Mathematics: Queen of the Sciences

[numan]

'
I think it would be nice to have a thread about mathematics....so I will start it out with a few quotations :

Pure mathematics is the poetry of logical ideas.
----Albert Einstein

Mathematics is the music of reason.
----James Sylvester

Mathematics, like the Nile, begins in minuteness but ends in magnificence.
----James Sylvester

How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?
----Albert Einstein

The concept "all numbers", though not itself infinitely complex, yet denotes an infinitely complex object. This is the inmost secret of our power to deal with infinity. An infinitely complex concept, though there may be such, certainly cannot be manipulated by the human intelligence ; but infinite collections, owing to the notion of denoting, can be manipulated without introducing any concepts of infinite complexity.
----Bertrand Russell, Principia Mathematica

Mathematics is like reading a beautiful and subtle book and discovering that you can change the letters by some rule, and in so doing, you have created a new and completely different work of art of comparable beauty and subtlety. Provided that you have some sense of the mysterious harmonies of mathematics, there is no end to this process.
----Ahem! Modesty forbids!

How happy is the lot of the mathematician. He is judged solely by his peers, and the standard is so high that no colleague can ever win a reputation he does not deserve.
----W. H. Auden, The Dyer's Hand

Mathematicians are a species of Frenchman : if you say something to them, they translate it into their own language, and Presto!! -- it is something entirely different.
----Goethe
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[numan]

For the delectation of the choicest of the choice spirits who constitute the inhabitants of this Emporium of Enlightenment, a peep into an approximation of the Mandelbrot Set, on a scale which is larger than the known universe :


Fractal Mandelbrot deep zoom to 2^316 (bigger than the universe!)


[ame=http://www.youtube.com/watch?v=_QskAoLIzuI]YouTube version[/ame]

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[numan]

'
Probably the most concrete introduction to the wonders of mathematics is the Mandelbrot Set. Created by a ridiculously simple recursive procedure, it is an object which is, quite literally, an infinite object, and one of breathtaking and supernal beauty. As such, it is an excellect metaphor for mathematics itself.

Now how is this possible? How can such awesome beauty arise from what is, after all, a mere game, a mindless procedure, an object which a machine like a computer can create by an infinitely tedious series of repetitive steps? This is the mystery which we must set out to understand.

As a hint, it can be pointed out that no contradiction is involved here. Mathematics can be a realm of Divine Beauty and Truth, and it can also be a totally mindless, repetitive procedure.

The two apparent opposites are enfolded together seamlessly, in a Unity that (almost) defies understanding. It is like a picture which, when you look at it one way, is a jolly, pretty lady, and when you look at it another way, is an ugly old crone. The picture is what it is; only your viewpoint changes.

The opposite of a correct statement is a false statement.
But the opposite of a profound truth may well be another profound truth.
---Niels Bohr
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[numan]

Ah, the logarithmic spiral ! the spira mirabilis !

The form of a mollusc which is growing continuously, always at the same rate.

e, the base of the natural logarithms, is defined by a bank account, which like the mollusc, is always growing at the same rate and compounding in ever shorter intervals, until it is compounding at every instant !

One example is an account that starts with $1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value is $2.00; but if the interest is compounded and added twice in the year, the $1 is multiplied by 1.5 twice, yielding $2.25. Compounding quarterly yields $2.4414..., and compounding monthly yields $2.613035....
This sequence approaches a limit for more and smaller compounding intervals. Compounding weekly yields $2.692597..., while compounding daily yields $2.714567..., just two cents more. Using n as the number of compounding intervals, with interest of 100%/n in each interval, the limit for large n is the number that came to be known as e. With continuous compounding, the account value will reach $2.7182818....

Ah, glorious, glorious 2.182818284590.... !!!

The gateway into an infinite variety of astounding beauties!!!
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[Rihhana_NZ]

Maths help to unwind the layers of mind to think. It has a direct impact on the logical power of brain. More the maths, more the brain performs.

 

[Old Rocks]

Calculus is a conspiracy of the coffee growers of the world to sell the total supply of their product. True story, personal experiance. As the consumer

 

[Saigon]

I have always hated Maths.

I think most of us see beauty in numbers or in words, and for me it was always the latter. I scraped through school maths for as long as I had to, but I could never see the appeal in it at all - although I admire those who are good at it.

 

[itfitzme]

For many people the thought of these kinds of problems is horrible. Painful, even. A study by psychologists Ian Lyon and Sian Beilock has shown that that’s not hyperbole — some people who dislike math do so because the thought of working out things with numbers is experientially similar to physical pain. For people with “high levels of mathematics-anxiety” (HMAs), maths hurts.

Math Problems Can Be Physically Painful | Wired Science | Wired.com

Personally, I find the opposite. Though there is work associated with it, mathematical models define the world and when they finally come to light, there can be a sense of euphoria. After all, at it's core, it is counting of real things. It describes the world with precision and accuracy, leaving no doubt to what is meant and what is not.

 

[itfitzme]

e^(i*pi)=-1

Euler's identity - Wikipedia, the free encyclopedia

In a world that constantly changes, these are the constants of change.

 

[Vikrant]

I would say mathematics is king of the sciences.

