Jules Verne’s scientific predictions for 2889

Jules Verne

In a science fiction story published in English in the United States in 1889, entitled In the Twenty-Ninth Century and subtitled One Day of an American Journalist in 2889, Jules Verne made several scientific predictions that, according to him, would take almost a millennium to be put into practice. Let us look at a few of the most interesting:

•         The average lifetime of the human population will have increased from 37 in 1889, to 68 in 2889. According to the UN, the average longevity in the world exceeded 68 years in the five-year period from 2005 to 2010, almost nine centuries before Verne’s forecast. Here, as elsewhere, he underestimated.

•         The land and sea voyages of the nineteenth century will have been replaced in the XXIX by air travel, or intercontinental underwater pneumatic tubes. At present, little more than a century after Verne’s story, although air travel has achieved great primacy, land and sea travel continue to exist, and for distances less than a thousand kilometers make a successful competition to air travel. Intercontinental pneumatic tubes, on the other hand, are still science fiction, although there some recent steps in this direction.

Irreversible processes

Those physicists who consider the arrow of time as an illusion have a problem: not all physics is compatible with a reversible time, as the equations and theories mentioned in an earlier article of this blog seem to indicate. The second principle of thermodynamics is known since the mid-nineteenth century (1850), when Clausius introduced the concept of entropy and it was proved that the value of this physical magnitude always increases, if it is measured in an isolated system that does not exchange matter or energy with its outside. Since the universe is an isolated system, we have at least one physical quantity that makes it possible to unequivocally signal the direction of time flow.

Aware of this problem, physicists in favor of the reversibility of time have answered in different ways: it has been said that the second principle of thermodynamics is a fictitious, subjective law that does not conform to reality; a mental illusion; an approximation; a consequence of the initial conditions of the universe. It has been hypothesized that, if the universe were cyclic, the arrow of time would be reversed during the contraction stage. (This theory has been abandoned). To escape the problem, Stephen Hawking proposed a universe without initial conditions in his book A Brief History of Time. It is curious, this desire to defend at any price the reversibility of time, when it was precisely Hawking who proposed the existence of an arrow of time in black holes, which rather than being permanent, would disintegrate.

In 1928, a year after inventing the term the arrow of time, Arthur Eddington challenged the physicists who defend the reversibility of time with the following devastating words: If your theory is found to be against the second law of Thermodynamics... there is nothing for it but to collapse in deepest humiliation (The Nature of the Physcal World, 1928).

Science and civilizations

In my book Biological Evolution and Cultural Evolution in the History of Life and Man, published in Spanish, I analyze the cultural history of 23 civilizations and compare their evolution. In the particular case of science, I wrote this:

...the first-generation civilizations (Mesopotamia, Egypt and the American) reached their maximum scientific development in mathematics and astronomy. Egypt and Mesoamerica added medicine to these sciences. A second-generation (Greco-Roman) and a third-generation civilization (Islam) also practiced the natural sciences. As for the West, it is a unique and unprecedented case, as its scientific development has been overwhelming.

...astronomy was the first cultivated science... Mathematics emerged in parallel... It was soon found that both sciences were related, for mathematics supported astronomy, making it possible to perform complex calculations and predictions.

The pagan religions... tried to predict the future, using for that purpose sacrificed animals, which led to an accumulation of anatomical knowledge, soon applied to man, which mixed with ancient knowledge about the properties of medicinal plants, led to the formation of a corpus of medical doctrines.

On the other hand, the development of the physical, chemical and biological sciences was less urgent... and so it was attempted only by civilizations that had freed from the necessities of survival an important part of human work... This happened for the first time in Greece, the cradle of philosophy and most of the modern sciences.

What is culture?

Politicians and the media do not seem quite clear about the meaning of culture. When people talk about the world of culture, they usually refer to issues as diverse as pop music shows, bullfighting, opera, theater, cinema, museums, university... This is an abuse of language that mixes four things quite different, though related: culture, shows, entertainment and education.

The Cambridge dictionary defines thus these four terms (in each case I have chosen the meaning closest to what I am speaking about, because there are others):

·      Culture: music, art, theatre, literature, etc.

·      Education: process of teaching or learning, or the knowledge you get from this.

·      Show: a theatre performance or a television or radio program that is entertaining rather than serious.

