Giuseppe Longo

BIG DATA AND ARTIFICIAL INTELLIGENCE: A LOOK INTO THE FUTURE To say or write something innovative on the ongoing revolution in the fields of Big Data and Artificial Intelligence is very difficult. The advent of these two new technologies is in fact among the most relevant events in human history since in a little more than a decade it will likely lead to the creation of the First Artificial Intelligence of the Fourth level: i.e capable to think and create autonomously. This is a strong statement but like it or not is also irrefutable and fully compatible with the current technological trends and with what is happening right now, in front of our eyes. For the time being, however, we are dealing just with a weak Artificial Intelligence deeply rooted in Big Data and Big Data Analytics. Even so, however, the epochal changes which are currently taking place in the social, economic, and political structure of our world, have no historical equivalent. In this essay, after a short introduction (for the nonprofessional) of Machine Learning and of the basics of the weak Artificial Intelligence, I will try to explain why Artificial Intelligence and Big Data inextricably connect to each other. I will also try to summarise some among the many positive and negative aspects of the problem.

In this paper, following the technical approach to biological time, rhythms and retention/protention in Longo and Montévil (Perspectives on organisms: Biological time, symmetries and singularities. Springer, Berlin, 2014), we develop a philosophical frame for the proposed dimensions and mathematical structure of biological time, as a working example of “theory building”. We first introduce what “theory building” means to our perspective, in order to make explicit our theoretical tools and discuss the general epistemological issue. Then, through a conceptual articulation between physics and biology, we introduce protention (anticipation) and retention (memory), as proper biological observables. This theoretical articulation, which we consider at the core of moving from physical to biological theorizing, allows us to use some of the properties of these observables as principles around which it is possible to outline a proper geometrical schema for biological time. We then philosophically motivate the analysis of “time” as an operator that acts in biological dynamics in a constitutive way. In other words, space and time become specials concepts of order, actively involved in the theoretical organization of biology, in contrast to existing theories in physics where they appear as parameters. In this approach, we first consider the usual dimension of an irreversible physical time. We then add to it a dimension specific to the internal rhythms of organisms. We motivate this dimensional extension by the relative autonomy of biological rhythms with respect to physical time. This second dimension of time is “compactified” in a simple but rigorous mathematical sense. In short, as soon as there are life phenomena, their rhythms scan biological time. We will consider such a statement as a starting point for an original notion of biological inertia.

The human attempts to access, measure and organize physical phenomena have led to a manifold construction of mathematical and physical spaces. We will survey the evolution of geometries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra in this transition and its Western symbolic development is emphasized. In this relation, we will also discuss changes in the ontological attitudes toward mathematics and its applications. Historically, the encounter of geometric and algebraic perspectives enriched the mathematical practices and their foundations. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematical analysis of the 19th century, led to the exclusion of “geometric judgments” from the foundations of Mathematics. After the success and the limits of the logico-formal analysis, it is necessary to broaden our foundational tools and re-examine the interactions with natural sciences. In particular, the way the geometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how the current notions of mathematical (phase) “space” should be revisited for the purposes of life sciences.

La cognition humaine paraît étroitement liée à la structure de l'espace et du temps relativement auxquels le corps, le geste, l'intelligibilité semblent devoir se déterminer. Pourtant, ce qui, après les approches physico-mathématiques de Galilée et de Newton, fut caractérisé par Kant comme formes de l'intuition sensible, n'a cessé au cours des siècles qui suivirent de se trouver remis en cause dans leur saisie première par les développements théoriques. En mathématiques d'abord, avec les géométries non-euclidiennes, en physique ensuite, où relativité générale puis théories quantiques et critiques ont dû remanier profondément l'objectivité de ces concepts pour en faire des catégories, certes toujours aussi essentielles, mais de plus en plus contre-intuitives, et maintenant en biologie où la temporalité, notamment, et la causalité se révèlent largement différentes de celles de la physique. C'est ce que nous tentons de présenter et de discuter dans ce texte en vue d'en dégager la pertinence pour la cognition elle-même.

This personal, yet scientific, letter to Alan Turing, reflects on Turing's personality in order to better understand his scientific quest. It then focuses on the impact of his work today. By joining human attitude and particular scientific method, Turing is able to “immerse himself” into the phenomena on which he works. This peculiar blend justifies the epistolary style. Turing makes himself a “human computer”, he lives the dramatic quest for an undetectable imitation of a man, a woman, a machine. He makes us see the continuous deformations of a material action/reaction/diffusion dynamics of hardware with no software. Each of these investigations opens the way to new scientific paths with major consequences for contemporary live and for knowledge. The uses and the effects of these investigations will be discussed: the passage from classical AI to today's neural nets, the relevance of non-linearity in biological dynamics, but also their abuses, such as the myth of a computational world, from a Turing-machine like universe to an encoded homunculus in the DNA. It is shown that these latter ideas, which are sometimes even made in Turing's name, contradict his views.

