Physics, Mathematics and Mathematical Physics

Eugene Wigner

Eugene Paul Wigner was a Hungarian physicist who received the Nobel Prize in Physics in 1963 for his contribution to the theory of the atomic nucleus and elementary particles. In a famous article published in 1960, Wigner said:

It is important to point out that the mathematical formulation of the physicist's often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena. (“The unreasonable effectiveness of mathematics in the natural sciences”. Communications on Pure and Applied Mathematics 13: 1-14).

Determinism or indeterminism?

Modern science has revolutionized our vision of the world. In the eighteenth century, Newton's theory of gravitation could be considered established, and gave rise to a materialistic-deterministic vision of the universe which can be personified in one of the most significant scientists of the time, Pierre-Simon, Marquis de Laplace (1749-1827), whose fields of study covered mathematics, astronomy, chemistry and biology. The success of his studies on the dynamics of the solar system moved him to state that, if we knew the exact initial conditions of the universe, it would be possible to predict all its past and future development. This gave rise to materialistic determinism, so successful in the nineteenth century and still a part of the popular vision of the world, in spite of the three devastating attacks it has suffered during the twentieth century.

Does the Moon influence people?

Astrologers have been answering this question in the affirmative since time immemorial. It’s true that astrology has been discredited since a few centuries ago, to the point that another name had to be found for the science dedicated to the study of the stars (astronomy = laws of the stars). Despite which, in this supposedly scientific age, the mainstream media dedicate significant space to horoscopes and other astrological products.

Sometimes the influence of the moon has been confirmed. Since ancient times it was observed that the tides are related to the position of the moon in the sky, although it was not known how this influence could take place. To support the theory of Copernicus against that of Ptolemy, Galileo formulated a theory, according to which the tides are not due to the attraction of the moon, but to the translational movement of the Earth around the sun. In this case Galileo was wrong, because the influence of the moon is real, although it was not explained until Isaac Newton formulated the theory of universal gravitation.

Did Dante anticipate Einstein?

A recent article has stated that Dante Alighieri's Divine Comedy offers a cosmology that closely resembles what Einstein expressed in his general theory of Relativity. Is there any truth in this?

In another post in this blog I summarized the history of cosmology, from the geocentric Greek version formalized by Ptolemy, to the modern version by Copernicus, Kepler and Newton. It is evident that Dante, who wrote the Divine Comedy at the beginning of the fourteenth century, could not know about modern cosmology, but he did know the Ptolemaic system, which he adopted in its entirety, with an important addition.

The relationship between the systems of Dante and Einstein was pointed out in an article published in Scientific American in August 1976, written by J.J. Callahan and entitled The curvature of space in a finite universe. This article compares Newton's universe (finite, non-homogeneous, Euclidean and with one center), Leibnitz's (infinite, homogeneous, Euclidean and without a center) and Einstein's (finite, homogeneous, non-Euclidean and without a center). By adapting to Euclid's plane geometry, the first two can be represented by graphic models as those in the attached figure.

The mystery of Planet X

Urbain Le Verrier

In 1845, some 60 years after the discovery of planet Uranus, the French astronomer Urbain Le Verrier tried to solve the problem posed by the discrepancies of a few minutes of arc between the observed orbit of this planet and the predictions made by applying Newton’s theory. Le Verrier thought the problem could be solved if there were another unknown planet beyond Uranus. On September 23, 1846, the German astronomer Galle discovered this new planet, which received the name of Neptune. The success of the prediction became top-notch scientific news and boosted Newton's theory of gravitation.

For decades, some unexplained irregularities in the orbits of Uranus and Neptune were attributed to the existence of another possible planet, located even further from the sun. In 1906, Percival Lowell undertook, at his private observatory in Falstaff, Arizona, a search program for the so-called Planet X, not because of its number in the list of planets (which would be 9), but because the letter X traditionally represents the unknown in a mathematical expression. In 1930, after Lowell's death, Clyde Tombaugh discovered Pluto, which came to be considered planet number 9, but as its small mass was insufficient to explain the discrepancies, the X in the name of planet X automatically came also to mean number 10, in accordance with the meaning of that letter in the Roman numeral system.

