The Golden Ratio

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.

Comments

The Da Vinci Code

[fee^]

On the fictional side of this type of thing, those of you into this kinda stuff (like me) should read Dan Brown's 'The Da Vinci Code'. I've read that and 'Angels and Demons'. Both fantastic reads. More Info Here

Re:The Da Vinci Code

[Anonymous Coward]

Fictional is absolutely correct, sir. Entertaining, perhaps even compelling, but in the end purely a creation of someone else's imagination. Those who accuse "The Passion of the Christ" of being anti-Semitic should also be willing to point their guns at Dan Brown for being anti-Catholic.

Re:The Da Vinci Code

[wfolta]

The Da Vinci Code is gripping fiction, but it's not in the same class as The Golden Ratio.

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:The Da Vinci Code

[Golias]

In the movie Pi, both Phi and Theta are referenced as possible evidence of an Order to the Universe. The protaganist, Max Coen, is trying to hack this order by examining the Stock Market for patterns (since the market is a subset of nature, and patterns are postulated to be everywhere in nature, he believes a pattern can be found in the market, which cna lead to a better understanding of how the world is put together.)

His university mentor, a Jewish concentration-camp survivor (Soviet, not Nazi), was performing a similar pattern-search using Pi as his data set. This is where the title of the movie comes from.

The plot thickens when a group of Hasidic fanatics who are searching for the name of God by scanning the Torrah for patterns recruit Max to help them, and Max's curiosity, along with his migrane-induced hallucinations, leads him to the blurry line between number theory and numberology.

It's probably one of my favorite movies of the last 10 years.

Re:The Da Vinci Code

[xgamer04]

spell it with me now...

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

The Galaxy??

[Gorimek]

The concept of math isn't even spread very far on this planet.

The Golden Ratio

[Anonymous Coward]

He who has the gold, makes the ratio.

Mathematics not universal?

[s20451]

Didn't read the book.

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.

Re:Mathematics not universal?

[kfg]

I offer this argument to those who state "You create your own reality."

I kick them in the shin.

Then say, "Why did you do that?"

KFG

Re:Mathematics not universal?

[Raindance]

Well, I wouldn't say that "if a different civilization has different mathematics, they have different physical laws as well."

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:Mathematics not universal?

[s20451]

Yes, but what would we have in common with them? We both exist in the universe, and can make observations about the universe. Thus, assuming they have a certain amount of technological advancement, we could communicate with them and ask, "How does your planet move around your star?". It would take a whlie for us to communicate our respective definitions of "ellipse" and "gravitation", but surely we would agree that the path is elliptical due to gravitation, even if they had three purple heads and were fifty feet tall.

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:Mathematics not universal?

[dabadab]

Mathematics clearly may not be universal as it is a purely human thing - it's not like physics or biology where you have to follow existing things. There are lots of things in mathematics that is/was completely "made up" (Boolean algebra springs to mind) - in the end, math is just as human as poetry.

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.

Re:Mathematics not universal?

[Ralph Wiggam]

Different civilizations use different math systems, but they're all related by basic concepts. Addition is addition, no matter how you write it down. Any two math systems can be translated back and forth.

-B

Re:Mathematics not universal?

[kallisti]

Boolean Algebra was indeed "made up" but just happened by an amazing coincidence to be a perfect model for creating digital circuits. Or is it a coincidence? Math is a game of sorts, you start with a set of rules and see what develops from them. The trick is, if you choose the initial rules carefully, you get results with real-world use. Minkowski space was a math oddity for years until Einstein realized it was a perfect model for General Relativity. It appears that an even odder model called Geometric Algebra gives even better models of physics. Are these all coincidences?

A branch of math called group theory models all the possible kinds of symmetry. Any study of symmetry will eventually lead to the same ideas that group theory is based upon. Doesn't that make group theory universal?

Yes, I'm a Platonist when it comes to math.

Re:Mathematics not universal?

[CGP314]

Even if the laws of mathematics are not universal, that doen't mean that the laws of the universe aren't... er... universal. There isn't any law that says `Nature follows math'. As far as we know, math is a useful tool that seems to be able to predict what will happen - but that doesn't mean that math is the cogs and wheels behind the scenes.

-Colin

Re:Mathematics not universal?

[KurtP]

I answer: "Clearly, I am a masochist."

Re:Mathematics not universal?

[wwest4]

a la "I refute it thus?" ... except Samuel Johnson's method was a lot more friendly than yours!

Re:Mathematics not universal?

[nerf]

I think if we look at other non-western civilizations here on Earth we might be able to see math as other non-humans might. For example, the Navahoe (sp?) native American indians had a different concept of time. I don't know enough about the specifics but their language evolved differently as a result of how they look at the universe. (which is why they made for good code talkers in WWII as their langauge was difficult to decipher to the japanese).

Obviously the laws of the universe are going to be the same, otherwise we wouldn't live in the same universe and we wouldn't be having this discussion. But the modeling might be different, either more efficent or just a different path to get from point A to point C.

Remember that movie Contact where the Vagans sent us a coded message that humans couldn't understand until we put the messages together in 3 dimensional space, instead of on a flat piece of paper.

I think IF there are other sentient species out there then it is possible that their view of the univierse has developed differently then ours, but they still must observe the same universe as ours. They might call gravity a hamburger for all we know, but a rose by any other name smells just as sweet.

So in effect, the mathmatical models may not be universal, but their results will be. I can't imagine being able to master space flight without simple mathmatical concepts like addition. Look at the universe, specifically how biology works and addition is every where.

my 2 cents. -- Jim

Re:Mathematics not universal?

[Mr. Slippery]

If mathematics are not universal, then the mathematical reasoning that can be conducted to deduce the laws of nature is also not universal.

You're assuming a relationship between mathematics and the "laws of nature" that isn't there. As Einstein put it, As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality."

Mathematics is as socially constructed as any other form of language. It is based on axioms and defintions, not observation of reality. We select those axioms and definitions in a way to be useful to us, just as we select for those lingustic constructs that are useful. But this selection is based on our desire to communicate with others - it is a social construct. Once upon a time if you asked mathematicians what nubmer, when squared, gave negative one, they'd say there was no such number; now, any bright middle school kids know it's i.

