June 26, 2026
Proof, but make it messy
Making Sense of Proof by Contradiction [pdf]
Math’s most confusing flex just got roasted, defended, and meme-ified by the comments
TLDR: The article says a famous math proof is a bad beginner example and argues teachers should use simpler ones first. Commenters turned that into a lively brawl over whether the method is elegant everyday reasoning, a mislabeled logic trick, or something mathematicians keep explaining badly.
A seemingly calm math paper about proof by contradiction turned into a mini comment-section soap opera, because apparently nothing gets people fired up like a way of proving something by saying, “Let’s pretend the opposite is true and watch it fall apart.” The article argues that students are often given a needlessly confusing first example—the famous proof that the square root of 2 can’t be written as a simple fraction—and suggests there are easier ways to teach the idea. In plain English: the author thinks math teachers may be opening with the hard mode tutorial.
The crowd? Very split, very passionate, very nerdy in the best way. One commenter fondly admitted they loved the method so much in college they tried to use it for everything on a logic test, which is honestly chaotic-good behavior. Another jumped in with the classic “actually...” correction, saying lots of what people call proof by contradiction is really a cousin method, and yes, the comments absolutely got pedantic fast. Then things got deeper: one reader said the whole idea only clicked after reframing it in a more logic-heavy way, while another complained that hidden assumptions are exactly what make students feel like the trick is being pulled behind their backs.
And then came the spiciest swipe: one commenter basically accused mathematicians of explaining this badly on a regular basis. So while the article wanted to demystify a classic math move, the community response was the real show: nostalgia, nitpicking, confusion, and a full-on debate over whether the lesson is elegant, misleading, or just dressed-up common sense.
Key Points
- •The article argues that proof by contradiction is common in everyday reasoning, despite often being seen as difficult by students.
- •It presents the standard proof that √2 is irrational and says this proof is subtle for beginners because it assumes a fraction in lowest terms to set up the contradiction.
- •The article suggests the √2 proof is better viewed as an instance of infinite descent rather than the best first example of contradiction.
- •It proposes the proof that log10 2 is irrational as a simpler introductory example because it reaches a contradiction through prime factors without requiring coprime assumptions.
- •The article explains that the key structural reason the log10 2 proof works is that 2 is not a rational power of 10, unlike pairs such as 2 and 8.