The Geometry of Algorithms – GoA

The GoA project starts from the observation that, contrary to a common assumption (especially in the context of mathematical logic), the work carried out in the 1930s by Alonzo Church, Alan Turing, Kurt Gödel, and Stephen Kleene does not provide a satisfactory formal definition of the notion of algorithm. Computability theory aims to characterize the functions that can be computed algorithmically, but it focuses on particular algorithms without specifying what an algorithm is in general or how two algorithms (calculating the same function) can be compared. It therefore provides neither criteria for the identity of algorithms nor a genuine ontological account of them.

Several authors have made similar observations and proposed formal frameworks for developing a genuine theory of algorithms (notably Andrey Kolmogorov and Vladimir Uspensky in the 1950s, and later Yannis Moschovakis and Yuri Gurevich between the 1990s and 2000s). These approaches nevertheless have two main limitations. First, they analyze algorithms primarily from the perspective of computability theory and theoretical computer science, while partly overlooking their connection with the notion of proof in mathematics. Secondly, they do not sufficiently distinguish algorithms from other fundamental notions, such as that of programs, computation, or model of computation. This is because they tend to adopt either an overly syntactic approach (the algorithm as a text) or an overly semantic one (the mathematical structure interpreting certain operations).

To overcome these limitations, the GoA project studies the notion of algorithm from a dynamic perspective (thereby moving beyond the traditional dichotomy between syntax and semantics). An algorithm is analyzed in terms of the actions that it makes possible, which can be studied formally as transformations generated by operations on a space (metrical, topological, etc.). This geometric approach gives the project its name and also makes it possible to establish a connection with the notion of proof. Indeed, it rests on techniques coming from proof theory, where proofs are studied as dynamic objects (that is, as objects upon which certain structural transformations can be performed).

Such an analysis nevertheless risks to bring to a definition that is too abstract and disconnected from the algorithmic methods employed in different disciplines. An essential objective of GoA is therefore to assess whether the proposed definition is compatible with different uses and analyses of the notion of algorithm, not only in the formal sciences (mathematics, logic, computer science), but also in the social sciences and humanities (law, sociology, history). The formal analysis conducted within GoA is thus constantly complemented by a conceptual analysis of this notion.

- Identification and analysis of the different meanings and conceptions of the notion of algorithm.

Papayannopoulos, P. (2023). « On Algorithms, Effective Procedures, and Their Definitions ». Philosophia Mathematica, vol. 31, n. 3, pp. 291-329.

- Conceptual distinction between the notion of model of computation and that of algorithm, together the proposal of a mathematical framework for defining these notions formally.

Seiller, T. (2026). « Mathematical Informatics: Algorithms ». In V. Brattka, H. Fernau et L. Galeotti (eds), Timeless Machines: Computability Across Eras. Proceedings of the 22nd Conference on Computability in Europe, CiE 2026. Trier, Germany, July 27–31, 2026, pp. 86-106. Berlin, Springer.

- Clarification of the relations between the notions of algorithm and proof.

Naibo, A. (2025) « Preuves et algorithmes ». In P. Wagner (ed.), Logique et épistémologie, pp. 27–52. Paris, Vrin.

- Analysis of the application of algorithmic methods (and, more generally, AI methods) in the search for mathematical proofs.

Dean, W. et Naibo, A. (2025) « Artificial intelligence and inherent mathematical difficulty ». Philosophia Mathematica, vol. 33, n. 3, pp. 283–329.

- Analysis of the epistemic dimension of the notion of algorithm as a problem-solving method.

Stephanou, H. (2025). Systems, Machines, and Problem-Solving. Berlin, De Gruyter.

(The works listed below are provided as examples and represent only part of the scientific output resulting from the GoA project.)

When the GoA project was conceived in 2020, AI techniques based on statistical learning already existed, but they had not yet acquired the visibility or importance they have today. The GoA project was therefore not specifically designed to provide a conceptual and formal analysis of these techniques. However, the analysis of the notion of algorithm carried out within GoA can now be brought to bear on the epistemological status of AI systems based on statistical learning.

One important point that emerges from the reflections conducted within GoA is that an algorithm is not merely a mechanical or blind computation: it has an epistemic dimension, insofar as it makes it possible to communicate, in a compact and synthetic form, information about how to solve a problem. An algorithm can thus be understood as a form of information compression that, in turn, makes it possible to perform an action aimed at producing a knowledge.

From this perspective, AI systems based on machine learning do not seem to qualify as genuine algorithms. An AI system is designed to solve a problem in a mechanical or automated way. However, this problem generally concerns objects for which no formal definition is available a priori. Consider, for example, a facial recognition system: a face is not an object that can be defined in purely mathematical and formal terms. The approach is instead to start from a large amount of data (images, for example) and use mathematical methods – the learning models – to identify regularities or "similarities" among these data, and then produce a program capable of determining, when presented with a new image, whether or not it contains a face.

However, if one wants someone else to be able to reproduce exactly the behaviour of this program, it is not enough to provide them with the learning model. They must also be given the entire set of initial data, together with the annotation and preprocessing work associated with it. There is therefore no way to communicate only the general idea underlying the behaviour of the program. This gives rise to a problem of information non-compressibility.

From this perspective, one could formulate the hypothesis that AI systems produce programs that do not genuinely implement algorithms. This would help to explain why AI represents a form of programming that differs from traditional programming, and why it is so difficult to explain how these systems work. In other words, the lack of explainability may be connected to the absence of a genuine algorithmic dimension.

Algorithms take nowadays a central place in the public debate: they structure our social interactions, they modify our work, our means of transport, and also our scientific and medical instruments. But who can say what exactly an algorithm is? Since there is no real consensus among the experts, it is important to give an epistemological foundation to this debate. By giving a precise mathematical representation to the algorithms, it becomes possible to assign them a genuine scientific status. In particular, the use of geometric tools will allow us to distinguish the notion of algorithm from other related and yet different ones, like that of computation or that of program. This analysis will thus open the way to develop an ontology proper to computer science, and to look at the latter as a mathematized theory, in the same way as physics.