Apr 14, 2021
Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses.
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Inside the symmetries of a crystal shape, a postdoctoral researcher has unearthed a counterexample to a basic conjecture about multiplicative inverses.
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From the About section: “I [Jean-Pierre Merx] initiated this website because for years I have been passionated about Mathematics as a hobby and also by ‘strange objects’. Mathematical counterexamples combine both topics.”
On this website, you find lots of ingenious counterexamples to mathematical statements which may sound reasonable but are really not true. For instance, can you find an unbounded positive continuous function with a convergent integral? Or did you realize that in general metric spaces, the closure of an open ball isn’t necessarily a closed ball with the same radius? Or that, even though matrices AB and BA always share the same characteristic polynomials, they do not necessarily have the same minimal polynomial? The list is long!
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#オイラー予想 の #反例 #counterexample to #eulersconjecture by #supercomputer #cdc6600 https://www.instagram.com/p/CS5Wy7Cp_J_/?utm_medium=tumblr
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One of the shortest serious math articles ever published, presenting a counterexample to Euler’s sum of powers conjecture, an unsuccessful attempt to generalize Fermat’s last theorem.
Disproof: Counterexamples and Terms
Disproof: Counterexamples and Terms
Aristotle proves invalidity by constructing counterexamples.
This is very much in the spirit of modern logical theory:
- all that it takes to show that a certain form is invalid is a single instance of that form with true premises and a false conclusion. [NB]
However, Aristotle states his results not by saying that certain premise-conclusion combinations are invalid but by saying that certain premise pairs DO NOT “syllogize”:
- that is, that, given the pair in question, examples can be constructed in which premises of that form are true and a conclusion of any of the four possible forms is false.
When possible, he does this by a clever and economical method:
- he gives two triplets of terms, one of which makes the premises true and a universal affirmative “conclusion” true, and the other of which makes the premises true and a universal negative “conclusion” true.
The first is a counterexample for an argument with either an E or an O conclusion, and the second is a counterexample for an argument with either an A or an I conclusion.
via: http://plato.stanford.edu/archives/spr2012/entries/aristotle-logic/