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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.
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.
Out of every 1000 slashdotters, 1.6180339887... will have had sex with a real woman.
Trolling is a art,
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
The concept of math isn't even spread very far on this planet.
What do you mean foldout? It struck me as more like a stretchout, or maybe a reamout? Inquiring minds want to know.
If you're trying to find a copy of the book, its available at Amazon. Hope that helps.
He who has the gold, makes the ratio.
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.
Toronto-area transit rider? Rate your ride.
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...
Comment removed based on user account deletion
The movie PI is also a very compelling watch for those who are interested.
Free XBox, PS2
I'd suggest The Da Vinci Code to people who are all sorts of interested in this sort of thing. Da Vinci played a small part in all this fun Phi stuff, and evidence of it can be found in his paintings. Besides, this is just a great book that everybody should read! They point out many places where one can find the "Golden Ratio" within this fine book.
Why the ads anyways? If I wanted the book, I'd be resourceful enough to find it.
Do they get referrer bucks or some other such lame innernet moneymaking scheme?
I don't need no instructions to know how to rock!!!!
With a title like that I was expecting 320 pages of this:
8 05 76286213544862270526046281890244970720720418939113 74847540880753868917521266338622235369317931800607 66726354433389086595939582905638322661319928290267 88067520876689250171169620703222104321626954862629 63136144381497587012203408058879544547492461856953 64864449241044320771344947049565846788509874339442 21254487706647809158846074998871240076521705751797 88341662562494075890697040002812104276217711177780 53153171410117046665991466979873176135600670874807 10131795236894275219484353056783002287856997829778 34784587822891109762500302696156170025046433824377 64861028383126833037242926752631165339247316711121 15881863851331620384005222165791286675294654906811 31715993432359734949850904094762132229810172610705 96116456299098162905552085247903524060201727997471 75342777592778625619432082750513121815628551222480 93947123414517022373580577278616008688382952304592 64787801788992199027077690389532196819861514378031 49974110692608867429622675756052317277752035361393 ...
1.618033988749894848204586834365638117720309179
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.
I have been pwned because my
I own the book, bought it a year ago myself. A good read.
If you're looking for something a bit along the same lines, but sprinkled with history, religion and conspiracy, I can recommend "the Da Vinci Code" by Dan Brown.
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.
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.
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.
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.
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...
for my english class. hope you enjoy it.
n quiry.asp?isbn=0767908155)
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.
Is covered many many many times in many other books (did I mention the word many).
I just finished the somewhat overrated, but entertaining Da Vinci Code which mentions this in addition to several other interesting. The presentation is that of fiction, which adds entertainment, but detracts from the believability.
Also the movie Pi, which I probably need not mention here, speaks of this to some length.
Final question being, does this book really add to my knowledge of the subject? I think I've heard all of the examples of where this ratio can be found in nature, is this guy just beating a dead horse? The review doesn't really imply that there's anything new here.
How hard was it to write this without mispelling ratio as ration?
"Draco dormiens nunquam titillandus."
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.
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?
"All great wisdom is contained in .signature files"
"Captain, I believe there is a 1.6180339887 percent probability that any security officers beamed down to the planet will survive."
A feeling of having made the same mistake before: Deja Foobar
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
This is more of a historical book than anything else. Since it is called the golden ratio, I expected more about "the golden ratio" that what it offered. I wish I had read the Amazon customer reviews before I wasted my money on this junk.
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.
---- El diablo esta en mis pantalones! Mire, mire!
Say you visit a planet where the dominant species, the one responsible for things like math and science, experiences everything double due to their funny optical and other sensory apparatus. How would you describe the concept of "one" to such an entity?
Dan Brown is the greatest writer of all time.
Provided we ignore EVERY OTHER WRITER EVER.
"After all, I am a Trekkie and this concept goes unstated throughout all four TV series." This is exactly how I feel about the Star Trek universe. All four TV series. You said it, my friend.
I don't have a concept of math, you insensitive clod!
Why should it be? The math we used was invented by us. It's a high level concept.
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.
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.
--
E_NOSIG
If the aliens were different (even far different) than we, they would indeed describe them differently. They would SEE ('see?') them differently.
What are the dimensions of this book? Is the size of the book pleasing to the eye? I always enjoy these things when someone takes a bit of a step back and relates the physical format of the book to the subject at hand.
We should be able to apply our mathematics to everything. However, if our thought process is drastically different, they ("aliens" if you will) may not recognize it as a corollary to their ideas. They may not even have a use for it.
For instance, math doesn't really need to add. The concept can be completely explained with the concepts of negative numbers and subtraction.
To get a real handle on the concept of different mathematics models, take the extremely difficult class of Abstract Algebra. It's called Algebra because it wraps itself around the ideas of open sets, closed sets, and operations as a generic concept.
Finally, Livio takes a shot at the idea that mathematics is a universal concept across the entire universe.
This seems like a tall, tall order. I've been into math/geometry/visual related software for years now and am now transitioning into making my living off it. However, the fact that there are still many fundamental mysteries in mathematics always raises doubt on the things like our origins, God, and the universe. Pi is the best example of that. It's no puzzle to me why countless minds have tried to be the hero (or the mathematician version of one), to unlock pi's mystery, but no one has yet to really break through. The film Pi is an excellent and enjoyable film, and considers the magnitude (as well as the price) of unlocking pi's mystery.
I'd like to day I'm open minded, but whew. Perhaps such things are more considerable when you start to consider all the various matter/energy theories floating around out there. There's still gigantic mysteries still out there for cosmology and physics (dark matter, open universe, dark energy, unification of gravity into the standard model), so I suppose we should never be too hasty to close the door on counterintuitive or far-fetched theories. I'd love to hear anyone who can paraphrase the thrust of this person's arguments, etc.
G-Force music visualization
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.
'In knowledge is power, in wisdom humility.'
Try Journey Through Genius by Will Dunham.
It covers a sampling of many of the great theorems and proofs of mathematics in a form that anyone with high school math can follow, as well as giving interesting insights into the personalities of the mathematicians (where this can be known). Most of them were, um eccentric. It is nice to know that Euler at least was well adjusted, if you couldn't exactly call him normal.
Euclid is represented twice here: once for his proof of the Pythagorean theorm and once for his proof of the infinitude of primes.
Post may contain irony: discontinue use if experiencing mood swings, nausea or elevated blood pressure.
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.
-Master Switch, one more element in the machine
If a creature can't distinguish between "more than" or "less than" of an object, it cannot be labeled "intelligent" to begin with.
show me math that has no connection with reality.. ask someone who can't see, hear, smell, or touch how mathematics works
or the quantum level? What if the aliens lived in different dimensions? Also, to say the laws of this backwater place in the immense universe are the same everywhere is a little arrogant
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.
HCG 50a = 2MASX J11170638+5455016
11h17m06.4s +54d55m02s
1.6180339887% of all statistics are made up.
Obviously the people that are truly running the show and know the answer to the meaning of life have 13 fingers, and use a base 13 numbering/math system.
6 X 9 = 42 in base 13.
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The golden mean (proportio divina or sectio aurea), also called golden ratio, golden section, golden number or divine proportion, usually denoted by the Greek letter phi, is the number phi = (1 + sqrt 5)/2 = approx. 1.618033 ... the unique positive real number with phi^2 = phi + 1 and the equally interesting property phi-1 = 1/phi.
...]
Two quantities are said to be in the Golden ratio, if "the whole is to the larger as the larger is to the smaller", i.e. if (a+b)/a = a/b. Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference: a/b = b/(a-b).
After simple algebraic manipulations (multiply the first equation with a/b or the second equation with (a-b)/b), both of these equations are seen to be equivalent to (a/b)^2 = a/b + 1 and hence a/b = phi.
The fact that a length is divided into two parts of lengths a and b which stand in the golden ratio is also (in older texts) expressed as "the length is cut in extreme and mean ratio".
