4 ms·

The 'Chomskyan Era' (2000)

19 points by GmeSalazar 12y ago

s_q_b 12y ago
>"Or, take the mathematical series called the 'Fibonacci series'. It shows up all over the place in nature; nobody knows exactly why. If you take a sunflower and you look at the flower, it has spirals that go in different directions. The number of parts that appear in adjacent spirals are related to one another as successive terms in the Fibonacci series. You find that kind of thing all over nature; it is not well understood why. There is something about the physical world that forces certain kinds of structures to emerge under particular conditions." So this is why I find Chomsky's conclusions to be suspect when he speaks about topics of which I have no understanding: he's often casually incorrect about topics in other fields. We understand why the Golden Ratio shows up all over the place, especially in biology: it doesn't. In reality, it's the logarithmic spiral that's common, and it's a due to logarithmic spirals being a necessary characteristic of certain structures that exhibit self-similarity. This quality was well known to Renaissance and Enlightenment era mathematicians and physicists, as well as to the ancients as the "spira mirabilis", or "marvelous spiral." But it seems to have lost favor in the modern era to a mythos surrounding the Golden Ratio Phi and the Fibonacci sequence. For example, Jacob Bernoulli, the famed mathematician and no stranger to the Golden Spiral and the Spira Mirabilis, requested the latter be placed above his headstone, with the inscription "Eadem mutata resurgo," meaning "Although changed, I shall arise again," a reference to the then well-understood ability of logarithmic spirals to change scale while preserving shape. Living organisms likely exploit this property of self-similarity for easy scaling. It's likely helpful to sunflowers to maximize the area of solar exposure, and snails to maximize living space. Once an organism evolves a roughly spiral structure, logarithmic spiral patterns are easy local maxima for the evolutionary algorithm to find, because it gives organisms the ability to scale aspects of their biology without major structural changes. Non-living systems exhibiting logarithmic spirals, such as certain galaxies, are a result of various (different) physical forcing functions that cause self-similarity to arise. One way to debunk the pseudo-mystical notions surrounding Phi is to very carefully measure the spirals themselves. What you'll actually find is not a series of systems all approximating the same Golden Spiral, but rather very different systems all exhibiting different logarithmic spirals, with the shared characteristic of self-similarity.
- zenogais 12y ago
  So far as I can tell your post is actually "casually wrong". The sunflower seed example given by Chomsky seems to be mathematically well supported as a representation of Phi/Fibonacci sequences [1][2]. There's no mystification going on here, Chomsky is simply saying we don't fully understand the morphogenetic systems that produce these results. You offered some speculation as to why this might be, but so far no formalized results. Rene Thom did some excellent early work in this area, and demonstrated that chaotic dynamics can help to model some of these issues, but the topic is far from fully explored [3]. [1]: http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html [2]: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci... [3]: http://www.amazon.com/Structural-Stability-Morphogenesis-Advanced-Classics/dp/0201406853 http://www.amazon.com/Structural-Stability-Morphogenesis-Adv...
  - Cowicide 12y ago
    Whoops, you must have written your response and submitted as I was typing mine. Thank you for saying this far more succinctly and intelligently than I could have (or tried to).
  - s_q_b 12y ago
    Fairly certain of the following: Nautilus shells and sunflower petals are consistently logarithmic spirals, not golden spirals, and deviate from the golden ratio. See e.g. Sharp 2002.[0] >"Starting with the observation that shell spirals are logarithmic spirals, many people automatically assume that, because the golden ratio can be used to draw a logarithmic spiral, all shell spirals are related to the golden ratio, when, in fact, they are not. Sharp's own measurements of nautilus shells also confirmed that the golden ratio rectangular spiral and the nautilus spiral 'simply do not match.' Nonetheless, many accounts still insist that a cross section of nautilus shell shows a growth pattern of chambers governed by the golden ratio. 'One of the amazing things about such misconceptions is that it is so widespread, even by mathematicians who should know better,' Sharp observes. 'It is a prime example of why geometry needs to be taught more widely and not only geometry, but the visual appreciation of shape and proportion.'" There is of course a mystification going on, which is exactly what Chomsky is trying to invoke. I am pointing out that these phenomena are far better understood than "oh, we don't know why there are spirals with a particular magic ratio all over the natural world." See e.g. Devlin 2007: "Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on."[1] There's only one leap I'm making here, but it's the only logical hypothesis I've heard: that the self-similarity of logarithmic spirals is the reason for their apparent prevalence. [0]https://www.sciencenews.org/article/sea-shell-spirals https://www.sciencenews.org/article/sea-shell-spirals [1]http://www.maa.org/external_archive/devlin/devlin_05_07.html http://www.maa.org/external_archive/devlin/devlin_05_07.html

