30, 2011  I posted  "A Mathematical Problem," in which " Coleridge uses verse to describe construction of an equilateral triangle; Coleridge introduces the poem with a letter to his brother telling of his admiration of mathematics, a view rather rare among poets.
On September 6, 2011 this blog posted a 4x4 symmetric square poem -- illustrating not only the best but the fewest words in the best order.  Here (via Mathematics for the Million by Lancelot Hogben (W W Norton, 1937)) is a 5 x 5 word square, this one based on the Latin sentence "Sator arepo tenet opera rotas' -- which might be translated as, "Arepo, the sower, uses the plough for his work."

S  A  T  O  R
A  R  E  P  O
T  E  N  E  T
O  P  E  R  A
R  O  T  A  S

In Wikipedia we learn that the earliest known appearance of the Sator Square was found in the ruins of Pompeii which was buried in the ash of Mt. Vesuvius in 79 AD.