This month's topics:
Numeral cognition and language
What is the relation between our concepts of number and the words we have in our language to express them? The old question was recently thrown into relief by Peter Gordon's report in Science (October 15, 2004) on the Pirahã, an extremely inscrutable Amazonian tribe whose language seems almost completely devoid of number-words. Gordon (Biobehavioral Sciences, Columbia) was categorical: "... the Pirahã's impoverished counting system limits their ability to enumerate exact quantities when set sizes exceed two or three items." Gordon was taken to task by Daniel Casasanto (Brain and Cognitive Sciences, MIT) who argues (Letters, Science, March 18, 2005) that "[the] results are no less consistent with the opposite claim [i.e., that they lack number words because they lack number concepts], which is arguably more plausible." The Pirahã controversy is the background for "The Limits of Counting: Numerical Cognition Between Evolution and Culture" (Science, January 11, 2008). The authors, Sieghard Beller and Andrea Bender (Psychology, Freiburg), focus on the evolution of numbering systems, for which they distinguish two properties: extent and degree of abstractness. They take their examples from Austronesian languages; Adzera is one of them. "Its number words for 1 to 5 are composed of numerals for 1 and 2 only: bits, iru?, iru? da bits (= 2 + 1), iru? da iru? (= 2 + 2), and iru? da iru? da bits (= 2 + 2 + 1)." This is a system with small extent. The authors contrast Adzera with Mangarevan, where besides a general counting sequence there is another one used for tools, sugar cane, pandanus (a fruit) and breadfruit, while ripe breadfruit and octopus are counted with a different sequence, and the first breadfruit and octopus of a season are counted with yet another. This system lacks abstractness. The point the authors emphasize is that both these languages "belong to the same linguistic cluster ... and inherited a regular and abstract decimal numeration system with (at least) two powers of base 10 from their common ancestor, Proto-Oceanic." As they state in their conclusion, "Numeration systems do not always evolve from simple to more complex and from specific to abstract systems.""The computational realization of gesture"
"The Art of Math" was an item on Andrew Sullivan's Atlantic Monthly-based blog for January 26, 2008. Sullivan: "Artist Thomas Briggs uses mathematics to create complex webs of color and form. Using tools more commonly applied to computational fluid dynamics, Briggs builds high resolution drawing-like images."
Thomas Briggs Veils #73. A larger image is available here but, as Briggs explains: "In order to represent these images on a web site they must be reduced in resolution by 99%. The works are a minimum of 3 feet square. The actual line weight is equivalent to that of a 0.2 - 0.3 millimeter pen nib, yet the large scale structure holds up when seen from a distance. This disparity of scale is an essential element of the experience of the works." Image used with permission.
On Briggs' website the artist details his methods, and the way mathematics enters into them: "The computational realization of gesture in my practice entails the construction of of a spatial field of action. In this space various mathematical functions which represent small aspects of movement are distributed. The sum of the various functions is recorded for millions of points in space. These data are collated and translated into thousands of drawing primitives which are written into an image file for printing and archiving."The mathematics of choosiness
"The coevolution of choosiness and cooperation," a Letter in the January 10 2008 Nature, describes a mechanism for the evolution of cooperative behavior. The Bristol-Debrecen team of John McNamara, Zoltan Barta, Lutz Fromhage and Alasdair Houston ran simulations of the "continuous snowdrift game," where in each round an individual, playing against one other, incurs a cost C(x) depending on its own cooperativeness x, and receives a benefit B(x+x') depending on the summed cooperativeness of both players. [The "snowdrift game" gets its name from an example where two drivers are stuck on opposite sides of a snowdrift, and have to choose between waiting in the car and shoveling]. Here are some details of the simulations: Along with cooperativeness (x), each player has a trait y called choosiness. Choosiness specifies the minimum degree of cooperativeness that the player will accept from its co-player.
Among the main conclusions of the experiments: "in a situation where individuals have the opportunity to engage in repeated pairwise interactions, the equilibrium degree of cooperativeness depends critically on the amount of behavioural variation that is being maintained in the population by processes such as mutation." Additionally, "The results suggest an important role of lifespan in the evolution of cooperation." The authors give heuristic arguments to interpret these results: in a uniform population nothing can be gained by being choosy, and therefore there is no incentive for individuals to be cooperative. "This situation changes profoundly if significant variation is maintained in the population by processes such as mutation." Moreover, high mortality counteracts the evolution of cooperation: "If the cooperative associations ... are soon disrupted by mortality, then establishing them is not worth the associated costs.""A Mathematical Gem"
is how Constance Holden (Random Samples, Science, January 18, 2008) describes this image, gleaned from the February 2008 issue of the AMS Notices.
The K-4 crystal is the maximal abelian covering of the tetrahedron, with the inherited geometry. For a larger and higher-resolution image, visit the February 2008 Notices. Image credit Hisashi Naito.
There it illustrates an article by Toshikazu Sunada, who shows that this crystalline structure shares with the diamond the "strong isotropy property," and that these are the only two such structures in three dimensions. (The strong isotropy property states that for any two vertices V and W of the crystal, any ordering of the edges adjacent to V and any ordering of the edges adjacent to W, there is a lattice-preserving congruence taking V to W and each V-edge to the similarly ordered W-edge). Sunada states that the K-4 crystal, beautiful as it is, is purely a mathematical object. Holden begs to differ: "In fact, it shows up in inorganic compounds, lipid networks, and liquid crystals and has been known for decades by other names."
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