Tuesday, October 04, 2011

Pilipinas Forum 14: Math, Infinity and Limit

Here is another mind-twisting thread in pilipinasforum yahoogroups made in early December 2001. There was no political discourse here. And that's my point in posting these exchanges in this blog: We need little or no politics (and government) if we wish to expand our individual imagination and knowledge of math, logic and their applications. So, enjoy the 10-pages long exchanges on the subject.

Related articles in this blog are (1) An ever-expanding universe, June 25, 2011. It's about a lecture by an American cosmologist in UP about the "Accelerated History of the Universe". Also, (2) Pilipinas Forum 12: Origin of Zero, Nothingness, Big Bang..., September 25, 2011

I was able to meet an old friend and a new friend - both Math professors in UP. My old friend, Fidel Nemenzo got his PhD in Sophia U. in Japan, and his specialization is "number theory", reputedly the purest of all math subjects.

Fidel's friend, Ernie, is really fantastic, academically i.e.: Monbusho scholar (like Fidel), finished MS and PhD Math in 5 years with "A" grades equivalent. At Sohia U, he was among the "wonder kids" by solving difficult math problems whle his Japanese classmates could only shake their heads. Undergrad, he finished BS Math in UP - Summa cum laude, in 3 years! High school, he came from Phil. Science HS; elem. he came from an obscure elem. school in an obscure town in Iloilo province. They're so poor that their house has no floor, that PSHS has to pay for his bus to Iloilo, his plane fare to manila. Yet this guy has no self-pity, nor hatred of the rich. He doesn't even see himself joining the corporate world, just a plain academic for the rest of his career. At 27, he's among the youngest professors in UP Math dept.

Fidel is moving to his friend's house, that of Mark Encarnacion, who took a temporary leave from UP for some stints in the US. Mark is another interesting kid. He got a PhD in "Computer Algebra" (!) from an Austrian university. The last time I talked to Mark in UP, about 4 years ago, he said he's the "only person in this country" who has that degree. Mark's father is UPSE's long-time dean, our dean that time, Dr. Jose Encarnacion. The Dean passed away about 3 years ago; his major works are often published in foreign academic journals, enough to be nominated for a possible Nobel prize in Economics.

OK, why am I sharing this? To show that there are such wonderful people in this country but our media and political and business leaders don't seem to notice. Every year we know who are the topnotchers in the bar exams, who are the valedictorians and salutatorians of the PMA, courtesy of the front page stories in our broadsheets and tabloids. But never has it occurred as far as I can remember, that Phil. broadsheets have given front page treatment to returning Filipino scientist or engineer or mathematician with distinguished PhD degree or astonishing research abroad. Which reflects the values and biases of Philippine media: future Filipino politicians and generals are hoorraayy! Future Filipino Einsteins are nobodies.

Of course I hope I'm wrong in this impression.

-- Nonoy Oplas

Inspired by Nonoy's post a few days back on our very own mathematicians in PilipinasForum, I decided to pick up a book, for leisurely reading, entitled, "A Tour of the Calculus" by David Berlinski. I wanted to be reminded as to why I fell in love with Math at one time in my life...

Math & life... the math of life...

From the book, "Some things were Greek to the Greeks. In the fifth century B.C., Zeno the Eleatic argued that a man could never cross a room to bump his nose into the wall...in order to reach the wall he would have first to cross half the room and then half the remaining distance again, and then half the distance that yet remains...this process...can always be continued and can never be ended..."

In the elevator, I always bump into the chief of another investment bank whose offices are in the same building as ours. He once asked me, "Gina, when do you think we can see the light at the end of the tunnel? You know, that one which is not from another train?" I answered, "Tomorrow!". Everytime I see him, we would look and smile at each other and we would, in perfect duet (seemingly practiced), say, "Tomorrow!"

In another instance, an analyst queried me, "Have we reached the bottom yet? How low can the market get?" Armed with calculus, I said, "We can never reach the bottom. The bottom will be forever unknown."

"...The plain fact is that we are capable of compressing those infinite steps..." and actually bump our noses into the wall. The truth is, for some businesses in the country today, tomorrow may not come. And I, a Trekkie, will uncharacteristically pass, on this voyage of discovery to the bottom of the Philippine markets. So much for infinity and continuity and the harsh concept of the limit.*

In case you have not noticed yet :-), this post has nothing to do with serious Math. I am just having fun with the concepts. May I end with this:

"The integral enables you
To do what you need not do. [ :-) ]
The theorem that will make this plain
Is one designed to spare you pain..."

