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).

Computational Intelligence and Consciousness

Eduardo César Garrido Merchán

In recent years there have been many advances in artificial intelligence, especially in the field of automatic generation of texts and images that sometimes compete successfully with human productions. In light of this, the media, and even some scientists, have rung the bells announcing that we are on the verge of creating conscious artificial intelligence, which would compete with human beings as our equal. But others believe that this goal, if it were possible (which is not clear), is much further away than some think.

In an article signed by Eduardo César Garrido Merchán and Sara Lumbreras and published in the journal philosophies with the title Can Computational Intelligence Model Phenomenal Consciousness, the authors review Bertrand Russell's analogy, which asserts that consciousness and intelligence are closely correlated. In other words, any entity that possesses consciousness will also possess a high level of intelligence, and vice versa. In a way, this analogy is similar to the Turing Test, which is much better known.

The limits of mathematics

Kurt Gödel

In the last decades of the nineteenth century, Friedrich Ludwig Gottlob Frege, a professor in the university of Vienna, undertook an ambitious goal: formalizing the arithmetic in a set of axioms and deduction rules, in such a way that every true theorem would be deductible from the axioms by a finite number of applications of the deduction rules. The result was a monumental book, Grundgesetze der Arithmetike (1893-1903), which introduced, among other things, a basic formalization of set theory and a cumbersome notation, quickly replaced by Peano’s, which we are using now.

Unfortunately for Frege, when the second volume of his book was about to be published, he received a letter from Bertrand Russell, proving that his formulation of set theory entails an inconsistency. In Frege’s set theory, some sets are not member of themselves (as the set of all integers, which is not an integer), while other sets are members of themselves (as the set of all infinite sets, which is an infinite set). Russell then defined this set: the set of all sets that are not members of themselves. It is easy to see that this set leads to a paradox: if it is a member of itself, it cannot be a member of itself, and vice versa. Russell’s paradox wreaked havoc with Frege’s work, who had to add a hasty appendix to his book and then abandoned his research on the fundamentals of mathematics.

Mathematical theology

Ernst Zermelo

Ernst Zermelo (1871-1953) was a famous mathematician of the early twentieth century. Among his achievements, the following can be mentioned:

• In 1899 he discovered Russell’s paradox, two years before Russell. Although he did not publish it, he did comment it with his colleagues at the University of Göttingen, such as David Hilbert. Russell’s paradox proved that Cantor’s set theory is inconsistent, since it makes it possible to build the set of all sets that don’t belong to themselves. There are sets that don’t belong to themselves, such as the set of even numbers, which is not an even number. Others do belong to themselves, such as the set of infinite sets, which is an infinite set. Now we can ask ourselves: Does the set of all the sets that don’t belong to themselves belong to itself? This question leads us to a paradox: if it does belong, it must belong; and if it doesn’t belong, it mustn’t belong.
• In 1904 he proved the well-ordering theorem as the first step towards proving the continuum hypothesis, the first of Hilbert’s 23 unsolved problems. The well-ordering theorem states that every set can be well-ordered, which means that any non-empty ordered subset must have a minimum element. To prove it, he proposed the axiom of choice, which we will discuss later.
• In 1905 he began to work on an axiomatic set theory. His system, improved in 1922 by Adolf Fraenkel, is a set of 8 axioms, which today is called the Zermelo-Fraenkel (ZF) system. Adding the axiom of choice to this system, we obtain the ZFC system, which is most used today in set theory.

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:

Four ideas by Alvin Plantinga about God and materialism

Alvin Plantinga

Taking advantage of the awarding of the Templeton Prize to the American philosopher Alvin Plantinga, this post will try to review a few of his thoughts in the debate between theism and materialism. As it is impossible to review all his work in detail, I will mention just four of his ideas:

1. The Mozart argument for the existence of God. Why are we able to appreciate beauty? According to the materialistic hypothesis, there is no explanation why evolution has led us to this, as it is difficult to see how this trait could be useful for our survival. Instead of good music, we should appreciate cacophony, which is more abundant in nature. If we assume that God exists, however, this fact is easy to explain, because God appreciates beauty (in fact, God is beauty). This argument, along with many others, is in this web address.

The myth of the Dark Ages

Bertrand Russell

Echoing the myth of the Dark Ages, a name for the European Middle Ages invented by the writers of the Enlightenment, Bertrand Russell wrote these words in his book Wisdom of the West (1959):

As the central authority of Rome decayed, the lands of the Western Empire began to sink into an era of barbarism during which Europe suffered a general cultural decline. The Dark Ages… It is not inappropriate to call these centuries dark, especially if they are set against what came before and what came after.

What came before was the Roman Empire; what came after the Renaissance.

The myth of the Dark Ages was invented by the writers of the first half of the eighteenth century to enforce another myth they had created, according to which at that time we were entering a new era of reason and knowledge, especially scientific knowledge, which they called by the name of the Enlightenment.

In the Espasa Dictionary, 1000 great scientists (1996) I proposed an objective procedure to quantify the relative importance of the various practitioners of science, using measurements such as the number of lines assigned to each scientist in encyclopedias of different countries (to avoid the bias in favor of countrymen). Later, in an as yet unpublished work (The quantification of history and the future of the West), I applied the same procedure to several branches of human creativity: science, philosophy, literature, the plastic arts and music. The next figure represents the resulting evolution of the Greco-Roman and Western science until the end of the Middle Ages. It can be seen that:

The god of the gaps

In 1977 Pergamon Press published a curious book called The Encyclopedia of Ignorance, which tried to collect, in a collection of articles written by specialists in different areas, most of the problems (then) unresolved in fields such as cosmology, astronomy, particle physics, mathematics, evolution, ecology, biological development, medicine and sociology. Some of these problems have not yet been solved, almost 40 years later; others, like the mystery of the missing neutrinos in the solar radiation, which I mentioned in the previous post, seem to be in the way of being resolved, although this has led to the emergence new problems, as often occurs in science.

Since the nineteenth century, one of the typical accusations of atheists against believers has been that they resort to the god of the gaps, i.e. to use God to explain those things we still don’t know about the structure of the world. We are still far from knowing everything, because science is (and probably always will be) incomplete: there will always be mysteries. Well, believers are accused to rely precisely on the mysteries (the gaps of science) to justify the existence of God. According to this view, God would be nothing more than the deus ex machina of the Greco-Roman drama, who appeared to solve the unsolvable problems where the playwright had entangled his characters. As science advances, the holes will be filled and the need to turn to God will get lower.