The ancient Egyptians and ancient Greeks already knew the number and, because they regarded it as an aesthetically pleasing ratio, often used it when building monuments (e.g., the Parthenon). The pentagram so popular among the Pythagoreans also contains the golden mean. It is also sometimes used in modern man-made constructions, such as stairs and buildings, woodwork, and in paper sizes, however it is a myth that the European formats (such as A4, which is actually cut to 4 decimal places of sqrt 2) are cut in the golden mean. Recent studies showed that the Golden ratio plays a role in human perception of beauty, as in body shapes and faces.
A possible reason for its supposed attractiveness is shown by the Golden rectangle, which is a rectangle whose sides a and b stand in the Golden ratio. If from this rectangle we remove a square with sides of length b, then the remaining rectangle is again a Golden rectangle, since its side ratio is b/(a-b) = a/b = phi. By iterating this construction, one can produce a sequence of progressively smaller Golden rectangles; by drawing a quarter circle into each of the discarded squares, one obtains a figure which closely resembles the logarithmic spiral theta = (pi/2 log(phi)) * log r.
Since phi is defined to be the root of a polynomial equation, it is an algebraic number. It can be shown that phi is an irrational number. Because of 1+1/phi = phi, the continued fraction representation of phi is 1+1/(1+1/(1+...)) = [1; 1, 1, 1,
The number phi turns up frequently in geometry, in particular in figures involving pentagonal symmetry. For instance the ratio of a regular pentagon's side and diagonal is equal to phi, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles.
The ratios of justly tuned octave, fifth, and major and minor sixths are ratios of consecutive numbers of the fibonnaci sequence making them the closet low integer ratios to the golden mean. James Tenney reconceived his piece For Ann (rising), which consists of up to twelve computer generated upwardly glissandoing tones, as having each tone start so it is the golden ratio (in between an equal tempered minor and major sixth) below the previous tone, so that the combination tones produced by all consecutive tones are a lower or higher pitch already, or soon to be, produced.
The explicit expression for the Fibonacci sequence involves the golden mean. Also, the limit of ratios of successive terms of the Fibonacci sequence equals the golden mean. From a mathematical point of view, the golden ratio is notable for having the simplest continued fraction expansion, and of thereby being the "most irrational number" worst case of Lagrange's approximation theorem. It is also the fundamental unit of the algebraic number field Q(sqrt 5) and is a Pisot-Vijayaraghavan number.
The golden mean has interesting properties when used as the base of a numeral
This explains a lot, especially Lincoln's distorted views of North versus South....
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.
---If you can't trust a nerd, who can you trust?
More here.
REM Old programmers don't die. They just GOSUB without RETURN.
Man where have I been... I guess I have to get this book now. I always thought it was 9:4 or 21/4 to 1. *shrug*
I think you've just stumbled on a way to get more guys to study math...
also quote possibly tied for the shortest title
For those who understand physics it is obvious
that all things in this universe
share a common set of rules and relations.
Nature proportionality is an emergent property of the laws of physics ruling the cosmos.
The numbers (mathematics) are just one way to
approach it.
Ill suggest to those who think mathematics
hold some intrinsic truth to learn more
about chaos theory, patterns and the limitations of reductionism.
Breaking things appart (staring at numbers) only goes so far.
Very complex systems can emerge from the most
simple set of rules and emerging patterns
may or may not tell us a damn thing about
the rules that generated them.
The real trick is not the analyze a system but
to create a new one from scratch that produces
predictable and not so predictable emergent
properties. So quit being so "amused" by numbers.
- these are not the droids you are looking for -
While he may have a point in suggesting that you can manufacture evidence of this ratio anywhere, it's also true that this ratio does appear in many great paintings and structures, because the creators used it on purpose. It's been taught to designers and artists for generations... and many of us use it.
Whether artists have used it instinctively because of its mathematical elegance, or it's merely a coincidence that works based on this ratio also tend to be visually pleasing, is kind of a causality/synchronicity chicken-and-egg argument. Coincidental or connected, conscious or not, a correlation does exist.
http://alternatives.rzero.com/
I can certainly see most people's point of view that math is some universal, even metaphysical, system. In some ways it certainly is. However, one only needs to study a bit of math history to see how much this viewpoint has changed from the time of Newton through Godel to the present. A very convincing book by Morris Klein called "Mathematics: The Loss of Certainty" is a really great read on these subjects. Klein is a great mathematical historian, and this book tracks the progress of both the theoretical and philosophical viewpoints of math throughout history. I think when people say that 'math is not universal', it means mostly what Klein's conclusion is. There is no 'right' system of mathematics.
Take geometry for instance, it 'seems' right and it works in lots of situations. But change one or two of your axioms, and you've got a completely different geometry, which is equally valid in theory and application. Math like anything, is about choosing the right tool for the job. Now geometry is one thing because we have a physical conception of it mostly, but as for other things, take number theory, I don't know. Are there different number theorys based on choosing different sets of axioms? Probably. So which one is the right one? Whichever one works for what you need it for. The fact that math is somehow 'out there' is a very Platonic concept. That's not to say it's wrong, before I read Klein's book, I was definitely a supporter of that viewpoint. But after that book, it seems a little shortsighted, and to tell you the truth, it takes a little of the mystery behind math away, which is too bad really.
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.
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.
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.
I haven't read the book, but I would like to know what the conclusion about mathematics not being an extra-terrestrially universal idea is based on. Here's another thought, not too original, but given what we know about Mathematics as a universal cultural phenomenon on Earth not a difficult one to arrive at: It has become apparent that every culture has in some sense created a system of mathematics, whether to organize dependencies, patterns, order, and tools for comparison of magnitude. Basically, cultures have tended to spawn organizing systems, which are in essence mathematical. The lack of such a system would suggest the lack of ability of abstraction of patterns, etc., which would suggest a creature without the means for formal reasoning. If we seek a dialogue between us and the inhabitants of another world, they will certainly have to decypher or reason out our messages (e.g. the golden record aboard the voyager 1), which would then suggest an ability to reason, and understand mathematical ideas. I suppose then we can't know for sure whether mathematics is universal or not until we hear back from whatever is out there. It just seems arrogant to assume that Mathematics is a uniquely human invention. Maybe someone who has read the book can provide some insight into its conclusions.
"Do what?"
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.
The previous sig has been removed due to
No mathematics has a connection with reality. If you decide to use a mathematical model of reality, you're not doing maths, you're doing physics or compsci or chemistry or something else.
Your deaf and dumb person could perfectly well do maths in their head. Their only problem would be communication.
Where x is the golden ratio:
1/x = 0.618...
x = 1.618...
x^2 = 2.618...
I didn't do it! Unless I was supposed to do it. . . (hmm. .
Maybe BN's warehouse got slashdotted.
.. but I did a report on this book back in the second grade. That was something like 18 years ago, and the book wasnt shiney new then either. I guess Im unclear about what the limits are of a book you can review.
"When life gives you lemons, don't make lemonade. Make life take the lemons back!" -- Cave Johnson
43 is very plain, bordering on the ugly.
--
E_NOSIG
Don't you know that Amazon.com is bad!!1! Your karma is going to get blasted so low for your post that you might as well just start over and create a new user account.
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.
You can have it fast, accurate, or pretty. Pick any 2.
a rose by any other name smells the same...
Say you visit a planet where the dominant species, the one responsible for things like math and science, experiences everything singly due to their funny optical and other sensory apparatus. How would you describe the concept of "half" to such an entity?
evil math within Nature's Cubic Creation!
Ummm... no. Physics is math.
Wish I could moderate that one. +5 funny! A new addition to my .sig file...
I am lucky - I am a geek with a geeky wife - go figure...
Reason is the Path to God - Anon
Seems to me that we base our math on the number ten for a pretty logical reason. I think it is easy to understand how we came up with that number.
But if we were born with 13 fingers on each hand, then how would we work math?
Is that what he means by different mathmetics? When I think of it that way I come to the conclusion that at the core everything is equal. Seems to me that no matter what number system you use you could always convert it.
On a side note, here is a question that I have never known the answer to.
Why do countries that have such dissimilar languages (the US and China or Russia for example) all use the same roman numeral numbers?
I can be looking at Russian text and it is all gibberish, but as soon as a number is inserted, bammm familiar territory.
