The Golden Ratio 676
raceBannon writes "The book surprised and fascinated me. I thought it was going to be solely about the Golden Ratio. Mario Livio does cover the topic but along the way he throws in some mathematical history and even touches on the idea that math may not be a universal concept spread across the galaxy." Read on for the rest of raceBannon's review.
| The Golden Ratio | |
| author | Mario Livio |
| pages | 320 |
| publisher | Broadway |
| rating | 7/10 |
| reviewer | raceBannon |
| ISBN | 0767908155 |
| summary | Through telling the tale of the Golden Ratio, Livio explains how this simple ratio pops up in all kinds of physical phenomenon and debunks the idea that the ratio is present in many famous man-made structures and art work. Along the way, he provides historical tidbits regarding some of the well-known and not so well-known mathematicians throughout the ages and he tells the story of some of the more famous and not so famous mathematical advances. Finally, he discusses the possibility that mathematics may represent some kind of global truth that exists throughout the cosmos. |
I have to admit that it is a little spooky to me that this ratio, this irrational number (1.6180339887...), pops up in many varied natural phenomena from how sunflowers grow to the formation of spiral galaxies; not to mention that the Golden Ratio and the Fibonacci Series are related. It makes you want to think that there is a God with a plan.
The Golden Ratio is defined as follows: In a line segment ABC, if the ratio of the length AB to BC is the same as the ratio of AC to AB, then the line has been cut in extreme and mean ratio, or in a Golden Ratio called Phi.
On the flip side, Livio squarely debunks the idea that the Golden Ratio is present in many famous paintings and architecture that has been postulated in previous books. He rightly points out that you can find the Golden Ratio in anything depending on where you decide to place the measuring tape. The idea that the Golden Ratio is such a symbol of universal beauty that it appears by accident in our great man-made buildings and artwork does not carry any weight. I think Livio makes his point.
He also uses the Golden Ratio as a framework to illuminate other historical tidbits about key mathematical figures, guys like Pythagoras and Euclid, who continue to affect the mathematical world to this day. I love this kind of stuff; the historical context of how and why these legends did what they did is very interesting to me. For example, I did not know that Euclid himself did not discover geometry or even make any great new contributions to the field in terms of ways to apply it. What he is famous for is organizing the information into a coherent fashion. He was a teacher of the highest order; so much so that Abraham Lincoln himself used Euclid's texts, unchanged after all those years, to learn the subject back in Lincoln's log cabin days.
The book is not all a philosophical discussion. Livio does use some simple math examples to make his points but it was at a level that I could follow. According to my college professor, I escaped College Calculus by sheer luck. Livio does provide the rigorous math examples in appendices at the end of the book (I did not bother with these).
Finally, Livio takes a shot at the idea that mathematics is a universal concept across the entire universe. To be honest, I have always assumed that it was. After all, I am a Trekkie and this concept goes unstated throughout all four TV series. The idea that mathematics is a human construction and probably holds no water in another civilization that grew up on the other side of the universe makes a lot of sense to me. I have to admit; I need to ponder that one for a while.
I recommend this book. If you like the history of science, your high school algebra class is just a little more than a dark fog in your memory, and you get a charge out of scientific mysteries, you will not be disappointed.
You can purchase The Golden Ratio from bn.com. Slashdot welcomes readers' book reviews -- to see your own review here, read the book review guidelines, then visit the submission page.
Mathematics not universal? (Score:5, Insightful)
If mathematics are not universal, then the mathematical reasoning that can be conducted to deduce the laws of nature is also not universal. Hence, if a different civilization has different mathematics, they have different physical laws as well.
This is basically a postmodern viewpoint, that reality is socially constructed. This viewpoint has been largely derided by the scientific community, and has failed to replace science because it hasn't really offered a compelling alternative. The only way I can see it being true is if other civilizations don't "exist" in the universe as humans do.
Do a google search for Alan Sokal for a scientist's viewpoint of postmodern scientific criticism.
Comment removed (Score:3, Insightful)
Why wouldn't math be known across the universe? (Score:5, Insightful)
Could there be some areas of mathematics that humans have discovered that has not been discovered by an alien race? Sure. Prior to Newton there was no calculus and so Kepler had to discover the period of planetary orbits using geometry and algebra. But this does not mean that Kepler would not have used calculus if it had been available to him, only that such a concept had not yet been thought of.