- Mathematicians' Doggerel (quoted from the same book; the smiley, mine)

-- Gina L.

Thanks, Gina -- for your post on time/math. Time has always been part of all mathematical if not philosophical equation. I was just going through Zeno of Elea's paradoxes when I came across your post. Here's his paradoxes including that which you mentioned on someone trying to get past half of a room and cannot reach the other end:

Zeno's paradoxes -- Do you have any answers?

In about 445 B.C., the Greek philosopher Zeno of Elea offered several arguments that led to conclusions contradicting what we all know from our physical experience. The paradoxes had a dramatic impact upon the later development of mathematics, science, and philosophy.

His most familiar paradox, the paradox of Achilles and the Tortoise, involves the fast-running Achilles and the slow-crawling tortoise. The tortoise has a head start. If Achilles hopes to overtake the tortoise, he must at least run to where the tortoise is, but by the time he arrives there, the tortoise has crawled to a new place. So, Achilles must run to the new place; but of course the tortoise isn't there, having crawled on to yet another place, and so on forever. Therefore, Zeno argues, good reasoning shows that fast runners never can catch slow ones. So much the worse for good reasoning. Notice that Zeno's reasoning rests on the assumption that time is continuous, that is, that time can be divided into infinitely many parts. We assume this continuity of time when we assume that a basektball dropped onto the court will bounce an infinite number of times before stopping.

In his Progressive Dichotomy Paradox, Zeno argued that a runner will never reach the goal line because he first must have time to reach the halfway point to the goal, but after arriving there he will need time to get to the ¾ point, then the 7/8 point, and so forth. If the distance to the goal is, say, 1 meter, then the runner must cover a distance of 1/2 + ¼ + 1/8 + ... meters. Zeno believed this sum is infinite and concluded that the runner will never have the infinite time it takes to reach this infinitely distant goal. Because at any time there is always more time needed, motion can never be completed. Worse yet, argued Zeno in his Regressive Dichotomy Paradox, the runner can't even take a first step. Any first step may be divided into a first half and a second half. Before taking a full step, the runner must have time to take a 1/2 step, but before that a 1/4 step, and so forth. The runner will need an infinite amount of time just to take a first step, and so will never get going.

(Does time really move? Please read on!)

Zeno's Arrow Paradox takes a different approach to challenging the coherence of the concepts of time and motion. Consider one instant of an arrow's flight. For that entire instant the arrow occupies a region of space equal to its total length, so at that instant the arrow isn't moving, he reasoned. If at every instant the arrow isn't moving, then the arrow can't move.

(The arrow paradox implies time as a roll of film where bits of oneself is imbedded as one moves thru time. Time travel if possible will allow one to return to his past and relive those moments imbedded in time. Really, now!

And here's Zeno's famous infinite half-the-time across anything paradox):

Yet another paradox created by Zeno attacks the notion that there are shorter and shorter times. Consider a duration of one second. It can be divided into two non-overlapping parts. They, in turn, can be divided, and so on. At the end of this infinite division we reach the elements. Here there is a problem. If these elements have zero duration, then adding an infinity of zeros yields a zero sum, and the total duration is zero seconds, which is absurd. Alternatively, if that infinite division produced elements having a finite duration, then adding an infinite number of these together will produce an infinite duration, which is also absurd. So, a second lasts either for no time at all or else for an infinite amount of time.

These paradoxes by Zeno can be considered to challenge the notion that time (and space) is continuous. Some of his other paradoxes, not discussed here, challenge the presumption that time might be discrete or discontinuous, with instants being like atoms of time.

Zeno's paradoxical arguments are valid, given his assumptions about space, time, motion and mathematics; and they reveal the underlying incoherence in ancient Greek thought, an incoherence that was not adequately resolved for 2,300 years. The way out of Zeno's paradoxes requires revising the concepts of duration, distance, instantaneous speed, and sum of a series. The relevant revisions were made by Leibniz, Newton, Cauchy, Weierstrass, Dedekind, Cantor, Einstein, and Lebesque over two centuries. The notion of infinite sums of numbers had to be revised so that an infinite series of numbers that decrease sufficiently rapidly can have a finite sum.

