Chapter 4, Longer 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 yesprovided Romeo can get to
the first or bottom-most step of the ladder. It is nootherwise.
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.
Mathematical Induction
The principle of mathematical induction stated below describes the Romeo
and Juliet 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.
We assume that when or if we have counted to any number n, we can count
to the next one as well. Just add one to the count n. This gives the
next number in our count which is written n+1. This offers a way to
begin counting all the whole numbers 1, 2, 3, 4 and so on.
Suppose or imagine for each whole number n, there is a situation
An. This gives a step on the ladder. Now the next whole number
after a whole number n is given by adding 1, that is n+1. So the next
step after An. is written as An+1. The principle of
mathematical induction says the following:
If
- for each whole number n, there is a situation An.
- every time the situation An. occurs, the next situation
Am. with m = n + 1 must also occur; and
- the first A1. situation occurs,
then all the situations An. (where n is a whole number)
occur.
The word occurs can be replaced by the expression can be
reached.The principle of mathematical induction is quite simple. It
requires the following: (1) there is a ladder; (2) on the ladder, from
each step we can reach the next; and (3) the first step is reachable.
When these three requirements are met, the principle of mathematical
induction says: all the steps can be climbed or reached. That is
all there is to this inductive principle.
Question. What can be said about the reach-ability of
Anwhere n > 4, if we find a ladder for which
requirements (1) and (2) are met, and we somehow know A4. is
reachable? Hint: Imagine a ladder where the first three steps are
broken, but the fourth is somehow climbable. Is the ladder climbable?
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
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Others are welcome to refine or exceed it. Please do.
Secondary
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See too, the BBC-Belgium story Texting and
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The Logic of Injustice:
How Texas sent
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justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
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May 2012, Composition Starting:
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Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
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the choice is theirs. But in retrospect, the selection does not
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Arithmetic
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Algebra
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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Flash
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They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
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if one or more explanations is not to liking, try another. It may
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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calculus and more generally in the first year of college. Bon
Appetite.
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