19 points by GmeSalazar 12y ago

s_q_b 12y ago

>"Or, take the mathematical series called the 'Fibonacci series'. It shows up all over the place in nature; nobody knows exactly why. If you take a sunflower and you look at the flower, it has spirals that go in different directions. The number of parts that appear in adjacent spirals are related to one another as successive terms in the Fibonacci series. You find that kind of thing all over nature; it is not well understood why. There is something about the physical world that forces certain kinds of structures to emerge under particular conditions." So this is why I find Chomsky's conclusions to be suspect when he speaks about topics of which I have no understanding: he's often casually incorrect about topics in other fields. We understand why the Golden Ratio shows up all over the place, especially in biology: it doesn't. In reality, it's the logarithmic spiral that's common, and it's a due to logarithmic spirals being a necessary characteristic of certain structures that exhibit self-similarity. This quality was well known to Renaissance and Enlightenment era mathematicians and physicists, as well as to the ancients as the "spira mirabilis", or "marvelous spiral." But it seems to have lost favor in the modern era to a mythos surrounding the Golden Ratio Phi and the Fibonacci sequence. For example, Jacob Bernoulli, the famed mathematician and no stranger to the Golden Spiral and the Spira Mirabilis, requested the latter be placed above his headstone, with the inscription "Eadem mutata resurgo," meaning "Although changed, I shall arise again," a reference to the then well-understood ability of logarithmic spirals to change scale while preserving shape. Living organisms likely exploit this property of self-similarity for easy scaling. It's likely helpful to sunflowers to maximize the area of solar exposure, and snails to maximize living space. Once an organism evolves a roughly spiral structure, logarithmic spiral patterns are easy local maxima for the evolutionary algorithm to find, because it gives organisms the ability to scale aspects of their biology without major structural changes. Non-living systems exhibiting logarithmic spirals, such as certain galaxies, are a result of various (different) physical forcing functions that cause self-similarity to arise. One way to debunk the pseudo-mystical notions surrounding Phi is to very carefully measure the spirals themselves. What you'll actually find is not a series of systems all approximating the same Golden Spiral, but rather very different systems all exhibiting different logarithmic spirals, with the shared characteristic of self-similarity.

zenogais 12y ago
So far as I can tell your post is actually "casually wrong". The sunflower seed example given by Chomsky seems to be mathematically well supported as a representation of Phi/Fibonacci sequences [1][2]. There's no mystification going on here, Chomsky is simply saying we don't fully understand the morphogenetic systems that produce these results. You offered some speculation as to why this might be, but so far no formalized results. Rene Thom did some excellent early work in this area, and demonstrated that chaotic dynamics can help to model some of these issues, but the topic is far from fully explored [3]. [1]: http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html http://www.popmath.org.uk/rpamaths/rpampages/sunflower.html [2]: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat2.html http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci... [3]: http://www.amazon.com/Structural-Stability-Morphogenesis-Advanced-Classics/dp/0201406853 http://www.amazon.com/Structural-Stability-Morphogenesis-Adv...
- Cowicide 12y ago
  Whoops, you must have written your response and submitted as I was typing mine. Thank you for saying this far more succinctly and intelligently than I could have (or tried to).
- s_q_b 12y ago
  Fairly certain of the following: Nautilus shells and sunflower petals are consistently logarithmic spirals, not golden spirals, and deviate from the golden ratio. See e.g. Sharp 2002.[0] >"Starting with the observation that shell spirals are logarithmic spirals, many people automatically assume that, because the golden ratio can be used to draw a logarithmic spiral, all shell spirals are related to the golden ratio, when, in fact, they are not. Sharp's own measurements of nautilus shells also confirmed that the golden ratio rectangular spiral and the nautilus spiral 'simply do not match.' Nonetheless, many accounts still insist that a cross section of nautilus shell shows a growth pattern of chambers governed by the golden ratio. 'One of the amazing things about such misconceptions is that it is so widespread, even by mathematicians who should know better,' Sharp observes. 'It is a prime example of why geometry needs to be taught more widely and not only geometry, but the visual appreciation of shape and proportion.'" There is of course a mystification going on, which is exactly what Chomsky is trying to invoke. I am pointing out that these phenomena are far better understood than "oh, we don't know why there are spirals with a particular magic ratio all over the natural world." See e.g. Devlin 2007: "Part of the process of becoming a mathematics writer is, it appears, learning that you cannot refer to the golden ratio without following the first mention by a phrase that goes something like 'which the ancient Greeks and others believed to have divine and mystical properties.' Almost as compulsive is the urge to add a second factoid along the lines of 'Leonardo Da Vinci believed that the human form displays the golden ratio.' There is not a shred of evidence to back up either claim, and every reason to assume they are both false. Yet both claims, along with various others in a similar vein, live on."[1] There's only one leap I'm making here, but it's the only logical hypothesis I've heard: that the self-similarity of logarithmic spirals is the reason for their apparent prevalence. [0]https://www.sciencenews.org/article/sea-shell-spirals https://www.sciencenews.org/article/sea-shell-spirals [1]http://www.maa.org/external_archive/devlin/devlin_05_07.html http://www.maa.org/external_archive/devlin/devlin_05_07.html