Induction (Mathematical vs. Argument by Induction)
Mathematical induction proves a statement for positive whole numbers by establishing a starting case and a general step to the next case. Empirical induction instead extends observations to unobserved cases without a deductive guarantee. Cheng’s cookie examples illustrate why a plausible step or a chain of small differences is not enough.
THE IDEA
Induction
Mathematical induction proves a statement through a starting case and a general rule for passing to the next case. It differs from empirical induction, which generalizes from observed cases to unobserved ones. Such a generalization may be well supported, but it is not a deductive guarantee. For example, “The sun has risen every day of my life so far, so it will rise tomorrow.”
Mathematical induction is logically secure, and is a bit like climbing up steps. Babies learn to climb up one step and then discover, with delight, that if they just repeat it they can go up whole flights of stairs, possibly all the way to the sky. All they need is for someone to put them at the first step (and for nobody to intervene and take them off the stairs).
Mathematical induction says if you know something is true for the number 1, and if you have a way of climbing up by 1, then you know it’s true for all positive whole numbers. If we apply this to small cookies, we’d say
• It’s fine to eat 1 cookie.
• If it was fine to eat some number of cookies, it’s okay to eat 1 more.
Therefore it’s fine to eat any number of cookies.
Mathematical induction is stated in terms of whole numbers n, and we say that we are trying to prove that some property P is true of each number n. So P(n) might be the statement “It is fine to eat n cookies.” Then the argument looks like this:
• P(n) is true at the starting value.
• For every positive whole number n, P(n) ⇒ P(n + 1).
Then by the principle of mathematical induction, P(n) is true for all positive whole numbers n. The logical form does not establish the cookie premise: we still need to prove that eating another cookie remains fine, however many have already been eaten.
This is fine for whole numbers, but it gets tricky if you’re trying to deal with a sliding scale that includes all the numbers in between, or even just all possible fractions. This is because there is no smallest unit of “jump” for us to take steps in.
We might try to apply this to a series of cookies that all seem to be “more or less the same size”. Perhaps this means that they are within 5g of each other’s weights. You might be happy thinking that a 50g cookie is more or less the same size as a 52g cookie, and that this is more or less the same as a 54g cookie, but after a few steps like that you will end up with a cookie twice as big. I have done this by handing out cookies to a class of 20 students without telling them what the point is. I ask them all to compare their cookie with the person next to them, and they are all happy that their cookie is more or less the same. But then I get the first and last student to compare cookies and we all collapse in giggles because the first one is tiny and the last one is enormous.
When to use it
Proving formulas or properties in mathematics (e.g., sum of first n natural numbers, divisibility).
Analyzing arguments in logic and critical thinking to distinguish valid inductive reasoning from fallacious hasty generalizations.
Identifying slippery slope arguments where each step seems reasonable but the final outcome is extreme (e.g., ethical debates, policy decisions).
Teaching the concept of threshold effects and why cumulative small changes can lead to large differences (e.g., weight gain, climate change).
FROM THEORY TO EVERYDAY LIFE
In practice
Empirical Induction: Predicting Sunrise
“The sun has risen every day of my life so far, so it will rise tomorrow.” This predicts an unobserved event from past observations. It is not a deductive certainty; its support depends on the evidence and assumptions behind the prediction.
Mathematical Induction (Secure) – Cookie Chain
• It's fine to eat 1 cookie. • If it's fine to eat n cookies, it's fine to eat n+1 cookies. Therefore, it's fine to eat any number of cookies. The form requires a true starting case and a step proved for every relevant positive whole number. The example has not established that eating another cookie is always fine.
Misapplication of Induction – Cookie Size Paradox
A class of 20 students each compare their cookie with the next student; each pair considers the cookies “more or less the same size.” However, the first cookie is tiny and the last is enormous, demonstrating that transitivity fails when the difference per step is small but cumulative.
A MOMENT TO REFLECT
Put it to work.
Can you think of a real-world situation where people mistakenly apply mathematical induction to continuous quantities?
How does the cookie size example illustrate the danger of assuming small differences add up to no difference?
What are the essential conditions for mathematical induction to be valid, and why are they not met in the cookie size example?
Often confused with
CONTINUE THE CONVERSATION
Source texts
- The Art of Logic How to Make Sense in a World that Doesn’t (Eugenia Cheng)