Although 1/2 + 1/3 + ¼ +... is infinite, the more rapidly decreasing series 1/2 + 1/4 + 1/8 +... is 1. The other key idea was to appreciate that durations and distances must be topologically like an interval of the linear continuum, a dense ordering of uncountably many points. Although individual points of the continuum have zero measure (that is, zero 'total length'), the modern notion of measure on the linear continuum does not allow the measure of a segment (continuous region) to be the sum of the measures of its individual points, as Zeno had assumed in his argument against plurality. With these contemporary concepts, we can now make sense of Achilles covering an infinite number of distances in a finite time while running at a normal, finite speed. The new concepts restore the coherence of mathematics and science with our experience of space and time, and they are behind today's
declaration that Zeno's arguments are based on naive and false assumptions.

- Lino Aldana

Dear Gina,

Your philosophical musings on the mathematics are amusing. Although math is considered as the most exact of all sciences, it is quite fascinating to read the concepts extending to the realms of other disciplines (and hopefully 'in the realm of the senses'; j/k). These concepts are pregnant with meanings and may be, by themselves, can be considered as pure poetry. Take the case of "asymptote" which can be a methapor for the fate of star-crossed lovers or the "east-west" idea of Rudyard Kipling. And "points of inflection" can be interpreted as the 'crossroads' in life and 'roads that are less travelled'.

And then, there's the concept of space-time that has spawned a number of Marvel comic issues, including crossover editions. 'Continuity' has also traveled into countless hyperrealities in the movies and literature.

-- Glenn de Guzman

Let me see if I still understand my math econ. Assume you are a guy, and there are 3 ladies that can steal your heart, named Aida, Lorna and Fe (iyong kanta ni Marco Sison yon di ba?).

You may have 4 behavioral axioms here.

1. Completeness: Aida is "weakly preferred" (>/) to Lorna and Lorna is "weakly preferred" to Aida, such that you may be indifferent whom to court. Notation: Aida ~ Lorna, or Lorna ~ Fe, or Aida ~ Fe
Sa tagalog: wala kang itulak-kabigin sa kanila, panalo ka sino man naging nobya or asawa mo.

2. Transitivity: If Aida is "better" than Lorna ("better" here is relative depending on your values, say better in intelligence, in physical beauty, in money, etc.), and Lorna is "better" than Fe, then Aida is better than Fe.
Notation: Aida > Lorna, Lorna > Fe, then Aida > Fe.

3. Continuity: Aida is "better" than Fe; however, Aida married another guy. Here comes Lorna you don't know before, but her qualities approaches that of Aida, you conclude Aida is "better" than Fe.
Notation: Lorna --> Aida, Lorna > Fe.

4. Nonsatiation: More is better than less. If you're Erap or Mr. Mumbo no. 5 (who has Erica, Ria, Sandra, Mary, etc.). Sa tagalog: mas mabuti kung 3 sila kesa 1 or 2 lang. (Ooppsss, feminists out there, don't hit me. No sexist joke, just plain examples, he he he).

Meron pa dapat pang-5th behavioral axiom, "strict convexity" pero di ko na ito maintindihan. katuwaan lang, mga katoto.

At sa mga gustong magbasa ng math & mathematicians. Ang title ng article ay "Mathematicians in our Midst" at sinulat ni Marivic.

Medyo kakatuwa ang istorya nitong 5 mathematicians, taga-UP Math Dept. yata lahat, below 40, with PhDs. Si Fidel, numbert theorist; si Noli, Fourier analyst; si Marian, spectral theorist; si Jimmy complexity theorist; at si Joey naman ay group theorist. Lahat itong mga specialization nila mga special branches ng math.

Dagdag pala ni Fidel, "I think Math is not as exact as most people think it is. A look at the history of mathematics would show that the notion of truth in mathematics is also context-bound. And to make matters "worse", one of the most important mathematical theorems of 20th century showed that mathematics is not free from contradictions, and that no finite set of axioms can sufficiently serve as foundations for mathematics. But I think this is a good thing, for reasons I'll tell you some other time. Mali si Bertrand Russell! Mathematics is not like a game of chess!"

As usual, I shortened this article for easier reading. If you want to read the whole article, go to http://www.legmanila.com//feature/article/943.asp

Here goes...

1. Fidel... wonders about prime numbers (numbers that are divisible only by itself and the number 1). In particular, he wants to know: how do prime numbers behave? How are they distributed in the number line? Fidel is endlessly fascinated with these questions. He is a number theorist, the only one in the country.

