years. This specific technique (not to be confused with scientific induction) is
widely used to prove statements involving whole numbers. It is especially useful
in graph theory, number theory, and computer science generally. As a practical
example, think of the problem of adding up the odd numbers. For instance, the
addition of the first three odd numbers 1 + 3 + 5 is 9 while the sum of first four
1 + 3 + 5 + 7 is 16. Now 9 is 3 × 3 = 3^2 and 16 is 4 × 4 = 4^2 , so could it be
that the addition of the first n odd numbers is equal to n^2? If we try a randomly
chosen value of n, say n = 7, we indeed find that the sum of the first seven is 1
- 3 + 5 + 7 + 9 + 11 +13 = 49 which is 7^2. But is this pattern followed for all
values of n? How can we be sure? We have a problem, because we cannot hope
to check an infinite number of cases individually.
This is where mathematical induction steps in. Informally it is the domino
method of proof. This metaphor applies to a row of dominos standing on their
ends. If one domino falls it will knock the next one down. This is clear. All we
need to make them all fall is the first one to fall. We can apply this thinking to the
odd numbers problem. The statement Pn says that the sum of the first n odd
numbers adds up to n^2. Mathematical induction sets up a chain reaction whereby
P 1 , P 2 , P 3 ,... will all be true. The statement P 1 is trivially true because 1 = 1^2.
Next, P 2 is true because 1 + 3 = 1^2 + 3 = 2^2 , P 3 is true because 1 + 3 + 5 = 2^2 +
5 = 3^2 and P 4 is true because 1 + 3 + 5 +7 = 3^2 + 7 = 4^2. We use the result at
one stage to hop to the next one. This process can be formalized to frame the
method of mathematical induction.
Difficulties with proof
Proofs come in all sorts of styles and sizes. Some are short and snappy,
particularly those found in the text books. Some others detailing the latest
research have taken up the whole issue of journals and amount to thousands of
pages. Very few people will have a grasp of the whole argument in these cases.
There are also foundational issues. For instance, a small number of
mathematicians are unhappy with the reductio ad absurdam method of indirect
proof where it applies to existence. If the assumption that a solution of an
equation does not exist leads to a contradiction, is this enough to prove that a
solution does exist? Opponents of this proof method would claim the logic is
merely sleight of hand and doesn’t tell us how to actually construct a concrete