"Reality" is also to a large degree socially constructed, since all can ever speak of is our observations, which are socially conditioned. You see what you expect to see or are trained to see. (You don't see the fnords, or Sombody Else's Problem, while the hypothetical planet Vulcan (the one inside the orbit of Mercury, not Mr. Spock's home) was observed several times, as were Blondlot's N-rays.) This is why double-blind protocols are used - though if everyone involved has an expectation, that doesn't help.

What we think of as "reality" is just a model that we mostly share. The electron, for example, is not a component of human experience but a component of a model that unifies and predicts many observations. That is a very good and useful model, but it is entirely conceivable that some extra-terrestrial civilization has (or some future human civilization will have) a model that is just as useful but doesn't contain anything like electrons. (Just like Chinese Medicine has a "patterne-thinking" model of the human being that is radically different than and incompatible with the reductionist model, yet is extremely useful.) What would such an electron-free model look like? I can't tell you, I'm too conditioned by the electron model.

Remember: for any set of observations, there are an infinite number of hypothesis to fit them. There's no end to the curves you can plot through any finite set of data points. We see the points and call them a line, but it ain't necessarily so. The best we can do is eliminate lines that don't go anywhere near the points.

Re:Mathematics not universal?

[Erich]

To those who would say "The sky is blue, and that's an absolute truth, whether I want it to be or not" ... How do you know what "Blue" is?

It's quotes like these that make me think postmodernism is based on pure stupidity, rather than any rational thinking system.

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:Mathematics not universal?

[NonSequor]

The fundamentals of logic are assumed to be true and it's very unlikely that any alien civilization would not use the same principles or at least principles equivalent to them.

However, it would be possible to derive mathematical systems very different from our own. It all depends on what one takes to be fundamental concepts. For example, we define functions in terms of sets, but we could also define sets in terms of functions.

We're not even certain that some of our own axioms are true. For example, the axiom of choice says that given any set of disjoint non-empty sets, there exists a set that contains exactly one element from each set. While most people will say that this seems to be a reasonable statement, if it is true, a number of counterintuitive statements are also true.

None of these things change the universe, only the way the universe is modeled. One might be able to come to some new conclusions and possibly even a few contradictory conclusions using a different form of mathematics, but all in all mathematics effectively is universal since there is no reason a mathematician from earth couldn't learn to understand alien mathematics.

Re:Mathematics not universal?

[photo storm]

You bring out a very subtle fallacy, and one that is tied to philosophical issues regarding mathematics. I bit of history is in order:

The fundamental question is this: is, or isn't, mathematics an extension of logic? A smart man named Frege (read about him here) said, yes, it is. He showed a way to connect formal logic with set theory, which is the basis for mathematics as we know it.

There was only one problem: Russell's Paradox. Bertrand Russell showed that, using Frege's axioms that defined set theory, we have a contradiction - Russell's Paradox. And as any student of logic knows, a contradiction can be used to prove anything at all, which means that mathematics as Frege defined it was not viable.

To make a very long and very interesting story short, Russell (with Alfred Whitehead) attempted to create a foundation for mathematics that would not give rise to Russell's paradox - the Principia Mathematica. And everyone thought the world was cool.

Then, in the 1930s, Kurt Godel came along and smashed a hole in Russell's approach by showing that, given a sufficiently powerful formal system, one will always find unprovable truths and irrefutible falsehoods. So mathematics was, by that line of reasoning, incomplete.

This leaves the door open to a variety of critiques, the most relevant of which is that it is automatically not universal. After all, how could it be - there are things missing! We can't prove everything that is true, and we can't disprove everything that is false!

Godel's argument tells us that we are unable to describe the universal laws of nature using non-universal and incomplete mathematics. That dosen't make mathematics useless - it just places a limit on what we can or cannot do. For instance, we cannot use deductive mathematics to describe the laws of nature in their entirety, because we know that any effort to be complete is doomed to failure - by Godel's theorems.

Also, there are some specific areas of mathematics that lead to direct examples of non-universal, but nonetheless consistent interpertations of nature. Take, for instance, Euclidean and differential geometry. Euclidean geometry is the geometry of flat planes, whereas differential geometry describes abstract mathematical notions. It was once thought that Euclidean geometry is "sufficient", and that it is the simplest way of representing spacial relationships. However, as it turns out, differential geometry is actually much more simpler when it comes to dealing with, say, the theory of relativity - even though it is not intuitively connected to our perception of the universe.

So in short, we have two different "geometries", each of which can, supposedly, explain spacial representation. Both are valid, but one is much more useful. Neither is universal. And yet, there is no contradiction.

I don't know about anyone else, but I think this stuff is interesting.

Re:Mathematics not universal?

[Junior J. Junior III]

If I see the color blue in a dream, where is the radiation?

Re:Mathematics not universal?

[OwnedByTwoCats]

Yes, but what would we have in common with them?

The Periodic Table.

The way to start communicating with an alien species is going to start with simple numbers and arithmatic, and then an important sequence will be:
(1,1) (1,2) (2,3) (2,4) (3,6) (3,7) (4,9) ...
the stable isotopes of Hydrogen, Helium, Lithium, Berylium, and so on up the periodic table.

Once two species share this information, then they can talk about stuff, literally. By adding unstable elements, they can talk about time.

Chris

Re:Mathematics not universal?

[Listen Up]

Wow, the shear ignorance in this entire article and book write-up is amazing. Not to truly upset anyone, but everyone here on Slashdot also appears to have a high school alegbra or entry level college mathematics background.

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.

Re:Mathematics not universal?

[Karhgath]

Different civilizations use different math systems, but they're all related by basic concepts. Addition is addition, no matter how you write it down. Any two math systems can be translated back and forth.

WRONG.

Let's take 2 valid mathematical system: Classical Mathematics(CM) and Intuitionist Mathematics(IM).

One thing that is provable in one system might not be provable in the other, or could even be wrong.

