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Guillermo Martínez & Gustavo Piñeiro

So far, in this Library section, we have only covered books in English. We have already discussed the hegemony of this language, and we think it is important to challenge it; so today we break the mould and introduce a book originally published in Spanish in 2009, “Gödel ∀ (para todos)” by Argentine mathematicians Guillermo Martínez and Gustavo Piñeiro, the former also a renowned fiction author.

The name of Kurt Gödel often appears in the pages of this magazine, and with reason. After all, the thought process that led to modern computing has Gödel as a major milestone and can be clumsily summarized as follows: Cantor ⇒ Hilbert ⇒ Russell ⇒ Gödel ⇒ Church ⇒ Turing ⇒ Von Neumann. This outstanding sequence of names happened in human history in a terrifyingly short span of time: roughly between 1880 and 1960. This means that in roughly 80 years, we went from Victorian-era steampunk machines to the IBM 1401 and FORTRAN. The speed of development of the computer (first as a concept and then as an industry) has no precedents in the history of Mankind.

It is hard to fathom the level of change experienced by society in such a short amount of time. We are used nowadays to change, or at least that is what John Kotter would like us to believe. The thing is, confusion and charlatans are well-known byproducts in situations of great change happening in a short period of time. As a result, Gödel’s incompleteness theorems got snatched and abused by philosophers, psychoanalysts, fiction writers, gurus, and pretty much anyone who aimed for a seat in the pantheon of LSD-fueled New Age pocket philosophy.

And this is where Martínez and Piñeiro’s book steps in to provide some much-needed context. Theirs is not only an explanation of Gödel’s theorem, its origins and structure down to its demonstration, or even its meaning; but also its impact in pop culture. This is where the book shines; highlighting all the occasions (and all the celebrities) that took the word “incompleteness” and turned it into something that it is not. The use of the mathematical sign “∀” in the title gives a clear idea of the target audience: this book is for everyone, or at least all those interested in understanding how the hype machine works in our psyche.

Let us enumerate some culprits guilty of extrapolating Gödel’s ideas into riskier fields, with more or less acumen: the psychoanalyst Jacques Lacan, the philosopher and politician Régis Debray, the philosopher Gilles Deleuze, the Bulgarian-French semiotician Юлия Стоянова Кръстева, and the architect Paul Virilio. To be honest, Gödel was not the only one undergoing such treatment; almost all big ideas of the twentieth century, including relativity and quantum physics, suffered from similar fates.

The book fulfills a double role; not only it constitutes a refreshing and correct description of Gödel’s incompleteness theorems themselves, it also debunks claims and folklore built upon decades of misinterpretation. We need more books like this, in many other fields.

Many more books have been written about Gödel’s incompleteness theorems; come to mind at least four. First, “Теорема Гёделя о неполноте” by Влади́мир Андре́евич Успе́нский, published by the venerable Mir Publishers in 1982. Then, “Gödel’s Proof” by Ernest Nagel and James R. Newman, originally published in 1958 and reprinted in 2001 with an introduction by Douglas Hofstadter. Of course, “Gödel, Escher, Bach” from the same Hofstadter, reviewed by Graham in a previous article in this magazine. Finally, Roger Penrose’s 1989 book “The Emperor’s New Mind” also contains important discussions about Gödel’s work (in particular a section in chapter 4) and rightfully belongs to this list.

But nothing beats going to the source. If you are drawn to this subject, you can read an English translation of the paper Gödel originally published at the Monatshefte für Mathematik und Physik, or watch a recent video about it on Veritasium, to understand its substance a bit better. Meanwhile, let us hope that an English translation of Martínez and Piñeiro’s book (and blog) will be available soon for those non-Spanish-speaking readers of this magazine.

Cover photo by the author.

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