Difference Paradox

Some of the most fascinating moments in mathematics come not from solving hard problems, but from stumbling into paradoxes. A mathematical paradox is a statement (or pair of statements) that seems perfectly logical on the surface, yet leads to a contradiction. You follow every step, nod along, and then realize the conclusion makes no sense. That discomfort is the point. It forces you to look closer and figure out where the reasoning went wrong.

The paradox I want to walk you through today is a beautiful example of this. It involves nothing more than two natural numbers and their difference. Both propositions below appear to have valid proofs, yet they directly contradict each other. Your job is to figure out which one is actually correct, and more importantly, why the other one fools you.

Setting Up the Problem

Consider two natural numbers $n_1$ and $n_2$, where one is twice as large as the other. We don’t know which one is bigger. It could be $n_1 = 2n_2$, or it could be $n_2 = 2n_1$. All we know is that the doubling relationship exists.

Given this setup, we can state two propositions about the difference between these numbers. Read them carefully, because both sound reasonable at first glance.

The Two Propositions

Proposition 1: The difference $n_1 – n_2$, if $n_1 > n_2$, is different from the difference $n_2 – n_1$, if $n_2 > n_1$.

Proposition 2: The difference $n_1 – n_2$, if $n_1 > n_2$, is the same as the difference $n_2 – n_1$, if $n_2 > n_1$.

These two statements are direct opposites. The difference is either the same or it isn’t. Both can’t be true. Yet both have what look like perfectly valid proofs. Let’s examine them.

Proof of Proposition 1

Let $n_1 > n_2$. Since one number is twice the other, this means $n_1 = 2n_2$. Therefore:

$$n_1 – n_2 = 2n_2 – n_2 = n_2$$

So the difference is $n_2$.

Now let $n_2 > n_1$. In this case, $n_2 = 2n_1$, which means $n_1 = \dfrac{1}{2}n_2$. Therefore:

$$n_2 – n_1 = n_2 – \dfrac{1}{2}n_2 = \dfrac{1}{2}n_2$$

So the difference is $\dfrac{1}{2}n_2$.

Since $n_2 \neq \dfrac{1}{2}n_2$ for any natural number, the two differences are not equal. $\Rightarrow$ Proposition 1 is true. $\Box$

Proof of Proposition 2

Let $n_1 > n_2$. Then the difference $n_1 – n_2 = n$, where $n$ is some fixed natural number.

Now if instead $n_2 > n_1$, then $n_2 – n_1$ again equals $n$.

Therefore, the difference is the same in both cases. $\Rightarrow$ Proposition 2 is true. $\Box$

Wait. Both proofs seem valid, but the propositions contradict each other. So which one is actually correct?

Resolving the Contradiction

If you’re leaning toward Proposition 2, I’d encourage you to slow down and re-read both proofs. Proposition 2 feels intuitively right because we’re used to thinking of “the difference between two numbers” as a single fixed quantity. But that intuition is exactly what the paradox exploits.

Let’s test it with actual numbers. Take $n_1 = 40$ and $n_2 = 20$. One is twice the other, so our setup holds.

If $n_1 > n_2$: the difference is $n_1 – n_2 = 40 – 20 = 20$.

Now flip the assumption. If $n_2 > n_1$, then $n_2 = 2n_1$, meaning $n_1 = 10$ and $n_2 = 20$. The difference is $n_2 – n_1 = 20 – 10 = 10$.

The differences are 20 and 10. They are not the same. Proposition 1 is true. $\Box$

Why This Paradox Works

The trick in Proposition 2’s “proof” is subtle and worth understanding. It treats $n_1$ and $n_2$ as if their actual values stay the same when you switch which one is larger. But that’s not how the problem works.

When we say “one of them is twice the other,” the actual values of $n_1$ and $n_2$ change depending on which case you’re in. In Case 1, where $n_1 > n_2$, you might have $n_1 = 40$ and $n_2 = 20$. In Case 2, where $n_2 > n_1$, the constraint forces different values: $n_1 = 10$ and $n_2 = 20$.

