maths4poets

Chaos ... What chaos?

2006-04-06T21:43:00.000+02:00

"The flapping of a single butterfly's wing today produces a tiny change in the state of the atmosphere. Over a period of time, what the atmosphere actually does diverges from what it would have done. So, in a month's time, a tornado that would have devastated the Indonesian coast doesn't happen. Or maybe one that wasn't going to happen, does."
(Ian Stewart, Does God Play Dice? The Mathematics of Chaos, pg. 141)

God of Chaos

2006-04-04T22:05:00.000+02:00

A Fractal by Ken Keller

15cents 4 your thoughts

2006-02-12T22:19:00.000+01:00

15cents 4 yours thought

After his death in 1955, Einstein became the epitomy of intelligence in general, not just of scientific genius. His image appeared on postage stamps and magazine covers. He became the subject of poetry, sculpture and painting. "Einstein advertisements" implied that buyers were smart to buy the product, or likely to become smart if they did. Conversely, the simplicity and ease of use of a product was endorsed by the assurance that you did not need to be an Einstein to use it. The most elaborate advertising campaign that exploited Einstein's image was the 1979 Pennaco Hosiery catalogue, intriguingly entitled "The Theory of Relativity — Fall-Winter 79". Women were exhorted to "Believe in the theory of relativity". "Fashion relates beautifully — making each part relative to the next — and thus creating fantastic total outfits. Thank you, Mr. Einstein, for the theory... and Happy 100th Birthday!"

A tour de force

2006-01-16T21:00:00.000+01:00

... do you recognize this formula?

Fomenko pictures

2006-01-11T22:48:00.000+01:00

"I do not really think of myself as an artist. I am a mathematician. To me, my drawings are photographs of some strange and interesting mathematical world" - Professor A.T. Fomenko of the Moscow State Unversity.

The history of the Fields Medal

2005-08-07T23:00:00.000+02:00

The history of the Fields Medal begins in the Committee of the International Congress set up by the University of Toronto in November of 1923, with the purpose of organizing the 1924 Congress to be held in Toronto.

Fields was its chairman, and his colleague J.L.Synge the secretary. Although Fields probably conceived of the medal at some earlier time, the first recorded mention of it is in the minutes of a meeting of that committee on February 24, 1931 where it is “resolved that the sum of $2,500 should be set apart for two medals to be awarded in connection with successive International Mathematical Congresses through an international committee appointed for such purpose initially by the executive of the International Mathematical Congress, but later by the International Mathematical Union”. The $2,500 was evidently the balance on hand after all expenses of the 1924 Congress had been met.

At the next meeting of the committee, in January 1932, Fields indicated that the idea of the medal had the support of the major mathematical societies of France, Germany, Italy, Switzerland and the United States, and he also outlined the principles behind the proposed medal. The genesis of the rule that it be awarded only to mathematicians no older than forty is evidently the statement that “… while it was in recognition of work already done, it was at the same time intended to be an encouragement for further achievement on the part of the recipients and a stimulus to renewed effort on the part of others”. Then he continued “In commenting on the work of the medalists it might be well to be conservative in one’s statements, to avoid invidious comparisons explicit or implied. The Committee might ease matters by saying that they had decided to make the awards along certain lines not alone because of the outstanding character of the achievement but also with a view to encouraging further development along these lines”. And mindful of the turmoil of ten years earlier (over the exclusion of the “Central Powers” from the 1924 Congress), he added “the medals should be of a character as purely international and impersonal as possible. There should not be attached to them in any way the name of any country, institution or person”.

Of course, in spite of Fields’s intentions, the medal became known as the Fields Medal when it was awarded for the first time in Oslo in 1936. It is interesting to note that, at the same meeting, it was decided that “the Chairman should see the Prime Minister of Canada to arrange if possible how permanence of capital and of interest of the fund might be assured”. Such an arrangement was apparently never made, and the monetary value of the Fields Prize is presently $15,000Can (about $9500US), hardly commensurate with its stature as the “Nobel Prize in Mathematics”.

It is not known why Nobel chose not to establish a prize in Mathematics, although there are several theories about the lack of one. Fields then proceeded with the planning of the award of the first medals, but fell ill in May of 1932 and died 3 months later. Just before his death, with Synge at his bedside, he made his will. It included an amount of $47,000 to be added to the funds for the medal. Synge carried Fields’s proposal to the Congress in Zürich in September of that year. It was accepted, and a committee consisting of G.D.Birkhoff, Carathéodory, E.Cartan, Severi and Takagi was formed to make the first awards at the Oslo Congress in 1936. They chose Lars Ahlfors of Finland and Jesse Douglas of the U.S.A.

Unfortunately, war again intervened, and the next ICM was not held until 1950, in Cambridge, Massachusetts, when Laurent Schwartz and Atle Selberg were selected as the Fields Medalists. A list of all Fields Medal winners (with a short description of their work) can be found here. An analysis by Michael Monastyrsky of the effect of Fields Medalists on 20th century mathematics and physics, delivered in a lecture at the Fields symposium “The legacy of John Charles Fields” held in Toronto in June, 2000, is available here. The medal itself Fields specified that the medals should “each contain at least 200 dollars worth of gold and be of a fair size, probably 7.5 centimetres in diameter. Because of their international character the language to be employed it would seem should be Latin or Greek”.
The medal does in fact meet these specifications (in 1933 dollars!). Its monetary value has at least on one occasion been of critical importance: in the turmoil at the end of World War II, Ahlfors became separated from his wife, and was allowed to leave Finland with only 10 crowns. He smuggled out his Fields Medal and pawned it, enabling him to reach his wife in Zürich. (He later retrieved it with the help of some Swiss friends). The medal, struck every four years in the Royal Canadian Mint, was designed by the Canadian sculptor R.Tait McKenzie. For the obverse, he chose a picture of Archimedes from a collection at Columbia University. The Latin inscription from the Roman poet Manilius surrounding his image may be translated “to pass beyond your understanding and make yourself master of the universe”. The phrase comes from Manilius’s Astronomica 4.392 from the first century A.D

Einstein and his blindfriend

2005-06-12T20:58:00.000+02:00

Albert Einstein (1879-1955)