·      Entertainment: public shows, performances or other ways of enjoying yourself.

Let us call things by their names. A cultural act should be a public celebration where attendees try to increase their culture, to get knowledge that will improve their critical judgment. A classical music concert, the presentation of a book, a visit to a museum, are cultural events. Conversely:

1. Except in a few cases, we do not watch a movie to increase our culture, but to enjoy ourselves (entertainment).
2. A pop festival or a bullfight are not cultural events, but shows.
3. We can go to the opera or the theater to improve our culture, but the performance itself may not be a cultural act, but a show, especially when the stage directors distort a classic work to express their originality or to shock the public.
4. University professors can be considered a part of the world of culture if they perform popularization, but that is not their main activity. Education and research are.

When the media talk about the world of culture and put there actors, pop musicians (some of whom confess that they do not know music), and even DJs, they are really talking about the world of entertainment.

Let us call things by their name.

The same post in Spanish
Thematic Thread on Science in General: Previous Next

Manuel Alfonseca

What would be an undeniable miracle?

In 1972, sci-fi writer and publisher Lester del Rey launched a challenge to three well-known authors of the genre: Poul Anderson, Robert Silverberg and Gordon R. Dickson. All three should write a novella on a specific topic: the effect of an undeniable miracle (the sun standing still) on human society. The three authors responded to the challenge, and the three stories were published jointly in a book entitled The day the sun stood still. In this post we shall consider the first of the three, written by Poul Anderson, whose title is A chapter of Revelation.

In a post in his blog, Pablo (a.k.a. sinopinionespropias) specifies which, in his opinion, should be the characteristics of an undeniable miracle:

1.      It must be a prophecy.

2.      Its materialization should not depend on people.

3.      Your probability must be negligible and calculable.

4.      It must be as concrete as possible.

5.      It must maintain the same demonstrative force with the passage of time.

6.      It must make sense at all times.

Split brain

Roger Sperry

During the 50s of the twentieth century, the American neurobiologist Roger Sperry performed various investigations on animals, and human epilepsy patients who suffered repeated attacks, intense and persistent. As a solution to these attacks, he used the somewhat drastic (but possibly necessary) technique of cutting the corpus callosum, a bundle of nerve connections that connect the two hemispheres of the brain. The treatment was successful and provided Sperry with a number of subjects with whom he could experience what happens when the two cerebral hemispheres are disconnected from each other.

In his experiments, Sperry proved that the two cerebral hemispheres can act independently. He also discovered that their function is different: the left hemisphere is usually the seat of aggressiveness, logical processes, and the interpretation of written and spoken word. The right hemisphere is responsible for short-term memory, global thinking, artistic activities (as our response to music) and the analysis of spatial relationships. These investigations shed light on diseases and behaviors such as autism, depression, schizophrenia and Alzheimer’s disease. In 1981, Sperry received the Nobel Prize in Physiology and Medicine, shared with David Hubel and Torsten Wiesel for a different research.

Physical Time and Inner Time

William Blake

We know that physical time goes on regularly, but inner time (our sensation of the passage of time) is very variable. The two times do not have to match. Sometimes, watching at our inner time, a minute can look like hours, while in other cases the hours fly away. An English poet, William Blake, expressed it well in a famous poem:

To see a World in a Grain of Sand 

And a Heaven in a Wild Flower, 

Hold Infinity in the palm of your hand,

And Eternity in an hour

(Auguries of Innocence, 1803?)

There is a long history of literary works, in which a character enters aesthetic or religious ecstasy, or simply falls asleep, and on returning to reality discovers that many years have passed, sometimes centuries. This subgenre (called by scholars sleeper legends) has representatives in many literatures. In Spanish literature, it is reflected in the legend of the monk and the little bird, associated with the monastery of Leire. In this legend, a monk who enters in ecstasy while a bird is singing, discovers upon awakening that three centuries have gone by. Among medieval French lays there is a legend about the knight Guingamor, who arrived in a wonderful city and stayed there for three days, but when he left, he found that three centuries had passed. And in the United States literature we have the famous story by Washington Irving titled Rip van Winkle, whose protagonist falls asleep one night and wakes up 20 years later.