...What's a number theorist? someone who studies the behavior or properties of numbers such as the integers (numbers as we know them, meaning 0, 1, 2, 3, ... and their negatives) and rational numbers (fractions...)

Number theory is considered one of the purest branches of mathematics and many of its areas of concern, up until the last few decades, had no direct and practical applications. But this is not an issue with Fidel. In fact, during his "purist" phase, to ask "but for what purpose?" was to risk being deemed a philistine in his eyes... "An elegantly proven theorem excites me in the same way that a good poem does," he matter-of-factly states.

But he also says this ivory approach in number theory no longer holds -- recent developments in information technology require number theory. "The vast amounts of information being transmitted over computer lines and wireless channels are encoded to make transmission safe and as free of errors as possible," Fidel claims, "and this requires a lot of number theory. Your ATM pin is safe from prying eyes because of math!"

2. Noli says "There is a lot of work for mathematicians in the real world," he asserts, "and doing mathematical modeling for many problems in the sciences like chemistry, biology, physics is enough to keep us busy."

Noli himself is a mathematician trained in Fourier analysis which, for the legions of the uninitiated, is the study of the mathematical aspects of signal analysis (radio waves) and image processing... He is currently doing research that may prove useful in the field of digital communications, which covers a wide range of applications from television to satellites to medical imaging. Noli is one of about five Fourier analysts in the Philippines.

Noli, however, has a mission to link mathematics with its applications. "To sell mathematics," he unabashedly states. He cites that in many developed countries, mathematical research flourishes because those in the airline, computer, and telecommunications industries, to name a few, hire
mathematicians to solve the mathematical problems which arise in their particular industry...

3. Marian... particular area is spectral theory, a branch of mathematics which can be used to model various phenomena, such as the movement of liquid. Other applications, according to Marian, can be found in mathematical physics and quantum mechanics.

...Marian admits that it's a constant struggle having to balance the discipline required of a mathematician, the long hours one has to put in to do research, and maintaining a vibrant social life. "I love to socialize, go out with friends, play golf and tennis, and just be with my daughter. I may be good in math, but talent isn't enough," she claims, "you need to be engrossed."

4. Jimmy... particular area of math is complexity theory, which is the study of the various levels of difficulty (or complexity) of algorithmic computation. An algorithm, by the way, is the step-by-step procedure for solving mathematical problems. Put simply, complexity theory provides the theoretical foundation of computer science. And Jimmy is the only trained mathematician in this area in the Philippines.

5. Joey, the group theorist. According to Joey, group theory is the kind of mathematics that studies symmetries, shapes, designs, patterns. "It's the math that explains Rubic's cube or classifies wallpaper," he says, "not to mention the symmetry found in the movement of electrons, molecules, or the structures of crystals." There are only a handful of group theorists in the country.

According to Joey, his romance with group theory began upon reading about the life of Evariste Galois who lived and died during the turbulent years after the French Revolution. He was jailed several times for his revolutionary beliefs and died before the age of 20 in a gun duel (over a woman, or so the legend goes). The night before he died, Galois wrote 60 pages of mathematics which laid down the foundations of group theory.

When asked to define mathematics, Joey almost rhapsodizes, "It's the study of numbers, shapes, patterns, abstraction, order, disorder, and logical processes... It's a way of thinking, a very human activity that involves creation and discovery."

Fidel, Noli, Marian, Jimmy, and Joey. Mathematicians no older than 40, with doctorates in their respective fields, a young discipline in the Philippines by international standards. In many ways, they can be seen as pioneers as they are among the first in the Philippines trained in their fields of specialization.

To them, the study of shapes, patterns, and ideas is math -- not the cut-and-dried collection of formulas and techniques most of us dread in the classroom. In their class, the shape of a doughnut and a coffee cup is a topic for mathematical inquiry... In their hands, prosaic formulas come alive as they are revealed to be part of some deep and wonderful mathematical concepts. Through their eyes, they show how math can be a meeting place between science and the humanities, even an object of beauty.

-- Nonoy

Dear Glenn, Lino and friends,

Glenn, you said: mathematical "concepts are pregnant with meanings and may be, by themselves...considered as pure poetry."