This story shows how complex Einstein could be. Not long after his arrival in Princeton he was invited, by the wife of one of the professors of mathematics at Princeton, to be guest of honor at a tea.-Reluctantly, Einstein consented. After the tea had progressed for a time, the excited hostess, thrilled to have such an eminent guest of honor, fluttered out into the center of activity and with raised arms silenced the group. Bubbling out some words expressing her thrill and pleasure, she turned to Einstein and said: "I wonder, Dr. Einstein, if you would be so kind as to explain to my guests in a few words, just what is relativity theory ? "
Without any hesitation Einstein rose to his feet and told a story. He said he was reminded of a walk he one day had with his blind friend. The day was hot and he turned to the blind friend and said, "I wish I had a glass of milk."
"Glass," replied the blind friend, "I know what that is. But what do you mean by milk?"
"Why, milk is a white fluid," explained Einstein.
"Now fluid, I know what that is," said the blind man. "but what is white ? "
" Oh, white is the color of a swan's feathers."
" Feathers, now I know what they are, but what is a swan ? "
"A swan is a bird with a crooked neck."
" Neck, I know what that is, but what do you mean by crooked ? "
At this point Einstein said he lost his patience. He seized his blind friend's arm and pulled it straight. "There, now your arm is straight," he said. Then he bent the blind friend's arm at the elbow. "Now it is crooked."
"Ah," said the blind friend. "Now I know what milk is."
And Einstein, at the tea, sat down.

Fractals and poems (VIII)

2005-06-11T21:52:00.000+02:00

Dragon Festival

Math jokes (V)

2005-05-24T23:13:00.000+02:00

Q. "What do you get when you cross an elephant with a banana?
A. Elephant banana sine theta in a direction mutually perpendicular to the two as determined by the right hand rule."

Q. "What do you get if you cross an elephant with a mountain climber?"
A. You can't do that. A mountain climber is a scalar.

Q. "Why did the cat fall off the roof?"
A. Because he lost his mu. (mew=sound cats make, mu=coeff of friction)

Q. "What do you call a teapot of boiling water on top of mount everest?"
A. A HIGH-POT-IN-USE

Q. "Why is it that the more accuracy you demand from an interpolation function, the more expensive it becomes to compute?"
A. That's the Law of Spline Demand.

Q. "How many seconds are there in a year?"
A. "Twelve; January second, February second, March second, ..."

Q. "What's the contour integral around Western Europe?"
A. Zero, because all the Poles are in Eastern Europe!
Addendum: Actually, there ARE some Poles in Western Europe, but they are removable.
Note: this is not a polish joke, so please, no Poles be offended. It is actually a mathematical joke--they are meant to be poles apart--oops, sorry about that.

Do you owe me something ... Mr. Nobel?

2005-05-16T22:54:00.000+02:00

Six Nobel Prizes are awarded each year, one in each of the following categories: literature, physics, chemistry, peace, economics, and physiology & medicine. Notably absent from this list is an award for Mathematics. Let me check briefly some facts:

Nobel prizes were created by the will of Alfred Nobel, a notable Swedish chemist.
One of the most common -and unfounded- reasons as to why Nobel decided against a Nobel prize in math is that [a woman he proposed to/his wife/his mistress] [rejected him because of/cheated him with] a famous mathematician. Gosta Mittag-Leffler is often claimed to be the guilty party. There is no historical evidence to support the story.
Gosta Mittag-Leffler was an important mathematician in Sweden in the late 19th-early 20th century. He was the founder of the journal Acta Mathematica, played an important role in helping the career of Sonya Kovalevskaya, and was eventually head of the Stockholm Hogskola, the precursor to Stockholms Universitet. However, it seems highly unlikely that he would have been a leading candidate for an early Nobel Prize in mathematics, had there been one - there were guys like Poincare and Hilbert around, after all.
For one, Mr. Nobel was never married.
There are more credible reasons as to why there is no Nobel prize in math. Chiefly among them is simply the fact he didn't care much for mathematics, and that it was not considered a practical science from which humanity could benefit (a chief purpose for creating the Nobel Foundation).
Further, at the time there existed already a well known Scandinavian prize for mathematicians. If Nobel knew about this prize he may have felt less compelled to add a competing prize for mathematicians in his will.

In my opinion, Nobel, an inventor and industrialist, did not create a prize in mathematics simply because he was not particularly interested in mathematics or theoretical science. His will speaks of prizes for those ``inventions or discoveries'' of greatest practical benefit to mankind. (Probably as a result of this language, the physics prize has been awarded for experimental work much more often than for advances in theory.)

However, the story of some rivalry over a woman is obviously much more amusing, and that's why it will probably continue to be repeated.

Fractals and poems (VII)

2005-05-07T23:30:00.000+02:00

Black Holes

As Poe said ... "A few words on secret writing"

2005-05-06T21:18:00.000+02:00

Despite its long history, cryptography only became part of mathematics and information theory in the late 1940s, mainly as a result of the work of Claude Shannon (1916-2001) of Bell Laboratories in New Jersey.

Shannon showed that truly unbreakable ciphers do exist and, in fact, they had been known for over 30 years. They were devised in about 1918 by an American Telephone and Telegraph engineer Gilbert Vernam and Major Joseph Mauborgne of the US Army Signal Corps, and are called either one-time pads or Vernam ciphers.

Both the original design and the modern version of one-time pads are based on the binary alphabet. The message, or plaintext, is converted to a sequence of 0's and 1's, using some publicly known rule. The key is another sequence of 0's and 1's of the same length. Each bit of the message, or the plaintext, is then combined with the respective bit of the key, according to the rules of addition in base 2:

0 + 0 = 0
0 + 1 = 1 + 0 = 1
1 + 1 = 0

The key is a random sequence of 0's and 1's, and therefore the resulting cryptogram - the plaintext plus the key - is also random and completely scrambled unless one knows the key. The plaintext can be recovered by adding (in base 2 again) the cryptogram and the key.

As an example, suposse a sender, traditionally called Alice, adds each bit of the plaintext (01011100) to the corresponding bit of the key (11001010) obtaining the cryptogram (10010110), which is then transmitted to the receiver, traditionally called Bob. Both Alice and Bob must have exact copies of the key beforehand; Alice needs the key to encrypt the plaintext, Bob needs the key to recover the plaintext from the cryptogram. An eavesdropper, called Eve, who has intercepted the cryptogram and knows the general method of encryption but not the key, will not be able to infer anything useful about the original message. Indeed, Shannon proved that if the key is secret, the same length as the message, truly random, and never reused, then the one-time pad is unbreakable. Thus we do have unbreakable ciphers!