Fred Saberhagen versus the Turing Test

Alan Turing

In 1950, the English mathematician and chemist Alan Turing tried to define the conditions so that it would be possible to affirm that a machine is capable of thinking like us. For Turing, this will be achieved when the machine is capable of deceiving human beings, making them think that it is one of them. This test is called the imitation game. I have talked about this in a previous post in this blog.

In 1956, Arthur Samuel of IBM built a program to play the game called draughts or checkers. The program kept information about the moves in the games it had played, which was used to modify its future moves (in other words, it learned). In a few years, after playing many games, the program was able to defeat its creator and played reasonably well in official championships.

That same year, during a summer course held in Dartmouth College, John McCarthy and other computer pioneers coined the term artificial intelligence. Getting their hopes too high, they predicted spectacular advances for the next ten years, which did not take place in the time envisaged, but much later. I have also written about this in another post.

In 1963, science fiction writer Fred Saberhagen published the first story of his famous series about the berserkers, autonomous and intelligent space fortresses created by an ancient extraterrestrial civilization to exterminate intelligent life wherever it appears in the galaxy. This story, titled Without a thought, is an answer to the Turing Test and a brake to the unbridled hopes of the inventors of the term artificial intelligence. This is the plot of the story:

The Hubble-Lemaître Law

Georges Lemaître

Let us look at a little history.

In various places in the sky, but especially in the constellation of Cepheus, where the first case was discovered, there are some stars whose light intensity varies regularly, and therefore are called variable Cepheids. In 1908, the American astronomer Henrietta Swan Leavitt discovered that the period of these stars is linked with their real luminosity: the greater the luminosity, the longer the period. Therefore, by measuring their period, their real luminosity can be deduced.

In 1913, the American astronomer Vesto Melvin Slipher obtained the spectrum of what was then called the Andromeda nebula (the giant galaxy closest to ours) and discovered a blue shift that indicated (according to the Doppler effect) that the nebula moves towards us with a speed of about 300 kilometers per second, much higher than expected. Slipher then studied the light of other spiral nebulae and made the unexpected discovery that most of them, unlike Andromeda, show redshifts, that is, they move away from the solar system with great speed. In fact, he measured speeds above 1000 kilometers per second.

In 1919, the American astronomer Edwin Powell Hubble used the Mount Wilson telescope to photograph several spiral nebulae, including Andromeda, and showed that, actually, they were not nebulae, as had been believed, but huge clusters of stars. From then on they were no longer called nebulae, but galaxies, in honor of our Milky Way, which also belongs to the class of spiral galaxies. Galactos in Greek means milk.

Measuring the Universal Gravitation Constant

Vertical section of Cavendish balance

In 1798, the English physicist and chemist Henry Cavendish was the first to measure Newton's universal gravitational constant (G) using a spectacularly ingenious method, which has been scarcely improved later. The method was devised by John Michell, who died without being able to carry it out, so Cavendish performed the experiment. In fact, his objective was not to measure the constant, but the mass of the Earth, but the value of the constant could be inferred from the result.

Cavendish’s instrument was a torsion balance from which two identical balls of lead hung. Next to these balls, one on one side and one on the other, hung two much larger lead spheres, 175 kg each, which attracted the first two, producing a slight twist of the balance, which Cavendish could observe by means of a small telescope located outside the enclosure, to avoid observer interference. He thus detected a displacement of about 4 mm, which he measured with a precision of ¼ mm. This allowed him to calculate that the density of the Earth is 5.448 times greater than that of water, from which it is possible to deduce the mass of the Earth and the value of G:

G=6.674×10^-11N.m²/kg²

This is the official value, which is known with quite a low accuracy (1 in 10,000), compared with other universal constants.

Science or philosophy

Pierre Teilhard de Chardin

In a previous post, speaking about intelligence, I mentioned that there are four incompatible philosophical theories that try to explain the phenomenon of human consciousness. I summarize them briefly here:

1.     Reductionist monism or biological functionalism: The mind is completely determined by the brain and by the network of neurons that makes it. The human mind is an epiphenomenon. Freedom of choice is an illusion. We are programmed machines.

2.     Emergent monism: The mind is an emergent evolutionary product with self-organization, which has emerged as a complex system from simpler systems made up by neurons. Some argue that the underlying structures cannot completely determine the evolution of the mental phenomena. These, however, would be able to influence the underlying structures.

