Algorithmic Randomness

[ The following pages are a translation of the book "Algorithmic Randomness", while I progressively prepare it for publication in Brazil. The work stems from a doctoral thesis submitted to the Department of Philosophy of Law at the University of São Paulo. It was evaluated summa cum laude. The English translation is mostly done by ChatGPT ].

Justice as Randomness

Maximum Algorithmic Randomness

This book argues for the introduction of the concept of algorithmic randomness into the theoretical discussion of justice, which also leads to its legal consideration. We know randomness as belonging to statistics. However, it takes on a new formulation in the 1960s and 1970s within the field of algorithmic information theory. In this context, randomness is related to language and refers to the absence of rules within a linguistic code.

Algorithmic randomness is encountered when examining combinations of language elements, such as alphabets or binary code. These combinations will be random whenever their redundancies are reduced to rules: from everything that is regular and repetitive within them, we can retain only the defining rules.

The idea originated in computing as a way to save space or memory. The goal was to store or transmit only the informative content in each message, what is new in it, including the rules for reconstructing the entire communicative system—and these more fundamental rules are also random by definition because there is no rule that constitutes them.

When working in this field, mathematicians eventually take it to an extreme. They imagine the totality of all possible arrangements in binary code. It would be the result of infinitely tossing a coin and marking zero and one for heads and tails, respectively. Upon careful examination, this totality is a fascinating object. Although it lacks immediate linguistic meaning because it is only an endless series of zeros and ones, considering that all languages have a digital structure that translates into binary code, we realize that this series encompasses the translation of any language. It is like a treasure trove containing everything that has been or can be said in any extinct, existing, or yet-to-be-created language. By reducing its redundancies, we achieve the maximum algorithmic randomness within this totality.

In the ensuing pages, I will present this maximum algorithmic randomness as the treasure of languages. Since it encompasses all possibilities of all languages, even those yet to be born, it can be seen as a resource for creativity. This is highlighted by one of its most expressive formulators, Gregory Chaitin. I propose that it should also be understood as the seat of maximum freedom of thought.

The repertoire projected within it is always subject to resignification. The semantically neutral arrangements of zeros and ones gain meaning, or not, depending on the language we are using. This semantic openness to an infinite repertoire makes us wonder if we are always capable of finding, in language, reconciling formulas and arrangements for any conflict. This may already allow us to glimpse the proposed connection between algorithmic randomness and justice.

Throughout this work, I present a genealogy of the concept of randomness, followed by an analysis of how it can be articulated with various senses of justice. At many points, the legal significance and potential uses of the concept in law are also evident. Ultimately, this book represents the first outline of a theory of justice as algorithmic randomness and presents a universal model for fair relations in language.

Expressions of randomness

Although the concepts of statistical randomness and algorithmic randomness are not completely separate and should be studied together, it is necessary to distinguish them from each other.

Statistical randomness dates back to the notions of probability formulated in the 17th century. It is defined as an equiprobable or uniform distribution of probabilities. Since its early conceptions, even before its modern mathematical formalization, it has been applied to matters of distributive justice. As a more recent expression, it is the mathematical framework of pure procedural justice described by Rawls in his "A Theory of Justice," where he proposes ensuring a "fair basic structure" (Rawls, 1971, 1999 edition, p.76) at the very least, to maintain the impartiality of law and distributive policies. It does not involve substantive considerations.

Superficially, algorithmic randomness can manifest itself in the form of statistical randomness. It is a digital and rigorous version of the latter. It emerged when computational, exact criteria for identifying randomness in statistical data were sought. Today, it allows us to delve deeper into addressing issues of justice than statistics alone.

As defined by Chaitin in the 1960s, algorithmic randomness is at the core of everything that can be discerned in language. It serves science and politics in a provocative and different way from statistics because it provides raw material for new classifications, rules, and formulas. It suggests that everything that has been named in science could be reconsidered. With its variety of constructions, it reveals a pluralism of possible methods for addressing any problem in language. It thus touches on epistemology, as Gregory Chaitin and Virginia Maria Fontes Gonçalves Chaitin realized in their collaborative work.

Those who understand algorithmic randomness, therefore, view the elements of statistical calculation as subject to deconstruction and reinvention. Politically, this implies that the distributive dilemmas addressed by statistics can always be questioned from their terms, the naming process that constitutes them. If there is algorithmic randomness, any impasse can be reconceived or reformulated. With it, the problem of justice reaches its ultima ratio in language. The evaluation of initial meanings. A deepened epistemic justice.

This is especially important in this historical moment when we delegate so many large-scale social attributions to statistics-based machines, to the so-called "artificial intelligence." By attempting to regulate it solely based on statistical criteria, we reinstall the impasses of statistics at the normative level. We do not escape the arbitrariness in the use of language.

Let us look at it from another, even more direct perspective. Statistics depends on a well-established communicative language. However, its practical application is always marked by semantic biases and hierarchies of communication. To work, the statistician needs to select the objects of calculation that seem most significant and reject alternatives.

Algorithmic randomness, on the contrary, reveals itself beneath and through communicative language. As it has as its object only the binary code, which is semantically neutral, it provides support for questioning every communicative choice. It embraces any language structure on equal terms, without partiality or bias. In it, no formulation is demoted or marginalized as an object of thought and valuation. No language, logic, or rationality is superior or inferior in principle. Its radical impartiality is a starting point for comparing or revising hierarchies established by statisticians. Algorithmic randomness suggests not only openness to a plurality of methods but also embraces the diversity of values and recognizes the genealogy of each of them in language. It can be perceived as a condition of possibility for the generation of knowledge and values. The site of their poiésis.