Why is that?
Okay lets see someone without a physical brain do math...
It's inescapable. The only reason we believe a mathematical truth is that our brains--physical objects--tell us that it is so.
no, you are misunderstanding what I'm saying.. my stupid rhetoric was supposed to demonstrate that a blind, deaf, dumb, mute, skinless whatever being is not capable of interacting with reality, and is therefore not capable of thinking at all, let alone performing formal mathematics in their head.. where do you think mathematical concepts form from? does the lord beam them down from on high?
Not enough can ever be written about the great mathematicians of history... Archimedes, Newton, etc. Such brilliance and distinction could not be captured in an inifinite numbers of words!
"Ain't I a stinka..." - Bugs
OK this is a serious question so please don't blast me, but what is wrong with Amazon?
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.
--- Ban humanity.
Ummm... back to school with you. Physics is physics. The rock will fall even if no one is there to calculate how long it took.
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.
"Giving money and power to governments is like giving whiskey and car keys to teenage boys." - P.J. O'Rourke
After the golden ratio was initially described, there was subsequent *deliberate* use of it by various artists of the time. It was also incorporated into some architecture.
--- Ban humanity.
Maybe counting/(add/sub) arithmetics might be 'universal', but as soon as you enter multiplications and divisions it starts showing problems. How do you multiply 3 apples with 2 apples? Oh yeah, to be precise, it's multiply 3 apples with the number 2(or else you have apples ** 2). But what is "number 2"? It's not a real thing. Then simple comparisons with things is much less direct now. We *define* multiplication as a way of simplifying repetitive additions (3+3+3+3 = 4*3), but when looking at it this way, it's not obvious than another race would do the same thing, or even think about the concept, especially if they have a system of 'numerals' that aren't suited for this(ever did multiplication/division in Roman or Egyptian numerals?? Try it, you'll understand. With those numerals, no one would have thought about multiplication or divisions).
Then you need to understand the concept of 0, which is central to our mathematics. Even in the roman language, I believe the word 'nothing' was based on the word 'something', because no one was able to think about nothing back in those days(... are we now more apt to do it? Ask someone in philosophy =). Then the Arabs(well, Indians if you want to be exact IIRC) changed this.
Then, for more complicated mathematics, you start having lots and lots of Axioms. An Axiom isn't TRUE or FALSE, it's just an Axiom. If you agree with the Axiom, then that's great. However, no universal concepts precludes you from not agreeing to an Axiom, and if you do you change a LOT of things, but it doesn't mean it's wrong.
For example, X ** 0 = 1. Or a division by 0 is impossible. Why? No reason, those are axioms, so you have to agree that it's that if you want to do lots of classical mathematics. You can 'Argue' axioms, but it's just that, arguying, you cannot prove without a single doubt an Axiom(beause then it's not an Axiom =)
Another example, in logic this time, is P = NOT(NOT(P)) and NOT(NOT(P)) = P. However, you could disagree with the second one and agree with the first and then(in additions to new and/or modified Axioms) you'd end up with a NEW logical system, vastly different from the classical logic: it's called Intuitionist Logic and isn't less valid than Classical Logic, just different.
One thing proven in one of the system might not be provable, or even could be false, in the other system.
This just is a small glimpse of why Math is certainly not universal. Simple counting(add/sub) MIGHT be universal, but I do believe it stops there if it does start there(you'd have to enter philosophical debates over this).
I think the longest movie ever would Return of the King
Or maybe Ghandi
"A great democracy must be progressive or it will soon cease to be a great democracy." --Theodore Roosevelt
Do a Google search for "Amazon" and "one-click" and "patent."
"I'm just here to regulate funkiness."
By definition, if you are interested (in the movie) it will be compelling.
However, it does not really say a whole lot about anything useful or even interestING: universality of mathematics, how to relieve tension headaches (!) or street-fighting tactics for use against a mob of Hassidic jews with bats. Least of all does it have anything useful to say about maintaining privacy in an antagonistic world. Even "Hackers" (!) was much more realistic here: at least once upon a time, red boxing worked. But I digress; after all, realism in this setting has been outmoded since Kafka, who apparently had never really seen a cockroach. Yes, I know it is really a "monstrous vermin", but in his notes he calls it a cockroach.
But especially, it has nothing to do with mathematics. I "do" mathematics. Many friends of mine "do" mathematics. We have each noticed independently that Pi is at most about human sanity. I am a degree more generous than my friends, in saying that it efficiently describes what it is like to be alienated in a complete and total way from your fellow meatbags. This is interesting & I believe that Pi did a brilliant job capturing this. However, the subject matter was mathematics mostly because the target audience (and arguably, director) is least familiar with it.
I have heard rumours that Aronofsky (s/nof/fon/?) is going to be doing Batman: Year Zero. Is this true?
Dude, you're so wrong. Math involves quantities. Quantities cannot exist without a reality to have things to count.
Reason, free market capitalism, and individualism
In Soviet Russia, he who has ratio makes the gold!
Fun problem:
A. Take a number. Add 1.
B. Take a number, square it.
For what number are the answers from (A) and (B) equal?
By now, you know the answer from the context of the question.
The book is a pretty good read, though it drags in a few places (the draggy places are still readable).
One other book I learned more from is called
An Imaginary Tale by Paul J. Nahin , which is the story of the imaginary number (square root of minus 1). It is written in clear language and is intended to shed light on the topic, rather than mysticize the "imaginary" phrase in its name.
I'm a physicist and I've been ridiculed by my philosophist friends for arguing against this point.
"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...
Any advise?
That's easy. Math is one of the many devices men have developed in order to subjugate womyn. In fact, the reason logic has a higher importance in our society than intuition is that men have no intuitive ability and have invented logic to maintain this unnatural barbaric societal structure.
(Think I'm joking? Such propaganda exists and quite possibly is a course of study at a college campus near you.)
Have a nice day.
If moderation could change anything, it would be illegal.
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.
Pleas address your complaints to the patent office not to Amazon.
That's like boycotting a company that's taking advantage of a loophole in tax-law. It's legal, so everyone's free to go for it.
All your talking about seems to be math in higher dimensions with a little bit of knot theory. Yeah perception changes, but mathmatical laws don't change just because perception changes. Things work differently in 2 dimensions than they do in 5, but that doesn't mean they don't work if you can't percieve it.
a la "I refute it thus?" ... except Samuel Johnson's method was a lot more friendly than yours!
Hear hear! Well quoted, sir.
-kgj
-kgj
Roman numerals are the same in different languages because they origionated from the Roman language. Example: MMIV is the current year.
The arabic numerals that we use today became standard mostly because of the need to trade currency. If there was a standardized number system, conversions would be easier. Arabic numerals were also superior to Roman numerals in that they could represent the number we know as "zero." (think why we need to all use metric in Science) Example: 2004 is the current year.
Install Ubuntu in Android
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.
This sig washed every five years whether it needs it or not!
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.
No, the hardcover edition is out, but the paperback is still available at bn and at amazon.
Cheers
E^(i*PI) + 1 = 0
Those three transendental numbers and one and zero are interrelated in a totally bizzare and inexplicable way. What is remarkable is that each of those numbers manifests itself in the natural world in independent ways. I'm not big on religion, but if you ask me for proof of a divine creator, there it is, buddy. And if you don't buy it I really doubt that you comprehend the burning intensity of just how inexplicable and irrational the realtionship between those numbers is.
Offer them to jump out of a tenth story window. If they indeed create their own reality, they won't go "splat".
Or something like that...
Manipulate the moderator system! Mod someone as "overrated" today.
The GR also has a direct relationship with biological systems, especially relating to growth.
Apart from the breeding of Fibonacci's Rabbits there are nice examples of Phyllotaxis and Sunflower Seed Patterns which exhibit the Golden Ratio.
Paul Gillingwater
MBA, CISSP, CISM
The Paradox of the Lie:
"This statement is false."
A textbook problem for philosophy, and it doesn't get dismissed as bogus by the pros.
This is not my sig.
Though nothing in our world exists as a half, there is always a single elemental particle at the very least.