2005-04-27T22:56:00.000+02:00

Andrei Nikolaevich Kolmogorov: a genius almost surely (and in law)

Math jokes (IV)

2005-04-21T14:21:00.000+02:00

"What is Pi?"
- A mathematician: "Pi is the ratio of the circumference of a circle to its diameter."
- A computer programmer: "Pi is 3.141592653589 in double precision."
- A physicist: "Pi is 3.14159 plus or minus 0.000005."
- An engineer: "Pi is about 22/7."
- A nutritionist: "Pie is a healthy and delicious dessert!"

A mathematician and a stock broker go to the races to bet on horses. The broker suggests a bet of $10,000. That's too much for the mathematician's taste: First, he wants to understand the rules, have a look at the horses, etc. "Don't worry", the broker says. "I know an empirical algorithm that allows me to find the number of the winning horse with absolute certainty." This does not convince the mathematician. "You are too theoretical!" the broker exclaims and puts his $10,000 on a horse. The horse comes in first - making the broker even richer than he already is. The mathematician is baffled. "What is your algorithm?" he wants to know. "It's rather easy. I have two children, three and five years old. I add up their ages and bet on that number." "But three plus five is eight - and that horse had number nine!" "I told you that you're too theoretical! Didn't I just experimentally prove that my calculation is correct?!"

A stats professor plans to travel to a conference by plane. When he passes the security check, they discover a bomb in his carry-on-baggage. Of course, he is hauled off immediately for interrogation. "I don't understand it!" the interrogating officer exclaims. "You're an accomplished professional, a caring family man, a pilar of your parish - and now you want to destroy that all by blowing up an airplane!" "Sorry", the professor interrupts him. "I had never intended to blow up the plane." "So, for what reason else did you try to bring a bomb on board?!" "Let me explain. Statistics shows that the probability of a bomb being on an airplane is 1/1000. That's quite high if you think about it - so high that I wouldn't have any peace of mind on a flight." "And what does this have to do with you bringing a bomb on board of a plane?" "You see, since the probability of one bomb being on my plane is 1/1000, the chance that there are two bombs is 1/1000000. If I already bring one, the chance of another bomb being around is actually 1/1000000, and I am much safer..."

Icosahedron

2005-04-13T22:26:00.000+02:00

Extruded Icosahedron Edges
by Michael Trott

The great and the good

2005-04-11T00:00:00.000+02:00

Norbert Wiener was renowned for his absent-mindedness. When he and his family moved from Cambridge to Newton his wife, knowing that he would be of absolutely no help, packed him off to MIT while she directed the move. Since she was certain that he would forget that they had moved and where they had moved to, she wrote down the new address on a piece of paper, and gave it to him. Naturally, in the course of the day, some insight occurred to him. He reached in his pocket, found a piece of paper on which he furiously scribbled some notes, thought it over, decided there was a fallacy in his idea, and threw the piece of paper away. At the end of the day he went home - to the old address in Cambridge, of course. When he got there he realised that they had moved, that he had no idea where they had moved to, and that the piece of paper with the address was long gone. Fortunately inspiration struck. There was a young girl on the street and he conceived the idea of asking her where he had moved to, saying, "Excuse me, perhaps you know me. I'm Norbert Wiener and we've just moved. Would you know where we've moved to?" To which the young girl replied, "Yes Daddy, Mommy thought you would forget."

The great Polish mathematician Waclaw Sierpinski was coincidentally also absent-minded and coincidentally also had to move house. His wife knew of his fallibility as they stood on the street with all their belongings, said to him, "Now, you stand here and watch our ten cases, while I go and get a taxi." She left him there, eyes glazed and humming absently. Some minutes later she returned, a taxi having been called. Sierpinski challenged her (possibly with a glint in his eye): "I thought you said there were ten cases, but I've only counted to nine." His wife insisted there were ten. "No, count them," replied Sierpinski, "0, 1, 2, ..."

The great logician Bertrand Russell once claimed that he could prove anything if given that 1+1=1. So one day, an undergraduate demanded: "Prove that you're the Pope." Russell thought for a while and proclaimed, "I am one. The Pope is one. Therefore, the Pope and I are one."

John von Neumann supposedly had the habit of simply writing answers to homework assignments on the board (the method of solution being, of course, obvious) when he was asked how to solve problems. Once, one of his students tried to get more helpful information by asking if there was another way to solve the problem. Von Neumann looked blank for a moment, thought, and then answered, "Yes."

Fractals and poems (VI)

2005-04-10T23:50:00.000+02:00

Candelabra

"Don't try to play with me, baby" (Game Theory I)

2005-04-03T13:48:00.000+02:00

A. Einstein y John von Neumann (1903-1957)

John von Neumann, one of the original six mathematics professors at the Institute for Advanced Study, left an indelible mark on the fields of mathematics, quantum theory, nuclear physics, and computer science. A pioneer of the field of Game Theory, Dr. von Neumann co-authored the book, THEORY OF GAMES AND ECONOMIC BEHAVIOR (1944), with his Princeton colleague, economist Oskar Morgenstern. The book is considered the seminal work in the field of Game Theory. Dr. von Neumann was also a pioneer of modern computing, devising the computer infrastructure that is now known as the "von Neumann Architecture." He perceived that a computer's program and the data that it is processing do not have to be fed into the machine while the computer is running, but rather can be stored in the computer's memory. This helped paved the way for the modern computing era.

The roots of game theory go back at least to the nineteenth-century economists Augustin Cournot and Francis Edgeworth. The formal theory was introduced in 1921 by Emil Borel, a French mathematician, who wrote about "la théorie du jeu" in a note which looked at the phenomenon of bluffing in poker. (Seven years later, von Neumann would arrive at game theory via the same route.) In addition to noting the possible applications of game theory to political and economic issues, Borel listed some basic questions: "[F]or what games is there a best strategy, and how does one find such a strategy?"

Von Neumann's 1928 contribution was to show that for two-person zero-sum games, a best strategy could indeed be found. He then extended the analysis to more general games, "[defining] three-person games and [suggesting] how to define a general n-person game." Games involving more than two players presented very different problems than did two-person games because of the possibility--and necessity--of alliances. "Conflicts of interest are less pat. What is good for player A may be bad for player B but good for player C. In such a situation, A and C might form an alliance. Such coalitions change a game radically."