In 1987, the search for planet X was still unsuccessful. From the analysis of the orbits of Pioneer X and XI, it was deduced that none of those space capsules had been subject to the influence of the mysterious planet X, so if this planet existed, it must be in a very elliptical orbit, inclined at least by 30º with respect to the plane of the trajectory of the two space capsules.

In 1992, it was proposed that planet X might not be a planet, but a swarm of bodies the size of Pluto (1,000 at least). In 1999, the analysis of the orbits of some comets gave rise to the proposal of the existence of a planet the size of Jupiter (or even larger) 25,000 astronomical units from the sun. An astronomical unit (1 AU) is the distance from Earth to the sun. This possibility was later rejected when NASA's WISE satellite ruled out the existence of an unknown body the size of Saturn less than half a light-year from the sun (about 30,000 AU). Nor can there be any Jupiter-sized body less than 1.5 light-years away (90,000 AU).

In 2001, the analysis of the orbit of Comet 2000 CR105 gave rise to the proposal of the existence of an unknown planet, intermediate in size between Mars and Earth, at the distance of the Kuiper Belt, an accumulation of objects beyond Neptune, between 30 and 55 astronomical units of the sun.

Pluto

On August 24, 2006, the International Astronomical Union decided that Pluto would no longer be considered a planet, moving it to the category of dwarf planet or plutoid, together with other bodies, such as Ceres, the largest of the asteroids, and Eris, more massive than Pluto and located farther away. With this change, automatically, planet X became known again with the alternative name of planet 9, and the double meaning of the letter X was no longer applicable.

Starting in 2016, the analysis of the orbits of several recently discovered bodies in the Kuiper belt and some simulations, suggested the idea that there could be an unknown planet up to 10 times larger than Earth, or another body of equivalent mass, at 500 or 600 AU from the sun. In comparison, Neptune's distance from the sun is less than 30 astronomical units, and its mass about 17 times that of Earth.

From Investigación y Ciencia, April 2016

The latest theory suggests that planet 9 might not be a planet, but a primordial black hole about 10 times more massive than Earth, located about 500 AU from the sun, which would have been formed shortly after the Big Bang, and which would be much more difficult to detect than a planet. However, there are proposals to locate its exact position, such as one, made in 2020, which would consist of sending a fleet of spacecraft weighing about 100 grams each, accelerated by laser beams to a speed of 300 km/second, with which they could reach that distance in about 10 years. They would send radio pulses that would make it possible to detect if the small probes had been subjected, somewhere in their trajectory, to the attraction of an unknown body. Edward Witten, one of the authors of the proposal, is skeptical and says that it is far from clear that this approach is practical, while other researchers, such as Mike Brown, assert that there is zero reason to think that Planet Nine is a black hole.

Meanwhile, there are other options. A theory dating back to 1984 argues that the sun could be part of a binary star system, and that its companion (nicknamed Nemesis, the Greek goddess of revenge) would have to move through an extremely elongated orbit that would take it to a maximum distance of the sun of 88,000 astronomical units (more than a light-year). When the star, which could be a brown dwarf (which would make it very difficult to locate), approached the sun once every 26 million years, its influence on the Oort comet cloud would cause many of them to rush over the inner solar system. Their impacts with the Earth would cause massive extinctions, similar to that at the end of the Cretaceous period that put an end to dinosaurs. This theory lost weight when it was found that the 26 million-year period of mass extinctions could, after all, be a statistical artifact.

The same post in Spanish

Thematic Thread on Space Exploration: Previous Next

Manuel Alfonseca

Scientific models: adjustment or validation?

Leonard Nimoy
as Mr. Spock

One of the ways in which science advances is by building models, which are often made up of more or less complex sets of mathematical equations, and trying to verify whether or not these models adapt to the functioning of the real world, as described by our senses and our instruments.