Numbers don't have to represent "counts" of anything, engineering-boy.
Math doesn't have to involve quantities!
;))
3 + 4 = 7
theres some math for you. Nothing says 3, 4 or 7 have to be quantities! Heck if we consider the possibility that im using the symbols 3, 4, and 7 as something other than the common usage (which isn't even the case in all parts of -this- world), they might not even be real numbers!
Want something even more interesting?
(sqrt(2)^sqrt(2))^sqrt(2) = 2 (try it
Have fun trying to associate sqrt(2)^sqrt(2) with any real-world quantities (not that I'm saying its impossible, but my guess is that it won't be an overly intuitive representation of the mathematics or vice-versa).
Communication of Mathematical Laws are based on a common perception and measurement system. If you and I are differing species, and I am trying to teach you to count in my system, and I show you a book, what you are referring to as one may in my number system be what I refer to as many, merely because of the perceptual differences. I'm not referring so much to the mathematical basis, but to the communication difficulties in assuming that what you percieve as singular I may not.
You can have it fast, accurate, or pretty. Pick any 2.
What is the square root of -1?
Mathematics is the language of reality, and like any language, it can be used to make meaningless phrases.
your sqrt(2)^sqrt(2))^sqrt(2) = 2 in math is no different than me saying:
twas brillig and mimsy were the borograves
in English.
They're both nonsensical phrases that sound and look cool.
You can tell a great deal about the character of a man by observing those who hate him.
I had a discussion with someone once about this stuff. I read amazingly fast but if I'm not interested I just put the book down and walk away, so I mostly absorb this stuff through assorted human filters, but one theory is that new universes are "created" all the time but certain constants to which we have become accustomed are set and fixed at the beginning (beginning being defined here as a very high state of energy; until things settle down a little bit and stop doing strange things, it's still the beginning) and if they don't end up with a usable combination, the universe just fizzles out and its energy goes back to wherever it goes, to create the next beginning.
Certain constants (like the strength or perhaps even existence of particular forces, like strong and weak attraction) must work with one another to create a usable universe. It's not necessarily that they have to have particular values, nor that they have to be in a certain range, but more that they have to be consistent to one another.
Therefore it might be posible to have a universe that works quite different from our own and in which many of our constants would be incorrect or just plain not useful. Mathematics would likely be quite different there. However, logic is unlikely to be affected, and as you can construct a complete system of mathematics from nothing but logic, someone who truly understands mathematics (as a system of logic) should be able to adapt to such a system.
"You're right," Fisheye says. "I should have set it on 'whip' or 'chop.'"
That's what SCO's stock price would be in US$ were it not for unfounded litigation and rampant FUD.
Take a piece of paper and a pencil and...
1) write down the month of your birthday (1-12)
2) write down the day of your birthday (1-31)
3) add 1)+2)
4) add 2)+3)
5) add 3)+4)
6) add 4)+5)
7) add 5)+6)
8) add 6)+7)
9) add 7)+8)
10) add 9)+10)
Divide 10) by 9)
Do it again with a different birthday
Do it again, but this time do 100 steps
Universality indeed
OK, everyone, go back to mom's basement and stop hassling Wind Walker. He's right and you're all stooges.
Superhypertechnobabble comes from Star Trek writers.
Just for when they find out why the consoles are on fire.
If you use it way too much then you just might be hired
As a superhypertechnobabble-using Star Trek writer!
The only reason we have the rights we have is that people just like us died to gain those rights. -- Cheerio Boy
How would the means by which humans experience things not be sufficient?
.. halves (or other such fractions).
If the alien makes an alien-pie for its alien-family, and there's only one alien-pie, and two people to eat it, they cut it and end up with a dual set of
Perhaps you'd care to better explain what you mean by experiencing things 'singly', since its pretty darn ambiguous as is?
Moo
Mathematics ... Universal? Mathematics is a human invention that we humans have used for a blink of time to try to explain "How the Universe works" to ourselves. Godel's Incompleteness Theorem demonstrates that we can't even use mathematics to prove to ourselves that mathematics is consistent!
Why would anyone think that one life form (us) on a small planet, at best a grain of dust in a parking space in a Universal parking lot, has found the ultimate description of the physical Universe in their formal system such that every other intelligent being in the Universe uses it? I dunno, but anthropomorphism comes to mind. Hell, not that long ago we thought we were the center of the universe and the whole thing revolved around us.
What's the old saying ? Oh yes ... "Learn from history, or you are destined to repeat it."
A finger pointing at the moon is not the moon itself. Don't confuse math with physics.
Something I like about the golden ratio is that it is the number that is exactly 1.0 greater than its reciprocal.
...
Another slick thing is that it is the limit of ratios of successive terms in a Fibonacci (or similar) sequence. Here's what I mean:
1. Take any two numbers (we'll use positive integers for this exercise, but they don't have to be), say, 5 and 9.
2. Add them up. 5 + 9 = 14
3. Now add the greater number (9) and the previous sum (14). 9 + 14 = 23.
4. Do this a bunch of times, and you get a nice sequence of numbers. 5, 9, 14, 23, 37, 60, 157
5. Take the ratio of any two successive numbers in the sequence, the greater over the smaller: 60/37 = 1.6216..., or 157/60 = 1.6185...
6. Notice that these ratios approach the golden mean (1.618033989...) as a limit. Be amazed!
The golden mean is also expressible as: (1 + sqrt(5))/2, which is nice and simple (as opposed to its decimal form, which is an infinite expression).
Belloc
I got more rhymes than Jamaica got Mangoes.
speaking of unavailable, I think http://www.wilwheaton.org/ isn't available right now - or it needs some editors.
I used amazon until they had ignored my privacy preferences and shared my info with partners a couple times. Their privacy policy is a bad joke and I refuse to do business with them because of it.
[Set Cain on fire and steal his lute.]
That would be "i" (or "j" for you EE majors). Simply set i = sqrt -1 and then you can solve for i: i^2 = -1
Very useful to get those imaginary problems solved. Wasn't this introduced in high school trig class?
Physical shmysical. As long as you can distinguish one thing or state from another (or even existence from non-existence) you can come up with math.
I realize for most of us this concept is old hat, since we know binary, or hex, or octal. However, this relates to the discussion of alien mathematics because here we have a system of non-traditional mathematics that was actually used and had its own structure. Many people argue that aliens may not understand concepts such as "prime numbers" or such. But even the Mesopotamians understood these concepts and had their own problems that we face in a base-10 number system.
For example, the Mesopotamians loved the number 60. After all, it is evenly divisible by 1, 2, 3, 4, 5, and 6, right off of the bat. But the number 7 always gave them headaches. Why? It's the first number where its reciprocal does not terminate (i.e. 1/4 = .25) or have a predictable pattern (i.e. 1/3 = .33333...). Just because the enumeration or base of a number system may be different, the rules of mathematics are still the same. And, as other Slashdot readers state, the very laws of nature and movements in the heavens will accurately adhere to different number systems.
Now, aliens may have a different way of looking at mathematics, or may have come up with different structures for representing mathematical ideas. Think of the Indian man who found a book on algebra and without any sort of education or outside intervention devised many proofs, some of which were old, some of which were new (I think his name was Ranmujon). Alien mathematics may not include things know, and vice versa. But I must agree with the general thought--Mathematics is universal.
--Chag
You've just fallen into the trap that numerologists live in - everything mathematical can be assigned signifigance if you look hard enough at it.
Given that all math depends on all other math, it isn't strange or inexplicable at all the these numbers have some sort of relationship. Rather, it's inevitable.
Another slick thing is that it is the limit of ratios of successive terms in a Fibonacci (or similar) sequence.
..., m, n, m+n, ...
In fact, this can be demonstrated.
At any point in the Fibbonaci sequence, the values are
Now, if you assume that the ratio of terms converges (proof left as exercise for reader), then, as the number of terms approaches infinity, the ratios of adjacent pairs of terms approaches equality:
n/m = (m+n)/n
or
n/m = m/n + 1
substitute x = n/m:
x = 1/x + 1
AHA! We're back to the equation representing the number that is 1.0 larger than its reciprocal! Solve the quadratic to get
x = (1 + sqrt(5))/2
The golden ratio.
