Variations on the theme of invariants: conceptual and mathematical dualities in physics vs biology1 Giulia Frezza* Giuseppe Longo Through the looking glass: could physics and biology reflect each other When one tries to elaborate a mathematical theory apt to explain some aspects of biology though starting from the clue of one and unique materiality anyhow one becomes aware of some peculiarities Especially during the elaboration of theoretical extensions of physics by new observables Bailly and Longo which gives an account in possibly new mathematical terms of living beings' singularity some characteristic polarizations have been enlightened and verified A key aspect of this approach is the claim of a duality: a conceptual opposition between some theoretical aspects of the two disciplines Table synthetically shows a representation of some conceptual dualities or could we say borrowing the term from biology a crossing over between physical and biological theories For example in our approach biological time and its irreversibility are viewed as constitutive operators of biological complexity while energy is analyzed as a parameter in contrast to the understanding of time as a parameter and energy as an operator in quantum physics Bailly and Longo As a matter of fact energy appears as a parameter in allometric scaling laws in biology Savage et al Moreover as a conceptual symmetry to entropy we also proposed in the same paper the notion of anti entropy as a measurable local reversal of physical entropy production corresponding to increasing biological Human Evolution vol n pp Dipartimento di Filosofia Università degli Studi di Roma TRE Université Denis Diderot Paris fr CNRS ENS Paris Dépt d'Informatique et CREA– Polytechnique fr

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1 Variations on the theme of invariants: conceptual and mathematical dualities in physics vs biology1 Giulia Frezza* Giuseppe Longo_ 1. Through the looking glass: could physics and biology reflect each other? When one tries to elaborate a mathematical theory apt to explain some aspects of biology, though starting from the clue of one and unique materiality, anyhow one becomes aware of some peculiarities. Especially during the elaboration of theoretical extensions of physics by new observables (Bailly and Longo, 2008, 2009), which gives an account in (possibly new) mathematical terms of living beings' singularity, some characteristic polarizations have been enlightened and verified. A key aspect of this approach is the claim of a duality: a conceptual opposition between some theoretical aspects of the two disciplines. Table 1 synthetically shows a representation of some conceptual dualities or, could we say borrowing the term from biology, a crossing over between physical and biological theories. For example, in our approach, biological time and its irreversibility are viewed as constitutive operators of biological complexity while energy is analyzed as a parameter, in contrast to the understanding of time as a parameter and energy as an operator in (quantum) physics (Bailly and Longo, 2009). As a matter of fact, energy appears as a parameter in allometric (scaling) laws, in biology (Savage et. al, 2004). Moreover, as a conceptual symmetry to entropy, we also proposed, in the same paper, the notion of anti-entropy as a (measurable) local reversal of (physical) entropy production corresponding to increasing biological

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Variations on the theme of invariants: conceptual and mathematical dualities in physics vs biology1 Giulia Frezza* Giuseppe Longo# 1. Through the looking glass: could physics and biology reflect each other? When one tries to elaborate a mathematical theory apt to explain some aspects of biology, though starting from the clue of one and unique materiality, anyhow one becomes aware of some peculiarities. Especially during the elaboration of theoretical extensions of physics by new observables (Bailly and Longo, 2008, 2009), which gives an account in (possibly new) mathematical terms of living beings’ singularity, some characteristic polarizations have been enlightened and verified. A key aspect of this approach is the claim of a duality: a conceptual opposition between some theoretical aspects of the two disciplines. Table 1 synthetically shows a representation of some conceptual dualities or, could we say borrowing the term from biology, a crossing over between physical and biological theories. For example, in our approach, biological time and its irreversibility are viewed as constitutive operators of biological complexity while energy is analyzed as a parameter, in contrast to the understanding of time as a parameter and energy as an operator in (quantum) physics (Bailly and Longo, 2009). As a matter of fact, energy appears as a parameter in allometric (scaling) laws, in biology (Savage et. al, 2004). Moreover, as a conceptual symmetry to entropy, we also proposed, in the same paper, the notion of anti-entropy as a (measurable) local reversal of (physical) entropy production corresponding to increasing biological 1 Human Evolution, vol. 25, n. 3-4, pp. 167-172, 2010. * Dipartimento di Filosofia, Università degli Studi di Roma TRE; Université Denis-Diderot, Paris 7; frezza@ens.fr # CNRS, ENS, Paris, Dépt. d’Informatique et CREA– Polytechnique; longo@ens.fr, http://www.di.ens.fr/users/longo/ 1

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