Proposition 2’s proof hand-waves over this by saying “the difference is some fixed number $n$” without acknowledging that the underlying values shift. It assumes the difference is an intrinsic property of the pair, independent of which number is larger. That assumption is false.

This is a common logical trap in mathematics. When you define a quantity using a conditional (“the difference, if $n_1 > n_2$”), the condition itself constrains the values. Switching the condition doesn’t just flip a sign. It changes what the variables actually represent.

Historical Context: Raymond Smullyan

This paradox comes from Raymond Smullyan, one of the most creative mathematical logicians of the 20th century. Smullyan (1919-2017) was an American mathematician, concert pianist, stage magician, and philosopher who spent his career making logic accessible and entertaining.

He’s best known for his puzzle books like What Is the Name of This Book? and The Lady or the Tiger?, which are packed with self-referential logic puzzles, knights-and-knaves problems, and paradoxes just like this one. His approach was always playful. He wanted you to feel the confusion first, then work your way out of it.

Smullyan taught at institutions including Princeton, Yeshiva University, and Indiana University. His academic work in mathematical logic, particularly on Godel’s incompleteness theorems, was serious and rigorous. But he believed that the best way to develop mathematical thinking was through puzzles and paradoxes that challenge your assumptions.

This “curious paradox” about differences is a classic Smullyan construction. It’s simple enough that anyone with basic algebra can follow along, yet the logical trap catches even people who should know better. That combination of simplicity and deceptiveness is what made Smullyan’s work so effective as a teaching tool.

Key Takeaway

The lesson here goes beyond this specific puzzle. In mathematics, whenever you work with conditional statements, you have to be extremely careful about what changes and what stays fixed when the condition changes.

Proposition 2 fails because it implicitly assumes that the values of $n_1$ and $n_2$ remain the same across both cases. They don’t. The constraint “one is twice the other” means that switching which number is larger actually changes the numerical values involved.

Here’s a good rule of thumb: whenever a proof feels too easy or too clean, check whether the variables mean the same thing in every step. Often, the “proof” quietly redefines what a symbol represents midway through the argument. That’s exactly what happens in Proposition 2.

Paradoxes like this one train you to read proofs critically, not just check that each individual step is valid, but verify that the steps are actually connected in the way the author claims. That skill is fundamental to mathematical maturity.

Frequently Asked Questions

What is a mathematical paradox?

A mathematical paradox is a statement or result that appears to follow logically from valid premises, yet leads to a contradiction or a conclusion that defies intuition. Paradoxes are valuable because they expose hidden assumptions in our reasoning and help us sharpen our understanding of mathematical logic.

Which proposition is correct in the difference paradox?

Proposition 1 is correct. The difference between the two numbers changes depending on which number is larger. You can verify this with a concrete example: if the numbers are 40 and 20, the difference in one case is 20 and in the other case is 10. Proposition 2 is false because its proof incorrectly assumes the variable values stay the same when the condition changes.

What is the flaw in Proposition 2’s proof?

The flaw is that Proposition 2’s proof treats the difference as a fixed number that doesn’t depend on the case. It says the difference is some number n, then claims the same n appears in both cases. But since the constraint is that one number is twice the other, changing which number is larger actually changes the values of both numbers, which changes the difference.

Who was Raymond Smullyan?

Raymond Smullyan (1919-2017) was an American mathematician, logician, philosopher, concert pianist, and stage magician. He was a professor at several universities including Princeton and Indiana University. He is best known for his recreational logic puzzle books such as What Is the Name of This Book? and The Lady or the Tiger?, which use paradoxes and knights-and-knaves puzzles to teach mathematical reasoning.

Why are paradoxes useful for learning mathematics?

Paradoxes force you to question your assumptions and read proofs more critically. When a seemingly valid argument leads to a contradiction, you have to identify exactly where the reasoning breaks down. This process builds the skill of careful logical analysis, which is essential in every area of mathematics. Paradoxes also make abstract concepts memorable because the surprise of the contradiction sticks with you.