Abstract
We provide a new perspective on the relation between the space of description of an object and the appearance of novelties. One of the aims of this perspective is to facilitate the interaction between mathematics and historical sciences. The definition of novelties is paradoxical: if one can define in advance the possibles, then they are not genuinely new. By analyzing the situation in set theory, we show that defining generic (i.e., shared) and specific (i.e., individual) properties of elements of a set are radically different notions. As a result, generic and specific definitions of possibilities cannot be conflated. We argue that genuinely stating possibilities requires that their meaning has to be made explicit. For example, in physics, properties playing theoretical roles are generic; then, generic reasoning is sufficient to define possibilities. By contrast, in music, we argue that specific properties matter, and generic definitions become insufficient. Then, the notion of new possibilities becomes relevant and irreducible. In biology, among other examples, the generic definition of the space of DNA sequences is insufficient to state phenotypic possibilities even if we assume complete genetic determinism. The generic properties of this space are relevant for sequencing or DNA duplication, but they are inadequate to understand phenotypes. We develop a strong concept of biological novelties which justifies the notion of new possibilities and is more robust than the notion of changing description spaces. These biological novelties are not generic outcomes from an initial situation. They are specific and this specificity is associated with biological functions, that is to say, with a specific causal structure. Thus, we think that in contrast with physics, the concept of new possibilities is necessary for biology.
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Notes
The mathematical space used to describe an object is the combination of all the quantities that are used to describe its state. For example, a cell population can be described by the number of cells n and the corresponding mathematical space is then the positive integers.
Here randomly means, for example, a number chosen randomly in a finite interval with the uniform probability distribution.
The axiom of choice illustrates this point. The axiom of choice enables the mathematician to pick specific numbers without an explicit method to do so. In this sense, the action of choosing a specific number becomes generic. An axiom is required for this operation because such a method cannot be constructed.
For example, this notion could be implemented in a similar way than Turing’s imitation game (Turing 1950), with listeners deciding whether a piece is admissible.
This is valid, for example, in a situation without energetic constraints and with an infinite number of particles.
Indeed, a definition is needed to talk about a set of pre-possibilities. As a result, pre-possibilities are defined for some operations.
Here, genetic determinism is postulated without an explicit generic causal rule linking dna to phenotypes. This epistemological status is to be contrasted with the deterministic frame of classical mechanics where determinism follows from the generic application of the Cauchy–Lipschitz theorem on the equations provided by the fundamental principle of dynamics, see Sect. 3.1.3.
Organisms are open systems, with fluxes of matter and energy that are not straightforward to describe in classical mechanics and pertain more to far from equilibrium thermodynamics. With the natural history in mind, the only relevant isolated system would be the solar system. Moreover, biology also involves chemical reactions which, in physics, pertain to quantum mechanics and not classical mechanics.
This line of reasoning is very general, but the corresponding randomness can be interpreted in different ways depending on the theoretical context. In classical mechanics and related deterministic frameworks, this randomness stems from measurement: the state of a system cannot be determined empirically with infinite precision which entails unpredictability, see Longo and Montévil (2017) for a general discussion.
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Acknowledgements
I am grateful to Ana Soto, Giuseppe Longo, Carlos Sonnenschein, Marc Godinot, Paul-Antoine Miquel, Arnaud Pocheville and the anonymous reviewers for their critical insights on previous versions of this article. I also would like to thank Guillaume Lecointre for helpful discussions and Jean Lassègue for pointing out the work of Leibniz.
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Montévil, M. Possibility spaces and the notion of novelty: from music to biology. Synthese 196, 4555–4581 (2019). https://doi.org/10.1007/s11229-017-1668-5
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DOI: https://doi.org/10.1007/s11229-017-1668-5