But counting and simple addition and subtraction are mathematical operations that are mastered even by animals. It is fairly condescending to assume that aliens could not even fathom those levels of mathematics.
Re:The Da Vinci Code (Score:2, Insightful)
Numbers are numbers (Score:5, Insightful)
By the same nature, prime numbers are always prime. There exist a certain number of things (5, 7, 11, etc) and cannot be evenly divided. Period. We call them prime numbers, and we use our base-10 radix. Aliens could call them Borgolsmocks in their base-182, but the fact still remains that a Borgolsmock cannot be divided evenly.
And I firmly believe that no intelligence would survive for long without a knowledge of mathematics. Counting the days for crop rotation, the ability to evenly divide food among the tribe, and communication of the number of animals in a herd... mathematics will be generated in the evolution of any intelligent species.
And it is truly universal.
Re:math is not universal? (Score:5, Insightful)
Ummm, no. That's not math. That's physics. Math is more abstract and one can do math without associating any of the concepts with "reality". One you use math to model reality, it becomes science and engineering.
From your english class? (Score:5, Insightful)
I presume you got an F. Since is a direct and obvious plagarism of the publisher's description of the book. (see: http://search.barnesandnoble.com/booksearch/isbnI
It's obvious, because it doesn't really say anything other than what can be related to the title of the book (which is not unusual for back-of-the-book descriptions)
It's direct, because, well -- I can search google for any sentence in your text and find it.
Lame.
How does one dispute math as a universal concept? (Score:3, Insightful)
I would think that math in some was is universal, in the sense that every sentient creature has to figure out a method of counting. Some creatures count in base 10, others base six, maybe base 12. Other creates could figure out a counting base we haven't thought of yet. However, if they have a method of counting and measuring, I'm sure we'd have a method of translating their mathmatical models to our own, without too much trouble.
Perhaps the definition of math here is different than mine? Thoughts?
First-contact scenarios? (Score:5, Insightful)
From what I understand, the vast majority of realistic first-contact scenarios postulated involve using mathematics as a common ground to bridge the language barrier. 1 + 1 equals 2 in every language on earth (except New Age holistic 1 + 1 = 3 crap). It makes sense and it works everywhere. It would be awfully damn hard to build a spaceship without mathetmatics, let alone trying to calculate launch trajectories or transfer orbits. Unless they had such an intuitive grasp of higher level mathematics that they don't even consider it worth talking about, I don't see how any species that had no concept of math could ever rise above the level of pointy sticks and sharpened rocks. And even then you'd probably want to keep track of how many rocks you had to make sure Lurg over there didn't *borrow* a few.
Re:The Da Vinci Code (Score:3, Insightful)
The Golden Ratio is carefully and deeply researched. The Da Vinci Codes is allegedly based on research, but the "research" behind it is recycling tired old conspiracy theories.
From his statements online and in his forward, methinks Dan Brown is trying to have it both ways: claim it's based on fact but use the plausible deniability of it being a fictional work. It is a gripping read, don't misunderstand me. But you have to remind yourself that it's totally fictional.
Re:Mathematics not universal? (Score:5, Insightful)
Rather, I'd set mathematics and logic equal (there's a respected tradition that does, see Bertrand Russell, Principia Mathematica). Then, to say that mathematics isn't the same across the universe, one would say that logic isn't the same across the universe.
Now, "Logic isn't universal" is a damn meaningless statement. It'd translate into "Logic cannot describe [timespace-area/context] X." Which is, of course, a logical assertion about X.
I think either the reviewer's portrayal of the argument or the argument itself is bogus.
RD
Re:Why wouldn't math be known across the universe? (Score:3, Insightful)
And it's fairly narrow-minded to assume that another life form in the galaxy has a frame of perception that's even remotely similar to ours. This is more than just saying "what if they see in infra-red!", but rather to say that we feel mathematics is the de facto language of the universe because it (as with terrestrial life on Earth) doesn't yet have a competitor.
math and humanity (Score:3, Insightful)
Could a race become spacefaring without math? Could they develop the radio communications we could use to detect them? I suppose they could if the circumstances of their environment or adaptation (Low-gravity, bio-radio communications) allowed it.