For example, if we take the mathematical subset of Logic we have Classical Logic(CL) and Intuitionist Logic(IT).

In CL, NOT(NOT(P)) |= P.
It is easy to see why.
Same with A OR NOT(A).

However, for IL, something is only True if and only if it's provable.

So, NOT(NOT(P)) |= P becomes:
If there is no proof that a proof of P is impossible, then P is provable. This is invalid. The absence of a counter-example doesn't prove the fact.

So we see that NOT(NOT(P)) doesn't imply that there is a proof of P.

Same for A or NOT(A), because we cannot assume that it's always possible to either prove A or it's negation.

One of the fundamental differences in the 2 math systems is that, in IM, it requires a constructive proof.

So, in IM, you cannot prove something like that:

Proof
(...)
Case1: A = X then (...)
Case2: A != X then (...)
(...)

This doesn't work, for the same reason as A or NOT A, you need to prove one or another, so you need to prove that A = X or that A != X.

Ok, the point is, these are 2 working, acceptable and valid mathematical systems, but they cannot be swapped, because CM != IM.

So, NO, two math systems CANNOT be translated back and forth. This is but the tip of the iceberg.

Re:Mathematics not universal?

[kfg]

No. I would agree he is a solopsist. I would contend there is a difference between one who believes there is an intangible ideal underlying perceived reality and one who believes reality is all in his own mind.

For instance, as a physicist I may admit to there being number under everything and idealized states that do not exist in reality. This does not, however, mean that I admit to the nonexistence of matter or objective reality at all.

As a Buddhist I might ask "Does a dog have Buddha nature?"

KFG

Re:Mathematics not universal?

[UserGoogol]

Language is... eh, very very flawed for these sorts of philosophical stuff. I'd say you don't see blue, you only think you do, but whether "blue" reffers to the wavelength of light or the feeling that the brain feels in response is not a particularly answerable question in English. This is because English, like most natural languages, only defines things to the degree that they have to be defined, and for most situations the two definitions are just as valid.

I personally think, however, that the definition leans towards the "wavelength of light" definition rather than the emotional definition.

Re:Mathematics not universal?

[lindsayt]

If your model and my model differ, then we disagree about reality, but we are probably not correct (either of us).

I agree with you completely, except that the very point of postmodernist approaches is that there is no such thing as correct or incorrect; there is observation and perception, which are sometimes shared and sometimes different. Science does NOT define reality; it simply provides a set of observations that are repeatable by anyone who follows the correct procedures and uses the right tools. These reproducible "facts" are of course of extreme use to our society, and I don't think postmodernists would claim that science is anything less than a boon to our society and our understanding of our universe. However, they would argue that there is no such thing as "correct" and "incorrect" because our understandings of such are merely anchored within the model we hold of the world around us.

So DJerman, I agree completely with you, and I think you agree completely with postmodernist critique as it is understood in the academy. The "Individual creates reality" kinds of statements are made by people who don't understand the inherent critique made by postmodernism.

Remember, postmodernism isn't really a philosophy per-se because even its strongest proponents (Michel Foucault should immediately come to mind) realized that it was not very useful for creating new interpretations of human nature. Rather, it is a tool for criticism and critique, a way of addressing the shortcomings of modernist notions of progress and the human condition. It was often said during Foucault's life that his own works of history were not really postmodernist, because in order to make a strong historical argument he had to betray many of the tenets of the postmodernist "we can't really know anything" dilemma.

Of course, I've been immersed in postmodernism for so many years that I'm not even sure I exist anymore. I don't call myself a postmodernist - I think the term has no meaning since postmodernism is not really an ideology (although it is taken to be one by people who have a little knowledge of it and think it sounds good); however, I think the postmodernist critique is an important one and very relevant for addressing many of the shortcomings of our perception of our condition.

And that's how we turn short observations into 400-page works in academia...

Re:Mathematics not universal?

[widmerpool]

"... But mathematics itself has an existance [sp] of its own, entirely unconnected with the physical universe."

Really?
Where?

Re:Mathematics not universal?

[efflux]

I'm going to take the time to respond to this post at some length, because I feel it raises certain issues that need to be addressed. I hope to address them in a manner conmensurate with their depth. Please bear with this analysis of your arguments.

Wow, the shear ignorance in this entire article and book write-up is amazing. Not to truly upset anyone, but everyone here on Slashdot also appears to have a high school alegbra or entry level college mathematics background.

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.

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.

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.

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.

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.

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.

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.

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 o

Re:Mathematics not universal?

[efflux]

Whose to say they couldn't have different argumentative rules for what "proves" a result?

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?

[DJerman]

Um, no. I argue that an objective reality is separate from perception, and a couple of things about terms. Whether those things make sense depends on the degree to which my fever is affecting my perceptions...

There is a sloppy argument in the parent that perception = reality. That is demonstrably false, if one imagines the "impossible box" illusion, for instance. Besides, it doesn't matter: if there is not a reality, we're both insane figments of the reader's imagination, because only the reader is creating reality (hi reader! keep thinking about me for a while!). After all, I know it's not you and you know it's not me...

A more rigorous reading would be that the process of perception creates my personal experience of reality. Well duh. So reality is distinct from the perception model -- I know that the process of perception is imperfect (via repeated and sometimes painful demonstration), and does not in fact create an accurate model of the empirical reality that thwacks me in the nose when I misjudge a softball catch. That the information in the model is incomplete or contradictory is demonstrated whenever you discover an illusion. That there is a reality is demonstrated when it bonks you.

BUT... That we have unique understanding of 'blue' does not practically prevent us from conversing constructively about 'blue' and having high confidence that what I recognize as 'blue' will be recognized by you and any other capable person as 'blue'. To the degree that we are specific about the method of measuring 'blue' it becomes more likely that we can agree. It doesn't matter that your 'blue' is related to 'sour' in your mind, as long as we agree that it is the color of the sky (when the other would label the sky 'blue').

Aside - Edwin Land showed that color perception is largely relative - blue light is always around 470nm but... the perception of the hue of a color depends largely on the relative intensity of other wavelengths also present. He was able to produce full-color images from grey-scale filters in two different-color light sources.

