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2.4
Induction
The process of generating a model and then proving it is useful (i.e. can be used
to make predictions) underlies the well known mathematical process of proof by
induction . For example, if we examine the sum:
1
+
3
+
5
+
...
of the successive odd numbers then we may notice that:
1 2
1
=
2 2
1
+
3
=
3 2
+
+
=
1
3
5
4 2
1
+
3
+
5
+
7
=
and so on. We can abduce (infer) the model that for every natural number n, the sum
of the first n odd numbers is n 2 . This is certainly true for all the first n odd numbers
from 1 to 4 we have observed so far, and we could continue in this vein until we
find an exception. For many scientific endeavours this may be the best we can do
but for mathematics this is not considered good enough. Since we have access to the
underlying foundations of mathematics the possibility of a sound proof is available.
Such a mathematical proof by induction follows the style: if we provide a general
form that shows this model to be true for any number n, then we are entitled to
suppose that it is true for any number less than n. We are also allowed to suppose
that we already know that the sum of the first n
1) 2 . The
sum of the first n odd numbers is obtained by adding the nth odd number, which is
(2 n
1 odd numbers is ( n
1). So:
1) 2
Sum of the first n odd numbers
=
( n
+
(2 n
1)
( n 2
=
2 n
+
1)
+
(2 n
1)
n 2
=
2 n
+
1
+
2 n
1
n 2
=
It should be noticed that:
1. First a proposition (i.e. hypothesis or model) must be proposed (insight). The
proposition comes from a set of concepts in which each proposition can be used
to generate a potential series through deduction. Usually the series here involve
sums rather than multiplication. Multiplication series are much more difficult to
prove.
2. Then the proposition is tested against observation (reason);
a process of
validation .
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