Previous: Chapter 6,
Chains of Reason
To induce means to extract. Induction here consists of
extracting conclusions from chains of rules and patterns, one after another,
perhaps without stopping or end. Another form or version of inductive reason
is concerned with the extraction of patterns from experience and observation.
See the last words of the previous chapter.
This chapter explains one version of
inductive reason: the recursive or repetitive approach to putting one-way
implication rules together, one after another. This chapter ends with a
description of the principle of mathematical induction – another method for
obtaining conclusions used only in mathematical arguments or computations. There
is more to mathematics than just doing arithmetic.
Recall that rules, which say that when a first situation occurs so should
a second, are called implication rules. Implication rules can be linked
together, one after another. A ladder-based story illustrates the underlying
idea. It is called induction. This story leads to the notion called mathematical
induction, a method of reason or logic used in mathematics after arithmetic to
get conclusions (or climb ladders). The method is described first with words, a
simple story, and then with some shorthand notation.
Romeo and Juliet
Imagine a hero, Romeo, riding a horse towards a tall building (a castle).
There is a ladder up the side of the building leading to the room where Juliet
lives. The bottom step of the ladder is two meters or more (several feet or
more) away from the ground. The ladder is not broken. It is in good condition. A
person getting to each step of the ladder can climb to the next. Question: Can
an able-bodied individual, Romeo, reach Juliet via the ladder? The answer is yes
provided Romeo can get to the first or bottom-most step of the ladder. It is no
otherwise. The main logic-related ideas in this brief story are as follows.
- There is a long ladder to be climbed.
- When any one step is reached, the next step can be reached. (The ladder
must be in good condition for this to hold).
- The first or bottom-most step can be reached.
This situation implies we (or Romeo) can reach each step of the ladder.
Note that the long ladder may have a finite number of steps, for example 183.
Then we (or Romeo) can with enough time and patience, reach the last one, or any
step in between.
On the other hand, we can imagine a ladder could have an infinite number of
steps. For each step we take, a next is possible. For instance, the whole
numbers we use for counting do not stop. Each whole number is followed by
another — just add 1.
Now suppose or imagine we have a sequence of steps, a ladder, which goes on
and on without stopping. Then with enough time and patience, we can reach anyone
you mention. An example is met in counting. We can begin counting with the
number 1, then 2, then 3 and so on.
When we begin to count, we may have only a finite number of objects to count.
With a long enough life, and enough patience, the count will end. But if we
count minutes there will always be one more to count. This minute count will
never end. More precisely, each of us counters may end, but the counting of
minutes in principle can continue. That is, this minute count can reach any
large number you specify in advance with or without you. In principle all
minutes after the beginning of the count will be met and counted.
To rephrase the above, on a ladder (or road) with finitely or infinitely many
steps, the first step needs to be reachable. And from each step, the next step
needs to be reachable. When this occurs, any whole number of steps along the
road or ladder in question is reachable.
[2] In practice, if each step takes time,
the number of steps reachable will depend on how much time is available.
CAUTION. The conclusion that all steps can be climbed or reached does not
follow from the principle of mathematical induction if the ladder is broken, or
if the first step is not reachable
or if a tornado comes along, or if you
break your ankle, etc.
Check for these nasty situations when you want to use this principle to get a
conclusion.
Reading Guide
The principle of mathematical induction stated below describes the above
ladder idea in the algebraic shorthand notation favored in mathematics. The last
part of this chapter will not make sense to you if you are not familiar with
this shorthand notation. If this is the case, you may skip this description of
mathematical induction.
Next: Mathematical
Induction, Chapter 7, part II
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