Quoteworthy


...quaecumque sunt vera, quaecumque pudica, quaecumque justa, quaecumque sancta, quaecumque amabilia, quaecumque bonae famae, si qua virtus, si qua laus disciplinae, haec cogitate.
-- Phil. 4:8
Showing posts with label mathematics. Show all posts
Showing posts with label mathematics. Show all posts

The Colour of the Bear

One of my favourite mathematical riddles:
An explorer travelled 5 km South from his camp, then 5 km East, then 5 km North. He found himself back in his camp and saw a bear rummaging through his food supplies. What is the colour of the bear?
Don't cheat! Answer is below -- you need to highlight it to see.
The explorer's path forms an equilateral triangle with all the angles equal to 90°. Obviously we cannot assume a Euclidean plane here, but curved plane like our very Earth. The only possible situation for the stipulated directions and path is when the camp is exactly at the geographical North Pole. So the bear was a polar bear. 
It was white.
The neat thing about this riddle is that it does not seem mathematical at all, what's with the question seemingly not related to the clues given. The additional deduction (North Pole --> polar bear) is witty, to say the least.

Consciousness and (again) dimensions

Reading this comment, I recall a very good description of pure consciousness that is independent of any platform.
"Look yonder," said my Guide, "in Flatland thou hast lived; of Lineland thou hast received a vision; thou hast soared with me to the heights of Spaceland; now, in order to complete the range of thy experience, I conduct thee downward to the lowest depth of existence, even to the realm of Pointland, the Abyss of No dimensions.
"Behold yon miserable creature. That Point is a Being like ourselves, but confined to the on-dimensional Gulf. He is himself his own World, his own Universe; of any other than himself he can form no conception; he knows not Length, nor Breadth, nor Height, for he has had no experience of them; he has no cognizance even of the number Two; nor has he a thought of Plurality; for he is himself his One and All, being really Nothing. Yet mark his perfect self-contentment, and hence learn his lesson, that to be self-contented is to be vile and ignorant, and that to aspire is better than to be blindly and impotently happy. Now listen."
He ceased; and there arose from the little buzzing creature a tiny, low, monotonous, but distinct tinkling, as from one of your Spaceland phonographs, from which I caught these words, "Infinite beatitude of existence! It is; and there is nothing else beside It." "What," said I, "does the puny creature mean by 'it'?" "He means himself," said the Sphere: "have you not noticed before now, that babies and babyish people who cannot distinguish themselves from the world, speak of themselves in the Third Person? But hush!"
"It fills all Space," continued the little soliloquizing Creature, "and what It fills, It is. What It thinks, that It utters; and what It utters, that It hears; and It itself is Thinker, Utterer, Hearer, Thought, Word, Audition; it is the One, and yet the All in All. Ah, the happiness, ah, the happiness of Being!"
"Can you not startle the little thing out of its complacency?" said I. "Tell it what it really is, as you told me; reveal to it the narrow limitations of Pointland, and lead it up to something higher." "That is no easy task," said my Master; "try you."
Hereon, raising by voice to the uttermost, I addressed the Point as follows:
"Silence, silence, contemptible Creature. You call yourself the All in All, but you are the Nothing: your so-called Universe is a mere speck in a Line, and a Line is a mere shadow as compared with —" "Hush, hush, you have said enough," interrupted the Sphere, "now listen, and mark the effect of your harangue on the King of Pointland."
The lustre of the Monarch, who beamed more brightly than ever upon hearing my words, shewed clearly that he retained his complacency; and I had hardly ceased when he took up his strain again. "Ah, the joy, ah, the joy of Thought! What can It not achieve by thinking! Its own Thought coming to Itself, suggestive of its disparagement, thereby to enhance Its happiness! Sweet rebellion stirred up to result in triumph! Ah, the divine creative power of the All in One! Ah, the joy, the joy of Being!"
"You see," said my Teacher, "how little your words have done. So far as the Monarch understand them at all, he accepts them as his own — for he cannot conceive of any other except himself — and plumes himself upon the variety of 'Its Thought' as an instance of creative Power. Let us leave this God of Pointland to the ignorant fruition of his omnipresence and omniscience: nothing that you or I can do can rescue him from his self-satisfaction."
-- Edwin A. Abbott, Flatland: A Romance of Many Dimensions
Flatland: A Romance of Many Dimensions by Edwin A. Abbott was published back in 1884, so it is available at public domain because the property rights have expired. I mentioned it a few times, here and here, when discussing about dimensions.
This was written because Abbott wanted to introduce the concept of higher dimensions, more than three, but of course we as the denizens of Spaceland (3-D world) find it difficult to imagine. So what he did was to go down one dimension and took the point of view from a being in Flatland (2-D world), a Square. Later in the story, Square is visited by Lord Sphere, a 3-D being from Spaceland. Of course, at first he is skeptical about Sphere and dismissed him as simply a Circle, but in the end he realises that his world is simply a flat plane. To make his point, Sphere actually brings Square to Pointland, which is 2 dimensions less, that is, no dimension. Sphere then asked Square to convince the God of Pointland that the world is not zero dimension. So that's the part in the excerpt above.
And why are higher dimensions important, you ask? Perhaps Abbott already foresaw a future research in Physics. You would have heard all the buzz of superstring theory by now, what is it all about? It is a promising Theory of Everything, but there's a catch: the equations only work out if we have 10 dimensions of space. So we imagine that our world may actually have extra dimensions, albeit tucked inside the fabric of space-time, curling and intertwining upon them themselves. Physicist Brian Greene offers the analogy of the power cables to make sense of these extra dimensions. Cables, from afar, look like they only have one dimension, i.e. length. But if you go down the scale as an ant, you would notice that the cables have thickness; an existing dimension, but often too small to observe. Similarly, our world may have 7 extra dimensions that are too ultra-microscopic to observe. This is why the concept of higher dimensions is very important.
This is also one of the purposes of the Large Hadron Collider: to detect the extra dimensions. When particles collide in certain manners, some of the energy may be ejected to the hidden dimensions. We can measure the energy of the particles before collision and compare it to the energy after. If there is energy loss, we may be able to conclude the existence of the extra dimensions.
I highly recommend watching the TEDTalk by Brian Greene on superstring theory (2005), embedded below:




