Is mathematical induction a deductive or inductive argument?

[jeremy22511]

Homework Statement

Is mathematical induction a deductive or inductive argument?
Would appreciate the help. Thanks.

Jeremy

Homework Equations

The Attempt at a Solution

Its name suggests that the process is inductive, yet I know all of mathematics depends on deductive reasoning...

 

[tjackson3]

Short answer: Your hunch was dead on. It's deductive reasoning (inductive is not accepted as a valid type of reasoning in most disciplines).

Long answer: The first step in mathematical induction is inductive. I'll use as an example the formula for summing all consecutive integers from 1 to n:

$$1 + 2 + ... + n = \frac{n(n+1)}{2}$$

The first step in proving this is to prove that it's true for n = 1. That is:

$$1 = \frac{2}{2} = 1$$

To stop there would be to use inductive reasoning - i.e., since it's true for n = 1, it must be true for all n. This is obviously not necessarily correct, and that's where the deductive part comes in. The purpose of a deductive argument is to prove that, given a hypothesis, its conclusion must be valid and follow directly from the hypothesis. That is, now that we know that the above is true for n = 1, we assume that it's true for some n (that's the hypothesis), and show that it must then be true for n + 1. Now that it's in general form like that, you've completed the deduction, and shown that it's true for all n in the domain of the problem (in this case, natural numbers).

 

[Architeuthis Dux]

Mathematical induction is a deductive argument. It is a form of proof used in mathematics to show that a statement is true for all values of a variable. It relies on a base case, which is usually a specific value of the variable, and an inductive step, where the statement is assumed to be true for a general value of the variable and then proved for the next value. This process continues until it can be shown that the statement is true for all values of the variable. Therefore, mathematical induction is a deductive argument as it starts with specific cases and uses logical reasoning to prove a general statement.