But how would you arrive at the necessary conclusions without an abstracted intellectual framework like math? Maybe progress would just be slower.
Hmmm... makes you wonder what we're still missing.
Don't confuse Syntax for Content (Score:5, Insightful)
Re:Mathematics not universal? (Score:4, Insightful)
Then again, if they have no concept of "time" or "movement", then I would argue that they don't exist in the universe in the same way we do.
Re:The Da Vinci Code (Score:1, Insightful)
And why shouldn't he be? Why shouldn't anyone? They were responsible for lots of bad shit back in the day. Some of the things that happened back then even have far reaching global consequences today.
Re:Mathematics not universal? (Score:4, Insightful)
Aliens does not have different rules of physics - but they probably have different models about it,but that should not come as surprise as we, humans had lots of different models of physics (and nature in general) throughout our history (Newton's mechanic view, quantum mechanics' probability approach, etc) - but it does not mean that Heidelberg did not exist as Newton did.
Sokal was basically trolling (to demonstrate that "postmodern scientefic criticism" is BS) - and probably that's what you do too.
Mathematics is an invented human language (Score:2, Insightful)
Re:Why wouldn't math be known across the universe? (Score:2, Insightful)
Godel's Theorem doesn't prove that mathematical systems are inconsistant. Rather Godel showed that a system can't be proved conistant within itself.
The diference is that Godel says you (in important casess) can't be sure math is consistant. He does not say that math is inconsistant.
Jeff
Re:Why wouldn't math be known across the universe? (Score:3, Insightful)
I see no reason to assume that an alien race has brains that function like that. They could function in any number of ways, I think they all could have an effect on how those creatures approached mathmetics.
Re:Mathematics not universal? (Score:4, Insightful)
-Colin [colingregorypalmer.net]
One small point (Score:3, Insightful)
Agreed (Score:2, Insightful)
My favorite part of the book is where he essentially disproves the many claims to the appearance of the golden ratio in aniquity and before. He tries to nail down the moment of when the golden ratio was actually discovered.
Re:A god with a plan? (Score:3, Insightful)
People like you seem to take for granted that the universe just exists. That spacial dimensions and time all sort of slid together to happen to be this way. Matter just so happens to work in such a way that that puddle can exist (not just one puddle, but any infinite number of similar but totally different puddles) and sustain an entire eco system.
We can simulate the entire puddle and ecosystem in a computer with mathetmatical models, but it still is not the ecosystem. The whole is more than the sum of its parts.
Re:Arguments against postmodern deconstructionists (Score:4, Insightful)
In a sense this is what science does. But my question is, who makes the rules? The postmodernist would argue that society does, that is, science is altered by our perceptions of it. The scientist (and, indeed, the philosophy of science) requires that the rules are cast by something external to humanity, that is, the rules are unalterable and are the same for each person. This is the basis of repeatable experimentation, and has proven itself far more successful than the alternative explanation.
I'm not totally unsympathetic to the postmodernists. You can look into the past and see where the "softer" sciences, especially psychology and medicine, have made errors based on prevailing social beliefs. But more fundamentally, I believe there are limits to scientific thought. For instance, perception is an important place to attack science, since science hasn't had much success in probing perception. This is probably because science deals in concrete concepts that can be described cleanly with language (e.g., the universe is expanding, the Earth goes around the sun every 365 1/4 days, every cell contains protein, etc.), but we can't describe perceptions in any way that is simple or concrete. For example, how would you explain the taste of an apple to someone who has only eaten meat?
Re:Mathematics not universal? (Score:5, Insightful)
Blue, obviously, is radiation in the wavelength of around 475 nm. It is measureable. When you look up at the sky, if light is primarily coming in at wavelengths around 475nm, the sky is blue.
On the other hand, if it is sunrise or sunset, or the end of the world or something, and the wavelength is much longer -- around 650 nm -- the sky is red.
If you are colorblind, it doesn't change the fact that the sky is, indeed, blue. And, even with colorblindness, you can measure the color of the sky using scientific instruments.
So, wake up, and enjoy the reality that is the universe.
Re:Arguments against postmodern deconstructionists (Score:3, Insightful)
Their point is difficult to refute because it's true, obvious, and pointless, all at once.