 

[numan]

'
Mathematics has traditionally been the Queen of the sciences for many centuries -- going back to the time when all science was written in Latin in Europe (indeed, all learned writing was in Latin).

In Latin, scientia is feminine gender; not only is mathematics feminine, but all the sciences are.
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[Vikrant]

A simple and easy to understand blog on regular language and its application in computer science:

All things that matter: Regular Language

 

[itfitzme]

How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?

Yeah, mathematics wasn't one of Einstien's "strong suits". It's amusing and not surprising he said that. He chose physics over mathematics as he didn't see a clear direction that he should focus attention. Physics had that vision for
him, and we are all the better for it. Besides, his theory of special relativity was just geometry and algebra. His talent rested in something far more significant. Whatever that talent was, it revolutionalized physics.

The foundation of mathematics is in counting objects. It is grounded in that single connection. It is quite remarkable though, how from that simple beginning, it expands so remarkably and sets up rules that when properly followed had so reveal insights in physics that would otherwise remained hidden.

 

[Vikrant]

^ Einstein's biggest talent was intuition and that applies to pretty much all great minds.

 

[westwall]

"Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean."— Godfrey Harold Hardy

 

[numan]

"Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean."— Godfrey Harold Hardy

There is a delightful little story by Karl Capek entitled, "The Death of Archimedes." It begins :

The story of Archimedes did not happen quite the way that has been written; it is true that he was killed when the Romans conquered Syracuse, but it is not correct that a Roman soldier burst into his house to plunder it and that Archimedes, absorbed in drawing a geometrical figure, growled at him crossly: "Don't spoil my circles!" For one thing, Archimedes was not an absent-minded professor who did not know what was going on around him....for another thing, the Roman soldier was not a drunken plunderer but the educated and ambitious staff centurion Lucius, who knew to whom he had the honor of speaking....

The Roman officer tries to convince Archimedes to put his skills to use in serving the Roman war machine, but Archimedes is uninterested. Finally, the Roman, in growing irritation, blurts out :

"Archimedes, aren't you attracted by the idea of winning world mastery with us? --- Why don't you answer?"
"I beg your pardon," grunted Archimedes, bending over his tablets. "What did you say?"
"That a man like you might win world mastery!"
"Hmm, world mastery," said Archimedes in a bored tone. "You mustn't be offended, but I've something more important here. Something more lasting, you know. Something which will really endure."
"What's that?"
"Mind ! Don't spoil my circles ! It's the method of calculating the area of a segment of a circle."

Later it was reported that the learned Archimedes had lost his life through an accident.
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[westwall]

"Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. “Immortality” may be a silly word, but probably a mathematician has the best chance of whatever it may mean."— Godfrey Harold Hardy

There is a delightful little story by Karl Capek entitled, "The Death of Archimedes." It begins :

The story of Archimedes did not happen quite the way that has been written; it is true that he was killed when the Romans conquered Syracuse, but it is not correct that a Roman soldier burst into his house to plunder it and that Archimedes, absorbed in drawing a geometrical figure, growled at him crossly: "Don't spoil my circles!" For one thing, Archimedes was not an absent-minded professor who did not know what was going on around him....for another thing, the Roman soldier was not a drunken plunderer but the educated and ambitious staff centurion Lucius, who knew to whom he had the honor of speaking....

The Roman officer tries to convince Archimedes to put his skills to use in serving the Roman war machine, but Archimedes is uninterested. Finally, the Roman, in growing irritation, blurts out :

"Archimedes, aren't you attracted by the idea of winning world mastery with us? --- Why don't you answer?"
"I beg your pardon," grunted Archimedes, bending over his tablets. "What did you say?"
"That a man like you might win world mastery!"
"Hmm, world mastery," said Archimedes in a bored tone. "You mustn't be offended, but I've something more important here. Something more lasting, you know. Something which will really endure."
"What's that?"
"Mind ! Don't spoil my circles ! It's the method of calculating the area of a segment of a circle."

Later it was reported that the learned Archimedes had lost his life through an accident.
.

I'll go with Plutarchs biography of Marcellus.......After all he was there shortly after...


"But nothing afflicted Marcellus so much as the death of Archimedes, who was then, as fate would have it, intent upon working out some problem by a diagram, and having fixed his mind alike and his eyes upon the subject of his speculation, he never noticed the incursion of the Romans, nor that the city was taken. In this transport of study and contemplation, a soldier, unexpectedly coming up to him, commanded him to follow to Marcellus; which he declining to do before he had worked out his problem to a demonstration, the soldier, enraged, drew his sword and ran him through. Others write that a Roman soldier, running upon him with a drawn sword, offered to kill him; and that Archimedes, looking back, earnestly besought him to hold his hand a little while, that he might not leave what he was then at work upon inconclusive and imperfect; but the soldier, nothing moved by his entreaty, instantly killed him. Others again relate that, as Archimedes was carrying to Marcellus mathematical instruments, dials, spheres, and angles, by which the magnitude of the sun might be measured to the sight, some soldiers seeing him, and thinking that he carried gold in a vessel, slew him. Certain it is that his death was very afflicting to Marcellus; and that Marcellus ever after regarded him that killed him as a murderer; and that he sought for his kindred and honored them with signal favors."



The death of Archimedes