Magazine article Word Ways

# Using Mathematical Parentheses in English

Magazine article Word Ways

# Using Mathematical Parentheses in English

## Article excerpt

Parentheses are used in mathematics to indicate that certain computations should precede other computations, and also to shorten expressions by virtue of the "distributive law" that says A x (B + C) = (A x B) + (A x C). Both of these uses can he applied to English text. A classic chestnut writes

((N + H)OW + (T + W)HAT)(I KNOW)

as shorthand for the introspective, epistemological statement:

NOW I KNOW HOW I KNOW THAT I KNOW WHAT I KNOW,

which expands the original 14 letters to 41 letters.

Smaller examples of this type of shorthand include: (D + T)OWN for DOWNTOWN; (T & G)OWN for TOWN & GOWN; (N + L)IGHT for NIGHTLIGHT; (F + W)INE for FINE WINE; etc. The reader is encouraged to find additional examples, and to look for phrases involving three or more "rhyming" words. (We only care about "eye rhymes", the kind where MATURE, NATURE, and STATURE are "rhyming" words.)

Most operations of ordinary arithmetic are associative: a + (b + c) = (a + b) + c and (a x b) x c = a x (b x c). However, in general, [a.sup.([b.sup.c])] [not equal to] [([a.sup.b]).sup.c]. For example, [([2.sup.3]).sup.2] = [8.sup.2] = 64, but [2.sup.([3.sup.2])] = [2.sup.9] = 512. The number of possible ways to insert parentheses grows exponentially with the number of letters in the exponential tower, following the sequence 1, 1, 2, 5, 14, 43, 132,... of the "Catalan numbers". This is mirrored by English phrases, which can have multiple meanings depending on where we put the parentheses, which are rarely written, but must be inferred.

For example, "A fine gold ring", if thought of as "A fine (gold ring)" means a gold ring that is fine, but "(A fine gold) ring" means a ring made of fine gold. With three words (ignoring the article "a"), the Catalan number 2 tells us we have two essentially different ways to insert parentheses. The next Catalan number is 5. …

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