Giuseppe Longo

La Redazione di Nóema ha invitato a una riflessione tre matematici contemporanei molto noti – non solo in ambito specialistico – che sono da sempre particolarmente attenti al dialogo con la filosofia. In queste pagine Giuseppe Longo (ENS, Parigi), Alessandro Sarti (EHESS, Parigi) e Fernando Zalamea (Università di Bogotà) rispondono ad alcune domande poste dai membri della rivista, interloquendo uno con l’altro e chiarendo come è da considerare, a loro modo di vedere, una ricerca che getti luce sulla genesi del senso matematico tenendo conto delle suggestioni che vengono da alcuni filosofi e delle operazioni condotte con uno sguardo sintetico, piuttosto che analitico e obiettivista. Nel primo contributo, Giuseppe Longo, rispondendo a domande sulla matematica e sul suo senso, accenna a come questa ridisegna il mondo a sua guisa, in effetti in guise differenti, seguendo storie diverse. L’invenzione di concetti e strutture ne è al cuore: l’audacia, ad esempio, di proporre “contorni”, “bordi” del mondo che non esistono, ma che lo ritagliano e qualificano con grande efficacia e stabilità concettuale. In fisica, nelle nostre culture, la matematica ha proposto un solidissimo impianto di tipo teologico ancor oggi attuale, ma certo inadeguato per parlare del vivente. Longo accennerà ad alcuni tentativi di superare tali schemi metafisici e cambiar di prospettiva. Nel secondo contributo, Alessandro Sarti chiarisce come la matematica non si limiti a stabilire regole e invarianti, ma apra anche nuovi campi di indagine e nuove possibilità. È possibile concepire un divenire differenziale eterogeneo nello spazio e nel tempo, dove causalità e composizione non siano in contraddizione? Si tratta di quelle dinamiche che Deleuze e Guattari hanno definito eterogenesi, in cui gli spazi di possibilità non sono invarianti ma parte del processo. Sarti propone un viaggio attraverso le dinamiche differenziali che spaziano dalla fisica allo strutturalismo dinamico fino all’eterogenesi, intesa come materialismo immaginativo. Queste “_mathématiques de la pensée_” sono strettamente intrecciate con la “_pensée des mathématiques_” di Longo e Zalamea. Nel terzo contributo, Fernando Zalamea illustra come la matematica sia una scienza complessa, che integra 1) immaginazione (abduzione nei termini di Peirce), 2) ragione (deduzione), 3) adeguazione (induzione). Il matematico ridisegna il mondo nel costante oscillare tra l’1-2-3 peirceano, applicato alle strutture matematiche. Una matematica rigida, che cerca solo i propri fondamenti, ha un senso molto ristretto, prezioso per studiare con estrema attenzione frammenti della logica classica, della teoria degli insiemi e dei numeri elementari, ma inutile per osservare la vera matematica in azione (teoria avanzata dei numeri, algebra astratta, topologia, variabile complessa, geometria algebrica, geometria differenziale, analisi funzionale). Per questo sono fondamentali altre prospettive, come le logiche non classiche, la teoria delle categorie, la teoria dei fasci, che servono da base per visioni filosofiche non restrittive e non analitiche.

The intellectual act of imposing borders to contain and delimit objects has been a constituent factor in physics since its origins, and is also fundamental for philosophical reflection. However, the characteristics of the conceptual universe thus constructed (tendency towards the ideal limit, invariance in variation, a conception of matter as residue, etc.) seem inadequate in biology. The essential characteristic of the living thing is, in fact, that of having a history: that is, of being the concrete trace of a memory. Making an epistemology of the living thing (and not of the inert) means identifying new categories which, being radically antithetical to disembodied notions like that of program and information, take account of the unpredictability and creativity which time introduces. This essay aims to bring some of these fundamental categories into focus.

Dans la mythologie grecque, Prométhée - le 'prévoyant' - déroba le feu aux dieux et le transmit aux hommes. Zeus, courroucé par ce transfert technologique, condamna Prométhée à un châtiment itéré à l'identique, à l'infini, algorithmique. Le problème que pose l'ancien mythe est celui des limites de la connaissance. En termes modernes, jusqu'à quel point pouvons-nous transformer la nature sans une connaissance des conséquences de nos actions sur elle? Cette question nous pousse à analyser les limites des sciences : à une techno-science devenue toujours plus invasive, la science répond en se donnant à elle-même des limites"--Page 4 of cover.

The reflections on the nature of time in Relativity Theory will be hinted in reference to the new bridges recently proposed by Connes and by Rovelli’s perspectival approach, two major steps towards a unification of quantum, thermodynamical and relativistic times. The so-called “time of philosophers”, a time of the cognizing ego, from Saint Augustin to Husserl and Bergson, is based on a different, but relevant perspective and it has been traditionally opposed to the “time of physicists”. In between these two approaches, we discuss a proper time of phylogeny and ontogeny, in biology, with their own rhythms and specific irreversibility. On the one side, biological time needs to be scientifically objectivised as an invariant of the knowing subject and thus move, as in physics, from the subjective-absolute to the objective-relative (Weyl’s approach, extended to time). On the other, we propose a “geometry” of life’s rhythms and an “extended present” that radically differ from the prevailing spatialization of physical time that Bergson soundly criticizes. The proper irreversibility and the central, “operatorial”, role of time in biology will be stressed, as nothing in biology can be understood except in the light of a temporal perspective, both evolutionary and organismal. In particular, today’s eco-systemic changes bring to the limelight some disruptions of the evolutionary fine-tuning of biological rhythms and physical clocks that may be better understood by highlighting their theoretical differences as well as their environmental interactions.