Math is poetry written in a language, which grammar and vocabulary is not quite easy to learn and understand. Many get naturally attracted to it, not just because it poses a great intellectual challenge but more importantly, it does present an opportunity to understand the world (and possibly, alternate and outer worlds) in a radically different language -- a language characterized by precision and clarity. It is ironic, though, that something so exact (or near-exact as Fidel might say) and logical be so obscure and esoteric to most of us. I guess, once we pass the regulation rite of passage, we find that math explains itself to us . . . and the most mysterious of intellectual disciplines reveals itself to us.

Somebody once said that this process is akin to a voyage which "begins with the unkown and ends with the unknowable". Lino, I am happy to note that you seem to be on this same cruise (for why else would you be going through Zeno of Elea's paradoxes?).

-- Gina L.

Gina (and others):

A most amusing post! Since you are one of those rare and fortunate people who appreciates math, I have a book suggestion for you: The Moment of Proof: Mathematical Epiphanies, by Donald C. Benson (Oxford University Press, 1999). It's one of the most well-written accounts I know (complete with intuitive but rigorous proofs) of things like Zeno's Paradox, the St. Petersburg Paradox, Fermat's Last and Fermat's Little Theorems (no enclosed proof of Fermat's Last Theorem, though; that's several hundred pages long and is already attributed to Andrew Wiles of Princeton), The Seven Bridges of Konigsberg, and other fun little puzzlements. I hope there is a bookstore in Manila that carries it or can obtain it for you.

Benson (an emeritus prof at UC-Davis) has a concise explanation of the "flaw" in Zeno's paradox (the arrow paradox, but also the other ones as well). And that is that Zeno assumed that the sum of an infinite series with positive terms must have an infinite sum. Well some do, some don't. For example, taking the sum (9/10) + (9/100) + (9/1000)+ .... is just like continually adding a nine to the end of 0.999.... We all know (now that we use decimal numbers that Zeno didn't have) that 0.999... is an infinite decimal for the finite number, 1. So Zeno's arrow would cover a finite distance in a finite (not infinite) amount of time (1 second, if the above series represented time lapsed in covering dissections of the arrow's path).

Still, a pretty paradox that is! On math and poetry, then, I end with a verse cited in the above book:

"Ah, but my Computations, people say,
Reduced the year to better reckoning?-- Nay,
'Twas only striking from the Calendar
Unborn To-morrow, and dead Yesterday."

---from the Rubaiyat of Omar Khayyam

-- Butch A.

Gina, seems like you're into numbers and concepts where others haven't gone before. Here's about Prof Todd Ebert's (of Univ Calif Irvine Info & Computer Science Dept) colored hat's puzzle, a variation of very popular hat puzzles that math enthusiasts have been pouring their minds over. Thought this might of interest to you and other PFer math-geeks! Enjoy: :)L

The Colored Hats Puzzle

If you are unfamiliar with this puzzle please read the article "Why Mathematicians Now Care About Their Hat Color", which appeared in the Science Times section of the NY Times, April 10th, 2001. The statement and solution of the same puzzle (although stated as the "seven-prisoners puzzle") can be found in last quarter's ICS 151 puzzle corner .

The colored hats version is stated as follows. Each player on a team of seven is randomly and independently assigned a colored hat (either red or blue) to wear, and then tries to guess the color of his hat by viewing the hats of the other teammates. No communication is allowed except for a strategy session before the game begins. The team wins a prize if at least one player guesses and all the guesses are correct. The team loses if no one guesses or some player guesses incorrectly. What strategy should the team adopt to maximize their chance of winning? I give a solution for the case of seven players.

I developed this puzzle as a math grad. student at UCSB. It was intended to be the finite layperson version of a rather surprising mathematical result regarding random infinite binary sequences (i.e. sequences that can be formed by tossing a fair coin an infinite number of times). The result goes something like this:

Given a random infinite binary sequence A (e.g. A= 1001010101010000111...), there exists a computer that can guess an infinite number of bits of A without ever making an error, despite the fact that the computer has only a 50% chance of guessing correctly each time, and that it will guess an infinite number of times. Indeed, for each bit i, the computer may examine the bits to the left of bit i and to the right of bit i, and then either guess bit i, or pass. In either case, its memory is erased and it proceeds to the next bit, and this continues ad infinitum. So in essence, the computer is playing the colored hats game, but plays an infinite number of them, and wins every time! What allows for the existence of such a computer is a famous result in probability known as the Borel-Cantelli Lemma. If you would like to learn more about this result, please read the paper "On the Autoreducibility of Random Sequences" .

