Exxplaining Induction pt 2

Question

What I am really looking for here is an explanation as to why this process is actually able to prove that a statement is true for all values of n. Try to make your example non-mathematical and explain how the process would prove what is being said.

For example, I can prove by induction that I can teach any size course.

1. I can teach a class with one student
2. It certainly is true that if I can teach a class with k students, I can teach a class with one more student.

Now why do those two taken together prove that I can teach any size class? How do I know, for example that I can teach a class with 5 students based on this information.

Answer 1

Actually it does not. Your scenario is a good examples of the frequent disconnect between the real world and the mathematics world.

 

According to induction:

1) Assume statement is true for one (you can teach a class of one person)

2) Prove that if you can teach "n" students, you can teach "n+1" students

Here is the conundrum. In the world of mathematics "n" is a variable (able to vary). It can stand for any number.

In the real world a class of students must be a constant number. Anything you prove would be true only for that number.

Now back to induction:

3) Since you demonstrated that the statement works for "n" it shall work of "n+1," you can say, well it works for 1 then it must work for 2, and if it works for 2 it works for 3, and so on just like those string openings on dog food bags.

 

The power of induction is that it works in the world of concepts and not physicals.

Answer 2

This is an interesting problem. What your teacher argument lacks is that teachers are not pure and perfect. Math and science are.

So here's an alternative. "Every living person has a heartbeat". This can be approved for the first person, assumed for and people, and then proven for n +1. It is also something that can be applied for all n.