3.     Neuro-physiologic dualism: Mind and brain are different, but they are so closely connected that they make up a unit, two complementary and unique states of the same organism.

4.     Metaphysical dualism: Mind and brain are two different realities. The first is spiritual and non-spatial, capable of interacting with the brain, which is a material and spatial substance. Both entities can exist independently of one another, although the body without the mind eventually decomposes.

Interview with Manuel Alfonseca in a Spanish Newspaper

On February 23, 2018, a Spanish Newspaper (La Opinión, El Correo de Zamora) published this interview with me, performed by Ana Arias, which I am now translating into English. The interview was re-published a few days later (March 10) in the website ReligionEnLibertad (ReligionInFreedom). This is the translation of the interview:

He took an interest in science since he was quite small, as he says. At age 16 he wrote a book of zoology in two volumes that was never published. Anyway, whenever he has to consult information about some little known animal, he consults his book. "And I can find almost everything there," he adds. Now, at 71, he is an honorary professor at the Autonomous University of Madrid.

He believes in science. And also in God. Under the sponsorship of the Caja Rural Foundation and the Science-Religion University Forum held yesterday at the University College, Manuel Alfonseca gave a lecture about The Faith of Contemporary Atheist Scientists.

What is the faith of those scientists?

That God does not exist.

Cyclic Time and Linear Time

Stephen Hawking

In an article published in 1999, in volume 879 of the Annals of the New York Academy of Sciences, Pier Luigi Luisi speaks about the two traditional models of time that have been considered by traditional philosophy and the mythologies of various historical civilizations. They must not be confused with the two philosophical models originated in the twentieth century, the time A and time B of which I spoke in another post of this blog.

• Cyclical time, predominant in Asian civilizations and the Greco-Roman world until the Christian world view took root there. The origin of this model is evident, for many natural phenomena are cyclical: sunrise and sunset; the phases of the moon; the annual movements of the stars, synchronized with the seasons and with many biological phenomena...
• Linear time, prevailing in the three religions who consider themselves descendants of Abraham: Judaism, Christianity and Islam. Linear time can be compared with the course of the life of a living being, which begins at birth, goes on with changes during a certain period, and ends with death.

What does science say about miracles?

Victor Hugo

In his work Quatrevingt-Treize, a title usually translated as “Ninety-three” (it refers to 1793, the year of the Reign of Terror and the Vendée), Victor Hugo introduces a character named Cimourdain, a former priest who has lost his faith because of science:

Science had demolished his faith; the dogma had vanished in him... He knew everything about science and he knew nothing about life.

Apart from the fact that the second sentence is quite debatable (no one can know everything about science), the first raises the confrontation between science and faith, which began with the Enlightenment and reached its maximum philosophical effect in the nineteenth century, Victor Hugo‘s time. He is probably projecting his prejudices into a historical anachronism.

Conan Doyle’s mistake

Sir Arthur Conan Doyle

Sir Arthur Conan Doyle is famous as the author of the character Sherlock Holmes, the detective who relies on logic to solve the most abstruse cases, as in the famous quote from the story The adventure of the blanched soldier, included in the collection The Case-Book of Sherlock Holmes:

When you have eliminated all that is impossible, then whatever remains, however improbable, must be the truth.

Apart from his mystery books, one of his science fiction novels is also very well known: The Lost World, published in 1912, whose protagonist is Professor Challenger, an unbearable scientist, who also appears in other stories by Doyle. This is the plot of The Lost World:

A group of explorers manages to reach an almost inaccessible mesa, lost in the Amazon rainforest, so isolated that dinosaurs and other extinct animals survive there, as well as two races of humans or primitive pre-humans (Pithecanthropus and Homo sapiens). After they manage to escape and return to England, Challenger gives a lecture about his findings, which nobody takes seriously until he exhibits a specimen of Pterosaur that he managed to take from the mesa in the form of an egg, later incubated.

Irreversible time: illusion, or simplification?

Ilya Prigogine

We know Einstein believed that the passage of time is an illusion. In a letter of condolence he wrote in 1955 he said: ...the distinction between past, present and future is only a stubbornly persistent illusion. To assert this, he relied on the fact that Newton’s equations of gravitation, his own equations of General Relativity, Maxwell’s equations (which apply to electromagnetic waves) and Schrödinger’s equation (which gives the wave function of a particle in quantum mechanics) are all symmetric with respect to time.

