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> Peluse answered that question in a counterintuitive way — by thinking about exactly what it would take for a set of numbers not to contain the pattern you’re looking for.

Proof by contradiction isn't that counter intuitive.

And it's utterly bizarre that this article doesn't explain that this technique is proof by contradiction, or give an example of what it is. The proof that the square root of two can't be expressed as a ratio of integers (hence is not-ratio-able, or irrational) is approachable by highschool freshmen, and would add so much to the article.

To me it does not seem like it is proof by contradiction, or maybe not in the straight forward way that contradictions are used in proofs. It just happens to think about the problem in a complementary (of complement sets ...) way.

When I was coming up with and solving problems with a math friend I learned that proof by contradiction was perhaps the most common way to prove things. It’s strange to people who don’t work with math everyday, to everyone else it’s secondhand.

It's not just math, it's a pretty basic concept in philosophy, or any area where critical thinking is required

> hence is not-ratio-able, or irrational


Interestingly, the etymology seems to go:

1. ἄλογος (alogos, Greek for unsayable or unreasonable)

2. irrationalis (Latin translation of ἄλογος)

3. rationalis (Latin backformation)

4. irrational/rational (English translation)

5. ratio (English backformation)

Steps 3 and 5, the backformations, are perhaps in the opposite direction one would expect.


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