So yes, 'blue' is an advanced concept that would have to be nailed down after months or years of discussion with the BEMs, possibly involving retinex algorithms to 'decide' if a thing is blue or not. There is, however, a 'blue' out there in reality to point at, and however they percieve it, we can explain to them that 'that' is 'blue'. Perception is relative, but reality is objective, for an agreed frame of reference.

A red stoplight indicates that you are not approaching it at a speed sufficiently close to the speed of light, but from the cop's point of view it's still red.

Re:Mathematics not universal?

[Nobo]

The arguement made by the original poster regarding blue is an argument that has been around for some time, and is not one that is directly related to postmodernism. Further, while it is my personal believe that postmodernism is in fact based on pure stupidity, that is tangential, as the question of blue is a legitimate and viable philosophical question.

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?

[efflux]

Thank you for pointing out some of the particulars of what I would have hoped to say, but lack the familiarity to speak about with much certainty. The PoEM is one such point of contention I had in mind, and (to some extent) illustrates my point.

This is where I wish to focus my discussion, as I see it primarily where we differ:

Constructivists and Classicists recognize that each other are doing Mathematics, getting the mostly same results, but by different rules, different means -- and cherishing where there are demonstrable differences in what can be proven.

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.

Something I learned from Martin Gardner...

[kzinti]

Something I like about the golden ratio is that it is the number that is exactly 1.0 greater than its reciprocal. This makes it easy to remember the exact value: just solve

x = 1 + 1/x

You'll get a quadratic with the solutions (1 +/- sqrt(5))/2, or 1.618... and -0.618...

Re:Something I learned from Martin Gardner...

[product byproduct]

A way more fun way to solve x = 1+1/x:

On a calculator:

1) start with any number
2) press [1/x] [+] [1] [=]
3) GOTO 2

In other words this converges to the golden ratio! It takes a while, so normally you do this when you're bored.

Re:Something I learned from Martin Gardner...

[tiled_rainbows]

Easier way:

1. Add two numbers together.
2. Add the result to the second (larger) number from step 1.
3. Repeat for a while.
4. Divide the last (biggest) result you get out by the second-last (second-biggest) result.

Example:
2 + 4 = 6
4 + 6 = 10
6 + 10 = 16
10 + 16 = 26
26 / 16 = 1.625
near enough.

Re:Something I learned from Martin Gardner...

[interiot]

byproduct's converges quite a bit faster though.

Re:Something I learned from Martin Gardner...

[PGillingwater]

If you're interested in convergence, try the Fibonacci sequences -- the ratio of successive terms converges to the Golden Ratio.

Years ago, I also made an analysis, and found the ratio in the trigonometry of a pyramid -- it's there if you look for it.

Algebraeically, try the square root of 5, + 1, divided by 2. i.e., (sqrt(5)+1)/2 = Phi.

Comment removed

[account_deleted]

Comment removed based on user account deletion

Movie

[savagedome]

The movie PI is also a very compelling watch for those who are interested.

Re:Movie

[ath0mic]

...which I think also holds the record for the "longest movie title in history."

:)

Why wouldn't math be known across the universe?

[ObviousGuy]

What reasons would there be for an alien to not understand or accept that one plus one equals two. Any being capable of human-equivalent level of thought would be able to count objects. Whether they did in this in base-2 or base-3 or base-10 or base-12, it doesn't matter because all these bases can be reconciled to each other.

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:Why wouldn't math be known across the universe?

[Sique]

With just adding and subtracting you get something called 'constructive Mathematics'. It's a subset of Mathematics, and it's missing some important axioms necessary for quite usual things like calculus (nonexistance of a supreme for any limited set of numbers) and algebra (no selection axiom, thus you can't prove that infinite dimensional vector fields have a base).

Re:Why wouldn't math be known across the universe?

[Apostata]

quote: "It is fairly condescending to assume that aliens could not even fathom those levels of mathematics."

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.

Re:Why wouldn't math be known across the universe?

[arbour42]

Prior to Newton there was no calculus

In a fascinating book, a Hindu scholar and monk, Sri Tirthaji, discovered in the Hindu Veda scriptures the basis for our math system. There he found shortcuts for most all our math work - easy ways to do difficult long divisions in a matter of seconds, quadratic formulas, PI to over 32 digits, the Pythagorean theorem (much before the Greeks), derivatives, calculus.

Our math is actually from the Vedas, and the Arabs got it from them, and then spread it through the Western world. The Vedas are at least several thousand years old.

The book is called Vedic Mathematics or Sixteen Simple Mathematical Formulae from the Vedas and can be found at amazon or used book stores.

It's one of the major works of genius of science. The first time i read it, it was shocking how advanced it was, and simple! Any division such as 1.748362 / 59487 can be done long handed (pencil and paper) in a minute.

Our math system, how it was discovered or invented, who knows and how far back, is absolutely brilliant.

Re:Why wouldn't math be known across the universe?

[fitzsimj]

You're an idiot. Take some upper level college math courses. Better yet, just read a few posts up where the guy points out that anything can count.

Re:Why wouldn't math be known across the universe?

[vontrotsky]

Also, you can use mathematical models to prove that math is not consistent... Read up on Godel's Theorems.

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?

[sllim]

I can take this a step farther. The human brain is set up to recognize patterns. In the same way that all computers come down to mathmetics and binary (no matter what it is you fed into it or what output you desire) the human brain breaks down into pattern recognition.

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:Why wouldn't math be known across the universe?

[DickBreath]

This is more than just saying "what if they see in infra-red!",

The reason we see in "visible light" is because that is the brightest radiation given off by most stars, especially our nearest one.

If an alien's local star gives off most of its radiation in a slightly different part of the spectrum, still mostly visible light, but let's suppose shifted more towards blue, then the alien's visual systems will be evolved to see that the best.

Re:Why wouldn't math be known across the universe?

[michael_cain]

No, but quantum behavior does. It is there, but not there, and ove there too... Also the problem is the info comes in quantas. If a truly alien race first dicovered this, and constructed their physics -> math based on this, then the concepts of integers for them is extremely alien.

