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Chapters and Appendices
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What is a Variable? |
Chapter 4. Longer Chains of ReasonPrevious 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.
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 JulietImagine 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.
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.
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 GuideThe 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 |
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