"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.
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].
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).
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.
Living organisms likely exploit this property of self-similarity for easy scaling.
Do note your usage of the term "likely". You don't know exactly why, just like no one else really does.
You're making a likely supposition, not a statement of fact. We have theories as to why and how the Fibonacci sequences are found in nature as you've posited, but that doesn't mean it's well-understood nor some sort of old and tired closed case.
pseudo-mystical notions
I don't see Chomsky espousing pseudo-mystical notions. He simply said it's found all over nature and it's not well understood why. By your insulting tone, I would surmise you have a grudge with Chomsky that goes well beyond anything to do with Fibonacci numbers.
Are you a neo-libertarian, perhaps?
We understand why the Golden Ratio shows up all over the place, especially in biology: it doesn't.
I don't think Chomsky meant for you to take "all over the place" so --literally--. It's a commonly understood expression that's not meant to be taken so literally unless one perhaps has a pedantic axe to grind.
The respected mathematician Arthur T. Benjamin would disagree with you that it's not an interesting fact when it shows up in nature:
Chomsky didn't say anything terribly problematic about Fibonacci numbers. You're frankly going to have to find a better example to discount his conclusions on other topics which you have no understanding (as you said).
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.
Yes, but why? Do you have the answer to that?
You've basically posited the same thing that Chomsky said when he said:
"There is something about the physical world that forces certain kinds of structures to emerge under particular conditions."
What you'll actually find is not a series of systems all approximating the same Golden Spiral
Correct, but one will also find otherwise. Also, often it's an approximation that's often not meant to be taken --literally--. It's just noting similarities in patterns that anyone (including laypersons) can observe and there's nothing wrong with doing that.
If Chomsky was writing a piece on Fibonacci numbers, I might agree with you that he didn't go into enough depth on the topic. But, it appears you missed his greater point while focusing on trite details and being too literal.
I don't see Chomsky espousing pseudo-mystical notions. He simply said it's found all over nature and it's not well understood why. By your insulting tone, I would surmise you have a grudge with Chomsky that goes well beyond anything to do with Fibonacci numbers.
Are you a neo-libertarian, perhaps?
Amusing, but no. I'm rather a fan of his political writings, and while he's extreme in his views, certain concepts like manufacturing consent appear to have validity. But, unlike his politics, I don't have a firm grasp of his linguistics, so in order to gauge his credibility, I'm looking toward a topic I know.
My grudge, if I have any at all, is not with Chomsky or his politics, but rather with the Myth of Magical Numbers, which has so entrenched itself that even mathematicians have fallen prey to it in one form or another for centuries, if not millennia. I'll point you back to Devlin[0][1] regarding the golden ratio claims.
I could go on, but you get the picture, I hope. I've done my bit once again, and attempted once more to right my own previous wrongs in inadvertently adding fuel to this curious forest-fire of myths.
Just as happened last time, I anticipate receiving some truly ANGRY emails from readers incensed that I should dare question their long cherished beliefs about this particular number. (Maybe that's why we say it is irrational?)
And there we scratch the surface of what I think is a fascinating aspect of human nature. People, at least many people, seem to really WANT there to be numbers with mystical properties. So much so that they are prepared to put aside their otherwise wise insistence on evidence or proof.
(Many of the emails I got last time demanded that I give proof that the golden ratio is not in the design of the Parthenon, for instance. Which is of course to get the scientific method completely backwards. The hypothesis in that case is that the GR did figure in the Greeks' design, and that is what needs justification - and does not get. Ditto all the other spurious GR claims.)
[0] See last post, Devlin 2007.
[1] Dr. Keith Devlin is a co-founder and Executive Director of the Stanford University's H-STAR institute, a co-founder of the Stanford Media X research network, and a Senior Researcher at CSLI. He is a World Economic Forum Fellow, a Fellow of the American Association for the Advancement of Science, and a Fellow of the American Mathematical Society. At the time of writing, he served as Executive Director of the Center for the Study of Language and Information at Stanford University and appeared as The Math Guy on NPR's Weekend Edition. http://web.stanford.edu/~kdevlin/ if you care for his credentials.
