most recent 30 from http://mathoverflow.net 2013-05-24T21:34:45Z http://mathoverflow.net/feeds/user/1023 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/27263#27263 Vania Mascioni 2010-06-06T18:31:16Z 2010-06-06T18:31:16Z

<p>A stunning, ignorance-based false belief I have witnessed while observing a class of a math education colleague is that there is no general formula for the n-th Fibonacci number. I wonder if this false belief comes from conflating the (difficult) lack of formulas for prime numbers with something that is just over the horizon of someone whose interests never stretch beyond high-school math.</p> <p>Behind a number of the elementary false beliefs listed here there is a widespread tendency among people to give up too easily (maybe when having to read at least to page 2 in a book), or to nourish an ego that allows to conclude that something is impossible if they cannot do it themselves.</p>

http://mathoverflow.net/questions/4994/fundamental-examples/5423#5423 Vania Mascioni 2009-11-13T19:11:22Z 2009-11-14T22:36:01Z

<p>In combinatorics there are very simple basic graphs from which a whole lot of theory came. For example the complete graphs <code>K_5</code> and <code>K_{3,3}</code> which alone provide the ground level for any non-planar graph according to Kuratowski's theorem. Another simple graph that gave rise to a huge amount of theory is Petersen's graph, which I like to think as the graph whose vertices are the ten two-element subsets of <code>{1,2,3,4,5}</code>, and for which two such vertices are connected iff they are disjoint.</p> <p>A link for Kuratowski's theorem is <a href="http://en.wikipedia.org/wiki/Kuratowski%27s%5Ftheorem" rel="nofollow">http://en.wikipedia.org/wiki/Kuratowski's_theorem</a></p>

http://mathoverflow.net/questions/4994/fundamental-examples/5208#5208 Vania Mascioni 2009-11-12T17:28:43Z 2009-11-13T13:02:45Z

<p>Tsirelson space (see the Wiki entry for quick facts) in Banach space theory was seminal, in that it generated a stream of further refined and specialized counterexamples (most notably in the work of Casazza, Odell, Schlumprecht, and culminating in the famous examples of hereditarily indecomposable spaces by Gowers and Maurey (apologies for the countless others not mentioned here, but the list would be too long). Several long-standing conjectures (even dating back to Banach) have been proved or disproved using these examples, and the flow is not over even now.</p>

http://mathoverflow.net/questions/915/is-there-a-high-concept-explanation-for-why-characteristic-2-is-special/2088#2088 Vania Mascioni 2009-10-23T14:02:39Z 2009-10-23T14:02:39Z

<p>I see binary arithmetic to be the natural companion to set theory, and this singles <code>F_2</code> out given that set theory is the core of mathematics. The basic idea being that all subsets of a finite set with n elements can be associated with a binary n-tuple (an element of <code>(Z_2)^n</code>). Or vice-versa, since we could as well consider set theory as the study of binary n-tuples. (Just an elementary example: the sum of two such n-tuples, using 1+1=0, corresponds to the symmetric difference of the two sets). The very fact that a set of n elements has <code>2^n</code> subsets reminds us of the core meaning of the powers of 2.</p>

http://mathoverflow.net/questions/1677/number-of-metric-spaces-on-n-points/2035#2035 Vania Mascioni 2009-10-23T04:30:42Z 2009-10-23T13:29:07Z

<p>I published a paper a few years ago that studies the question for n=3 and n=4 points, and maybe there are connections with what you are discussing here.</p> <p>Vania Mascioni: On the probability that finite spaces with random distances are metric spaces, Discrete Mathematics 300 (2005) 129-138.</p> <p>My approach was to first count all the metric spaces arising from distances chosen from the pool of integer values in {1,2,...,n}, and then take the limit to infinity to obtain the ratio of metric (real) distances versus all possible distance choices. </p> <p>So, for n=3 points, if the distances are real and set at random, the probability that the triangle is metric will be 1/2 (this can also be handled as an easy calculus exercise, if you like integrals). This is a very easy result.</p> <p>For n=4 points the counting task was much more involved, but the real limit is easily stated: with four points and with the six (real) distances among them chosen at random, the probability that they form a metric space is exactly 17/120.</p> <p>I lost patience and gave up when it came to five points, though the paper above contains an estimate (weak) of what to expect when the number of points is allowed to get large.</p> <p>[Edit: I had mistyped, sorry, the probability I had typed in, 103/120 was for a 4-point set with random distances <em>not</em> to be metric. The metric probability decreases sharply with the size of the space, roughly with the order of c^{M^2}, where c is a constant (between .7 and .9?) and M is the number of points in the space. Also, as some comments below state, indeed the problem has similar nature and motivation to the one of counting finite topologies (see the work by Kleitman and Rothschild to get started), but the nuts and bolts are necessarily different.]</p>

http://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics/27262#27262 Vania Mascioni 2010-06-06T20:53:08Z 2010-06-06T20:53:08Z

This seems related to the widely held belief that 1+1=2, (and how could 2 be equal to 0 ?!). On a psychological level this may connect with the assumption we all tend to make that a is different from b when asked to count the elements of set {a,b} (a related entry is somewhere else on this page), or that in general two different symbols are instinctively taken to denote two distinct objects.