most recent 30 from http://mathoverflow.net 2013-05-22T08:41:02Z http://mathoverflow.net/feeds/user/1859 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7584/what-are-the-most-misleading-alternate-definitions-in-taught-mathematics/7867#7867 Christos 2009-12-05T15:05:40Z 2010-05-27T13:17:00Z

<p>A simple example is the two definitions for independence of events:</p> <ol> <li>A and B are independent iff $P(A\cap B) = P(A)P(B)$</li> <li>A is independent from B iff $P(A\mid B) = P(A)$</li> </ol> <p>Some presentations start with Definition 1, which is entirely uninformative: nothing in it explains why on earth we bother discussing this. In contrast, Definition 2 says exactly what "independent" means: knowing that B has occured does not change the probability that A occurs as well.</p> <p>A reasonable introduction to the subject should start with Definition 2; then observe there is an issue when P(B)=0, and resolve it; then observe independence is symmetric; then derive Definition 1.</p>

http://mathoverflow.net/questions/24526/good-papers-books-essays-about-the-thought-process-behind-mathematical-research/24769#24769 Christos 2010-05-15T16:09:43Z 2010-05-15T16:09:43Z

<p>Manindra Agrawal has talked about the story behind the primality testing algorithm: <a href="http://www.cse.iitk.ac.in/~manindra/presentations/GodelTalk.pdf" rel="nofollow">http://www.cse.iitk.ac.in/~manindra/presentations/GodelTalk.pdf</a></p>

http://mathoverflow.net/questions/9073/when-does-positive-imply-sum-of-squares/9156#9156 Christos 2009-12-17T06:17:40Z 2009-12-17T06:17:40Z

<p>I believe <a href="http://mathoverflow.net/questions/9089/what-was-hilberts-geometric-construction-in-his-17th-problem" rel="nofollow">another recent question</a> contains an answer: </p> <blockquote> <p>Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. </p> </blockquote>

http://mathoverflow.net/questions/7584/what-are-the-most-misleading-alternate-definitions-in-taught-mathematics/7868#7868 Christos 2009-12-05T15:26:51Z 2009-12-05T15:26:51Z

<p>Another simple example is the definition for equivalence relations:</p> <ol> <li>R(.,.) is an equivalence relation iff R is reflexive, symmetric, and transitive.</li> <li>R(.,.) is an equivalence relation iff there exists a function f such that R(a,b) iff f(a)=f(b).</li> </ol> <p>Most presentations start with Definition 1, which contains no hint as to why we bother discussing such relations or why we call them "equivalences". In contrast, Definition 2 (along with a couple of examples) immediately tells you that R captures one particular attribute of the elements of the domain; and, since elements with the same value for this attribute are called "equivalent", R is called an "equivalence".</p> <p>A reasonable introduction should start with Definition 2, then go on to prove Definition 1 is a convenient alternative characterization.</p>

http://mathoverflow.net/questions/7584/what-are-the-most-misleading-alternate-definitions-in-taught-mathematics/7868#7868 Christos 2009-12-17T06:26:33Z 2009-12-17T06:26:33Z

I first saw this as Definition 2.1 in <a href="http://courses.csail.mit.edu/6.042/fall05/ln4.pdf" rel="nofollow">courses.csail.mit.edu/6.042/fall05/ln4.pdf</a> .