An interesting idea; under what kinds of conditions might it be possible? An electron may be both here and there, but a sheep (to pick a particular object) is either here or there. Technically, I suppose that this sheep in front of me is both here and on the other side of the hill with different probabilities, but the probability for it's being anywhere else is so small that I'm unlikely to observe it in the next several billion years. Postulate some sort of quantum intelligence; it seems to me that one of two things must be true: (a) it occurs on a scale so small that we are unlikely to encounter one another or (b) it has in some fashion managed to extend quantum effects to a scale more like our own. Starting from the latter perspective, they would certainly be alien! Would they exist in some fuzzy quantum state so that they are always both here and there? Can they choose where to be (just as we can "choose" where electrons are in electron-slit experiments by observing them)? Integers would certainly be a strange concept for them -- a fuzzy being postulating that an object be in a single place, not infinitely many? Physics concepts like "speed of light" would also be alien to them, if they could coherently be here at one instant of time (quantum time?) and there at the next instant.

Numbers are numbers

[Wind_Walker]

Regardless of what radix used (10-based, hex, octal, etc) "6" of something is still "6". Call it whatever you want, some alien name. There are still 6 items there.

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:Numbers are numbers

[vingilot]

Very intersting about the radix used. Is there a mathmatical proof for that? I have often thought that perhaps we find some numbers unique or special because we tend to use base 10, 16,8,2).

For instance 1/3 in base 10 is (.33333333->infinity)
but with a radix of 3 is .1 which unlike a radix of 10 is discrete. (Not that 1/3 is that unique or special, its just nondiscrete)

Re:Numbers are numbers

[Lemmeoutada Collecti]

Borgolsmock is divisible using a thersian constant. To assume that just because your single dimensional mathematics cannot divide a borgolsmock does not similarly limit n-dimensional. Why the mere act of division is a limitation of linearly conecived time, a limitation we have never faced.

Aside from which, where you see a single item, I percieve an infinite semi recursive series. There are more than one apple in that one apple. There are an infinite (using your limited numbering) number of apples. That apple you call 'one' in fact contains the entirety of it's temporal measurement, which is a bounded infinite series. So now tell me about this concept you call singularity?

Re:Numbers are numbers

[Wind_Walker]

And maybe if you inverted the warp plasma through the deflector dish, you could create an inverse tachyon field that could disrupt that thersian constant! Make it so, Number One!

You see, I was limiting myself to thinking of... what's the word... reality. I was talking about plausable scenarios of intelligent extra-terrestrial life. You've obviously spent too much time reading Sci-Fi novels and not enough time in what we like to call the "real world" - and it's not that shitty MTV show you watch.

Re:math is not universal?

[greatmazinger]

Math is just a way of describing objects, forces, and interactions..

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.

A god with a plan?

[mindstrm]

Why does this make you think there is a supreme being, with a plan? Just beause things work out?

The balance and beauty of nature and all that?

OF COURSE there is a pattern, and things work out. Look at evolution.

You take a puddle in the middle of nowhere.. it has an ecosystem in it with a perfectly balanced population (too many, it dries up, too few, they reproduce...). Would these little creatures say "Oh wow! Look how there is JUST enough water for each of us! There must be a GOD!".... silly, right?

Nature seems balanced in the world, becuase that world produced nature... they are intertwined, infinitely.

Irrational numbers only seem strange because of the way we choose to look at things... the fact that it doesn't reduce to some fraction in our counting system doesn't *mean* anything holy or significant....

The fibonacci series and the golden ratio are related? Sure are.
(The ratio of successive numbers in the fib. series approaches the golden ratio as you go upwards)

But it's not so weird, is it? A sunflower.. the way it grows, it builds on itself.. in a spiral... naturally following this series.

Is it some grand creator that made it that way, or is it just the plain, obvious way for something to grow?

What would be evidence of a creator would be if things did NOT follow what was natural and obvious. If these things did NOT follow the golden ratio and other straight math.. if we could find no explanation for why things had a weird ratio, or weird behavior.. no explanation from the current or possible past enviroment to explain how something evolved.... come to me with that, then we can talk about a creator.

Until then, i'ts just nature.

Re:A god with a plan?

[Geckoman]

What would be evidence of a creator would be if things did NOT follow what was natural and obvious. If these things did NOT follow the golden ratio and other straight math.. if we could find no explanation for why things had a weird ratio, or weird behavior.. no explanation from the current or possible past enviroment to explain how something evolved.... come to me with that, then we can talk about a creator.

If God had intentionally inserted all these frequently recurring constants and ratios everywhere, then they, like the Babelfish, would be proof of God's existence. That would defy faith, and He would disappear in a poof of logic.

Then, unfortunately, I'm afraid we'd all get hit by a bus....

Re:A god with a plan?

[danmitchell]

Since I'm taking a class on general relativity this semester, I'll weigh in with a quote of Albert Einstein.

What I'm really interested in is wheter God could have made the world in a different way; that is, whether the necessity of logical simplicity leaves any freedom at all.

In other words, maybe nature is what it is because God created it that way, or maybe it is what it is because it has to be.

Re:A god with a plan?

[scorp1us]

I tend to agree with you. But I think that the use of "God with plan" actually weakens the argument.

Assuming God is all powerful, as is the usual definition of God, then God would not need to follow any plan. Things would just be. 1+1=2, 1+2=3, etc until you try to do math with a number that God had not created yet. Then thigns would break down.

Of course, that is the plan - to keep things consistant so they scale and continue to work. x1+1=x2 ... x(n-1)++1=xn gives a number line that is infaninte in size. Things since all things are mathematical in nature, and constructed from the same elements of the universe, everything shares a common mathmatical foundation. When that is the case, of course a few of the same numbers pop up everywhere. Pi, e, and the Golden Ratio to name a few.

Once you realize that God is slave to math and rules, then you must comclude that math is more powerful and absolute than God. Therefore your old notion of a traditional God should be superceeded byt the ultimate one - mathematics.