Causality would tend to preclude this type of sensory perception.
Please, let's confine our hypotheticals to reality and not fantasy.
I swear, you people watch way too much Star Trek.
You can tell a great deal about the character of a man by observing those who hate him.
There are people who understand the math, people who don't, and then perhaps nay-sayers, who really don't understand anything. If we can establish that math is a language to describe our universe (roughly), would it surprise anyone that a different race would have a different language? Whether mathetmatics is universal is not a relevant question. Whether the universe works for everyone may be.
I wish I didn't have to use 8x11.5 and 11x17 standard here. I would ideally prefer we'd use the European A1, A2, A3, A4 standard for paper sizes. Cut an A1 paper down the middle across the width and you get A2. You can keep going all the way down to the A8 size.
Now they need to implement that for monitors. A 1920 by 1200 monitor could display two pages at once. Great for desktop publishing. Or tiled displays breaking up tasks with multiple GPUs with a ratio you can keep splitting down and scale easily....next universal adoption of the metric system, if only that would happen in my life time !!!
So, the Real Ultimate answer to Life, the Universe and Everything is... 1.6180339887?
A fascinating natural topic - see more here where the Golden Angle and its close relationship to the Golden Ratio (Mean) is described.
As long as you can distinguish one thing or state from another (or even existence from non-existence) you can come up with math.
Or astrology. Or Tarot. Or any pseudo-science with its own set of made up rules. Difference being that they all fall apart when you try to get predictable, provable results in the real world with them. Math holds up.
Overcaffeinated. Angry geeks.
uh.. let me be the first to say you completely miss the point
Say you visit a planet where the dominant species, the one responsible for things like math and science, experiences everything singly due to their funny optical and other sensory apparatus. How would you describe the concept of "half" to such an entity?
That doesn't really make sense? If they had two apples in front of them, they would seem them as one? If they had a tree and an ocean in front of them, they couldn't distinguish between the two? It seems if that was the case then their "eyes" wouldn't be anymore developed than the eyespots of a protista. Explaining math to a species that primitive would be as pointless as explaining art to a door knob.
We hope your rules and wisdom choke you / Now we are one in everlasting peace
Personally I prefer 5 * 9 = 42. In base 13, this is true. In base 10 it is not. How does that mean anything in reality?
7 November 2006: The day Americans realized corruption and incompetence weren't addressing 11 September 2001
When you have property, you need to know what you've got. So you need to have a number system for counting. If you acquire someone else's property, you need to recount what you now own... until you discover the shortcut of addition. Likewise if you sell some of your property, you need to recount what you have, until you discover the shortcut of subtraction.
If you become a big buyer, and acquire the same amount from three different people, you're then stuck with a lot of adding to do... until you eventually discover that additions can be strung together to form multiplication. On the flipside, if you have offfspring or take on partners, and want to give equal shares of your property among them, you're going to have a big problem counting the number of beings you're giving to, and then counting and recounting the amounts you give to each, until you either try doing multiplication backwards, or study the relationships between the numbers and discover that they work like that.
Their higher math may differ, but their lower math will be the same, because the lower math contains the base and relationships all else is built off of.
*honk*
This is my sig. It's prescription, I swear. I need it for reading things... on the other side of things
ask someone who can't see, hear, smell, or touch...
You haven't left the poor guy/gal much to work with.
How would we do that? With a symbolic language based on varying degrees/amounts of duration, balance, and/or taste? I'd rather save that for the bedroom.
I had never thought about this before. But generally, a half dozen is about the point where people do lose the concept of "one unit". Interestingly, the limit of human short-term memory is five plus-or-minus two discrete objects. So maybe six is a good average "memory chunk".
So fundamental mathematical concepts might be contingent on how the human brain is wired, just as our decimal system is based on the number of digits some amphibian happened to evolve.
I could see some alien creature with a short-term memory capacity of a hundred, or a million, or a gorf or whatever they call it out there. And that in turn might affect how their "mathematics" or "philosophy" works.
We might also conceive of an alien race that, due to its "wiring", thought entirely in binary with a totally boolean philosophy. In fact, I suspect several members of that race now work in my computer center...
"Obviously, I'm not an IBM computer any more than I'm an ashtray" (Bob Dylan)
For some reason I've always been fascinated with the history of how mankind discovered various mathematical concepts. Awhile back I saw a book with a title like The History of i which was actually about how i (the imaginary number component, or square root of -1) was postulated and developed, but I didn't get a chance to order the book. Anyone know what its real title is, and/or how I might find it?
Secession is the right of all sentient beings.
Let me rephrase then. Math represents concepts that are involved in the real world. Negative numbers are merely an abstraction of quantities where the number line has two sides. Imaginary numbers actually prove me right, since a bunch of our mathematical functions require positive values, i . e. quantifiable values. Decimals and all sorts of other numbers are merely scales of actually quantifiable things. (where 3.67 is 367 X 10^-2)
Not only that, but math requires a reality insomuchas you need pencils and paper, which are known to be universally reality based.
On a serious note, though. Math was developed in order to measure and predict physical behaviors. Mathematics isn't some etherial spiritual language handed down from druids or anything. it is completely reality based
Reason, free market capitalism, and individualism
Then how do classify Euclidian geometry? Where is two dimensional flat space found in nature/reality?
Or do you mean that integers make no sense to you unless you imagine them as amounts? Where is -1 found in nature/reality?
Just because He established rules that the *universe* is bound by doesn't mean that He is bound by them. He is only bound by the rules He decides He's bound by. Math is simply God's design pattern that we're forced to follow since we were designed based on it.
We're not supposed to kill but he has no moral dillema when He does.
That's what makes him God. He can choose which rules to be bound by. We have a very very limited ability in that area.
If God wants to run off a cliff naked and not fall down He doesn't have to. If you run off a cliff naked you have no choice in whether or not you fall.
And that whole walking on water bit. The only reason we can make choices is because God withholds his ability to force us to do His will.
Ben
Work Safe Porn
In Proverbs, Wisdom is the first of Creation.
Are we slaves to the CPUs we create, though they slavishly and regularly run our code?
And what of what we call miracles? God has a 'debugger' to correct certain problems with our 'code' that cannot easily be explained by the normal operation of the 'CPU' so we may only hypothesize about unknown code or unknown conditions we don't have the means to duplicate.
I refer you to Lourdes, France. There is much written of that place.
Deus ex machina... ironic, no? I don't mean the game. At least, not that one.
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
I got more rhymes than Jamaica got Mangoes.
The flaw in your thinking is that this quantum mechanical being can perceive me as how I expect to be perceived (i.e., within time and space.) If they cannot then I cannot communicate with it anywise. True that it may exist, but how should I know unless it can effect my current environment, and even if it can, it must be able to some way decipher a change that I made to my environment as a response. The mere fact that it wishes to communicate shows that it perceives a difference between us. If it realizes that I am not it and that it wants to communicate then it must go through a "comparison" thought process, which is the basis to mathematics. Through that comparison which we both agree that we are not the same, we have agreed on a basis for mathematics.
or they wouldn't have spent so much energy fighting over who invented it first.
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?
are all derived from the exponential series. The exponential series can be used to derive exponents to any base (e being the "simplest"), and trigonemtric identies by substituting various coefficients in the summed series.
So you expand e^(i*x) and discover what you get is actually an intertwining of the expansion of -sin(x)*i and cos(x). It's not that amazing... and by making x = Pi the sin terms will sum to 0 and the cos term is -1 by defintion... and -1 + 1 = 0.
So there.
THIS THING CAN TURN ON A DIME, MACROSSZERO STYLE ALSO FUCK BETA, ~NYORON
Mod parent up as +5 funny!
A real Trekkie would know that there have been five live-action Star Trek television series and one animated one. :-)
"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. "
Uh, yeah. I guess if a Klingon on TV can do math, then anyone or anything in the real universe can.
Or something. heheh, heheh, hehe.
