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       4. Romeo and Juliet

Chapter 4. Longer Chains of Reason

Previous Chapter: Chains of Reason


This chapter explains one version of inductive5 reason: the recursive or repetitive approach to putting one-way implication rules together, one after another.

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 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.

  1. There is a long ladder to be climbed.
  2. When any one step is reached, the next step can be reached. (The ladder must be in good condition for this to hold).
  3. 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.

In practice, if each step takes time, the number of steps reachable will depend on how much time is available. Reach-ability here does not take into account the amount of time available, albeit people doing numerical computations on electronic computers must consider the latter.

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. If you read it, and you find that you do not understand it, you could return to it later after you have seen the following chapters on algebra. They explain the use of shorthand in algebra.

Next Section: 4. Induction Mathematical

Next Chapter: 5. Islands and Division of Knowledge, How knowledge divides

 

Three Skills
For 
Algebra

understanding & explaining
Reason and Math
Volume 2
Printed in Canada
ISBN 0-9697564-2-9
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Chapters and Appendices

Home
Postscript: The 4-th Skill For Algebra
Foreword
1. Introduction
2. Implication Rules
3. Chains of Reason
4. Romeo and Juliet
4. Induction Mathematical
5 Knowledge Islands
6  Old Language
7  Arith Skill Check
7. The Next Chapters
8 The Three Skills
8 VNR-Concise-Encyclopedia
PS. What is a Variable
9. Algebra Talk
10 Two More Skills
11 Why Shorthand
12 Shorthand Usage
13 What's Next
14 Compound Interest
15 Linear Equations
PS I.  Distributive Law
PS II. Polynomials
16 Painless Proofs
17 Pythagoras
18 Rules of Algebra
19  Functions & Sets
20 Degrees & Radians
21 What's Next
22. Arith & Geometric Sums
23 Summation Notation
24 Your Money
25 Induction & Recursion
26 What's Next
27 Pronouns in Logic
28 Occurrence Tables
29 Contrapositive
30 Truth Tables
31 Indirect Reason
A. Advice For Learning

Words Before Symbols: 
What is a Variable?
Introduction
Variation between Examples

Variation of Letters

A letter denotes a variable

Cases of Double Variation

Three Notions of a Variable

Constants, Parameters
& Variables

Talking about numbers
Dependent or Independent
Variable, a Matter of Choice


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