Morgenstern, who had fled Nazi persecution, had arrived at Princeton in 1939. He and von Neumann began work on the application of game theory to economics in that year and continued intermittently until 1943; the book was completed shortly before von Neumann was called away to England on a war assignment. Between 1939 and 1943, von Neumann was consulting extensively with the U.S. government on war work, and therefore much of his work was done in Washington, D.C. The Theory of Games and Economic Behavior, which "describes the mathematics of the theory of games in great detail and applies it to many different economic problems, including exchange of goods between n parties, monopolies and oligopolies, and free trade," was an enormous success (by academic standards)--much to the surprise of its authors and publisher. Although von Neumann did not return to his study of economics, he remained interested in game theory and saw its applicability to a broad range of public activities--including military strategy. Thus, von Neumann's association with RAND after the war came about because RAND was convinced of the importance of game theory for strategic purposes in the Cold War. But despite his work with RAND, von Neumann spent little time on game theory after the Second World War, turning his attention instead to the emerging disciplines of computer science and atomic energy.

Fractals and Poems (V)

2005-03-27T21:21:00.000+02:00

Fractals and Poems (V)

... no comments ...

2005-03-25T19:27:00.000+01:00

A mathematician who is not also something of a poet will never be a complete mathematician.

Karl Weierstrass

Sorry Wolfram, but always exists a smarter one

2005-03-25T18:51:00.000+01:00

Born in Milan, Italy, Dr. Bombieri earned his doctoral degree at the University of Milan in 1963. He taught at the University of Pisa and at the Scuola Normale Superiore at Pisa before joining the Institute for Advanced Study in Princeton, in 1977.

Enrico Bombieri is the IBM von Neumann Professor of Mathematics at the Institute for Advanced Study in Princeton, New Jersey. Professor Bombieri is regarded as one of the world's leading authorities on number theory and analysis. In 1974, when he was a visiting Member of the Institute for Advanced Study, he received the Fields Medal, the highest award given in the mathematical sciences. He was cited for his work in both number theory, the study of integers and their relation to one another, and minimal surfaces, the study of multidimensional surfaces. While his interests range throughout mathematics, Professor Bombieri has been described as a "problem-oriented" scholar - one who tries to solve deep problems rather than to build theories. According to a colleague, his analytical ability, combined with great powers of innovation, "enable him to recognize elements of a solution that may already be present and to mobilize techniques and results from other fields to reach a final conclusion."

In addition to the Fields Medal, his awards include the Feltrinelli and Balzan Prizes. He is a member of the National Academy of Sciences, the American Academy of Arts and Sciences, the Accademia Nazionale dei Quaranta in Rome, the Accademia Nazionale dei Lincei, the European Academy of Sciences, Arts, and Humanities, and a foreign member of the French Académie des Sciences, the Royal Swedish Academy and the Academia Europaea.

Check that CV ... WOW (opss!)

2005-03-23T21:56:00.000+01:00

Born in London in 1959, Stephen Wolfram was educated at Eton, Oxford, and Caltech. He published his first scientific paper at the age of 15, and had received his Ph.D. in theoretical physics from Caltech by the age of 20. Wolfram's early scientific work was mainly in high-energy physics, quantum field theory, and cosmology, and included several now-classic results. Having started to use computers in 1973, Wolfram rapidly became a leader in the emerging field of scientific computing, and in 1979 he began the construction of SMP--the first modern computer algebra system--which he released commercially in 1981.

In recognition of his early work in physics and computing, Wolfram became in 1981 the youngest recipient of a MacArthur Prize Fellowship. Late in 1981 Wolfram then set out on an ambitious new direction in science aimed at understanding the origins of complexity in nature. Wolfram's first key idea was to use computer experiments to study the behavior of simple computer programs known as cellular automata. And starting in 1982 this allowed him to make a series of startling discoveries about the origins of complexity. The papers Wolfram published quickly had a major impact, and laid the groundwork for the emerging field that Wolfram called "complex systems research."

Through the mid-1980s, Wolfram continued his work on complexity, discovering a number of fundamental connections between computation and nature, and inventing such concepts as computational irreducibility. Wolfram's work led to a wide range of applications--and provided the main scientific foundations for such initiatives as complexity theory and artificial life. Wolfram himself used his ideas to develop a new randomness generation system and a new approach to computational fluid dynamics--both of which are now in widespread use.
Following his scientific work on complex systems research, in 1986 Wolfram founded the first research center and the first journal in the field. Then, after a highly successful career in academia--first at Caltech, then at the Institute for Advanced Study in Princeton, and finally as Professor of Physics, Mathematics, and Computer Science at the University of Illinois--Wolfram launched Wolfram Research, Inc.

Wolfram began the development of Mathematica in late 1986. The first version of Mathematica was released on June 23, 1988, and was immediately hailed as a major advance in computing. In the years that followed, the popularity of Mathematica grew rapidly. Wolfram has been president and CEO of Wolfram Research since its inception, and continues to be personally responsible for the overall design of its core technology.

Sir Roger Penrose, the tiling problem and AI

2005-03-21T22:02:00.000+01:00

A problem from recreational mathematics to which Roger Penrose has made a significant contribution is the tiling problem. The tiling problem is this: given a collection of polygonal shapes, is it possible to cover the whole plane using just these shapes, with no overlaps? Such an arrangement of shapes is called a tiling. Tilings are said to be periodic if they are exactly repetitive in two different directions (a direction and its opposite are not counted as different!).

At first sight, the problem of whether one can find a tiling of the plane by shapes which will only tile non-periodically, seems lighthearted. In fact, although it is "fun" mathematics, it has a philosophically deep aspect - it is part of the area of mathematics known as non-recursive.
A class of mathematical problems is called recursive if there is an algorithm for finding the answer in each individual case. It is called non-recursive if there is a yes-or-no answer to each individual problem, but there is no algorithm for deciding whether the answer is yes or no in each individual case.

It has been known for about forty years that there is no algorithmic way of deciding whether a given collection of polygonal shapes will tile the plane, that is, the tiling problem is non-recursive.

The way in which this was originally proved was interesting. First it was shown that if, whenever a collection of polygonal shapes, or "tiles" could tile the plane in some way, the same collection could tile the plane periodically, then the tiling problem had to be recursive. The next step was that the tiling problem was proved not to be recursive. This meant that there had to be a collection of polygonal shapes that could tile the plane, but only non-periodically. The first such collection of tiles that was found was was absolutely enormous - it contained 20,426. Various mathematicians found smaller collections, culminating in Penrose's discovery of a pair of tiles that tile the plane only in a non-periodic way. In fact, there are more than one such pair - two are shown below.