How then can we explain the fact that it seems so obvious that time goes from the past to the future? Usually, physicists who believe that time is an illusion explain it by saying that, at the microscopic level, time is actually reversible, but when we move to the macroscopic level, new, emerging phenomena appear, one of which is the irreversibility of time. Let's give an example:

According to the usual theories, the movement of the molecules of a gas is perfectly reversible. If we reverse the direction of time, all the particles behave exactly the same and continue colliding with each other, only they would move in the opposite direction. However, when we consider all the trillions of particles that make up a gas, we see irreversible phenomena arising, such as the fact that the gas always tends to occupy as much space as possible, while its accumulation in a corner of the container is much less likely.

The problem is that our physical theories are based on approximations. Mathematics is a very important tool for physics, but in mathematics there are several kinds of very different problems, which differ in their difficulty to be solved. Let us look at a few:

Freedom and prior censorship

Wikipedia Logo

On July 5, 2018 the European Parliament rejected, by 318 votes against 278, the proposal for the Copyright Directive in the European Single Market. In the days leading up to this vote, there were many public and private activities in favor and against the proposal, which after this defeat will have to be debated again in committee, probably with amendments. The most controversial points of the proposal, those that gathered most rejection, were incorporated in two articles of the regulation:

• Article 11: Establishes what has been popularly called the Google tax. It makes it compulsory, for those responsible for web pages, to request permission, and if the copyright owners wish, to pay a fee, for including a link to a news or copyright owner that has appeared in any of the media. The most favored by this article are not individual authors, but mass media (especially the press on the Internet), the main defenders of this measure.

The MEPs who defended this article argue that it does not affect individuals or the Wikipedia, although the latter felt so threatened, that it declared a strike for the first time in its history, so that access to the Spanish, Italian and French versions of the Wikipedia was closed during the day before the vote. The problem is, this article may be expressed so ambiguously that, although just now may not apply to individuals or to Wikipedia, there are no guarantees that in the future this cannot be done.

Zero risk does not exist

No entry: radiation risk

We would like to live in a world where we run no risks, but that is impossible. Whenever we get into a car, cross the street, turn on the gas, or play sports, we run a risk. The most elementary acts of our life entail a risk: breathing polluted air; getting exposed to the natural radioactivity in buildings; passing under a roof just when a tile is falling down... We have always known that life is synonymous with danger, and we have adapted to that. In our time, however, it seems that the threshold of risk we are willing to tolerate has fallen down. In other words: we are now more cowardly.

The media are largely to blame. Trying to attract readers and increase their profits, they often encourage states of opinion close to panic. We can see it in the way many news are presented, especially those affecting health (mad cow syndrome, bird flu, SARS, type A influenza, whatever...); the viability of human life on Earth (global warming, collision with an asteroid); or the economy (times of crisis). Many of these threats are real, but they are systematically exaggerated.

Exaggeration of scientific news: superconductivity

Levitation of a superconducting sheet

In 1986, a team from the IBM research center in Zurich discovered high-temperature superconductivity. Until then, this phenomenon, well known since it was discovered by Heike Kamerlingh Onnes in 1911, only occurred at temperatures close to absolute zero. Thanks to the use of ceramic materials made with copper and rare earths, the critical temperature rose first to 35K, and soon after to 92K (Kelvin, or degrees above absolute zero). As a comparison, take into account that the fusion of ice into water takes place at about 273K.

Immediately the media announced this discovery as the door to a new technological revolution. Among the revolutionary applications announced were nuclear fusion, high-speed trains and ships that would move in levitation, the lossless transmission of electrical energy over long distances, supercomputers, and many more. The “fever” of the media grew even more when Bednorz and Müller, members of the team that made the discovery, were awarded the Nobel Prize in just one year, in 1987.

Has research on the human genome stopped?

We all know the Human Genome project, officially launched in 1990, although it had been working partially since 5 years earlier. Its purpose was to identify and decipher all the genes in human DNA in 15 years. The project was completed in 2003, within the foreseen term, although in the year 2000 partial results were published. From the scientific point of view, the project was a success, but perhaps for a part of the public it can look like a failure, as the exaggerated expectations aroused by some media have not been fulfilled.