I figure out how American government works and who to vote for next year by watching West Wing. Dean in 2004!, er, or whatever year we vote.
*scratches head*
So you're saying that if I'm receeding from the light source at 0.303c my perception of blue would acutally be red due to relativistic doppler shift? That's great! Red RULEZ!! Blue SUCKS!! Yea!
The series of number, in your example 2,4,6,10... can be described recursivly as:
f_0 = 2
f_1 = 4
f_n = f_(n-1)+f_(n-2)
If we assume that f_(n+1)/f_n will indeed become closer and closer to a given value, then we get the equation:
f_(n+1)/f_n = f_n/f_(n-1)
But the left hand side can of course be rewritten to 1 + f_(n-1)/f_n (just use the recursive definition above)
If we set x = f_n/f_(n-1) we get the equation: 1 + 1/x = x with the roots Phi and -1/Phi.
Look at it this way. Ask yourself why human beings whose major pre-occupation for 95% of its evolution was being a hunter-gatherer should somehow have the brain hardware to indpendently discover mathematics in Europe, the Americas (Maya), India and China and be able to therefore design spacecraft etc. It doesn't make sense unless evolution supplies these skills as part of the normal repertoire of a successful hunter-gatherer. Evolution is convergent, if skills are useful to us in this environment there is a very good chance they will be selected on the other side of the galaxy as well. I don't even think that mathematical reasoning is unique to humans you would suspect that the roots of it are probably older.
Bitter and proud of it.
Consciousness, is about awareness of the environment. If any kind of an intelligent being can perceive the various forms of matter in the universe, then one would expect that a system for relating to another being that you had two atoms in your pockets, as opposed to one, would be fabricated.
Math is a system for dealing with quantites. Unless you stumble upon a doped up alien race than cannot perceive that the universe is made up of separate entities (i.e. I am the universe, the rock is me), then there will be a math equivilent.
Here is another sequence you can make with the golden ratio.
....
...76, 123,....
phi = 1.6180339887....
phi^1 - (1/phi)^1 = 1
phi^2 + (1/phi)^2 = 3
phi^3 - (1/phi)^3 = 4
phi^4 + (1/phi)^4 = 7
phi^5 - (1/phi)^5 = 11
if x = even then add phi^x + (1/phi)^x
if x = odd then subtract phi^x - (1/phi)^x
it will generate the sequence 1,3,4,7,11,
now if you divide the a number in the sequence by the previous number you approach the limit phi
3/1 = 3
4/3 = 1.3333333...
7/4 = 1.75
11/7 = 1.5714...
18/11 = 1.6363...
29/11 = 1.6111111...
.
.
.
123/76 = 1.61842...
I wonder how many other sequence may be related to phi
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
Prime numbers are exactly what Alan Greenspan says they are -S. Minsky
I believe widescreen TVs, movie theater screens, and even Credit Cards also follow the golden ratio (loosely).
The math behind this often-passed rumor is easy to do, so here it is:
phi: ~= 1.618
Credit card: 86mm:54mm ~= 1.593 (1.5% off phi)
Normal TV: 4:3 ~= 1.333 (way off phi)
Widescreen TV: 16:9 ~= 1.778 (10% off phi)
And as the author of the book points out, if you drag a tape measure all over -any- painting, you're going to find ANY proportion your're looking for all over the place.
The analogy can be extended. Like the beautiful woman (or this particular one, for most of us), we can't have the real thing. I find difficult enough to estimate my own variability, let alone simulate that of anyone else. In order to get closer to her, we approximate. Perhaps you get bad black-and-white pron, move up to clearer pictures, maybe find color ones. Maybe she was in a movie, and so you get that. Maybe someday you even meet her, fall in love, marry, have children, etc. Each time we move closer, we gain information about her - what she likes, what she looks like, etc. We make successive approximations, modeling her behavior as we go. We never have her inside, just a model containing things we know and deductions from them. The woman is still alive, and more unpredictable than our model allows. We don't know how she'll behave in circumstances none of us have seen, nor do we know the things she hasn't told us. Everything we know of her helps us to build a model, but the model isn't the object (the person), and never will be.
Because the GR is irrational, we will never have it exactly. Higher precision numbers give us a better picture of what the GR is. While I like the compactness and beauty of your expression of the GR, the fact is that I don't know sqrt(5) exactly and never will. Successive decimal approximations give me a more exact picture of the GR, even at the cost of beauty. Perhaps the added precision is useless, or the error in the expression makes it incorreect, but those are different issues.
I see the whole PI thing as evidence of a hoax. Sagan new that PI contained all possible messages, and that a sufficiently clever mathematician could trick people ala the bible code.
I still expect something like this to happen one day, for there are ways of searching mathematical formulas for sets of strings, rather than specific ones. Thus, you describe something similar to what you want, and set a supercomputer out for a year or two finding the best matches, then, sort through them for one that you can trick people with, and construct an outrageous story to explain how you arrived at the idea.
I think it's something like where 1/(1+x) = x (or if you like) x^2 + x = 1 (or roughly)2 0309179805 76286213544862270526046281890244970720720418939113 74847540880753868917521266338622235369317931800607 66726354433389086595939582905638322661319928290267 88067520876689250171169620703222104321626954862629 63136144381497587012203408058879544547492461856953 64864449241044320771344947049565846788509874339442 21254487706647809158846074998871240076521705751797 88341662562494075890697040002812104276217711177780 53153171410117046665991466979873176135600670874807 10131795236894275219484353056783002287856997829778 34784587822891109762500302696156170025046433824377 64861028383126833037242926752631165339247316711121 15881863851331620384005222165791286675294654906811 31715993432359734949850904094762132229810172610705 96116456299098162905552085247903524060201727997471 75342777592778625619432082750513121815628551222480 93947123414517022373580577278616008688382952304592 64787801788992199027077690389532196819861514378031 49974110692608867429622675756052317277752035361393 62107673893764556060605921658946675955190040055590 89502295309423124823552122124154440064703405657347 97663972394949946584578873039623090375033993856210 24236902513868041457799569812244574717803417312645 32204163972321340444494873023154176768937521030687 37880344170093954409627955898678723209512426893557 30970450959568440175551988192180206405290551893494 75926007348522821010881946445442223188913192946896 2200230144375
0.6180339887498948482045868343656381177
(more or less)
...it's so easy to get them riled up, and so funny too.
No postmodern thinker is denying that there is great agreement within our cultural context about the color of the sky.
We talk about "the sky", and say "the sky is blue".
But to say "the sky is indeed blue", or "the sky is really blue" either makes no sense or is redundant.
"indeed", "really", and "actually" are, I must presume, nothing more than an expression of your own insecurity, your own anxiety, your need to force your opinions upon us, to silence us, and to lord power over us.
I will not submit to you. I am free.
from 0.1% to 0.13%?
Bravo!
A most excellent post. It gets right at the heart of what is "truth" and what we fool ourselves into believing is true.
Free yourself. Everything else will follow.
You say that they can only percieve countable objects. So their mathematical knowledge, such as it was, is on the integers, let's say. So we would probably agree on such operations as addition, subtraction, and multiplication on the integers, since the integers are closed under those operations (e.g., integer + integer = integer). Since subtraction is the inverse of addition, we can argue that there exists an operation -- call it "division" -- that is the inverse of multiplication. However, big surprise, the integers are not closed under division, because (as you say) 1/2 is not an integer. So we have to expand our knowledge of numbers into numbers that can be expressed as one integer divided by another -- the rational numbers. This is exactly how our own mathematics works -- take some mathematical construction, find where it breaks down, and extend it. This is where we get our own complex numbers.
Your alien mathematician would even have the satisfaction of realizing that the set of rational numbers is countable (i.e., isomorphic to the integers)!
Toronto-area transit rider? Rate your ride.
Actually, that's pretty idealistic -- have you ever taught math in high school or junior high? I have, and decided it wasn't for me. Why? Students, parents, other teachers, the school administrators, and the government all expect that the students will be "taught to the test." You would not believe how frustrating it was to try to teach the underlying fundamentals to students who wanted nothing more than "just how to get the right answer" so they could get the problem done, or to flirt with the girl/boy in the seat next to them, or to just doodle.
