Timeline for Examples of common false beliefs in mathematics
Current License: CC BY-SA 2.5
13 events
| when toggle format | what | by | license | comment | |
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| Apr 21, 2016 at 14:37 | comment | added | ACL | @Michael Wasn't it the sum of the empty set of half-roots that was equal to $1/6$? This is at least what Demazure writes after writing Kac's character formula for the group $U(1)$. But his witty addition that “this would have unexpected consequences, especially regarding the teaching of mathematics in kindergarten” should not be have been taken seriously! | |
| Dec 3, 2013 at 1:11 | comment | added | Michael | My first confusion about empty sets came from Seminaire Bourbaki expose on Macdonald polynomials that established that the sum of a certain empty set is $\pi^2/6$. On the same page the author added that those who have a problem with the above result belong in a kindergarten, which only compounded to my feeling of inadequacy. | |
| Mar 14, 2012 at 22:01 | comment | added | Tom Goodwillie | I once taught abstract algebra from a book that adopted the artificial convention that the domain of a map of sets must be nonempty. I eventually figured out that the reason was in order to be able to say that every one-to-one map has a left inverse. And I have many times taught topology from a book that adopts the artificial convention that when speaking of the product of two spaces we require both spaces to be nonempty. I eventually figured out that the reason was in order to be able to say that $X\times Y$ is compact if and only if both $X$ and $Y$ are compact. | |
| May 10, 2011 at 4:03 | comment | added | roy smith | congratulations if you convinced them the empty set is a (minimal) spanning set? | |
| Apr 4, 2011 at 9:46 | comment | added | Toby Bartels | @ Todd: That's just because logicians don't know mathematics. | |
| Mar 31, 2011 at 14:26 | comment | added | Todd Trimble | Probably many mathematicians in the twentieth century didn't accept that the empty set "existed", including for example R.L. Moore. And to this day, a lot of people have trouble comfortably dealing with it, as witnessed in the WP article en.wikipedia.org/wiki/First-order_logic#Empty_domains | |
| Dec 16, 2010 at 23:08 | comment | added | Qiaochu Yuan | @kow: I disagree. That is the "wrong" definition of transitivity for empty G-sets. See the discussion at qchu.wordpress.com/2010/12/03/empty-sets . | |
| Dec 14, 2010 at 18:05 | comment | added | kow | @ACL Of course it is transitive - given any x,y in the empty set, the identity element sends x to y. It is even n-transitive for all n! | |
| Dec 1, 2010 at 22:53 | comment | added | ACL | Thierry: of course it makes sense. But the action is not transitive. | |
| Aug 31, 2010 at 2:24 | comment | added | Thierry Zell | I can combine Qiaochu's and Victor's remarks in this memory I have of a coffee break conversation between two colleagues, who were arguing on whether it made sense to say that the 1-element group acts on the empty set. I wisely decided to stay out of the controversy... | |
| Jul 9, 2010 at 4:12 | comment | added | Victor Protsak | For most of the history of civilization, zero was very controversial... | |
| Jul 7, 2010 at 23:56 | comment | added | Qiaochu Yuan | Bjorn Poonen once gave a lecture at MIT about the empty set; it really opened my eyes. If someone wrote a textbook or something on the matter I think everyone would be a lot less confused. | |
| Jul 7, 2010 at 23:47 | history | answered | Kurt Luoto | CC BY-SA 2.5 |