Euclid's Elements?
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Euclid's Elements?
It is related, in the 15th century commentary of the arab mathematician Qaadiizaade arRuumii on a work called the Ashkaal atta' siis (fundamental propositions) by the geometer Ashraf Shamsaddiin asSamarqandii (c. 1276) that over the porch of Plato's Academy at Athens was the following inscription:<br /><br /> Let no one unversed in geometry enter my doors.<br /><br /> From another source we are told that this was the inscription above Plato's Academy<br /> _________________________________<br />   <br />  Let no one come to our school, <br />  who has not first learned <br />  the elements of Euclid. <br /> ________________________________ <br /><br /> And I was wondering whether this was the reason why nobody comes to the Academy to talk about beautiful and noble things. Is the reason then for the mournful silence of the Academy that you all remain at home sharpening the instrument of reason through in depth study of Euclid's Elements in preparation for grander discourse? Well, that I do hope, for in my next post I shall raise a matter for discussion.<br /><br />S.<br /><br /><br />
The Beauty of Mathematics...
Some are surprised that I can use the words 'beauty' and 'Mathematics' in relation to one another, but it is indeed a beautiful area of knowledge...<br /><br />Actually, I prefer to speak of Mathematics as being elegant. In essence, it really just involves saying the same thing in different ways... like those (horrid to do, lovely to behold) trigonometric proofs I had to get through in my IB years... but what I like most of all is how something extremely complex can often be rendered in a simple formula.<br /><br />As a simple example, take the statement:<br /><br />62=36<br /><br />...and an attempt to explain it in normal English:<br /><br />six groups of six items gives a total of thirtysix items<br /><br />Doesn't the English look ungainly next to the short and precise mathematical sentence?<br /><br />My favourite example, though, is the general differentiation formula  when you see how it was derived (from a lengthy method called First Principles), you can appreciate the elegance of that short formula which can be applied almost without thinking:<br /><br />f/(x): axn = naxn1
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Re:The Beauty of Mathematics...
[quote author=Raya link=board=13;threadid=97;start=0#415 date=1052401832]<br /><br /><br />Actually, I prefer to speak of Mathematics as being elegant. [/quote]<br /><br /><br />From what I have observed through talking with mathematicians (okay, physicists, who are a subspecies of mathematician), they prefer the word "elegant" as well. I always found that somewhat unexpected  for whatever reason, considering nuances of language isn't something that associates well with physicists. On the other hand, they are  or tend to be, from observation... my father, brother and sisterinlaw are all physicists  very precise with terminology, which, I suppose, falls out of the mathematics side of things. I suppose, as well, that the precision is what ties mathematics to philosophy, which is trying to define and otherwise pin down ideas about living (or how the world works). (My exposure to philosophy has been much more limited than my exposure to mathematics, but I have to admit that reading Cicero is causing me to be very intrigued by ancient philosophy, anyway.) I do find it interesting that similar logic theory is taught by both disciplines.....<br /><br />Kilmeny
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The Art of Math, Astronomy & the Description of the Cosm
Mathematics may be elegant or clumsy depending on how it is performed. Just as painting cannot be said to be elegant in itself, but only that painting that has been executed by an elegant hand. I would even venture to call Mathematics an art more than a science. It is used by it's devotees to describe the landscape; Van Gogh with the same purpose in mind used oil on canvass; Herodotus too had done the same with words on parchment, and no art is in some aspect entirely free from mathematics. Paintings obey laws of perception, symphonies those of harmony, the style of Herodotus and of any other ancient writer is today meticulously studied by philologists trapped in dusty libraries, coughing ink, and crunching numbers.<br /><br /><br />I am fascinated by it. It is as moving as any other form of art can be once one learns to appreciate it. It is all a language of it's own, and one that I have not practised enough. I wasn't good enough to read Einstein's theory of relativity without really working on my calculus first, and well, now that I remember, I think first I had to understand Maxwell's electromagnetic wave theories. I loved that class, I wrote a really great electromagnetic wave theory paper, which today I am utterly incapable of following. That's how high my skill in math once rose, and subsequently fell. When I read Einstein's Relativity, I remember the difficulty. My roommate and I would talk about how our brains were going to crack trying to figure it out, and suddenly, just like that, one day I saw it. I'm getting chills now typing this, and it's funny because I no longer remember what the hell I saw, it was a glimpse of something beautiful, but the vision was fleeting and soon lost. I just remember that sense of wonder when I had looked upon the face of some mystery, now lost again behind a veil of darkness. <br /><br /> Sometimes I still wonder whether Einstein was really right? It explained everything, it made sense to me, but there was also a day that Ptolemy had explained everything, and for hundreds of years people navigated according the system of epicycles upon epicycles expounded in his Almagest and accurately predicted the motions of the planets or the wandering stars as they called them, against the backdrop of the sphere of the fixed stars. <br /><br /> Copernicus came later with a theory that turned Ptolemy inside out. No longer was the Earth considered the centre of the ko/smoj, but was now described as another one of the wanderers floating through the vast sea of space, on elliptical orbits round the sun. <br /><br /> For a time men fought about which of these two models was the true representation of the heavens, both of the systems described the observations and predicted events and the positions of the planets perfectly, how could we tell? Ptolemy had the weight of tradition behind him, but with time the balance began to tip in favour of Copernicus and Ptolemy