When building and using a model we must consider two distinct phases:

• Model adjustment: it consists of assigning values ​​to the parameters of the model to ensure that it fits the data we already have about the real world. A model not adjusted to such prior knowledge would be totally useless.
• Model validation: it consists of using the model to make surprising predictions that nobody could have foreseen without the help of the model. If these predictions are confirmed, they become surprising accurate predictions, validating the model. However, the validation is never final, for a new surprising inaccurate prediction could invalidate it in the future.

Let's look at a few examples:

Why we have no great men today

G.K. Chesterton

First, a clarification: I won’t let myself be dragged by political correctness. I’m not going to change the title of this post to “great human beings.” For me, the word “man” (equivalent to the Latin homo) still has a main generic meaning, different from the meaning whose Latin antecedent is vir (male), opposed to woman or female.

The absence of great men is a common place today and affects almost all fields:

Five years in PopulScience

Albert Einstein

This week we celebrate a small anniversary: five years since this blog was created. In this time, 245 posts have been published. The Spanish version of the blog is a little older: it was created 30 weeks before, in January 2014, and has published 257 posts.

To mark the date by some kind of celebration, I have decided to compute the list of people most mentioned in the blog in these five years. The following table shows the names of the ten people most quoted and the number of times their name has been quoted:

Name	Times quoted in PopulScience
Albert Einstein	42
Isaac Newton	33
Stephen Hawking	20
C.S. Lewis	20
Aristotle	17
Charles Darwin	14
Isaac Asimov	14
Richard Dawkins	12
Plato	10
Ptolemy	9

Travelling to the past?

S.Augustin, by Louis Comfort Tiffany
Lightner Museum

In his Confessions (Book XI, chapter 14), St. Augustine wrote these words, still valid today:

What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know.

In the current situation of our scientific and philosophical knowledge, we still don’t know what time is.

·         For classical philosophy and Newton’s science, time is a property of the universe. Therefore, time would be absolute.

·         For Kant, time is an a priori form of human sensibility (i.e. a kind of mental container to which our sensory experiences adapt).

·         For Einstein, time is relative to the state of repose or movement of each physical object. There is, therefore, no absolute time.

·         For the standard cosmological theory, there is the possibility to define an absolute cosmic time for every physical object, measuring the time distance since the Big Bang to the present.

·         For the A theory of time (using J. McTaggart’s terminology) the flow of time is part of reality. The past no longer exists. The future does not yet exist. There is only the present. If the A theory is correct, travel to the past is impossible, because you cannot travel to what does not exist.

·         For the B theory of time, the flow of time is an illusion. Past, present and future exist simultaneously, but for each of us the past is no longer directly accessible, and the future is not yet accessible. Einstein adopted the B philosophy of time. In a condolence letter written to someone who had lost a beloved person, he wrote the following:

The distinction between past, present and future is only a stubbornly persistent illusion.

Anti-realist answers to the no-miracles argument

Hilary Putnam

The previous post described the no-miracles argument, proposed by Hilary Putnam. The article ended thus:

What do anti-realists answer to this argument? Are they convinced?

I guess the readers have deduced that the answer to the second question must be negative, otherwise the debate between realism and anti-realism would have ended. Let us look, therefore, at the answer to the first question. Faced with the abductive argument of no-miracles, anti-realists answer in two different ways:

1.      Bas van Fraassen is an anti-realist American philosopher who criticizes Putnam’s argument, arguing that scientific theories are successful because unsuccessful theories have been eliminated by natural selection (i.e. scientists have ruled them out). Therefore, asking why science is successful is similar to asking why basketball players are tall: because they have been selected. Let us see how Fraassen describes his theory, which is called constructive empiricism:

The debate of realism and anti-realism

Gottlob Frege

The secular debate between realism and nominalism (or anti-realism, its now preferred name), has been expressed in a few new theories of the so-called analytical philosophy, whose origin dates from the early twentieth century, with Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein, the Circle of Vienna and several philosophers of the last fifty years, especially in the Anglo-Saxon world.