The media hailed the project as the door to a new medical revolution. Among the revolutionary applications announced were: gene therapy to prevent or correct genetic diseases; premature diagnosis of actual or potential diseases, even from the embryonic stage; or personalized medicine, which would adapt treatments of diseases to the ailing person. Possible dangers were also discussed, such as the manipulation of human embryos to adapt their genes to the wishes of parents or dictatorial governments; or the use of genetic data to select personnel, or to grant or deny insurance and credit...

Extraterrestrials in literature

Extraterrestrials can only appear in two types of literary works: in essays, or in novels, and in the latter only in the genre of science fiction. If an extraterrestrial appears in any novel, the novel automatically becomes science fiction.

Science fiction literature shows very many types of extraterrestrials:

• Fully humanoid, such as the red men in the Mars series by Edgar Rice Burroughs, who are so humanoid that they are even fertile when mating with terrestrials, as shown by the two sons of John Carter and Dejah Thoris, despite the fact that Martian women are oviparous (!!!) To this group also belong the aliens of The People series by Zenna Henderson, who are only different from us by their mental abilities, and those of Perelandra by C.S.Lewis, also titled Voyage to Venus.
• Partially humanoid, such as those in Star Ways by Poul Anderson, whose women are also capable of falling in love with terrestrials. This novel develops a typical Anderson argument: extraterrestrials who differ culturally from us in their ecological view of the world, but who are fated to be defeated when confronting terrestrials, who are much more active and aggressive than they are.

Is Physics science or literature?

Freeman Dyson, who proposed a way
to extract energy from stars

We usually assume that physics is the most rigorous of the experimental sciences, the closest to mathematics, which serve as the fundamental basis for all sciences. However, some recent developments raise doubts about this. In other articles I have spoken of a few: the theories of the multiverse, time travel, that usually provide appealing headers in the media, but cannot be considered scientific theories, not because they cannot be verified, but because they cannot be proved false.

A recent article published in the high-profile journal Science News can be classified within this group, and in my opinion adds fuel to the fire, endangering the prestige of physics as a rigorous science and turning it into science fiction literature. This publication refers to an article recently published in arXiv, whose title is quite indicative: Life versus Dark Energy: How an Advanced Civilization Could Resist the Accelerating Expansion of the Universe. This article has been classified in the category Cosmology and Nongalactic Astrophysics.

What is a good historical novel

Battle of Borodino, by Louis-François Lejeune

In the previous post I mentioned War and Peace by Leo Tolstoy as a paradigmatic example of a good historical novel. In my opinion, the three golden rules for good historical novels are the following:

1.      The main characters are fictitious: In the case of War and Peace those characters are Pierre, Natasha, Prince André and their relatives, friends and spouses.

2.      The real historical characters are secondary: In War and Peace the historical characters are Napoleon, Alexander I of Russia and General Kutuzov. These characters act in the novel exactly as they did in reality. Regarding them, facts are not invented, they are interpreted.

3.      The lives of both groups of characters are intertwined perfectly.

Is History science or literature?

Allegory of Science, by Sebastiano Conca

There are different kinds of sciences. Some are rigorous and have great predictive power, others less, others practically none. Let us look at a classification of the sciences:

·         Formal sciences: They start from axioms or postulates, more or less unassailable, and use deduction as the main method of reasoning. A few play the role of fundamental basis for other sciences. In this group, we can include mathematics, logic and theoretical computer science.

·         Natural sciences: They use induction as the main method of reasoning. Their objective is to discover fundamental laws that explain the working of the world. They rely more or less on logic and mathematics. These sciences include (in order of decreasing rigor) physics and astronomy; chemistry; biology, geology and paleontology.

·         Social Sciences: They use abduction as the main method of reasoning (see an earlier article in this blog). Their object of study is man or society. These sciences include psychology, economics, sociology, anthropology, politics, archeology and history.

·         Finally, the applied sciences, whose objective is to develop practical applications from the theoretical knowledge provided by the experimental and social sciences. They usually associate under the name of technology, although there are some disciplines that fall outside that umbrella, such as legal sciences, applied economics, medicine, or applied psychology.