Two Pairs of Penrose Tiles

It's interesting to think about the problem of inventing a tiling in the light of Penrose's thoughts on AI, consciousness and inspiration, because the problem seems exactly like the sort of thing a computer couldn't do. Penrose says that people ask him where he got the inspiration, whether it came to him suddenly. "It wasn't like that, but, you see, it's never like that. What it's like is that one has been thinking about a problem for a long time and getting very familiar with it. After a while, you may be worrying about some particular aspect of the problem, and then it's quite possible, even when you're thinking about something else, that an idea may come to you, and you realise that things fit together in some way, with the accompanying realisation that this must be right. Of course sometimes one may be mistaken in such things!"

"You see, I was already aware of certain nonperiodic patterns of pentagons and other shapes which are just nice to look at. What you might call the inspiration was to realise that, by modifying the shapes in different sorts of ways, by putting knobs on like jigsaw pieces, you could force the pattern by local matching rules. As a pattern, I was aware of it before, but the thing that required a little bit more, something for me to guess or something to come from somewhere else, was that realisation."

As Penrose says, one of the interesting features of the tiling problem is that it is noncomputable. "It is possible to explain noncomputability in relatively simple terms, but people almost always get the wrong end of the stick! They say, 'show me a problem you can't solve', but it's never like that! It's a class of problems which have no systematic solutions. With the tiling problem, I sometimes say in talks, 'well it depends on the existence of nonperiodic tilings', and give an example of such a tiling. And then at the end of the talk people come up to me and say, 'how do you know that that tiles when the problem is noncomputable?' but that's not the point. The point is that in a particular case you may have a way of seeing the solution, but there is no systematic procedure that could be put on a machine, which requires no more thinking."

A fractal is not a simple draw ...

2005-03-20T13:01:00.000+01:00

A detail from the "Lyapunov Exponent" by Andy Burbanks

A fractal is usually defined as a mathematical structure that exhibits some sort of "self-similarity", meaning that if you zoom in on one, the same type of structure will keep appearing. For example, a fern leaf is built up of leaflets and fronds with the same reiterative structure.

They are also often defined as shapes that have fine detail at all scales, and it is this combination of repetition and intricacy that makes them so aesthetically interesting.
When the sensitivity of a complex system to small changes is mapped on the plane, the result is often a fractal. When this map is colour-coded, the result can be strikingly beautiful. The "Lyapunov Exponent" fractal on the top is one example.

Fractals and poems (IV)

2005-03-17T21:55:00.000+01:00

Fractals and poems (IV)

The Tragic Story of Schrodinger's Cat

2005-03-11T21:00:00.000+01:00

Schrodinger's cat

Schrödinger's cat is a famous illustration of the principle in quantum theory of superposition, proposed by Erwin Schrödinger in 1935. Schrödinger's cat serves to demonstrate the apparent conflict between what quantum theory tells us is true about the nature and behavior of matter on the microscopic level and what we observe to be true about the nature and behavior of matter on the macroscopic level.

Here's Schrödinger's (theoretical) experiment:
We place a living cat into a steel chamber, along with a device containing a vial of hydrocyanic acid. There is, in the chamber, a very small amount of a radioactive substance. If even a single atom of the substance decays during the test period, a relay mechanism will trip a hammer, which will, in turn, break the vial and kill the cat. The observer cannot know whether or not an atom of the substance has decayed, and consequently, cannot know whether the vial has been broken, the hydrocyanic acid released, and the cat killed. Since we cannot know, the cat is both dead and alive according to quantum law, in a superposition of states. It is only when we break open the box and learn the condition of the cat that the superposition is lost, and the cat becomes one or the other (dead or alive). This situation is sometimes called quantum indeterminacy or the observer's paradox: the observation or measurement itself affects an outcome, so that it can never be known what the outcome would have been if it were not observed.

We know that superposition actually occurs at the subatomic level, because there are observable effects of interference, in which a single particle is demonstrated to be in multiple locations simultaneously. What that fact implies about the nature of reality on the observable level (cats, for example, as opposed to electrons) is one of the stickiest areas of quantum physics. Schrödinger himself is rumored to have said, later in life, that he wished he had never met that cat.

Ode to Fermat ( see theorem III )

2005-03-08T23:05:00.000+01:00

Pierre Fermat

In around 1640, Fermat,
upon his reading of Diophantus

Was led to a romantic assertion that would
From that point on entrance us

Never did he dream that a few words
in a margin could make him a hero

As he wrote that certain equations had no
solutions other than zero.

Many searched for a proof and there may
have been the rumor,

It looks a complete solution will
be found soon by Kummer.

This was not the case and a proof would
not appear out of thin air,

but perhaps using the curve of Frey and
and conjecture of Serre

This strategy indeed works as proven by
Ribet, Taylor, Wiles, and Wiles

Causing the mathematical community at large
to don tearful smiles

Thus although now the spirit of Fermat
is finally content

He is shocked that the proof is not just a
simple method of descent.

Author: Sharon Ann Kineke

Math jokes (III)

2005-03-04T00:00:00.000+01:00

... check the following theorems (?):

Theorem (I): All numbers are equal to zero.
Proof:
Suppose that a=b.
Then a = b
a^2 = ab
a^2 - b^2 = ab - b^2
(a + b)(a - b) = b(a - b)
a + b = b
a = 0

And furthermore if a + b = b, and a = b, then b + b = b, and 2b = b,which mean that 2 = 1

Theorem (II): 1 + 1 = 2
Proof:
n(2n - 2) = n(2n - 2)
n(2n - 2) - n(2n - 2) = 0
(n - n)(2n - 2) = 0
2n(n - n) - 2(n - n) = 0
2n - 2 = 0
2n = 2
n + n = 2
or setting n = 1
1 + 1 = 2

Theorem (III): x^n + y^n = z^n
has no non-zero integer solutions for x, y and z when n > 2.
Proof:
I have discovered a truly remarkable proof which this margin is too small to contain.

Well ... in fact, the last theorem is one of the oldest math jokes (around 1630).

Fractals and poems (III)

2005-03-03T23:39:00.000+01:00

Roger Penrose fractal

"Give me another problem, please" Hilbert said (I)

2005-02-27T21:30:00.000+01:00

David Hilbert (1862-1943)

David Hilbert studied at the University of Königsberg under Lindemann. He was friends with Hurwitz, who taught there, and Minkowski, who was also a doctoral student. Both of them were to strongly influence Hilbert's mathematical progress. Hilbert got his doctorate in 1895, and taught at Königsberg for 9 years. In 1895, Hilbert was appointed to the chair of mathematics at the University of Göttingen, where he continued to teach for the rest of his career.

