Mathematical Debates
Marc West
Friday, 30 November 2007 11:45 UTC
Mathematical debates are something that we like to highlight in Plus. Please feel free to post mathematical debates, controversies and hearty discussions here.
$50000 Turing Controversy
To kick things off, one of the most current mathematical debates is over the concept of a Universal Turing Machine. The 50-year old mathematical problem has thrown up enormous debate in the last month, and its resolution is worth no less than $US 25000.
The half-century quest to find the simplest possible universal Turing machine was thought solved by 20-year old Birmingham undergraduate Alex Smith, who provided a 40-page proof that Wolfram’s 2,3 Turing machine is indeed universal.
However, this result, and the $25,000 Wolfram 2,3 Turing Machine Research Prize, is now being seriously debated.
Professor Emeritus Vaughan Ronald Pratt, one of the earliest pioneers in the field of computer science, has questioned the proof in a non-to-subtle way by saying:
“Not wanting to push my luck I’ll settle for one question. How did an argument containing such an elementary fallacy get through the filter? [...] Had I pushed my luck my second question would have been, who has verified this proof that has taught an automata theory course at a suitably accredited institution?”
You can read more on Plus in this article on the topic
Updated 30 November 2007 16:44 UTC
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Rollicking Pollock Debate
Publications in Nature and Plus have explored the proposition that fractal structures within the drip paintings of abstract expressionist Jackson Pollock can be used to identify the painter by his “fractal fingerprint”.
But research led by Case Western Reserve University physics doctoral student Kate Jones-Smith suggests that fractal analysis should not be considered a foolproof technique for authenticating works by Pollock and opened up quite a debate.
In 1999 and 2000, University of Oregon physicist Richard Taylor and colleagues proposed that fractal analysis could be used to discriminate between a Pollock drip painting and one by an impersonator.
It seemed that by 2006 public interest in the matter had waned, but this changed when Taylor was invited by the Pollock-Krasner Foundation to determine the authenticity of paintings found by Alex Matter, a close friend of Pollock’s, who claimed that the paintings were created by Pollock. Taylor claimed, using his criteria for determining if a painting is by Pollock, that they were not indeed Pollock paintings.
With interest in the topic rekindled, Jones-Smith along with Harsh Mathur, recently contended that:
“Several problems must be addressed before fractal analysis can be used to authenticate paintings.”
When coming to her conclusions, Jones-Smith came up with creation in Adobe Photoshop in a matter of minutes and applied Taylor’s fractal authentication criteria – it was found to be a genuine Pollock!
Part of the debate lies with the fundamental aspect of defining a fractal and its fractal dimension.
You can read more on Plus in this article on the topic
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Cubic and Quartic Equation Debate
Lodovico Ferrari was an Italian mathematician famed for solving the quartic equation. Ferrari was born in 1522 in Bologna and at the age of 14 became the servant of Gerolamo Cardan, a celebrated Italian Renaissance mathematician, physician, astrologer and gambler.
Ferrari showed mathematical promise at a young age, and at the age of 20 became a public lecturer in geometry. He was also a player in a great mathematical controversy of the time – who should get credit for the development of solutions for the cubic and quartic equations.
For more on this debate, read the full Plus story
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