My grudge, if I have any at all, is not with Chomsky or his politics, but rather with the Myth of Magical Numbers, which has so entrenched itself that even mathematicians have fallen prey to it in one form or another for centuries, if not millennia.
I see your point, but I'm not sure Chomsky was espousing anything magical (as in pseudo-science) by saying the Fibonacci sequence is found in nature and isn't well understood why. Present company excepted, perhaps.
I also try to keep in mind that the term "magical" is often rather amorphously applied.
Some call pseudo-science "magical" due to ignorance.
Some call some science "magical" due to ignorance.
However, some call certain science "magical" not due to ignorance, but as a respectful nod towards the vast complexities involved that we still don't know (and perhaps never will) that surround the complex "nature"[0] of science.
If you haven't already read it, I think this is a pretty interesting take on it (in a sense):
I see your point as well, and it's quite well taken.
On second reading Chomsky's quote doesn't seem as hand-wavy about the Fibonacci Sequence as it initially appeared. It's one of those topics that has such a myriad of falsehoods perpetuated around it, that perhaps I was too primed to see it here.
Some call pseudo-science "magical" due to ignorance.
This was the sense in which I used magical, although you are right. "Magical" can be used to evoke either the vision of pseudo-scientific nonsense or "magical" in the sense of "insufficiently understood, complex, and thus awe-inspiring."
If you haven't already read it, I think this is a pretty interesting take on it (in a sense):
I devoured A New Kind of Science when it was released because I was fascinated by emergent systems and cellular automata at the time, and have read it again about a year ago, and found it didn't hold up as well. Nevertheless excellent book, if a bit bold in its claims.
I haven't seem the follow ups, but thanks for pointing me toward them! I'll be sure to give them a read.
Apologizes if I pushed back a little hard, but rest assured my only grudge here is with misuse of the Golden Spiral, not with Prof. Chomsky and certainly not with you.
Comments
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.
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
[2]: http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci...
[3]: http://www.amazon.com/Structural-Stability-Morphogenesis-Adv...
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).
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]
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
[1]http://www.maa.org/external_archive/devlin/devlin_05_07.html
Do note your usage of the term "likely". You don't know exactly why, just like no one else really does.
You're making a likely supposition, not a statement of fact. We have theories as to why and how the Fibonacci sequences are found in nature as you've posited, but that doesn't mean it's well-understood nor some sort of old and tired closed case.
I don't see Chomsky espousing pseudo-mystical notions. He simply said it's found all over nature and it's not well understood why. By your insulting tone, I would surmise you have a grudge with Chomsky that goes well beyond anything to do with Fibonacci numbers.
Are you a neo-libertarian, perhaps?
I don't think Chomsky meant for you to take "all over the place" so --literally--. It's a commonly understood expression that's not meant to be taken so literally unless one perhaps has a pedantic axe to grind.
The respected mathematician Arthur T. Benjamin would disagree with you that it's not an interesting fact when it shows up in nature:
http://www.ted.com/talks/arthur_benjamin_the_magic_of_fibona...
Chomsky didn't say anything terribly problematic about Fibonacci numbers. You're frankly going to have to find a better example to discount his conclusions on other topics which you have no understanding (as you said).
Yes, but why? Do you have the answer to that?
You've basically posited the same thing that Chomsky said when he said:
"There is something about the physical world that forces certain kinds of structures to emerge under particular conditions."
Correct, but one will also find otherwise. Also, often it's an approximation that's often not meant to be taken --literally--. It's just noting similarities in patterns that anyone (including laypersons) can observe and there's nothing wrong with doing that.
If Chomsky was writing a piece on Fibonacci numbers, I might agree with you that he didn't go into enough depth on the topic. But, it appears you missed his greater point while focusing on trite details and being too literal.
