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.
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.