Algorithmic randomness lies at the intersection between computability theory and probability theory. In order to fully explore this interaction, one naturally needs a computable version of measurable functions. While several such notions appear in the literature, most of them do not interact well with algorithmic randomness because they are only defined up to a null set. Therefore, we need a computable notion of measurable function which is well defined on algorithmically random points, and this is what layerwise computability precisely does. This article is a survey about this notion. We give the main definitions, the most important properties, and several applications of this notion. We prioritize motivating this framework and explaining its salient features.

The field of algorithmic randomness studies the various ways to define the inherent randomness of individual points in a space (for example, the Cantor space or Euclidean space). Classically, this quest seems quixotic. However, the theory of computing allows us to give mathematically meaningful notions of random points. In the past few decades, algorithmic randomness has been extended to make use of resource-bounded (e.g., time or space) computation. In this survey we survey these developments as well as their applications to other parts of mathematics.

Algorithms are often disparaged, but they are also sometimes overrated. To understand their place in our world it is also important to understand their limits.

In the age of algorithms, new fears have arisen, including the fear that one day we will be overtaken, or even enslaved, by new beings, transhumans favored by natural selection, but also, more simply, by computers or algorithms more intelligent than us. This leads us to a question that computer scientists have been asking since the 1950s: Can an algorithm be intelligent? This question brings up two others. What does the adjective intelligent mean? Can we create an intelligent being?

Algorithms have become an essential component of our professional lives and social interactions, in health care, transportation, commerce, industry. Algorithms are transforming the natural sciences, social sciences, and the humanities, and in doing so, enrich our knowledge. They allow technology to continually push beyond the limits of the possible.

We used to believe that intelligence, like speech, culture, and self-awareness, made us unique. However, diluting intelligence across a variety of faculties contributes to blurring the boundary that separates man and machine. Man is better at speaking Japanese, but machines are better than man at playing chess. Perhaps one day machines will also be better at speaking Japanese. The difference between man and machine seems to be more a matter of degree than of nature, a distinction that enabled us to conceive of the idea of augmented man.

What could we still say was unique about us in the twentieth century?

Algorithms transform the relationship between human beings and nature, and in doing so, transform nature itself. This leads us to examine the relationship between the digital revolution and another transformative factor in our world today: the ecological transition.

The community is usually defined socially as a group of human beings whose life together is made possible by respecting a certain number of rules that define the rights and obligations of each of them. According to this definition, the members of the community are women and men. However, a gradual evolution of this concept has led us to consider that groups of human beings, for example, companies, and associations, can also have rights and obligations and, therefore, can also be considered members of the community.

In using the example of the professions of driver and translator, we implicitly assumed that drivers and translators would always exist. However, it is also possible that these professions may one day disappear if, at some point, algorithms for driving a car or translating a text perform as well as, or even better, than a human. This is also true for many other professions. Of course, this transition also paves the way for new professions to design, implement, and accompany all of these algorithms, but in the age of algorithms, much less work may be required to provide the same goods or services as before.

We have been using symbolic algorithms since the advent of writing, five thousand years ago. How is it then that this concept has suddenly become such a hot topic in the public sphere today? To explain this, we need to look into objects other than algorithms – computers and programs.

In 2002, during a performance/installation, 35 Hours of Work, Benjamin Sabatier sharpened pencils seven hours a day for five days. Sharpening pencils in this way for thirty-five hours is a deviant act, because custom dictates that we use a pencil sharpener for a few seconds, to sharpen a pencil, after which we put it away in a drawer until we need it again. As a consequence, a pencil sharpener is used only a few minutes per year.

Algorithms and computers help with everything. But what tangible purpose do they serve? The remarkable diversity of their uses derives from their universality.

Why is the notion of privacy at the core of the issues emerging in the age of algorithms?

We have certain expectations of the algorithms we use. For example, we would like them to be fair. These properties are essential for the peaceful coexistence of humans and algorithms, and for establishing a climate of trust in the community. They are even more critical when these algorithms exercise a certain power, such as when an algorithm makes the decision to approve or refuse a bank loan. What exactly are these expectations?

A scientific revolution does not only create new knowledge. It also generates new ways of thinking, new ways of asking questions, and new ways of answering them. Before the scientific revolution at the beginning of the seventeenth century, when questions were raised, for example, about whether or not blood circulated in the body, people looked for answers in the ancient texts. Aristotle and Galen taught that blood did not circulate in the body; therefore, the question had been answered. How did Aristotle and Galen know what they knew? That question was not posed. They were more knowledgeable than us, and that was sufficient.