When you pray, you pray that the maths of the universe work out in your favor. Since we mathmatically backtrace events, we know that God has not suspended reality, but you have mathmaticaly evaluated the likely outcomes and calulated the propability of your favored action to be within the realm of mathematical rendering. So you pray. Had it been clear cut you would not have wasted your time.

Math is the CPU in wich the universe runs.

Re:A god with a plan?

[zapp]

And who is to say that God didn't design the system such that it balances itself in such a way?

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.

I rememeber this from...

[gpinzone]

Donald in Mathmagic Land. It was a great little video Disney produced back in 1959 with Donald Duck. The narrator goes off the topic at times, but the overall animated descriptions of the golden ratio and its related golden values were awesome. Unfortunately, this Disney short is not available on VHS or DVD currently. Look to eBay to find a long lost copy of it.

Furniture design

[hulap0pr]

The golden ratio concept is a big part of furniture design. Case pieces (boxes, bureaus, etc...) appear more balanced and pleasing to the eye when the golden ratio is followed. Go home and measure your highboy...

From your english class?

[iota]

for my english class. hope you enjoy it.

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/isbnIn quiry.asp?isbn=0767908155)
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.

universal math?

[mblase]

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.

Yes and no. Mathematics is just a way of modelling things abstractly. Even things like counting from one to ten is a model for concrete objects, and provides a way of, say, making sure the number of cows you have today is the same as the number of cows you had yesterday. At the higher level, mathematics lets you model shapes, motion, acceleration, and gravitational collapse of entire stars.

The most common types of mathematics we use include decimal arithmetic (trading with money), algebra (solving for unknown quantities), and geometry (simplifying the world into abstract shapes). Hundreds of other branches of mathematics exist to model different things in different ways, and none of them are "right" -- they all are optimized for particular problem-solving.

With that in mind, I find it inconceivable that advanced civilizations on other planets would not have some kind of mathematics, and at least share the natural numbers with us (not necessarily base ten, though). If all you're doing is raising food for your family, then even arithmetic may be more than you need to bother with. But anything that involves abstract problem-solving, measurement, and/or exchange of goods for trade is going to need some kind of math. The models they use may bear no resemblance to the ones we use, but that doesn't mean it's not math.

How does one dispute math as a universal concept?

[hellfire]

I'd be fascinated to hear more about this. I want to get the book but I'm impatient and want to discuss it now! :)

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?

Opinion, Mr. Spock?

[ackthpt]

"Captain, I believe there is a 1.6180339887 percent probability that any security officers beamed down to the planet will survive."

Definition FYI

[Prince Vegeta SSJ4]

Golden Ratio:

The number 1.618..., which is half the sum of one plus the square root of five (1+SQRT(5))/2. This number was known in ancient times, and has many interesting properties in many fields. In Fibonacci series, the higher one goes in the series, the closer the ratio between a number and it's predecessor comes to the Golden Ratio.

From "The Technical Analysis of Stocks, Options & Futures" - William F. Eng

Geez, I never thought my Trading and /. would come together. Then again it is delving into the Uber Math Geek world - lol

First-contact scenarios?

[bravehamster]

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. ... 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.

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: Da Shitty Code

[Anonymous Coward]

Dan Brown is the greatest writer of all time.

Provided we ignore EVERY OTHER WRITER EVER.

Intelligent marketting strategy for stupid ideas

[dtfinch]

He probably just caught on to the idea that if you say something outrageous enough in your books, like math is wrong, people will buy them.

Phi

[Rupert]

I hate it when people use extreme amounts of decimal precision when talking about irrational numbers. Really, is 1.6180339887 (or 1.6180339887498948482045868343656) much more informative than 1.618? If you're going to do calculations with it, use the exact value:

1/2 * (sqrt(5) + 1)

and sort out the irrational bits at the end, rather than introduce rounding errors at the beginning.

That's just a rationalisation, of course. My real reason for complaining about decimals is that it feels wrong. 1.6180339887 does not look like a profound number. It's like the number is a beautiful woman, and the decimal representation is the pornographic pictures she posed for when she was young and needed the money. Yes, it looks like her, and it may even be useful. But the real thing is *so* much better.

Re:Phi

[pomakis]

I hate it when people use extreme amounts of decimal precision when talking about irrational numbers. Really, is 1.6180339887 (or 1.6180339887498948482045868343656) much more informative than 1.618? If you're going to do calculations with it, use the exact value:

1/2 * (sqrt(5) + 1)

and sort out the irrational bits at the end, rather than introduce rounding errors at the beginning.

This is sound advice for hand calculations, but of course it doesn't apply for floating-point calculations performed on a computer. The two statements:

a * (sqrt(5) + b)

and:

a * (2.23606797749978969640 + b)

are exactly equivalent computationally using double-precision floating-point arithmetic on a 32-bit processor. In fact, the second statement will execute more quickly because there's no need to perform an expensive sqrt() operation.

The reason that these statements are equivalent is that floating-point calculations can't deal with irrational numbers. Heck, they can't even deal with most rational numbers. A lot of people don't realize this, but the even simple fraction 1/3 can't be represented exactly as a floating point number. All the function call sqrt(5) does is calculate the floating-point number that's closest in value to the square root of 5. If you can provide that to the program as a constant, then you save an unnecessary computation step.

Obviously for readability you'd define a named constant (e.g., SQRT5) rather than just using the number in place explicitly, but my point is that you're still better off defining the value of that constant as 2.23606797749978969640 rather than as sqrt(5).

math and humanity

[jstoner]

It's an interesting question: how far could a civilization get without math? IANA historian, but it seems to me the more sophisticated a (human) civilization, the better its mathematics. The Aztecs did develop a fair amount of math completely independently of Eurasian civilizations.

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

[Master Switch]

While the syntax we use for mathmatics is culturally defined, the content beneath them is not. We humans discover, not invent mathmatical constructs. As much as we would like to think we create, we do not. We iterate and find the best fit solutions.

Debunking constants

[hcg50a]

Livio squarely debunks the idea that the Golden Ratio is present in many famous paintings and architecture that has been postulated in previous books.