The Colored Hats Puzzle: Revised

Given your knowledge about how to solve the colored hats puzzle, now apply it to solve the following revised version. For the team to win, *all* the players must guess correctly, but now there are two rounds of guessing. In other words, those players who guess during round one leave the room. The remaining players must guess in round two. What strategy should the team adopt to maximize their chance of winning?

"Red Hat Story"

This puzzle comes from the diary of Walter Wesley Winters (1905-1973), who began recording puzzle folklore while working as a mechanical engineer for Curtis Wright Aircraft during WWII.

Three men (one of them blind) are taken into a room containing a table upon which five hats are placed. Three of the hats are white and two are red. The lights are turned out and each man selects and puts on a hat. The remaining hats are removed and the lights are turned back on. Each man is then asked to look at the other two and determine what color hat he has on.

The first man (sighted) says, "I do not know."
The second man (sighted) says, "I do not know."
The third man (blind) answers, "I have on a white hat."

Is he correct? If so, how did he deduce his answer?

Hope you enjoyed this -- ho ho ho!

-- Lino Aldana

The first man (sighted) said I do not know. Means either that he saw the sighted man and the blind man wearing both white hats or either wearing alternately white or red hats.

The second man (sighted) said I do not know. Same with first.

The blind man said, "I'm wearing a white hat."

First conclusion from the 1st statement: They are not both wearing red hats. It is either he (blind) or the second sighted man wears a red or white hat.

Second conclusion from the 2nd statement: The blind man is wearing a white hat because he can "see" that in both 1st and 2nd statements the sighted men conclude that they don't know what they are wearing. If blind man wears a red hat it follows that if one of the sighted men wears also a red hat
then one of them must conclude that he wears a white hat. But no one concludes, so the blind man concludes he must be wearing a white hat.

-- Joey Sescon

Pareng Joey,

3 white hats, 2 red hats.

Parang kulang ang permutation mo pareng joey to arrive at the answer. Tingin ko lang kasi, doon sa first part ng first conclusion mo, tama yung they are not both wearing red hats, otherwise, the first sighted man would conclude he is wearing a white hat. Pero, if the 2nd sighted man and the blind man are wearing a red and white hat, pwedeng its the blind man who is wearing red and the sighted man is wearing white.

kaya pwedeng mali si blind man when he said he is wearing a white hat.

Kalas-kalasin natin.

Sighted 1 (S1)
Sighted 2 (S2)
Blind Man (B)

First Sighted Man

S1 sees S2 is red and B is red: S1 concludes I'm white
S1 sees S2 is red and B is white: I don't know
S1 sees S2 is white and B is red: I don't know
S1 sees S2 is white and B is white: I don't know

Second Sighted Man

S2 sees S1 is red and B is red: Concludes, I am white
S2 sees S1 is red and B is white: I don't know
S2 sees S1 is white and B is red: I don't know
S2 sees S1 is white and B is white: I don't know

In both S1 and S2, only in a situation that they see two red hats already worn can they
conclusively say they are wearing white. Hey buhey.



S1 concludes he is white,
and S2 concludes he is white,
therefore Either S1 or S2
is lying, and B can't
make a conclusion


S1 says he is red
and S2 says he is red
then B will think that
the only way for both
of them to say they are red
would be by way of a
non-conclusive statement.
So again he can't make a conclusion

But if S1 says i don't know, and S2 says i don't know, could mean:

S1 saw S2 red and B white
S1 saw S2 white and B red
S1 saw S2 white and B white

Same goes for S2

Therefore, for B to say he is wearing white is not foolproof. However, it is a better answer than to say he is wearing red becasue from above, there is two out of three situations that he may be white. and only one out of 3 situations that he may be red.

Di ba po?

Ngay-on, komo mathematical daw itong problem na ire, ay subukan niyo po uling basahin ulit mula sa taas, right after the line :kalas-kalasin natin….

O, tapos nyo nang basahin? o hindi ga ay poem ang inyong binasa at hindi mathematical equation?

Conclusion: Tama si Gina L., ang mathematics nga ay pwedeng poetry.

Another conclusion: If the blindman is white, he is either Anglo-Saxon or Albino. If he is red, he must be an American Indian. If he is black, he is Stevie Wonder.

Joke lang po. Please don't play me the race card. TY

-- Ozone Azanza

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