All of the axioms of natural science are based on our observations. All observations depend on sensory input. Since our senses can be fooled, so can natural science. Ultimately, the only thing you can be sure of is "Cogito Ergo Sum," as old Rene once said. Everything else requires faith in the correctness of our perception.
However, it's a pointless observation. If we reject the input of our senses, we have nothing at all to go on which establishes even the existence of anything, yes. However, there is no way to demonstrate the total falsehood of observation because we have nothing else to go on.
Re:Mathematics not universal? (Score:3, Insightful)
Re:Mathematics not universal? (Score:5, Insightful)
To start with, Mathematics is not just as human as poetry. Where do you get that idea? Yes, pure mathematics (which is my passion in life) is essentially pure thought. BUT, nothing in mathematics is just 'made up'. All mathematics is based on fundamental, logical axioms (truths), and if anything were to violate those axioms, or the completely logical conclusions drawn therefrom, it would not be mathematics. You can think of mathematics as a grand puzzle, with each discovered piece and each mathematical truth found spelling out a larger picture. You can create bogus logic, bogus mathematical problems but it does not make it true mathematics.
You are also confusing human representation with mathematics in your other statements. On a fundamental level, a law is a law, mathematically/physically/logically/universally. The universe is not ruled by human imagination (i.e. completely imaginary human created friend(s) as in religions) and therefore the system to understand our universe has to follow the same sets of rules as the universe (even rules involving possible pure chaos, as in some areas of quantum theory). Without mathematics, our universe and all that lies within it could only be understood on a physical observation level. Mathematics is the language of the universe, it is the language of physics.
For a slightly deeper explanation, let me explain that Mathematics does not involve physical representations as you were taught in HS and earlier. For example, the number 1 as opposed to a capital S to complete addition (which is a logical law) means that 1+1=2 is the same as S+S=* because the system is beyond the physical characters used to represent the logic. The logic would not be different in an alien society. The laws of the universe do not change, therefore the same logic would be implemented. Using a 1 or an S would make no difference.
There is sooooo much more, but just reading this story and people's posts makes me sad on a certain level. One of the oldest "truths" in the world...The person who is always least understood is not an artist, it is a mathematician.
be more cynical (Score:1, Insightful)
Perhaps it was precisely to obscure the actual origin of the work that they spent so much energy fighting over who "invented" it first.
Bluster and a big show are a common technique used to take credit for someone else's work. Ever been to a meeting with your manager and your manager's manager?
Re:Mathematics not universal? (Score:2, Insightful)
Really?
Where?
Re:The Da Vinci Code (Score:2, Insightful)
the T-O-R-A-H
yes that's the book of greats
i stand alone on the word of g-d
the T-O-R-A-H
btw, i'm not jewish
Re:Mathematics not universal? (Score:4, Insightful)
Personally, I was rather surprised at the book review myself. I have found, however, many fine folk here on slashdot who appear to have a solid theoretical background in mathematics, perhaps they are hiding out on this one.
I find this is an odd thing for a self-styled mathematician to say. First, I would not label axioms as "truths" but rather as "putative truths". That is their purpose (to be putative). They are styled from not only as "seeming to be true" or appealing to intuition, but also in their ability to form a basis of thought, their simplicity, a small number is required for important results (as in defining a system), and their seeming irreducibility. Axioms are usually formed "post hoc", with an idea of the desired results in mind. Don't you find it disconcerting that in Topology the definition for an open set in a metric is a union of open balls, but an open set in a topologcial system is definied simply as that which exists in the topology? It's a function of how strong of results you have, and later developments are "force fit" into prior studies. Much like topological systems are "force fit" into Metric systems by how they are to be constructed, thus claiming itself an abstraction of the latter.
Furthermore, you claim that any results that contradict the "fundamental" axioms, is not properly called mathematics? You are aware of the results of Godel and his Incompleteness Theorem? It's more than the latest rave, it has results that bear directly on what you are oh-so-devoutly avering. In this light, what is mathematics, when it is either incomplete or contradictory? And quite often it is contradictory? Which set of axioms is the more fundamental one you wish to choose? And how do you choose it? These are problematic questions, with unforthcoming answers. Modern mathematics is rattled as it has been faced with it's own, ultimate, incompetence. It describes nothing but itself.
