Tuesday, June 15

The Rehearsal: From meta-theater to the fundamentals of mathematics | Coffee and theorems

Cuqui Jerez, one of the creators of 'The Rehearsal', in the Parque de Enrique Tierno Galván.
Cuqui Jerez, one of the creators of ‘The Rehearsal’, in the Parque de Enrique Tierno Galván.KIKE TO

What is the metatheater? We add the meta- prefix to an object, discipline or field of human creation when we want to emphasize that the self-reference is part of it. In the case of a meta-theater work, elements appear that refer to the representation itself. Self-reference thus creates different levels of meaning that are related to each other, suggesting a world of connections with philosophy, logic and mathematics.

Some of these interesting connections can be found in The Rehearsal, a work created by the sisters Cuqui and María Jerez, Cristina Blanco, Gilles Gentner and Amaia Urra, and premiered in 2008. According to Cuqui Jerez herself, “The Rehearsal it is a fiction within a fiction within a fiction … and so on ad infinitum […] A rehearsal is led by a director when another director interrupts and directs the director who is leading the rehearsal. Another director interrupts and directs the director who directs the director who directs the rehearsal ”.

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Self-reference is ubiquitous in all arts, be it plastic, from the painting within the painting, represented, among others, in Las Meninas from Velázquez, to Escher’s most conceptually complex works; literary, like Don Quixote of La Mancha, by Cervantes, or the vast universe of Borges; and also musical ones, like Bach’s canons, or Rosalía’s sound selfies.

In the case of the performing arts, we can find metatheatrical elements in classical Greek tragedies and, more obviously, in The life is dream of Calderón de la Barca or Hamlet Y Enrique IVby Shakespeare. However, it is usually considered Six characters in search of author, by Luigi Pirandello, as the first modern metatheatrical work. Released in 1921, the play places us in a theater where actors, director and stagehand are rehearsing. His work is interrupted by six characters who claim to have been created by an author who never wrote his work, and seek to convince the director to put his drama on stage. Soon, all six characters will end up mocking the actors’ ridiculous attempts and inability to portray them.

The incompleteness theorem states that all formal language rich enough to refer to itself has formulas that cannot be proved or disproved

Also in the 1920s, the logician and mathematician Kurt Gödel was conducting his doctoral thesis in Vienna, in which he studied formal languages, specifically, the relationship between notions such as consistency, demonstrability and completeness. In 1931 he published the demonstration of his famous incompleteness theorem, which asserts that all formal language rich enough to refer to itself has formulas that cannot be proved or disproved. Self-reference is fundamental in Gödel’s ideas, since his theorem starts from constructing a mathematical sentence that affirms its own indemonstrability. Essay Gödel, Escher, Bach: an eternal and graceful loop, by Douglas Hofstadter, presents the suggestive interactions between Gödel’s work in logic, Escher’s drawings, and Bach’s canons and fugues, with self-reference as one of the main threads.

A similar idea, albeit on a more elementary level than Gödel’s works, gave rise to the so-called Russell paradox, which caused a crisis in the foundation of set theory at the beginning of the 20th century. Remember that the mathematical edifice, on which so many advances in the different current sciences and technologies are based, rests on a foundation formed by a few logical rules and axioms of the set theory. In this rationale, one must clearly define what is a “set” (and what is not) to avoid ambiguity. Set theory, in the naive version that was considered at the end of the 19th century, proposed calling any definable collection a “set”.

We say that the elements that form a set “belong” to it. For example, we could consider the set of sentences in this text. Thus, it would be correct to say that this phrase belongs to that set. We could also consider the collection of all sets that do not belong to themselves – a clearly self-referential definition; let’s call C. Note, for example, that the set of sentences in this text does not belong to itself, because a set of sentences is not a sentence. The key question is, can it be C a set? If it is, then either it belongs to itself or it does not; if it belongs to itself, then – by definition of C– cannot belong to C; but if it doesn’t belong to itself, then it belongs to C. Therein lies the paradox – C belongs and does not belong to himself – which comes from supposing that C it is a set. Therefore, the definition of the set itself had to be revised.

The use of repetition and self-reference to imply the idea of ​​something infinite is masterfully employed in ‘The Rehearsal’

Self-referencing is also a useful tool for making precise sense of the notion of infinity. In fact, we can understand infinity in terms of repetition: to build the natural numbers we start with 1, and from it, adding one, we build the next number (which we call 2), and from 2, the next (or be, 3), and from 3, the next … and the key is that forever I will be able to construct the next number.

The use of repetition and self-reference to imply the idea of ​​something infinite is masterfully employed in The Rehearsal; in all its approach, but especially, in its conclusion (which we will not reveal to avoid spoilers to potential viewers). The Rehearsal It constitutes a sublime exercise in dissemination, an invitation to reflection and self-reflection, recommended for any lover of theater, mathematics or philosophy.

Pedro Tradacete He is a researcher at the Higher Council for Scientific Research (CSIC) at ICMAT

Coffee and theorems is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”

Editing and coordination: Agate A. Timón G-Longoria (ICMAT).

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