Currently, the two camps, realist and anti-realist, agree on one thing: science works. But although this is considered an incontrovertible fact, very divergent positions are posed to explain it.

As it has always happened throughout history, neither of the two fields is united. Both realism and anti-realism are divided into two branches, at the least.

Let us start by describing the realist position:

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.

What really happened in the history of cosmology

To complete last week’s post, I will offer here a summary of the history of Cosmology, from the Greeks to the paradigm shift that took place in the sixteenth and seventeenth centuries.

The basic elements of Ptolemaic astronomy, showing a planet on an epicycle (smaller dashed circle), a deferent (larger dashed circle), the eccentric (×) and the equant (•).

• Greek cosmology (with the exception of Aristarchus of Samos) put the Earth at the center of the universe. Plato and, above all, Aristotle established the idea that, since the sky is perfect, the orbits of the planets must be exactly circular, because, for them, the circumference is the most perfect curve of all.
• The Greek model explained well the movements of the sun and moon, and therefore made it possible to predict eclipses, but had a problem with the retrograde movements of the planets then known (Mercury, Venus, Mars, Jupiter and Saturn). Three centuries before Christ, Apollonius of Pergamum proposed that the orbits of these planets are epicycles, circumferences centered on another circumference (the deferent), which in turn revolves around a point located near the Earth, but apart from its center (the eccentric).

Was physics wrong in Ptolemy's cosmology?

1919 solar eclipse

A recent article in the journal Science News has this title: Eclipses show wrong physics can give right results. It claims that Ptolemy’s physics was incorrect, because he assumed that the Earth was at the center of the universe, and yet Greek science was able to predict the dates of eclipses.

According to the article, Ptolemy’s physics was less correct than the physics of Copernicus, who fourteen centuries later proposed that it was not the Earth, but the Sun, at the center of the universe.

The analysis in this article in Science News is completely wrong. Ptolemy’s physics was exactly the same as the physics of Copernicus. Copernicus did not propose a change in the physical theories that had governed classical astronomy since Hipparchus (2nd century BC). Copernicus just showed that, with a change in the coordinate system, and applying the same physics, the calculations are easier to perform. Logically, the same results are obtained.

Newton, the greatest scientist of our civilization

Isaac Newton

As I said in the previous article, in my biographical dictionary 1000 great scientists (1996) and an unpublished book, I proposed an objective quantification of the importance of different scientists, using measures such as the number of lines that various encyclopedias assign to each. Six scientists, one Greek (Aristotle), of whom we have already spoken, and five from the West (Descartes, Newton, Darwin, Freud and Einstein) were tied with the highest score in these studies. Among these five, is there one who can be considered the greatest scientist of our civilization?

In 1964 Isaac Asimov conducted another study (The Isaac Winners) on the relative importance of men of science, which resulted in a list of the 72 best scientists of all time, in his opinion. This list is simply qualitative and does not establish a relative order among the names that appear in it, although Asimov (again in his opinion) asserts that Isaac Newton, who happened to be his namesake, was the greatest scientist of all time.

The debacle of determinism

Isaac Newton

By the end of the eighteenth century, Isaac Newton’s theory of universal gravitation was well established. As this theory makes it possible to predict very accurately the orbits of the bodies in the solar system, the French astronomer Pierre Simon de Laplace believed he had sufficient reasons to say the following:

An intelligence that knew all the forces that animate nature, as well as the respective situation of the beings that make it... could cover in a single formula the movements of the largest bodies of the universe and those of the lighter atom. Nothing would be uncertain and both the future and the past would be present before his eyes.

This assertion became the dogma of deterministic materialism, a philosophical (not scientific) doctrine asserting that only matter exists (taking the term broadly) and that the whole history of the universe is determined. Therefore there is no human freedom, nor intentionality, nor final causes in nature. There are just efficient causes.

Laplace’s statement can be expressed in more modern terms:

If we knew the position and the momentum of all the particles of the universe at a given instant, we could predict all their past and future development.

