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Exploration in Modern Mathematics

Mike Janssen

Section 2.1 Inductive Reasoning

Motivating Questions.

In this section, we will explore the following questions.
  1. What is truth? What is fact? Is there a difference?
  2. What is inductive reasoning? When is it helpful? What are its shortcomings?
We’ll begin by attempting to define truth and fact, and compare them. We’ll then explore inductive reasoning in depth, considering its benefits shortcomings.

Activity 2.1.2. What’s my world?

In the game What’s my World?, one person thinks of a single law that defines a hypothetical world (e.g., “My world does not contain things that start with the letter “C”.”). The other players attempt to guess this governing law by taking turns asking questions of the creator, such as “Does your world contain dogs? Does it contain cats?” and so on. When one of the guessers believe they can identify the governing law, they ask the creator.
Play a few rounds of this game, taking turns in the role of creator. Other than asking the creator directly, how could you be certain that you had determined the law of the world?

Definition 2.1.3.

Inductive reasoning is the process of drawing general conclusions from particular instances, generally known as data.
The work of the guesser in Activity 2.1.2 is to employ inductive reasoning to determine the general law put in place by the creator. This is the heart of the scientific endeavor: looking at the world in its orderliness and using our curiosity and creativity to infer larger governing principles.
But how does this work in mathematics?

Exploration 2.1.4.

Let’s play a game with dots and lines. We’ll start with at least two dots (though you’ll probably want to increase this number pretty quickly). The rules are:
  • Split your dots into two groups, group \(A\) and group \(B\text{,}\) and draw each group on its own line.
  • Connect (some of) the dots from \(A\) to (some of) the dots in \(B\) with lines. The lines don’t have to be straight —they can curve in any way you want! —but each line should connect precisely two dots: one from \(A\) and one from \(B\text{.}\)
  • Each dot should be connected to at least one other dot —no lonely dots!
So, if I label four dots as \(X, Y, Z, W\text{,}\) one possible drawing is given in Figure 2.1.5.
Figure 2.1.5.
However, there is a problem with this drawing: the lines cross! I know, this wasn’t one of the rules above, but let’s add it.
  • The lines must be drawn so that none of them cross.
Now consider the following questions.
  1. Redraw the picture in Figure 2.1.5 so that none of the lines cross.
  2. Give a name to drawings of figures like Figure 2.1.5 which can be drawn so that none of the lines cross.
  3. Which (non-crossing!) drawings are possible with two or three dots?
  4. What other non-crossing drawings are possible with four dots? Five?
  5. Based on your work here, do you think it will always be possible to draw these pictures so that none of the lines cross? Explain your thinking.
In this section, we explored inductive reasoning. We saw that inductive reasoning is the process of drawing general principles from data. It is generally the case that, the more evidence we have for a conclusion, the more sure we can feel about it. We note that, in mathematics and elsewhere, our inductive conclusions are tentative, always subject to new data for which we must account.

Exercises Exercises

1.

Do some internet research on the Twin Prime Conjecture. What is it? When was it first formulated? Is it true? Likely true?

2.

The process of accounting for new data which challenges an accepted perspective is often messy and exciting. Find a time in scientific history in which new data and an accompanying theory challenged accepted understanding, and write two or three paragraphs about it. What was the status quo and how was it revised in light of new information?
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