Hilbert's first work was on invariant theory, and in 1888 he proved his famous Basis Theorem. Hilbert submitted a paper on the subject, and despite objections from Gordan, the world expert on invariant theory, it was accepted. He expanded on his methods in a later paper, and Klein, after reading the manuscript, wrote "I do not doubt that this is the most important work on general algebra that the [journal] has ever published."

From 1893 to 1897, Hilbert worked on a book on algebraic number theory. This was a brilliant synthesis of the work of Kummer, Kronecker and Dedekind but contains a wealth of Hilbert's own ideas. The beginnings of Class field theory are all contained in this work.

Hilbert's work in geometry had the greatest influence in that area since Euclid. A systematic study of the axioms of Euclidean geometry led Hilbert to propose 21 such axioms and he analysed their significance. He published a paper in 1899 putting geometry in a formal axiomatic setting. The book continued to appear in new editions and was a major influence in promoting the axiomatic approach to mathematics during the twentieth century.

Hilbert gave a famous speech, "The Problems of Mathematics" to the Second International Congress of Mathematicians in Paris (1900). In it he presented 23 unsolved problems he felt were fundamental to mathematics. Hilbert's problems included the continuum hypothesis, the well ordering of the reals, Goldbach's conjecture, the transcendence of powers of algebraic numbers, the Riemann hypothesis, the extension of Dirichlet's principle and many more. Many of the problems were solved during the twentieth century, and each time one of the problems was solved it was a major event for mathematics.

Hilbert contributed to many branches of mathematics, including invariants, algebraic number fields, functional analysis, integral equations, mathematical physics, and the calculus of variations. Hilbert's work in integral equations began the field of functional analysis. This work also established the basis for his work on infinite-dimensional space, later called Hilbert space. Making use of his results on integral equations, Hilbert contributed to the development of mathematical physics by his important memoirs on kinetic gas theory and the theory of radiations.

Among Hilbert's students were Hermann Weyl, the famous world chess champion Lasker, and Zermelo. His first student, Otto Blumenthal, wrote "Insofar as the creation of new ideas is concerned, I would place [others] higher, but when it comes to penetrating insight, only a few of the very greatest were the equal of Hilbert".
Hilbert received many honors. In 1905 the Hungarian Academy of Sciences gave a special citation for Hilbert. In 1930 Hilbert retired and the city of Königsberg made him an honorary citizen of the city.

Hey, Gödel ... what is really truth? (II)

2005-02-24T23:54:00.000+01:00

Kurt Gödel and Einstein in Princeton (1950)

Gödel's theorem is surrounded by many fallacies. People have drawn many conclusions from it that are incorrect but never the less pervade popular thought.

One such fallacy is that Gödel's theorem limits what scientists can know about the world. It is true that Gödel's theorem restricts what we can derive from a set of axioms, the presumption in this argument is that axiomatic deduction is the only method science uses for acquiring knowledge. The theory of scientific method is quite vast and there are several notable contributions which pertain to this misconception. The philosopher Carl Popper proposed that science is a combination of deductive logic, to which Gödel's theorem applies, and falsification. It is the falsification which serves to provide us with axioms. From these axioms we derive properties of the world. Although this viewpoint succumbs to Gödel's theorem it does not admit ultimate defeat. As opposed to pure mathematics, science has the luxury of reality, in particular Realists cite the existence of ultimate truth as the attainable goal of science. Specifically Gödel's theorem does not taint the notion of truth, just that a complete set of truths cannot be derived automatically from a set of axioms.

Another fallacy is that the notion of artificial intelligence should be dismissed due to Gödel's theorem, that machines cannot think simply because they can be "Gödel'd". Abstractions of computers are equivalent to axiomatic systems. Turing machines are an often cited example of the theoretical abstraction of computers. The importance of these abstractions is that they can be shown to be equivalent to any computer, that is, they are capable of evaluating anything that any computer can evaluate. They also have the same limitations. Gödel's theorem then states that there exist true propositions which cannot be obtained by a given computer, but which we can obtain. Firstly, this doesn't free humans from Gödel's theorem. The axioms used by a computer may be different from those which are equivalent to the human axiomatic system. This justifies the possibility that a human can arrive at a truth unattainable by a computer. If humans can be reduced to an axiomatic system then clearly artificial intelligence is possible, all that would be required would be for a computer to implement a human's axioms and we would have an artificial human intelligence.

Many fallacies arise through misuse of Gödel's theorem, especially in trying to apply its implications to non-formal systems. For example many have questioned the implications of the Incompleteness theory in reference to language, the bible or quantum mechanics. This can not be done, as none of these examples are formal systems.

Just for computers ...

2005-02-20T22:49:00.000+01:00

Chess problem ... probably, only for a computer

Einstein on the Brownian beach

2005-02-20T22:27:00.000+01:00

"One thing I have learned in a long life: that all our science, measured against reality, is primitive and childlike - and yet it is the most precious thing we have."

Albert Einstein (1879-1955)

Albert Einstein, in his doctoral dissertation, submitted to the University of Zurich in 1905, Einstein developed a statistical molecular theory of liquids. Then, in a separate paper, he applied the molecular theory of heat to liquids in obtaining an explanation of what had been, unknown to Einstein, a decades-old puzzle.

Observing microscopic bits of plant pollen suspended in still water, English botanist Robert Brown had noticed in 1828 that the pollen seeds exhibited an incessant, irregular "swarming" motion, since called "Brownian motion." Although atoms and molecules were still open to objection in 1905, Einstein predicted that the random motions of molecules in a liquid impacting on larger suspended particles—such as pollen seeds—would result in irregular, random motions of the particles, which could be directly observed under a microscope. The predicted motion corresponded precisely with the puzzling Brownian motion! From this motion Einstein accurately determined the dimensions of the hypothetical molecules.

By 1908 the molecules could no longer be considered hypothetical. The evidence gleaned from Brownian motion on the basis of Einstein's work was so compelling that Mach, Ostwald, and their followers were thrown into retreat, and material atoms soon became a permanent fixture of our knowledge of the physical world. Today, with the advent of scanning tunneling microscopes, scientists are nearly able to see and even to manipulate actual, individual atoms for the first time—a circumstance that would satisfy even the most entrenched Machian skeptic.

