Timeline for Examples of common false beliefs in mathematics
Current License: CC BY-SA 2.5
15 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 6, 2020 at 14:30 | comment | added | Jim Oldfield | Side note: I agree it is clearest to prove $I$ by first proving $F$ totally independently, rather than starting the whole thing with "assume there are infinitely many primes…" as most proofs do. But you still need proof by contradiction to make that last little step | |
| Feb 6, 2020 at 14:30 | comment | added | Jim Oldfield | I don't claim to know whether Euclid only proved $F$ or also went on to prove $I$, but in the first case he didn't prove the infinitude of primes and in the second case must have used proof by contradiction, so (by the law of excluded middle!) either way he did not prove the infinitude of primes without proof by contradiction. I know it seems pedantic but I see this myth a lot of places online and it seems people really think statement $I$ doesn't need proof by contradiction; certainly a lot of people reading posts like that must surely get that impression. | |
| Feb 6, 2020 at 14:30 | comment | added | Jim Oldfield | @MichaelHardy This "myth", exactly as stated, is itself a myth. It conflates two different statements: In your article you repeatedly use the phrase "Euclid's proof of the infinitude of primes" for a proof of the statement "given any finite set of primes, there exists a prime not in that set" (let's call this statement $F$). But the phrase "infinitude of primes" means, of course, that there are infinitely many primes (call this statement $I$). Admittedly $F$ and $I$ appear almost trivially similar, but to actually prove $I$ from $F$ you need to use a proof by contradiction. | |
| Mar 10, 2016 at 5:15 | comment | added | Michael Hardy | @roysmith : Euclid didn't even consider $1$ to be a number. $\qquad$ | |
| Mar 18, 2013 at 0:04 | comment | added | Toink | @roy: if there are no primes, there are finitely many a fortiori. So there's nothing to check there. | |
| May 9, 2011 at 2:27 | comment | added | roy smith | Since Euclid omits to check that there exists at least one prime, his induction must begin with the case n = 0, hence his argument seems to require the fact that the product of an empty set of primes equals 1. | |
| Jul 20, 2010 at 22:43 | comment | added | Michael Hardy | Actually I think the use of three letters was just a notational device. He clearly meant an arbitrary finite set of prime numbers (if he hadn't had that in mind, he couldn't have written that particular proof). | |
| Jul 20, 2010 at 14:51 | comment | added | Pietro Majer | Note indeed the original Euclid's statement: Prime numbers are more than any previously assigned finite collection of them (my translation). This reflects a remarkable maturity and consciousness, if we think that mathematicians started speaking of infinite sets a long time before a well founded theory was settled and paradoxes were solved. Euclid's original proof in my opinion is a model of precision and clearness. It starts: Take e.g. three of them, A, B and Γ . He takes three prime numbers as the first reasonably representative case to get the general construction. | |
| Jul 8, 2010 at 20:29 | comment | added | BlueRaja | Yes, Euclid sorry :) | |
| Jul 7, 2010 at 21:55 | comment | added | Michael Hardy | @ BlueRaja: I'm assuming "Euler" is a typo and you meant Euclid. Euclid said if you take any arbitrary finite set of prime numbers, then multiply them and add 1, and factor the result into primes, you get only new primes not already in the finite set you started with. The proof that they're not in that set is indeed by contradiction. But the proof as a whole is not, since it doesn't assume only finitely many primes exist. | |
| Jul 7, 2010 at 21:36 | comment | added | BlueRaja | So what was Euler's true original proof, then? | |
| Jun 12, 2010 at 1:38 | comment | added | Rob Harron | wow! I tend to take historical statements in math books/papers with a grain of salt, but this makes me think more salt might be required... | |
| Jun 7, 2010 at 3:28 | comment | added | Michael Hardy | Actually, if you read our paper on this, you'll find that I won't be surprised at all. (BTW, my first name is spelled in the usual way, not the way you spelled it.) | |
| Jun 7, 2010 at 0:07 | comment | added | The Mathemagician | And you'd be surprised how many quite knowledgable PHD's spend decades repeating this mistake to thier students,Micheal. | |
| Jun 6, 2010 at 19:52 | history | answered | Michael Hardy | CC BY-SA 2.5 |