Dark matter or a new theory

Urbain Le Verrier

Science studies facts and tries to explain why they occur. Scientific theories are the more credible, the more facts they explain or predict. A single fact in opposition to a theory, or a single unconfirmed prediction, is enough to make us consider revising the theory. With the scientific method, theories are never final and facts must always take precedence.

We have a classic historical example in the theory of universal gravitation, which allowed Newton to explain events like the fall of bodies and the movement of planets and satellites. Its first achievement, by Newton himself, was the mathematical deduction of Kepler’s three experimental laws, obtained empirically from the observation of the orbits of the planets. But the greatest success of the theory was a correct prediction when discrepancies were detected between the orbit of Uranus deduced from the theory and the observed orbit. When something like this happens, the problem can be solved in two ways:

The hollow Earth in pseudoscience and science

Cyrus Reed Teed (Koresh)

1870 saw the first appearance of a curious variant of the hollow Earth theory outside the literary field. The American Cyrus Read Teed proclaimed his belief that the Earth is hollow, but (here is the difference with previous theories) we live inside. Although the sea surface has been known for over two thousand years to be convex, and in spite of the arguments that led the Greek philosophers to assign the Earth a spherical shape, with ourselves on its outer surface, Teed was convinced that the Earth is really concave. The apparently infinite outer space would be a hollow bubble inside a universe made of rock. Teed changed his name to Koresh and founded a religion (Koreshanity) which reached several thousand followers, although they were scattered after his death in 1908.

Soon after, a German aviator named Bender, a prisoner in France during the First World War, read Teed publications and believed them. Bender developed these theories and asserted that the universe is an infinite mass of rock surrounding a bubble 13,000 kilometers in diameter, in whose inner surface we live. The atmosphere, 60 kilometers thick, thins up to the central vacuum, where three bodies move: the sun, the moon and the ghost universe, a ball of gas with shining points of light: the stars. When the ghost universe passes before the sun, it causes the alternation of day and night in the various regions of the inner surface of the Earth.

The fallacy of the invisible cat

Isaac Newton

In Chapter 1 of his book Astrology, science or belief? published in 1992, Manuel Toharia writes:

However wise they can be about certain subject matters, there is always some element that contradicts the myth of the perfect genius. For example, it is well-known that Newton was an angry man, terribly unfriendly and probably a repressed homosexual. Lest there be any misunderstanding, we must add immediately that what we find wrong with this alleged homosexuality of the English genius is its repression, which certainly made him a bitter person, no doubt with a minimal dose of self-esteem.

Probably a repressed homosexual? And how can we know this, if it is true that Newton repressed it? Or did Toharia (or whoever was his original source) have inside information, or perhaps he came to this conclusion because he knows that Newton suffered at least two psychic crises in his life, and believes that their cause must have been his repressed homosexuality? Observe the use of the qualifiers certainly and no doubt. If so, his argument would be a textbook example of the fallacy of the invisible cat:

The celebration of Christmas

Earth lighting in the winter solstice

The time of the winter solstice was the occasion for major celebrations by nearly all ancient peoples. It represents the time when the sun, after losing height for six months, begins again to recover its upward movement. The ancients had always the fear that some year the sun could fail to recover, and would continue down until disappearing forever, a catastrophe for humanity.

In the Roman Empire, the Saturnalia played that role, for Saturn was the god of agriculture and the recovery of the sun was a sine qua non for the success of the next harvest. Also, just by that time the sun entered the sign of Capricorn in the zodiac, which was astrologically linked with the planet Saturn. The festival, which began on December 17, lasted for several days, until the 23rd. During these days, banquets were held, gifts were distributed, and masters served their slaves.

One of the traditional deities of ancient Indo-European peoples, Mitra, had a varied fate, according to the particular people we are talking about. Thus, in Vedic India he was one of the chief gods, along with Varuna and the other asuras, but came to play a secondary role, almost demonic, when in Hindu India prevailed the devas, another group of gods including Siva and Vishnu.