zero:
one:
two:
three:
etc.o
50
550
5550The symbol 5 has an interpretation--"the successor of". Hence, the in-
terpretation of 550 is literally "the successor of the successor of zero".
Strings of this form are called numerals.
Variables and TermsClearly, we need a way of referring to unspecified, or variable, numbers.
For that, we will use the letters a, b, e, d, e. But five will not be enough. We
need an unlimited supply of them, just as we had of atoms in the Proposi-
tional Calculus. We will use a similar method for making more variables:
tacking on any number of primes. (Note: Of course the symbol "'-read
"prime"-is not to be confused with prime numbers!) For instance:are all variables.e
d'
e"
b'l'
a' I"In a way it is a luxury to use the first five letters of the alphabet when
we could get away with just a and the prime. Later on, I will actually drop b,
e, d, and e, which will result in a sort of "austere" version of TNT-austere
in the sense that it is a little harder to decipher complex formulas. But for
now we'll be luxurious.
Now what about addition and multiplication? Very simple: we will use
the ordinary symbols '+' and" '. However, we will also introduce a paren-
thesizing requirement (we are now slowly slipping into the rules which
define well-formed strings of TNT). To write "b plus e" and "b times e", for
instance, we use the strings(b+e)
(b 'e)There is no laxness about such parentheses; to violate the convention is to
produce a non-well-formed formula. ("Formula"? I use the term instead of
"string" because it is conventional to do so. A formula is no more and no less
than a string of TNT.)
Incidentally, addition and multiplication are always to be thought of as
binary operations-that is, they unite precisely two numbers, never three or
more. Hence, if you wish to translate" I plus 2 plus 3", you have to decide
which of the following two expressions you want:(^206) Typographical Number Theory