If you teach geometry, you cover two types of reasoning. Inductive reasoning is based on patterns or probabilities and its conclusions aren't certain. We saw an example of that in this post. It looked obvious that the next number in the sequence 2, 4, 6, 8, 10, . . . had to be 12 because it was "obvious" what the pattern was: 2n for n a positive integer. Obvious and wrong; we saw any real number can be justified as the next number.

Inductive reasoning is the reasoning of science. Use experimentation to find a relationship/fact and then test it. If it works over and over and over then eventually it becomes a scientific law, like the Law of Conservation of Mass. If further experimentation shows that the Law of Conservation isn't always correct then the law will be scrapped or modified to account for the exceptional cases.

Deductive reasoning separates math from other sciences because deductive reasoning is infallible. Yes, people can make mistakes in reasoning but if a statement is true because it follows logically from other facts then the conclusions must be true. The most amazing example I know is the De Broglie Wavelength: there is a wave-particle duality to matter because the equations say so.

I've put together a simple example to explain the difference between them. By noticing

1+2+3 = 2(3) 2+3+4 = 3(3) 3+4+5 = 4(3) 4+5+6 = 5(3) 5+6+7 = 6(3)

we can use inductive logic to form a conjecture. But to prove that pattern is always true takes deduction. Now a computer or calculator can help you verify the formula is correct for a finite number of cases but there are an infinite number of equations that need to be checked. With deduction, we can prove the conjecture will be true for an infinite number of equations in a finite amount of time. That makes deductive reasoning and math very special. I've written up the details for this example and it's posted on the Handouts page.