In the course of his fundamental work on applications of statistical methods to the random motions of Newtonian atoms, Einstein discovered a connection between his statistical theory of heat and the behavior of electromagnetic radiation—the first step toward his hoped-for unification of these two fields.

Einstein obtained a mathematical expression for the fluctuations, or oscillations, in the average energy of any system, using his statistical theory of heat. He applied this expression to the average energy of thermal radiation—the electromagnetic waves given off by glowing bodies—in a perfectly reflecting box (often called "blackbody radiation"). He obtained results in close agreement with experimental observations. This connection, he declared in obvious understatement, "ought not to be ascribed to chance."

For a physicist like Einstein interested in uniting perspectives, the connection provided an extraordinary opportunity. Einstein's fundamental papers on relativity and quantum theory, also submitted in 1905, may be seen as far-reaching explorations of the connection.

Moire Trefoil

2005-02-17T22:19:00.000+01:00

Moire Trefoil by Michael Trott.

La Géometrie

2005-02-16T21:28:00.000+01:00

This image is taken from a copy of the first edition of Descartes' La Géometrie (appendix to Discours sur la Méthode) located at the Thomas L. Fisher Rare Book Library of the University of Toronto. It was made from a wood cut, probably one carved by Frans van Schooten the Younger.

... if you really want to change the world, never forget this

2005-02-12T21:38:00.000+01:00

"Imagination is more important than knowledge."

"Logic will get you from A to B. Imagination will take you everywhere."

Albert Einstein (1879-1955)

Hey, Gödel ... what is really truth? (I)

2005-02-12T21:07:00.000+01:00

The Liar's Paradox is a simple statement which raises the all important question, does a person lie when he or she says they are lying? It was first presented by Epimenides in the sixth century BC He, being a Cretan himself, stated that "All Cretans are liars". If this is indeed true, then Epimenides is himself a liar and therefore the statement is untrue, and so there is a contradiction. However if the statement is untrue, then all Cretans are not liars, and hence we have another paradox.
200 years later Eubulides furthered this argument by asking whether it is true or false for him to suggest "I am lying". Again we arrive at a paradox similar to the one suggested by Epimenides.
A third example of the liar paradox is attributed to Socrates who once stated "One thing I know is that I know nothing".

Although these examples appear trivial, they exemplify a larger question, what is truth? Even after 2600 years, there is still no formal theory to explain a way out of the liar paradox, not through lack of trying. Many believe the closest we have come to explain what truth is was by Aristotle, who proposed essentially as follows: "A declarative sentence is true if and only if what it says is so"

In 1928, David Hilbert asked the question, "Is it possible to prove everything in mathematics"? Most of the mathematics of today was worked out by the 1800's but by the 1900's mathematicians were coming up with theories that had paradoxes in them, even when they followed correct rules of math, they were finding answers that did not make sense.

Most mathematicians of the time believed that it was possible to (eventually) solve everything in mathematics, however in 1931 Gödel's Incompleteness theorem proved that there are some things in mathematics that can never be proven, even though we may possibly know that they are true.

Gödel's theorem of incompleteness is basically a glorified version of the Liar Paradox. He clearly showed that within a system, certain clear-cut statements can neither be proved or disproved.

Top Ten excuses for not doing Maths homework

2005-02-12T00:08:00.000+01:00

I accidentally divided by zero and my paper burst into flames.
Isaac Newton's birthday.
I could only get arbitrarily close to my textbook. I couldn't actually reach it.
I have the proof, but there isn't room to write it in this margin.
I was watching the World Series and got tied up trying to prove that it converged.
I have a solar powered calculator and it was cloudy.
I locked the paper in my trunk but a four-dimensional dog got in and ate it.
I couldn't figure out whether i am the square of negative one or i is the square root of negative one.
I took time out to snack a doughnut and a cup of coffee. I spent the rest of the night trying to figure which one to dunk.
I could have sworn I put the homework inside a Klein bottle, but this morning I couldn't find it.

It's easy to see that ...

2005-02-09T22:23:00.000+01:00

It's easy to see that a cylinder of cheese can be cut into eight identical pieces with four straight cuts.

Can this be done with only three straight cuts?

Answer

The importance of understanding the infinitesimal ...

2005-02-08T10:18:00.000+01:00

Calculus has practical applications, such as understanding the true meaning of the infinitesimals. (Image concept by Dr. Lachowska.)

Attraction and Repulsion ?

2005-02-05T21:47:00.000+01:00

FieldLines of Two Oppositely Charged Spheres
Created by: Michael Trott

Who is this guy?

2005-02-03T22:39:00.000+01:00

Photograph of Henri Léon Lebesgue (1875-1941),
who laid the groundwork for modern measure theory and remade the theory of integration (if you are interested in his bio, try the link).

(Courtesy of The MacTutor History of Mathematics Archive , University of St. Andrews.)

Math jokes (II)

2005-01-30T22:25:00.000+01:00

An assemblage of the most gifted minds in the world were all posed the following question: "What is 2 + 2 ?"

The engineer whips out his slide rule (so it's old) and shuffles it back and forth, and finally announces "3.99".
The physicist consults his technical references, sets up the problem on his computer, and announces "it lies between 3.98 and 4.02".
The mathematician cogitates for a while, oblivious to the rest of the world, then announces: "I don't know what the answer is, but I can prove an answer exists!".
Philosopher: "But what do you mean by 2 + 2 ?"Logician: "Please define 2 + 2 more precisely."
Accountant: Closes all the doors and windows, looks around carefully then asks "What do you want the answer to be?"

A mathematician, a physicist and an engineer are given an identical problem: Prove that all odd numbers greater than 2 are prime numbers. They proceed:

Mathematician: 3 is a prime, 5 is a prime, 7 is a prime, 9 is not a prime - counterexample - claim is false.
Physicist: 3 is a prime, 5 is a prime, 7 is a prime, 9 is an experimental error, 11 is a prime, ...
Engineer: 3 is a prime, 5 is a prime, 7 is a prime, 9 is a prime, 11 is a prime, ...
Computer Scientist: 1 is a prime, 1 is a prime, 1 is a prime, 1 is a prime, ...Yes, they're all primes.