The ongoing MOOC revolution is bound to change the academic world on an unprecedented scale. It is in fact very likely that in the coming decades universities all over the world will shrink in size and number, while professors will assume more and more the role of specialized tutors looking after lectures delivered by a small number of world known academic “superstars”. In what follows we shall analyze some aspect of the phenomenon, focusing on the reasons why, like it or not, the academic world cannot avoid a radical rethinking of its role and goals.

The dependence on history of both present and future dynamics of life is a common intuition in biology and in humanities. Historicity will be understood in terms of changes of the space of possibilities as well as by the role of diversity in life’s structural stability and of rare events in history formation. We hint to a rigorous analysis of “path dependence” in terms of invariants and invariance preserving transformations, as it may be found also in physics, while departing from the physico-mathematical analyses. The idea is that the invariant traces of the past under organismal or ecosystemic transformations contribute to the understanding of present and future states of affairs. This yields a peculiar form of unpredictability in biology, at the core of novelty formation: the changes of observables and pertinent parameters may depend also on past events. In particular, in relation to the properties of synchronic measurement in physics, the relevance of diachronic measurement in biology is highlighted. This analysis may a fortiori apply to cognitive and historical human dynamics, while allowing to investigate some general properties of historicity in biology.

This short note develops some ideas along the lines of the stimulating paper by Heylighen (Found Sci 15 4(3):345–356, 2010a ). It summarizes a theme in several writings with Francis Bailly, downloadable from this author’s web page. The “geometrization” of time and causality is the common ground of the analysis hinted here and in Heylighen’s paper. Heylighen adds a logical notion, consistency, in order to understand a possible origin of the selective process that may have originated this organization of natural phenomena. We will join our perspectives by hinting to some gnoseological complexes, common to mathematics and physics, which may shed light on the issues raised by Heylighen

The “DNA is a program” metaphor is still widely used in Molecular Biology and its popularization. There are good historical reasons for the use of such a metaphor or theoretical model. Yet we argue that both the metaphor and the model are essentially inadequate also from the point of view of Physics and Computer Science. Relevant work has already been done, in Biology, criticizing the programming paradigm. We will refer to empirical evidence and theoretical writings in Biology, although our arguments will be mostly based on a comparison with the use of differential methods (in Molecular Biology: a mutation or alike is observed or induced and its phenotypic consequences are observed) as applied in Computer Science and in Physics, where this fundamental tool for empirical investigation originated and acquired a well-justified status. In particular, as we will argue, the programming paradigm is not theoretically sound as a causal(as in Physics) or deductive(as in Programming) framework for relating the genome to the phenotype, in contrast to the physicalist and computational grounds that this paradigm claims to propose.

The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition.

The proofs of universally quantified statements, in mathematics, are given as “schemata” or as “prototypes” which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of λ-calculus and act as “proof-schemata”, as for universally quantified types. We examine here the critical case of Impredicative Type Theory, i. e. Girard's system F, where type-quantification ranges over all types. Coherence and decidability properties are proved for prototype proofs in this impredicative context

The first part of this paper highlights some key aspects of the differences in the use of mathematical tools in physics and in biology. Scientific knowledge is viewed as a network of interactions, some than as a hierarchically organized structure where mathematics would display the essence of phenomena. The concept of "unity" in the biological phenomenon is then discussed. In the second part, a foundational issue in mathematics is revisited, following recent perspective in the physiology of action. The relevance of the historical formation of mathematical concepts is stressed in several parts

We informally discuss some recent results on the incompleteness of formal systems. These theorems, which are of great importance to contemporary mathematical epistemology, are proved using a variety of conceptual tools provably stronger than those of finitary axiomatisations. Those tools require no mathematical ontology, but rather constitute particularly concrete human constructions and acts of comprehending infinity and space rooted in different forms of knowledge. We shall also discuss, albeit very briefly, the mathematical intelligence both of God and of computers. We hope in this manner to help the reader overcome formalist reductionism, while avoiding naive Platonist ontologies, typical symptoms of Godelitis which affected many in the last seventy years

How do we prove true but unprovable propositions? Gödel produced a statement whose undecidability derives from its ad hoc construction. Concrete or mathematical incompleteness results are interesting unprovable statements of formal arithmetic. We point out where exactly the unprovability lies in the ordinary ‘mathematical’ proofs of two interesting formally unprovable propositions, the Kruskal-Friedman theorem on trees and Girard's normalization theorem in type theory. Their validity is based on robust cognitive performances, which ground mathematics in our relation to space and time, such as symmetries and order, or on the generality of Herbrand's notion of ‘prototype proof’