Very large databases are a major opportunity for science and data analytics is a remarkable new field of investigation in computer science. The effectiveness of these tools is used to support a “philosophy” against the scientific method as developed throughout history. According to this view, computer-discovered correlations should replace understanding and guide prediction and action. Consequently, there will be no need to give scientific meaning to phenomena, by proposing, say, causal relations, since regularities in very large databases are enough: “with enough data, the numbers speak for themselves”. The “end of science” is proclaimed. Using classical results from ergodic theory, Ramsey theory and algorithmic information theory, we show that this “philosophy” is wrong. For example, we prove that very large databases have to contain arbitrary correlations. These correlations appear only due to the size, not the nature, of data. They can be found in “randomly” generated, large enough databases, which—as we will prove—implies that most correlations are spurious. Too much information tends to behave like very little information. The scientific method can be enriched by computer mining in immense databases, but not replaced by it.

Zalamea’s book is as original as it is belated. It is indeed surprising, if we give it a moment’s thought, just how greatly behind schedule philosophical reflection on contemporary mathematics lags, especially considering the momentous changes that took place in the second half of the twentieth century. Zalamea compares this situation with that of the philosophy of physics: he mentions D’Espagnat’s work on quantum mechanics, but we could add several others who, in the last few decades, have elaborated an extremely timely philosophy of contemporary physics (see for example Bitbol 2000; Bitbol et al. 2009). As was the case in biology, philosophy – since Kant’s crucial observations in the Critique of Judgment, at least – has often “run ahead” of life sciences, exploring and opening up a space for reflections that are not derived from or integrated with its contemporary scientific practice. Some of these reflections are still very much auspicious today. And indeed, some philosophers today are saying something truly new about biology...

The rich blend of theories and experiences that made the history of physics possible still now enlightens the scientific method. We stress the need to learn from this method the force of making its principles explicit, while developing a rich diversity of theories, which are often incompatible. Unity is preserved by common founding principles and their mathematical form, such as the understanding of conservation properties (energy, momentum etc.) in terms of symmetries. When moving from the inert to the living state of matter, new challenges are posed, beginning with biological “heterogenesis”, as “genesis of and from diversity” in a changing space of pertinent observables and parameters: the Darwinian ecosystem. The question that is posed is how we may consistently embed the theories of the inert into biology. By naturalization we mean an analysis of physics as part of the sciences of nature, not as the science governing them all. In particular, the founding symmetry principles of physical theories, often used to “naturalize” (but, actually, to “physicalize”) other sciences, will instead be framed in more general dynamics which deal with fundamental changes of symmetries, as they apply, in our views, in all historical sciences, beginning with biology. This paper will accordingly explore the notion of (non-)conservative extension of theories in a precise mathematical sense. We stress a perspectival epistemology that promotes a dialogue of theories, in search for bridges or even unity, inspired by the method of “unification” at the core of major theoretical inventions in physics. Our main motivation is the need to go beyond the strong dualistic separation of space (or of the more general “phase space”) as a pre-given container of the dynamics of matter, that biased physics from Aristotle to Newton and, in a technically different way, even Einstein.

We propose a reflection on the construction of scientific knowledge and in so doing an image of this knowledge. This will allow us to develop a comparative analysis of some of the main principles underpinning the constitution of the different sciences. We will highlight the role of critical thought in science, or even “negative results,” which pose limits and hence open new trajectories. In particular, we will address a misleading point of view, based on some informal concepts taken from computer science, which dominates the biological imaginary. An “ethics of knowledge” will follow, as a critical and open perspective, capable of taking on the well-intentioned responsibility of proposing universal, though not absolute, principles of knowledge.

Biological evolution is a complex blend of ever changing structural stability, variability and emergence of new phe- notypes, niches, ecosystems. We wish to argue that the evo- lution of life marks the end of a physics world view of law entailed dynamics. Our considerations depend upon dis- cussing the variability of the very ”contexts of life”: the in- teractions between organisms, biological niches and ecosys- tems. These are ever changing, intrinsically indeterminate and even unprestatable: we do not know ahead of time the ”niches” which constitute the boundary conditions on selec- tion. More generally, by the mathematical unprestatability of the ”phase space”, no laws of mo- tion can be formulated for evolution. We call this radical emergence, from life to life. The purpose of this paper is the integration of variation and diversity in a sound concep- tual frame and situate unpredictability at a novel theoretical level, that of the very phase space. Our argument will be carried on in close comparisons with physics and the mathematical constructions of phase spaces in that discipline. The role of symmetries as invariant preserving transformations will allow us to under- stand the nature of physical phase spaces and to stress the differences required for a sound biological theoretizing. In this frame, we discuss the novel notion of ”enablement”. Life lives in a web of enablement and radical emergence. This will restrict causal analyses to differential cases. Mutations or other causal differ- ences will allow us to stress that ”non conservation princi- ples” are at the core of evolution, in contrast to physical dynamics, largely based on conservation principles as sym- metries. Critical transitions, the main locus of symmetry changes in physics, will be discussed, and lead to ”extended criticality” as a conceptual frame for a better understanding of the living state of matter.