Cubes Collapsing Inward

2005-01-30T22:14:00.000+01:00

Series of Frame Cubes Collapsing Inward. Created by: Sandor Kabai

Fractals and poems (II)

2005-01-29T22:51:00.000+01:00

Fractals and poems (II)

MIT OpenCourseWare project: try it !

2005-01-29T21:33:00.000+01:00

How many amongst us are old enough to remember the days when the only source of material to help in the production of lecture notes was from our own library's finite resources? Nowadays, an appropiate phrase entered into Google (or Google scholar) reveals a plethora of academic material that can be used by the students or researchers directly from the Internet, or can be edited to produce to produce one's own set of course handouts.

But from 2002, a project from Massachusetts Institute of Technology has resulted in MIT's educational material being laid before the world as totally free Internet resource for anyone to use. This is the OpenCourseWare (OCW) project.

Through the OpenCourseWare (MIT OCW) initiative, MIT makes its faculty's core teaching materials openly available for anyone, anywhere in the world with access to the Internet. We have found that, with 900 of MIT's classes available, the project can help -- educators, enrolled students, and self-learners around the world -- by providing a high-quality publication of MIT's core teaching materials.

MIT OCW provides users with open access to the syllabi, lecture notes, course calendars, problem sets and solutions, exams, reading lists, even a selection of video lectures, from 900 MIT courses representing 33 academic disciplines and all five of MIT's schools. The initiative will include materials from 1800 courses by the year 2008.

MIT OpenCourseWare (MIT OCW) is a remarkable story of an institution rallying around an ideal, and then delivering on the promise of that ideal.

THANKS MIT !!!

Math jokes (I)

2005-01-28T21:22:00.000+01:00

"A mathematician is a device for turning coffee into theorems." (P. Erdös). So while you're waiting for the coffee to take effect, look through these...

Q: How many mathematicians does it take to replace a lightbulb? A: Ten: One to do it and eight to watch.
Q: How many numerical analysts does it take to replace a lightbulb? A: 3.9967: (after six iterations).
Q: How many topologists does it take to change a lightbulb? A: Just one. But what will you do with the doughnut?
Q: How many analysts does it take to screw in a lightbulb? A: Three: One to prove existence, one to prove uniqueness and one to derive a nonconstructive algorithm to do it.
Q: How many professors does it take to replace a lightbulb? A: One: With eight research students, two programmers, three post-docs and a secretary to help him.
Q: How many university lecturers does it take to replace a lightbulb? A: Four: One to do it and three to co-author the paper.
Q: How many graduate students does it take to replace a lightbulb? A: Only one: But it takes nine years.
Q: How many maths department administrators does it take to replace a lightbulb? A: None: What was wrong with the old one then???

Fractals and poems (I)

2005-01-28T21:12:00.000+01:00

Fractals and poems (I)

Complexity, Theology and Maths.

2005-01-26T22:39:00.000+01:00

Both complexity and order in the natural world have been cited as evidence for an intelligent creator. Early mythologies most often assume that the universe started in chaos, with a supernatural being adding order, then creating a series of specific complex natural systems.

In Greek philosophy it was commonly thought that the regularities seen in astronomy and elsewhere (such as the obvious circular shapes of the Sun and Moon) were reflections of perfect mathematical forms associated with divine beings. About complexity Aristotle did note that what nature makes is "finer than art", though this was not central to his arguments about causes of natural phenomena.

By the beginning of the Christian era, however, there is evidence of a general belief that the complexity of nature must be the work of a supernatural being - and for example there are statements in the Bible that can be read in this way. Around 1270 Thomas Aquinas gave as an argument for the existence of God the fact that things in nature seem to "act for an end" (as revealed for example by always acting in the same way), and thus must have been specifically designed with that end in mind.

In astronomy, as specific natural laws began to be discovered, the role of God began to recede somewhat, with Isaac Newton claiming, for example, that God must have first set the planets on their courses, but then mathematical laws took over to govern their subsequent behavior.

Particularly in biology, however, the so-called "argument by design" became ever more popular. Typical was John Ray’s 1691 book The Wisdom of God Manifested in the Works of the Creation, which gave a long series of examples from biology that it claimed were so complex that they must be the work of a supernatural being.

By the early 1800s, such ideas had led to the field of natural theology, and William Paley gave the much quoted argument that if it took a sophisticated human watchmaker to construct a watch, then the only plausible explanation for the vastly greater complexity of biological systems was that they must have been created by a supernatural being.

Following the publication of Charles Darwin’s Origin of Species in 1859 many scientists began to argue that natural selection could explain all the basic phenomena of biology, and although some religious groups maintained 'strong resistance', it was widely assumed by the mid-1900s that no other explanation was needed.

In fact, however, just how complexity arises was never really resolved...

Some quotes from lecturers at Cambridge University, England

2005-01-26T11:40:00.000+01:00

A slight difficulty occured with geometry in an Engineering lecture oneday:"This is the maximum power triangle." said a lecturer, pointing to a rectangle.

This year the Computer Scientists seem to be in the running for the Honesty Award:"Sorry, I should have made that completely clear. This is a shambles."

From a Computer Sciences Protection lecture:"Who should be going to this lecture? Everyone...apart from the third year of the two-year CompSci course." "I don't want to go into this in detail, but I would like to illustrate some of the tedium."

Oh those poor CompScis...."I'm not going to get anything more useful done in this lecture, so I might as well talk."later followed by ..."Well, there you are, one lecture with no useful content."

Three from a NatSci Physics lecturer:"You don't have to copy that down -- there's no wisdom in it -- it only repeats what I said. ""We now wish to show that they are not merely equal but _the same thing_.""And before I leave this subject, I would like to tell you something interesting."

From a first year chemistry lecture some personal problems of the lecturer:"Before I started this morning's lecture I was going to tell you about my third divorce but on reflection I thought I'd better tell my wife first."

From a single research seminar at the King's College Research Centre:"I'm sure it's right whether it's valid or not.""WARNING: There is no reason to believe this will work."

La mejor forma de empezar ... (the best possible begining)

2005-01-26T10:39:00.000+01:00

A LA MEMORIA DE FISCHER BLACK
(in F. Black memory)

"In the end, a theory is accepted not because it is confirmed by conventional empirical tests, but because researchers persuade one another that the theory is correct and relevant. "

Fischer Black (1986)

Relativicemos la verdad. Aceptemos los errores como posibilidades.
Creer que los axiomas son probables pero nunca ciertos en todas partes... y entonces ¿que nos queda?

HUMANITY (maybe) ...