This doesn't sound exactly right.

I think it may be the case that writers have attributed the use of phi in art when there was no such intentional use by the artist.

But the very nature of phi makes it unlikely not to appear in certain contexts.

Same with pi.

The thing I love about math is that it has utterly nothing to do with reality or the universe or anything at all.

Typically, however, physicists make assumptions that match, more or less closely, to what is happening in the real world, so the conclusions from such assumptions match, more or less closely, to what is actually happening in the real world.

But there is no reason why some utterly alien intelligence can't make a set of assumptions that match their reality, which would be utterly alien to us, yet still valid, and still recognizable by mathematicians, if not physicists.

This is the giant flaw at the end of the book Contact, by Carl Sagan. Ellie discovers a message in the constant pi, placed there by an intelligence. If this were a constant of physics, that would imply the existence of some incredibly advanced intelligence that engineered the universe to contain a constant with precisely that value. This is somewhat plausible, and I believe it was Sagan's intent.

But he picked pi, which actually has nothing at all to do with this or any other universe.

What kind of incredibly advanced intelligence can possibly engineer that? I can only think of One.

Re:Debunking constants

[Angst Badger]

This was actually the most tedious part of Livio's book. His argument, essentially, is that if a) the artist does not explicitly say that he used the Golden Ratio, or b) the Golden Ratio appears in the work to an accuracy of the third decimal, then the Golden Ratio was not used.

Aside from conveniently (and fallaciously) proving a negative, the first condition is highly unlikely to be satisfied in any premodern work, and the second condition borders on the absurd. In one case, he takes a discrepancy of less than a quarter inch in a painting measuring more than four feet high as a "disproof" of the use of the Golden Ratio.

In fact, the majority of the book is devoted to such sloppy debunking. The remaining fraction of the text -- which actually touches on real mathematics -- is quite interesting, but comprises perhaps fifty pages at most, and probably could have been condensed into a longish magazine article.

Pi the movie

[cryptochrome]

I literally watched that movie 2 nights ago. Spooooooky....

Not bad (aside from one glaringly obviousl mathematical error). The thing that I mulled over the most was the proposition that a large integer could be a number of fundamental significance. In the movie it was 216 digits long. I had always figured all the really fundamental numbers were irrational. After thinking about it and looking up on the internet it seems there are actually only 6: pi, e, i, 1, 0, and phi (and arguably, -1). And the first five can be directly related with the equation:

e^(pi*i) + 1 = 0

phi is not directly related to the others in such a manner (In the movie the god number is somehow tied to both pi and phi). Although pi and phi both happen to be ratios that are also irrational. But to get back to my original point, the suggestion that any number of a truly fundamental significance besides 0 and 1 would be not only rational but an integer seems improbable.

Re:Pi the movie

[kallisti]

suggestion that any number of a truly fundamental significance besides 0 and 1 would be not only rational but an integer seems improbable

What about the Monster?
This is the largest "simple" group which doesn't fit into any group category. What this means is rather hard to explain in simple terms, but this group has lots of mysterious connections to other maths. The order is 2^46 * 3^20 * 5^9 * 7^6 * 11^2 * 13^3 * 17 * 19 * 23 *29 *31*41*47*59*71.

1.61803399

[karmaflux]

More here.

also

[Savatte]

also quote possibly tied for the shortest title

The Golden ratio and the fibonacci numbers

[Den_onda_kotten]

There is a really great page that explains the relation between the Golden ratio and the Fibonacci numbers here
The fibonacci number is the series 1,1,2,3,5,8... where every number is the sum of the two numbers before it. What does this have to do with the golden ratio? Everything! Just check it out, you'll be amazed.

Mathematics is an invented human language

[pkcyll]

It may be shocking to some, but mathematics is an invented language. It is used to describe physical events around us. But invented it is. When we state that 1 + 1 = 2, we already make assumptions (such as the + and = operators are neutral) and we know that in the mathematics of quantum mechanics 1 + 1 is not two because "adding" injects its own effect and that "equal" depends on the situation (is it a wave or a particle - it depends on the experiment.) So is mathematics an invented language, yes. Is it a language that waited to be discovered, well, that is the question.

Re:math is not universal?

[Timmeh]

HEY! Don't push your modernist science bullshit on me, PATRIARCH. My goddess awakening mentor told me about you so-called intellectuals. Using 'symbols' and 'information' is just another form of OPPRESSION.

To which the sage answer

[Sunnan]

"Do what?"

Re:To which the sage answer

[criordan]

And they respond by kicking you in the shin again.

Why do we need cardinality?

[Xoder]

I see a lot of nay-sayers in this thread talking about "How could some alien not understand that one of one thing and two more of that same thing make three?"

You are assuming that everyone has a concept of cardinality. Realistically, people don't have much of one beyond the number six (yes, there are outlyers for whom eight objects in a group is eight objects not one-two-three-four-five-six-seven-eight objects). If a being had no concept of cardinality, that would make many things more difficult, but many others much easier. This organism would not think of a system as the sum of its parts, but rather as a cohesive whole (or rather the cohesive whole). It is likely that they would be philosophical geniuses compared to us. There are creatures of this type toward the end of Calculating God by Robert J. Sawyer (See your favorite bookseller and/or your local library), and their possible existance is not implausable.

Re:Why do we need cardinality?

[greatmazinger]

True. Although very hard to imagine, I guess it is possible for an advanced alien civilization to develop mathematics that does not have the concept of cardinality. It may just be based on continuity or graph theory. It may be something that has never been conceived by human mathematicians. But the optimist that I am likes to believe that the logic underlying the alien math/science is something a human mind can understand given the proper axioms and rules for derivation.

Re:How does one dispute math as a universal concep

[Lemmeoutada Collecti]

The flaw in that line of thinking, which many on /. are making, is assuming that what we percieve singularly is similarly percieved by another species. Let's take a thought walk for a moment... First, we percieve an object, say a book, at a single position in time/space. While we can percieve the entirety of it in space, our perceptions cannot percieve simultaneously the entirety of it's temporal measurement. Therefore, we see on book in the now.