Are you a neo-libertarian, perhaps?
Amusing, but no. I'm rather a fan of his political writings, and while he's extreme in his views, certain concepts like manufacturing consent appear to have validity. But, unlike his politics, I don't have a firm grasp of his linguistics, so in order to gauge his credibility, I'm looking toward a topic I know.
My grudge, if I have any at all, is not with Chomsky or his politics, but rather with the Myth of Magical Numbers, which has so entrenched itself that even mathematicians have fallen prey to it in one form or another for centuries, if not millennia. I'll point you back to Devlin[0][1] regarding the golden ratio claims.
I could go on, but you get the picture, I hope. I've done my bit once again, and attempted once more to right my own previous wrongs in inadvertently adding fuel to this curious forest-fire of myths.
Just as happened last time, I anticipate receiving some truly ANGRY emails from readers incensed that I should dare question their long cherished beliefs about this particular number. (Maybe that's why we say it is irrational?)
And there we scratch the surface of what I think is a fascinating aspect of human nature. People, at least many people, seem to really WANT there to be numbers with mystical properties. So much so that they are prepared to put aside their otherwise wise insistence on evidence or proof.
(Many of the emails I got last time demanded that I give proof that the golden ratio is not in the design of the Parthenon, for instance. Which is of course to get the scientific method completely backwards. The hypothesis in that case is that the GR did figure in the Greeks' design, and that is what needs justification - and does not get. Ditto all the other spurious GR claims.)
[0] See last post, Devlin 2007.
[1] Dr. Keith Devlin is a co-founder and Executive Director of the Stanford University's H-STAR institute, a co-founder of the Stanford Media X research network, and a Senior Researcher at CSLI. He is a World Economic Forum Fellow, a Fellow of the American Association for the Advancement of Science, and a Fellow of the American Mathematical Society. At the time of writing, he served as Executive Director of the Center for the Study of Language and Information at Stanford University and appeared as The Math Guy on NPR's Weekend Edition. http://web.stanford.edu/~kdevlin/ if you care for his credentials.
I see your point, but I'm not sure Chomsky was espousing anything magical (as in pseudo-science) by saying the Fibonacci sequence is found in nature and isn't well understood why. Present company excepted, perhaps.
I also try to keep in mind that the term "magical" is often rather amorphously applied.
Some call pseudo-science "magical" due to ignorance.
Some call some science "magical" due to ignorance.
However, some call certain science "magical" not due to ignorance, but as a respectful nod towards the vast complexities involved that we still don't know (and perhaps never will) that surround the complex "nature"[0] of science.
If you haven't already read it, I think this is a pretty interesting take on it (in a sense):
http://www.amazon.com/New-Kind-Science-Stephen-Wolfram/dp/15...
Series of followups 10 years later:
http://blog.stephenwolfram.com/2012/05/its-been-10-years-wha...
[0] https://www.youtube.com/watch?v=Bo3R3LBjDek (Love And Rockets - "No New Tale To Tell")
I see your point as well, and it's quite well taken.
On second reading Chomsky's quote doesn't seem as hand-wavy about the Fibonacci Sequence as it initially appeared. It's one of those topics that has such a myriad of falsehoods perpetuated around it, that perhaps I was too primed to see it here.
This was the sense in which I used magical, although you are right. "Magical" can be used to evoke either the vision of pseudo-scientific nonsense or "magical" in the sense of "insufficiently understood, complex, and thus awe-inspiring."
I devoured A New Kind of Science when it was released because I was fascinated by emergent systems and cellular automata at the time, and have read it again about a year ago, and found it didn't hold up as well. Nevertheless excellent book, if a bit bold in its claims.
I haven't seem the follow ups, but thanks for pointing me toward them! I'll be sure to give them a read.
Apologizes if I pushed back a little hard, but rest assured my only grudge here is with misuse of the Golden Spiral, not with Prof. Chomsky and certainly not with you.