Mistakes in popular science in the media: Stephen Hawking didn’t discover everything

Stephen Hawking

Stephen Hawking has been in the last decades a scientific icon for the media. His painful personal situation turned him into a celebrity who inevitably attracts attention. Therefore, the media have a tendency to exaggerate his scientific work, attributing to him achievements that weren’t his, which he would be the first to repudiate, if he were still among us.

For example, on the occasion of his death, the following headlines appeared in several media:

•         AlDiaNews.com: Stephen Hawking, author of the Big Bang theory dies at 76

•         ElTiempoHoy: Creador de la teoría del Big Bang y los agujeros negros: fallece Stephen Hawking a los 76 años. (Creator of Big Bang’s theory and black hole theory: Stephen Hawking dies at 76).

•         Gizmodo Univisión: Del Big Bang a la Teoría del Todo: el increíble legado científico de Stephen Hawking (From the Big Bang to the Theory of Everything: Stephen Hawking’s incredible legacy).

Gödel and realism

Kurt Gödel

Kurt Gödel (1906-1978) was one of the most important mathematicians of the 20th century. In 1931, when he was 25, he rose to fame with his mathematical proof that the attempt to build a complete axiomatic system, from which one can deduce all the arithmetic of natural numbers or any equivalent system, is doomed to failure.

His first incompleteness theorem says the following:

Every consistent formal system as powerful as elementary arithmetic is not complete (it contains true undecidable propositions).

Let us look at a simplified informal demonstration:

Let theorem G say the following: This theorem G cannot be proved from the axioms and rules of system S.

• If we assume that Theorem G is false, system S is inconsistent, since a false theorem can be proved from the axioms and rules of S.
• Then if S is consistent, G must be true, and therefore cannot be proved from the axioms of S.
  

Gödel’s theorem shows that every axiomatic formalization of arithmetic is either inconsistent (it allows false theorems to be proved), or incomplete (it contains true theorems that cannot be proved).

The 528th digit of Pi

Gotfried Wilhelm von Leibniz

Two posts ago I mentioned that the best simple fractional approximation of the value of p is 355/113 = 3.14159292..., which was discovered in the West in the 16th century. Later, better approximations were obtained, but no longer in the form of a fraction, rather as the sum of a series. Several infinite series of terms are known whose sum is p. So it is enough to add a sufficiently large number of terms to obtain as many digits of p as we want, as long as we have time to do the sums. The first to propose one of these series was the French mathematician François Vieta. As his series was quite complicated, we give here the much better known series proposed in 1673 by the German mathematician and philosopher Gotfried Wilhelm von Leibniz:

The more terms we add of this series, the closer we will come to the value of p. The following table shows the advances made over time in the calculation of the successive approximations of this number, using different series, formulas or procedures.

The mystery of the Great Pyramid

The Great Pyramid of Giza, also called Pyramid of Cheops or Pyramid of Jufu, was built to be the tomb of the pharaoh Jufu (called by the Greeks Cheops), of the fourth dynasty, the high point of the Ancient Egyptian Empire. The reign of Jufu is usually dated in the 26th century before Christ, over 4500 years ago.

The current height of the Great Pyramid is 138.8 meters, but the pyramid is truncated, having lost its top. It is easy to calculate that its original height was about 8 meters higher: 146.7 meters, or 280 Egyptian cubits. The base of the pyramid is a square with a side of 230.34 meters, or 440 Egyptian cubits.

Observe a curious point: the semi-perimeter of the pyramid (twice the side of the base) is equal to 880 cubits. If we divide it by the height of the pyramid, we get the following:

(880/220) = (22/7) = 3.142857...

Are the digits of Pi real?

Martin Gardner

In an article published in Discover magazine in 1985, Martin Gardner wrote this:

As it happens, the thousandth decimal of pi is 9... The question: Was [this assertion] true before the 1949 calculation? To those of the realist school, the sentence expresses a timeless truth whether anyone knows it or not... [Others] prefer to think of mathematical objects as having no reality independent of the human mind.

This problem is quite old, as we have been discussing it for over two thousand years. The question about whether mathematical objects really exist or are a pure creation of our mind is a particular case of another problem, much more general, that debates whether ideas and concepts (like the dog species) really exist, or just this dog and that dog exist. This is the problem of universals, famous in the Middle Ages, which has not yet been solved to everyone’s satisfaction. In fact, at present, this debate is more virulent than ever.