Now another lifeform comes along, one which can percieve the entirety of the book in time/space. They percieve not only a different book than we are capable of, but further, they may percieve each temporal book as a seperate item, just as we percieve spacially translated objects as seperate. So where we see a single book, they see an infinite number of books. We can only assume that their method of counting would differ from ours, or that we would be unable to correlate ours to theirs because we can not percieve the many, only the one.

Assuming that another specias percieves the universe the way we do is the height of hubris, and the largest flaw in alien contact scenarios. Our mathematical beauties when percieved on a larger scale may be no more than a mere curiosity, instead of the vaunted unchanging laws.

Just a thought.

Other ancient number systems

[HarveyBirdman]

10 is not a given. Other number systems used in ages past include base 5 and base 20. The Babylonians used base 60. I believe that's where we get our time system from (60 seconds per minute, 60 second per hour).

Two other interesting books: Zero: The Biography of a Dangerous Idea by Charles Seife.

Trigonometric Delights by Eli Maor.

Both books cover the a lot of historical ground in mathematics.

Now it all makes sense.

[Talinom]

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.

So that is why all of those UFOs are crashing all over the place.

One small point

[Atario]

Irrational numbers only seem strange because of the way we choose to look at things... the fact that it doesn't reduce to some fraction in our counting system doesn't *mean* anything holy or significant....

Well...not holy, but yes they are significant. They can't be expressed as a fraction in any counting system. (Unless, of course, you cheat and decide to "count in units of" the exact irrational you're trying to express, in which case it's going to be "1" -- you've just replaced one symbol (pi, e, phi, etc.) for another.)

Agreed

[DougMackensie]

This book is absolutely excellent. Its aimed at everyone from a passing interest in math and up.

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.

Fibonacci

[Lewie]

This book was a great light read, the math is not difficult and some of the classic paintings and such were really cool to see.

The most interesting part of the book for me was the correlation between Fibonacci and the Golden ratio. As I read it, as you ascend the Fibonacci sequence the ratio between the current number and the one before it converges on the golden ratio. F20 divided by F19 is as near the golden ratio to as many decimal places as any of us have use for, probably.

An interesting "party trick" was also mentioned that I remember vividly. Take any two numbers and add them, then take the new number and the larger of the first two and add them, then take the new sum and the old sum and add, ala Fibonacci. Continue for twenty or so iterations and the 20th number divided by the 19th will be damn close to the golden ratio. This is, I think, because any such construction is a linear multiple of the base Fibonacci set (see prev. paragraph). When you divide, the common multiple falls off and you still get Phi. I thought that was pretty cool. :)

anything similar

[dominator2010]

If this is anything similar to the book "A history of Pi" by Petr Beckmann then I'm all over it. Petr Beckmann was able to put down some ideas and link them together in that book that I was very captivated by. It was technical enough to hold it's own, but he explained it well enough for most people to understand. At least explain it to the people who had enough interest and brain capacity to be interested in the subject and read the book.

The Golden Ratio

[t0ny]

The Golden Ratio- He who has the gold makes the ratio.

Or something like that...

Re:Arguments against postmodern deconstructionists

[s20451]

"What else is natural science than a common set of rules for perception" is their answer and I can't answer it.

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:Arguments against postmodern deconstructionists

[Golias]

"What else is natural science than a common set of rules for perception" is their answer and I can't answer it. I believe my inability to refute their point is simply because the point they make is so idiotic, but still...

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:I wrote a review..

[belloc]

This curious mathematical relationship, widely known as the "Golden Ratio," was defined by Euclid more than two thousand years ago because of its crucial role in the construction of the pentagram, to which magical properties had been attributed.

Funny, because there's not a single pentagram anywhere in Euclid's Elements. Care to research your plagiarees a bit further?

Belloc

Some criticisms of the book

[Charles Dodgeson]

I read the book a few month ago, and I largely agree with the review, but I have a few more criticisms as well.

First let me highlight one of the really nice points that the author makes (with many well-researched examples in the book). Recently created myths about things long ago can easily be mistaken has ancient stories. It was interesting to learn that the Renaissance fascination in art and architecture was basically a 19th century invention. For me, the most interesting thing about the book is its debunking of similar historical myths, always working to show what grain of truth their might be to them.

One minor gripe I have is in the context of the praise above. While debunking historical myths, the book reinforces the myth that Einstein's theory of Special Relativity was primarily motived by the Michelson-Morley experiments.

For me, the both the most interesting thing and the most disappointing thing about the book is that the history of the Golden Ratio isn't all that interesting. What turns out to be most interesting is the history of the myths about the Golden Ratio.

This is not to say that the Golden Ratio isn't interesting itself. It's relation to fractals, repeated fractions and parallel curves is interesting, but I guess I would have preferred a "happy ending" where it would play something likes its reputed role in psychology/aesthetics. Of course it is hardly the fault of the author that it doesn't have such an ending

Vedic Math and Indian Math.

[The Cydonian]

I read the book in question a couple of years back. Just would like to say this:- you'd like to distinguish between arithmetic and mathematics, as also, perhaps, between actual scientific history and propaganda.

Vedic 'math' is mostly arithmetic; it's about how to multiply numbers faster (cool method that; helped me throughout most of school) and, like you said, long form division. Even in that, I doubt it was from the Vedas themselves; I remember reading about those 'tricks' (using the term in a broad sense; not a negative connotation) even before I read Tirthaji's book in an old book published in 1936. The book claimed it was a translation of an even earlier Sanskrit book on mathematics (an absolutely fascinating treatise called Leelavati Ganitham); don't quite think it mentioned any Vedic references.

Indian mathematics, OTOH, was mostly from the Medieval Ages, between 5th and 10th centuries CE, when mathematicians such as Bhaskaracharya and, of course, Aryabhatta, wrote their treatises. The reason, apparently, was astronomy and trade; when you are the center of a globalised trade in gems and spices, you want to get your math right.

Quite possible that ancient India knew about calculus, but it's more likely than not that it was a result of a gradual excellence in the sciences, not something that's been left to us automagically by our Vedic ancestors.