Disregarding your mixed metaphor "spelling out a larger picture", this doesn't make much sense to me. I guess I don't understand how you are differentiating bogus mathematics and true mathematics.
On what fundamental level? What is a law? How is one to determine what is a law? And what is derived from a law? Or what is contradictory to a law? I do agree that the grandparent poster did mix things up when he contrasted mathematics to physics et al.
Re:Mathematics not universal? (Score:3, Insightful)
Or a different philosophy for what is a sound, provable result. I think there is more politics in mathematics than you'd wish to admit.
Re:Mathematics not universal? (Score:2, Insightful)
Your answer to that question is a nearly canonical definition of physicalism -- This is the belief that everything there is to know about the world is contained in the physical description of the world.
This specific claim is answered by Nagel in his paper titled, "What is it like to be a bat?" and by Jackson in his paper, "What Mary Didn't Know."
Nagel argues that if you knew everything there was to know about neurology and biochemistry and every other part of how a bat works, you would still not know what it was like to be a bat. This lack of knowledge is not explained in physicalism.
Jackson tells the thought experiment of a girl named Mary who is born in a black and white room, is educated through black and white books and TV and film, and learns everything there is to know about physics, optics, the structure and workings of the eye, the optic nerve, the brain, etc. (Including of course that 'blue' has a wavelength of 475nm.) She is then let out of this room, and sees a blue light, and a wavelength meter that indicates the light is 475nm. Does she recognize the color blue as such, before she reads the indication on the meter? Does she respond by saying, "This is just as I have been taught, this is blue?" Then, knowing from her color-blind education that a sunset is considered by most people to be an intrinsically beautiful thing, she sees a sunset for the first time, the whole sky aglow in red and orange. Does she learn anything when she sees this? Does it surprise her? Specifically, before she leaves the room, she has an idea of what people experience when tey see a sunset. Do these ideas change after she has seen it? The key thing that mary learns in this experiment is non-physical information, called qualia. Defined loosely, qualia is the "What it's like" knowledge you have. And if you tell a colorblind person that blue is 475nm and red is 650nm, you haven't given them a shred of this qualia. They have no way to relate to your ideas, and have gained nothing from your descriptions but a few numbers to toss about in their heads. Their perceptions of the world will be fundamentally different. Their understanding of beauty will be based on things like proportion, or form, or contrast. What would a field of tulips look like to a colorblind person?
To quote: If you are colorblind, it doesn't change the fact that the sky is, indeed, blue.
No. If you are colorblind, it doesn't change the fact that the sky is, indeed, illuminated with a light with a wavelength of 475nm. That is a wholly different thing from "Blue." We, as humans, use words to describe experiences and sensations, not physical facts. You mistakenly equate the two.
Re:Mathematics not universal? (Score:3, Insightful)
This is where I wish to focus my discussion, as I see it primarily where we differ:
Certainly, they recognize each other as "doing" mathematics. Just as we might recognize another species or such as "doing" mathematics even if it does appear radically different from ours. I appologize if you found my use of the word "politics" inflammatory, perhaps I was simply using it in looser manner than you might expect, I also don't wish to single out mathematics in this charge--I believe much of what we hold as acceptable means for establishing "truth" is indeed "politically" motivated. Or to put it another way, is ultimately determined through appeals to power. Certainly we cannot appeal to reason in establishing what reason is, as it is not yet properly established (or would you say we can as this argument itself is based on reason--eh? You see the difficulty? How do we escape what we've already assumed?). Now, you would be right to say that I'm using an exlcuded middle within this argument (in a rather fallacious way). I would like to hear any ideas from you regarding how we might go about establishing rules of inference (not that if you failed it would prove my point). I am genuinely interested.I would go even further to say that they idea itself that we can infer anything through any rules (or at least, assuredly infer) is itself a politcally motivated idea. I don't buy that we have such a gurantee, or that another "intelligent" species would perceive things in the same manner that we do.
This is the social construct that is mathematics. That said, I do not believe that thinking this way about mathematics diminishes mathematics in any way, nor is it my intention to use this push some other agenda. Mathematics is a usefull tool, one that we have created. I only think it imperative to recognize how we intimately relate to this tool.