Start teaching the basic concepts behind the Peano axioms to anyone not in college, without cloaking it very carefully, and you'll have parents calling in, and students who think its not important since it's not going to be on the SATs. Another issue is politics within the school -- newer teachers (with the most drive often) can get stuck with the remedial arithmetic and algebra 1 courses, not being given the option to teach "the real fundamentals behind math", which is a course that doesn't exist in standard schools anyway.
Unfortunately, the curriculum within schools is kinda stuck. Arithmetic, then Algebra, then Geometry, then Algebra 2/Trig, then Pre-calc, then Calc, with a couple other variants. All of those classes have stuff that students are required to know, and there's not enough time in the day, or energy in a teacher to a) teach the required stuff, b) teach the important stuff, and c) deal with a classroom with 30+ kids with understanding levels that span the bell curve. The required stuff is universally accepted as required, or the kids don't get into college X, and that's why the ideals go out the window.
Which is too bad... Where this kind of stuff really needs to change is at the grade school level. Lots of bright kids sit dormant with dumb arithmetic being repeated instead of getting stuff like this. Write yr congressperson or school superintendent, I guess, or if you're a parent, get active with your school district.
my two cents.
*** once i really listened, the noise just went away. -liz phair
1.618033988749894848204586834365638117720309179805 76286213544862270526046281890244970720720418939113 74847540880753868917521266338622235369317931800607 66726354433389086595939582905638322661319928290267 88067520876689250171169620703222104321626954862629 63136144381497587012203408058879544547492461856953 64864449241044320771344947049565846788509874339442 21254487706647809158846074998871240076521705751797 88341662562494075890697040002812104276217711177780 53153171410117046665991466979873176135600670874807 10131795236894275219484353056783002287856997829778 34784587822891109762500302696156170025046433824377 64861028383126833037242926752631165339247316711121 15881863851331620384005222165791286675294654906811 31715993432359734949850904094762132229810172610705 96116456299098162905552085247903524060201727997471 75342777592778625619432082750513121815628551222480 93947123414517022373580577278616008688382952304592 64787801788992199027077690389532196819861514378031 49974110692608867429622675756052317277752035361393 6
From excellent karma to terible karma with a single +5 funny post...
Our math is only our own attempt to describe reality as we have perceived it.
Math is a language. Conversation about "laws of mathematics" might thus be compared to a discussion about "laws of the Esperanto language" or "laws of the game of Chess." It's our own construct, although a very useful one.
Neither do the continually revised "Laws of Physics" fully encompass any objective laws of Nature. Rather, they describe behavior we have observed in Nature. As such, they are rooted in human subjectivity and limited by our perceptions.
A different species may have a different experience of reality, with different things to express in a common language. There would likely be radically different assumptions, not at all compatible with our own system of mathematics. It would be a different language, optimized for saying different things, some of which might not ever be possible to express in our system.
so that's the golden ratio? I figured that number out years ago, just for kicks. Were my thoughts guided by a supernatural force? Nahhhhh... Seriously, math is NOT magic. Anyone who claims otherwise is selling something...
One is still one... However, if they do not understand the concept of finite objects, then there might be a problem.
However, since they still percieve objects as objects, they might choose to count timelines rather than an object-time coordinate. Speaking of which, they still need to have a coordinate system to identify a position in space-time...
If there is a sentience that percieves absolutely everything... Hmm, it is possible that it does not understand the concept of finite numbers.
1+1=2?
In a digital perception, yes. That is what we have built up around us. But not everywhere. We have a digital interpretation of an analog world. We can define an analog expression in a formula, but but don't really think in it. Yes/No, Black/White, that is how we think. When we go between, we still try to digitize it. We have our monitors up to millions of colors to represent the infinite. Anyone know pi? Someone out there knows figured the digital representation out to millions/billions of digits, but do they think in terms of pi? Of the equation that defines it?
Many of the assumptions here have been that math is fundamental, 'we know it when we see it'.
But what about light diffraction patterns? When two lightwaves interfere, 1+1 = somewhere between 0 and 2.
When we grab an apple, point to it and say 'one', do they see the apple as a whole? Do they 'see' the most significant wavelength where we define 'red'? Do they 'see' our hand moving at a specific frequency? Do they see the heat energy transference? The potential energy change? All of these are different perceptions of the same event, and each of these as a different starting point in mathematics.
I would have to say that while mathematics may be fundamental, it is also infinite. What is the probability that we and an alien society have a starting point in common that we recognize?
Tom
Euclid himself did not discover geometry or even make any great new contributions to the field in terms of ways to apply it.
That was left to his much smarter but not quite as well known cousin, Noneuclid.
How does one dispute the assertion that math is a universal concept?
By exposing the argument as a tautology!
Should we find a race of highly civilized (but otherwise intelligent) beings on Mars, and these hypothetical beings are determined to have a functional language with which they communicate concepts which are alien to us and which defy expression in our own system of mathematics, then what we arrogant humans would conclude is that the Martian culture has not yet discovered mathematics as we know it... and that the possibly superior system they are using is not really a system of mathematics AS WE WOULD DEFINE IT.
Thus since we narrowly restrict our definition of math to that which we have already invented, then we create a tautological argument in our application of this definition. We arrogantly define our concept as universal, with the qualification that not all species in the universe, nor even on this planet, have discovered it.
In the alternative, should our definition of what constitutes mathematics be sufficiently loose that we are able to say, "the Martians have an object-oriented mathematical system which includes 'time' as the first two dimensions and has no operators," then we have allowed ourselves to redefine this word, mathematics.
Thus even if we are more liberal in our definition of what consitutes "mathematics," we do also encounter a tautology in the sense that our definition of mathematics changes whenever we encounter something which we hadn't anticipated when we made the original statement that "mathematics is universal."
The statement gains its truth value by definition, by semantics. Because of how we would apply such a principle when dealing with other species, the statement effectively says nothing.
"It makes you want to think that there is a God with a plan." @ For about a second or two... and then you'd realize that these things would exist with or without us to observe them or a deity to put them there. For instance, a^2 + b^2 = c^2 would be true if the universe NEVER EXISTED. 2 + 2 = 4 would be true after the universe ceased to exist. The golden ratio might pop up so often because that is what was necessary to make something work! @ This sort of thinking is a really weak effort to support the claim that an invisible pink unicorn farted the universe one day. Given any serious thought it is just plain wishful thinking. http://en.wikipedia.org/wiki/Invisible_pink_unicor n
Utilizing the synergization of benchmark e-solutions to pre-workaround action items!
> math may not be a universal concept spread
> across the galaxy...
For those who aren't reading between the lines...
Mathematics is one of the very strong arguments
for a Creator. Math didn't "evolve". The idea
that mathematics in not universal is an attempt
to argue against the idea (or implication) of
a Creator. The basic idea being that math
"evolved" in ways peculiar to the local
environment much as biological organisms (read
Humans) have evolved in ways peculiar to the
local enviroment.
I read this book earlier this year. I liked it, but there were parts that really put me to sleep. I was a math major, and I have a master's in CS. I liked very much the history and the theory. I just thought the endless thorough debunking of rediculous claims regarding the golden ratio were, although perhaps someone had to do it, really boring.
What really fascinates me is that all of these methods are closely intertwined.
You just described the generation of an alternate, arbitrary Fibonacci sequence. The standard Fibonacci sequence starts with 1,1 and and then applies the same algorithm you just mentioned. In the end, your algorithm applies the fact that many on this thread have pointed out: the ratio of sequential numbers in the Fibonacci sequence converges to phi.
What your sequence just made me realize, though, is that this is much more generic:
The ratio of sequential numbers in any Fibonacci-generated sequence converges to phi.