The number system is like human life

"The foundation of mathematics is numbers. If anyone asked me what makes me truly happy, I would say: numbers. Snow and ice and numbers. And do you know why?"
He splits the claws with a nutcracker and pulls out the meat with curved tweezers.
"Because the number system is like human life. First you have the natural numbers. The ones that are whole and positive. The numbers of the small child. But human consciousness expands. The child discovers longing, and do you know what the mathematical expression is for longing?"
He adds cream and some drops of orange juice to the soup.
"The negative numbers. The formalization of the feeling that you are missing something. And human consciousness expands and grows even more, and the child discovers the in-between spaces. Between stones, between pieces of moss on the stones, between people. And between numbers. And do you know what that leads to? It leads to fractions. Whole numbers plus fractions produce the rational numbers. And human consciousness doesn't stop there. It wants to go beyond reason. It adds an operation as absurd as the extraction of roots. And produces irrational numbers."
He warms French bread in the oven and fills the pepper mill.
"It's a form of madness. Because the irrational numbers are infinite. They can't be written down. They force human consciousness out beyond limits. And by adding irrational numbers to rational numbers, you get real numbers."
I've stepped into the middle of the room to have more space. It's rare that you have a chance to explain yourself to a fellow human being. Usually you have to fight for the floor. And this is important to me.
"It doesn't stop. It never stops. Because now, on the spot, we expand real numbers with the imaginary ones, square roots of negative numbers. These are numbers we can't picture, numbers that normal human consciousness cannot comprehend. And when we add the imaginary numbers to the real numbers, we have the complex number system. The first number system in which it's possible to explain satisfactorily the crystal formation of ice. It's like a vast, open landscape. The horizons. You head towards them and they keep receding. That is Greenland, and that's what I can't be without! That's why i don't want to be locked up."
I wind up standing in front of him.
"Smilla," he says, "Can I kiss you?"
-- Peter Høeg, Miss Smilla's Feeling for Snow, pp. 101-102
After hearing that kind of that beautiful metaphor, I think I want to kiss her also. The excerpt is my favourite among others in the TOK textbook. It's a very good example of an allegory, an extended metaphor.
Why post this now? Because I just finished the book and the movie. The former is highly recommended, the latter not at all.
Despite that, this is the scene from the movie, although it is significantly different from the book:



The Curious Case of Infinity

I remember in high school when the teacher didn't allow us to write '1/0=...' . I protested. Isn't the answer infinity? No, there is no answer, you cannot even put the equal sign there, because it is undefined.
OK, that's an exaggerated version, but looking back in retrospect, it is now easy to see why. Calculus provides the answer. '1/0' is undefined, but the limit of 1/x as x goes to 0 is indeed, infinity. Conversely, the limit of 1/x as x goes to infinity is zero. I felt cheated, but it is brilliant cheating nonetheless.
What is this 'infinity ' anyway? Well, the younger and less wise me thought that it was intuitive that if some number is divided endlessly, in the end it must be zero, and conversely so. That magic denominator is infinity.
As things go by, it is clearer (or less so) that zero and infinity are problems.
Why zero you ask? "I understand that infinity is pushing the limit of human mind, but why zero?" Ah, if you consider that the earliest numeral system did not include zero, isn't that proof that zero is so elusive a concept? Let's see. The Greeks even rejected the idea of nothingness and adopt 'zero' thanks to the adoption of Arabic numeral system. The Arabs got it from the Indians. So the Arabs got nothing from the Indians (If you are not laughing, then you miss the joke, sad case). Well, enough of that, 'zero' is another story. For now let's consider infinity.
Let's do a warm-up: what's infinity minus infinity?
If you answered 'zero' with a great deal of suspicion, yes, you are correct, the answer is not that straightforward. Let me try to retell Dave Hilbert's Paradox of the Grand Hotel without sounding boring:

Suppose that there is this famous Grand Hotel. Why is it famous? Because it has infinite number of rooms, aha! On weekend you want to see this hotel for yourself. You go to the receptionist to check in but unfortunately the rooms are all full. The manager came out and tell you not to worry. He told you he would move the guest in Room 1 to Room 2, the guest in Room 2 to Room 3, and so forth. You got to stay in Room 1.
Satisfied with the hotel's excellent service, the following weekend you brought an infinite number of friends to the hotel. Again, the hotel was full. But the manager was unfazed. He moved the guest in Room 1 to Room 2, the guest in Room 2 to Room 4, the guest in Room 3 to Room 6 and so forth. Since there is infinite number of even-numbered rooms, all the guests are accounted for. The manager then put you and your retinue in the infinite number of odd-numbered rooms. Everyone is happy.
Room service is still excellent though. You know, they've got infinite number of employees.

This illustrates how normal mathematical operations don't usually work when infinity is involved. Have yourself a set of infinite integers. Take away the infinite set of odd numbers and you are left with the infinite set of even numbers. Infinity minus infinity can be infinity.
Very curious, isn't it?

Mathematical Tragedy

As lines, so love's oblique, may well
Themselves in every angle greet :
But ours, so truly parallel,
Though infinite, can never meet
-- Andrew Marvell in Definition of Love

Oh my, that's a clever one. I never see Euclidean fifth postulate in such romantic (albeit tragical) way.