was forgotten. What was the reason for this? <br /><br /> Some say that Copernicus's system was more elegant in that it was simpler; it did not require manifold epicycles positioned at different angles and turning at different rates to describe the retrograde motion of Mars, and hence was an easier system to work with. And I might agree today that simplicity in mathematics is elegance, but back then when I was reading the astronomers I did not. Maybe I've always been pretty complicated myself and the intricacy of all the epicycles in the machinery of the ko/smoj as Ptolemy had described it appealed more to me than the munditia simplex of Copernicus. Maybe I was arrogant and egocentric and demanded that the Universe remain geocentric and fashioned after my own image, and so for about a year while the rest of my companions were circling the sun for me it continued to rise and set, and no one could prove that the world was not the way I preferred to see it, until the day when we began to read the Principia Mathematica.<br /><br /> Newton destroyed the Ptolemaic model with such elegance, and with such beautiful skill using accepted geometrical proofs and assigning planetary bodies to points, orbits to lines, time to area, so that when the geometry worked, the physical description of the ko/smoj Newton had described emerged upon a solid foundation, because the Universe itself was the geometry. He has since then become my favourite, though there are so many others whom I have not read, and I confess that I look forward more now to rereading Kepler than Newton.<br /><br /> But still, how do we know that Newtons description is right? Everything is hanging together by the tiny thread of Euclid's 5th Postulate, which may not hold if space is curved. There are so many wonders in the world, and I feel that life is best when one is contemplating them, whether the answers come,....or not.<br /><br /><br />valete bene,<br /><br />Seba
A Theory on Theories
It's interesting how people will accept theories as 'truths' when, by definition, a theory can never be proven. In fact, the only time truth can be derived from a theory is when the theory is disproved  then you know that the thing you were trying to explain doesn't happen in the way that you thought it might, so it's time to try another train of thought...<br /><br />I wish I could call this my own line of thought  well, actually it IS  but, annoyingly enough, some other guy by the name of K. Popper published it first. ;D
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Re:Euclid's Elements?
Fortunately, a complete and thorough knowledge of the finer points of mathematics isn't required to participate in Textkit discussion! For, while I readily admit the beauty of the study, I have difficulty with mathematics, and I still can't do some of the formulasjust as I admire the beauty of a sunrise, although I can't make one. <br /><br />Unlike those of you who are fully immersed in the elegance of higher mathematics, I am still delighting in the "challenge" part. Sometimes it feels like Mount Everest; clean cut, clear and starkly beautiful against a bright blue sky, but oh! so very difficult to climb! <br /><br />Which is, of course, just another way of telling people not to wait to post in the Academy until they completely understand the intricacies (or the simplicities!) of Euclid. <br /><br />Keesa
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Re:A Theory on Theories
[quote author=Raya link=board=13;threadid=97;start=0#445 date=1052589038]<br />It's interesting how people will accept theories as 'truths' when, by definition, a theory can never be proven. In fact, the only time truth can be derived from a theory is when the theory is disproved  then you know that the thing you were trying to explain doesn't happen in the way that you thought it might, so it's time to try another train of thought...<br /><br />I wish I could call this my own line of thought  well, actually it IS  but, annoyingly enough, some other guy by the name of K. Popper published it first. ;D<br />[/quote]<br /><br />Yes, but Karl Popper was talking about scientific theories, not mathematical ones, which are provably true or false.
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Re:The Art of Math, Astronomy & the Description of the C
[quote author=Elucubrator link=board=13;threadid=97;start=0#439 date=1052530691]<br /> But still, how do we know that Newtons description is right? Everything is hanging together by the tiny thread of Euclid's 5th Postulate, which may not hold if space is curved. [/quote]<br /><br />Yeah, I think a Swiss gent by the name of Einstein had something to say about this..... perhaps it's time to study German so you can read his papers.
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Re:Euclid's Elements?
Wow....before I read these postings. I hated math. I am young and not even CLOSE to the higher mathematics you all were talking about (Honors PreCalculus) I really think that this year I will go back to school and see math and all challenges as good challenges that MUST be interesting if I look at them the right way. Thank you everyone for opening my eyes. ALso, that kinda sucks that entire TEXTBOOKS (chemistry, biology, etc...) are all based on THEORIES. Science is looked at by most people (that aren't scientists) as infallible. "The Scientists say this" is often an irrefutable argument. Why is there not anything more concrete than this.....WHY??
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Re:A Theory on Theories
[quote author=Lex link=board=13;threadid=97;start=0#3106 date=1060344601]Yes, but Karl Popper was talking about scientific theories, not mathematical ones, which are probably true or false.[/quote]<br /><br />But *I* wasn't referring to scientific theories, and I wasn't referring to Popper either  the only reason I mentioned him was anticipating anyone who was going to accuse me of using his ideas (or something very similarsounding) as my own without acknowlegement.<br /><br />However... let us consider the idea that mathematical theories can be known to be true or false, whereas scientific ones cannot. I would think it's because science speculates about the world around us  which exists in all its complexity whether or not we do  whereas mathematics is largely (if not entirely) human invention and does not speculate about anything new (if you think about it, mathematics is just different ways of saying the same thing: 2 is just another way of saying 1+1 or 6/3 and so on).
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