Now, for the (1/x)+1 algorithm in the grandparent, play with the algebra for a bit by replacing [x] with [(1/x+1)] a few times. It turns out that the successive iterations of the fraction (in simplest terms) follow the following pattern: the next numerator is the old denominator. The next denominator is the sum of the old numerator and the old denominator. If we ignore the division for a second, we're generating a sequence of numbers where each number is defined as the sum of the previous two. So, essentially we're generating a Fibonacci sequence out of the numbers (1/x + 1) and (1/x + 2), but with the ratio (division) baked in.
So I guess another interesting question would be: why do the ratios of all Fibonacci-generated sequences converge to this golden ratio ((sqrt(5) + 1)/2) ?
It all goes downhill from first post
Mathematics is a language for describing phenomenae. We'll have better reason to believe that the phenomenae are consistent across the universe, like circle rolling once around three times as far as its width, when we've measured it outside our possible local "consistency bubble". We've already seen consistency break down between the femtoscopic (nuclear) and mesoscopic (baseball) scales, with baseballs statistically blurring the counterintuitive behavior of their atomic components. Other scales of distance might also behave differently.
The mathematics, of course, remains consistent. It's just our description of the phenomenae, and we tweak the mathematics whenever we need a new paradigm to explain new data. So while we might share phenomenological experience with distant intelligences, the possibility of communicating to them about it is slim. Our electromagnetic senses, our oversimplified imagination of the sensorium, our facile self-referentiality that exceeds the other animals, but is hardly complex in the astronomical scale, all drown the hope of communicating, even with the handily manipulable tool of language. However, maybe intelligence is actually so abundant in the Universe, once interstellar distances are transcended, that we'll find among it an intelligence sympathetic enough to communicate with us.
--
make install -not war
Cursive's a hair different---reading cursive is HARDER than reading printed letters.
Cursive is about speed - you don't have to ever lift your pen/pencil/quill.
If you were to sit beneath a palm tree on a beach and write the great american novel with a pen and pad, you'd do it in cursive. Print would cramp your style.
The point of courtesy here is that it's for _you_ to read. If you need to write for others, always print. For substantially alphanumeric writing, always use smallcaps, to avoid numeral/letter confusion.
My God, it's Full of Source!
OUTSIDE_IP=$(dig +short my.ip @outsideip.net)
The thing about Contact wasn't that the message was in there, but that it was so close to the beginning. As in, something chimps with primitive computers could find in a couple years. Your life in Krazy Kat style on DVD would probably take a very very very long time to find.
What Sagan was trying to say was, "the universe has an easter egg," left for sufficiently advanced life to find quite easily once they were advanced enough.
The _really_ interesting question that raises is, "advanced enough for what?"
Contact 2.
My God, it's Full of Source!
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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.
More than mere navel gazing.
So close to right that I can't resist replying... The problem is that if lifeform X perceives the book as some kind of collection over time (or something), it still has a concept of `the book'. Even if X can't differentiate between `the book now' and `the book always' (or something), it can still count by noting the existence of `the book' and `the other book' or `the book now' and `the book a little while ago'. It doesn't matter if X and you or me think that `the book' is the same thing; we're still enumerating.
In fact, as long as lifeform X can determine that [something] exists and [something else] exists (or even may exist), maths sneaks back in. If X cannot distinguish objects, the question of how it can interact with the world becomes much more interesting.
And this leads to a problem that's been bugging me for years: What is an object anyway? What I think of as `my desk' is just a local irregularity in the chemical soup on the surface of what I like to think of as a `planet' which is just...
.evom ton seod gis eht
Take any 2 numbers (lower values will make this easier). Produce a fibonacci sequence with them: Write down the first number, then the second, calculate the 3rd from adding the 1st and 2nd, then the 4th from the second and third, etc, etc. When you get 20 numbers calculate the ratio of the 19th and 20th (20th divided by 19th). That is the golden ratio.
one really practical use of
the "golden ration" you can find
in two stroke engines. good
for high RPMs.
definetly look out for more
"golden ratio" eplications
in all kind of maschinery
(DIN, ANSI, etc. anyone?)
it might be that mathematics and
logic are universal and that other
intelligent beings in the universe
are using it.
what definetly is not universal
are the signs humans use for
displaying mathematical operations.
mebelieves that there exists
TRUE forms for mathematical operations.
if you show a TRUE form of a sign
for a mathematical operation even
someone not "fluent" in math will
immidiately understand what it
means. this is not true with
the mathematical operations used
today.
considering that one cannot get rich
with mathematics there seems to
be a strong incentive to obscure
math. spreading confusion normally will
get you rich ("Size matters! get
you're sexual organ enlargment kit
here", etc.)
Mathematics and Logic are just models, founded on postulates assumed to be true. An entirely different civilaztion may have developed their own math and logic using entirely different axioms, or they may model their universe in a completely different manner.
Since *our* mathematics relies on the set theory, we should realize that we can never truly define the universe (universal set) with mathematics (a subset). The best we can do is increase the scope of the mathematics in a way that it accurately models a good portion of the universe for enough applications.
Thank you, that is a more felicitous phrasing than "politics", especially in a debate where the audience are largely likely to use the engineering mis-definition thereof. This is indeed where we disagree. The Formalist claim that mathematics was doable strictly formally without recourse to social process was disproved, within mathematics, around these questions of Axioms and Proof, before "Social Contract" theory was the rage; you are effectively correct, in that it was demonstrated that there was a matter of taste in deciding what was formal enough. Thus, Peer Review is always required; machine verification of proof will always require at least peer review of the machine, and likely a Peer Review that what the Machine reviewed matched the theorem claimed.
I gather you are fond of post-modern social-relativist deconstruction of everything to morally equivalent social constructs. This may be invogue in the liberal arts. It is not a mere coincidence that when Trudeau/Doonesbury poked fun at this sort of thing it was a Math professor who was charged with cultural insensitivity because a student claimed 2+2 wasn't 4 in his culture.
There is a social contract in doing mathematics, but it does not follow that all of mathematics is a social construct ; Mathematics is discovered, not created. Our matters of taste (over how it is done, our "rules") will affect the texture of the mathematics found, how it's organized, and what is found early or at all; it affects what of True Platonic Mathematics we know or accept to be true.
If Mathematics were as you claim a "Social Construct", there could be more than one such, with mutually incompatible conclusions. This is not so. It is demonstrable that no consistent logic can prove all true statements -- that means no logic can prove all true statements and only true statements. Thus each Mathematics(as we know it) for different "rules" will have a different subset of all true theorems proved, grow asymptotically closer annually -- but none will have things that the others prove false. As described in my prior post, the varying philospophies of mathematics admit more or fewer theorems, based on what is available for proof by their evidence, but never do different philosophies get different answers -- only sometimes only some get no answer or assert no answer is possible. (Differing Geometries differ from differing philosophies of mathematics, since one must switch back and forth at will, they are not matters of taste, it is imperative to explore all 3+N geometries and catalog the differences.)
That all Mathematics is intrinsically the same and discovered can be seen in a simpler experiment, at the level of Arithmetic -- the mathematics of the Babylonians and the Mayans and each other civiilzation which developed their own enumerations are commensurable, some of more power and expression, but all equivalent -- in each of the 5 or so different formal schools of "doing" mathematics.
We definitely have parted company long since if you think this applies to Mathematics.
There is be power in editing journals and chairing departments, a power particularly strong in some academic fields. When new theories must wait for the Chairs and Editorships to be relinquished by the heirs of the theory to be displaced before it can be published, one can speak of Truth by Power. This can be the case in fields that work by approximation, as in the hard sciences, and is very much the case in "soft science" fields where "theory" is fashion, not tested. I won't cast aspersions on certain so-called sciences with this sort of cycle by naming names, but we know who they are.
Mathematics does not create new truth by tearing down old, but by building upon old, so Truth by Power
Where nothing seen is ever true,
For then you will have no relief,
And beauty will be forever beyond you.
Rather pursue balance in your life,
Accept that you are what you make yourself;
Realize that knowing comes from strife,
Not from books found on a library shelf
Always struggle to understand,
But take time to notice beauty;
Not all of life is planned,
And not all of life is duty.
-- Michael Murphree
(copyright 2004 (with apologies to the
I don't know what it was, but I couldn't resist responding to this thread in this manner.