The physical singularity of life phenomena is analyzed by means of comparison with the driving concepts of theories of the inert. We outline conceptual analogies, transferals of methodologies and theoretical instruments between physics and biology, in addition to indicating significant differences and sometimes logical dualities. In order to make biological phenomenalities intelligible, we introduce theoretical extensions to certain physical theories. In this synthetic paper, we summarize and propose a unified conceptual framework for the main conclusions drawn from work spanning a book and several articles, quoted throughout.

n this text, we revisit part of the analysis of anti-entropy in Bailly and Longo (2009} and develop further theoretical reflections. In particular, we analyze how randomness, an essential component of biological variability, is associated to the growth of biological organization, both in ontogenesis and in evolution. This approach, in particular, focuses on the role of global entropy production and provides a tool for a mathematical understanding of some fundamental observations by Gould on the increasing phenotypic complexity along evolution. Lastly, we analyze the situation in terms of theoretical symmetries, in order to further specify the biological meaning of anti-entropy as well as its strong link with randomness.

This paper proposes an abstract mathematical frame for describing some features of biological time. The key point is that usual physical (linear) representation of time is insufficient, in our view, for the understanding key phenomena of life, such as rhythms, both physical (circadian, seasonal …) and properly biological (heart beating, respiration, metabolic …). In particular, the role of biological rhythms do not seem to have any counterpart in mathematical formalization of physical clocks, which are based on frequencies along the usual (possibly thermodynamical, thus oriented) time. We then suggest a functional representation of biological time by a 2-dimensional manifold as a mathematical frame for accommodating autonomous biological rhythms. The “visual” representation of rhythms so obtained, in particular heart beatings, will provide, by a few examples, hints towards possible applications of our approach to the understanding of interspecific differences or intraspecific pathologies. The 3- dimensional embedding space, needed for purely mathematical reasons, allows to introduce a suitable extra-dimension for “representation time”, with a cognitive significance.

Symmetries play a major role in physics, in particular since the work by E. Noether and H. Weyl in the first half of last century. Herein, we briefly review their role by recalling how symmetry changes allow to conceptually move from classical to relativistic and quantum physics. We then introduce our ongoing theoretical analysis in biology and show that symmetries play a radically different role in this discipline, when compared to those in current physics. By this comparison, we stress that symmetries must be understood in relation to conservation and stability properties, as represented in the theories. We posit that the dynamics of biological organisms, in their various levels of organization, are not just processes, but permanent (extended, in our terminology) critical transitions and, thus, symmetry changes. Within the limits of a relative structural stability (or interval of viability), variability is at the core of these transitions.

This article proposes an abstract mathematical frame for describing some features of cognitive and biological time. We focus here on the so called “extended present” as a result of protentional and retentional activities (memory and anticipation). Memory, as retention, is treated in some physical theories (relaxation phenomena, which will inspire our approach), while protention (or anticipation) seems outside the scope of physics. We then suggest a simple functional representation of biological protention. This allows us to introduce the abstract notion of “biological inertia”.

This authored monograph introduces a genuinely theoretical approach to biology. Starting point is the investigation of empirical biological scaling including their variability, which is found in the literature, e.g. allometric relationships, fractals, etc. The book then analyzes two different aspects of biological time: first, a supplementary temporal dimension to accommodate proper biological rhythms; secondly, the concepts of protension and retention as a means of local organization of time in living organisms. Moreover, the book investigates the role of symmetry in biology, in view of its ubiquitous importance in physics. In relation with the notion of extended critical transitions, the book proposes that organisms and their evolution can be characterized by continued symmetry changes, which accounts for the irreducibility of their historicity and variability. The authors also introduce the concept of anti-entropy as a measure for the potential of variability, being equally understood as alterations in symmetry. By this, the book provides a mathematical account of Gould's analysis of phenotypic complexity with respect to biological evolution. The target audience primarily comprises researchers interested in new theoretical approaches to biology, from physical, biological or philosophical backgrounds, but the book may also be beneficial for graduate students who want to enter this field.

This short paper is meant to be an introduction to the ‘Letter to Alan Turing’ that follows it. It summarizes some basic ideas in information theory and very informally hints at their mathematical properties. In order to introduce Turing’s two main theoretical contributions, in Theory of Computation and in Morphogenesis (an analysis of the dynamics of forms), the fundamental divide between discrete vs. continuous structures in mathematics is presented, as it is also a divide in his scientific life. The reader who is familiar with these notions, and is convinced that they (and their differences) are relevant in the mathematical understanding of phenomena, may skip this introduction and go directly to the Letter.