In the rich and novel Galilean relationship between experiments and theories, physical theorizing is meant to provide intelligibility of phenomena as well as predictability: one first observes and measures, then the theory should produce a prediction capable of confirming it.

The scientifically expected future was set at the core of the understanding of modern science. And prediction is done in the space and time of physical events, mathematically described by the Cartesian analytic representation of space enriched by Galileo's relativity: the modern space-time of phenomena is born by an analysis on how to go from one (Cartesian) reference system to another and preserve the physical laws, inertial movement in particular. In the pre-given space-time of possible trajectories, the invariants are described as symmetries by Galileo's group.

I would ascribe, though, the turning point towards the myth of a scientific expectation of a possible (and predictable) future to the early Italian Renaissance.

The rational insight into the future, within a given space of possibilities, goes back to the appreciation of progress, and of possible estimates of it, in Italy, in the XIV and XV century. This is when both modern technologies began to change the world and *bank credit* was invented.

Lending money was finally allowed, in particular under the form of the "letters of credit" or early paper money. No more a sin, one could bet on possible future progress, obtain money from a bank, then invest, expect the return of the money, plus interest, and also obtain personal gain.

It was an economic and a conceptual revolution.

There was no magic in the expectation of progress and a foreseeable future, but rational knowledge. Of course, hazards are possible, but within a perfectly pre-given space of possibilities: like throwing dice – it is a risk, but within the six possible outputs, no more, no less. betting is rational: one can compute the probabilities and evaluate the risk.

And the society of an expected future progress, in a *predetermined list *of possible worlds, begins. The society where one can dare to borrow and lend money as well as construct scientific knowledge within mathematically pre-determined space-time; a science, where it is possible to predict, by a scientific theory, the output of an experiment.

Later on, as Stuart Kauffman observed in last week's blog post, Newton and Laplace gave us the mathematics of modern "state determined systems". Indeed, by solving equations, Laplace says, "one must be able to predict all future event of mechanics" (celestial mechanics, but he thinks to the entire physical world).

Pascal's and Laplace's analysis of probabilities scientifically deal also with unpredictability, but randomness is extraneous to the mathematical determination. Anyway, for them, unpredictable events happen within the pre-determined Cartesian space of all possible trajectories and facts.

Much later, quantum mechanics will integrate randomness in the theory, under the form of intrinsic indetermination. Yet, the space of possible trajectories and events will still be mathematically predetermined, whether they be infinite or even infinite dimensional — Hilbert and Fock — spaces. By a finite axiomatization we give them *a priori* and they may accommodate the most unpredictable quantum event. Now, the *finite description* of these possibly infinite spaces, from Descartes to quantum spaces, is made possible by their regularities: they are given in terms of mathematical symmetries (as sets of invariants and invariant preserving transformations).

I think that this is where we are stuck now: in the analysis of the living, both as biological and as societal entities, we are understanding that there is no way to (mathematically) pre-determine the very space of possible evolutions, an idea hinted at 11 years ago by Kauffman (and proposed also in 2006, in my book with Bailly, following a different conceptual path). Let's try to further specify this intuition.

The randomness of dice or coin flipping, of a quantum event, as I said, takes place is a pre-given spaces of possible dynamics and symmetries govern these spaces. In contrast to this, there is no way to predetermine the space of possible future phenotypes (biological forms) along evolution.

In no way was there a sign of the nose of mammals in the bacterial DNA of 600 million years ago. But even next century's list of possible biological events, eukaryotes' forms for example, is not in mathematically pre-given spaces: along evolution, phenotypes and ecosystems *co-constitute* themselves and *jointly produce* space of possibilities.

The symmetries that beautifully ruled physics, are continually changed: biology is a *never identical* iteration of a morphogenetic process, which simultaneously shapes the ecosystem. Structural stability preserves some global symmetries (eg. basic bodily bauplans), but each mitosis is a symmetry change: the two novel cells are never identical, not even to the mother cell. And this is fundamental to understand variability and diversity that are at the core of evolution and ontogenesis. The permanent change contributes to the very robustness of life, as adaptability and as modifying force of the very ecosystem.

Mathematics is a science of invariants and invariant preserving transformations, thus of symmetries. Shall we be able to invent new mathematics to deal with continual symmetry changes? Why not? The founding fathers invented their tools, the mathematics of invariance. Anyway, we need to dare, in order to deal with life as well as with economics, far away from the absurd theories of equilibrium. It does not exist in an ecosystem nor in a society, unless everybody is dead. And the need for a change comes also from the crisis of the bank lending system which started the whole story six centuries ago.

These audacious and once fruitful bets on a foreseeable future, in a mastered list of possibilities, have now become the pure transfer of richness towards the richest, by the refined mathematics of finance. Its aim is not prediction, but to construct new possibilities for bets, to shape the unforeseeable markets and to distribute the risk maximally, so that the Chinese workers will buy the debts of the bets on a totally unpredictable risks taken by American finance.

The theoretical challenge is thus to invent tools for understanding, not necessarily for prediction (Darwin's evolution predicts nothing — yet it is knowledge). Qualitative estimates on the effects of an activity may allow us to act on the world, if these estimates are grounded on criteria of robustness of development, as (increasing) diversity and adaptability. These words, in a societal context, mean democracy and justice.

As for biology, in a book and in several downloadable papers, with Bailly and Montévil, we hinted at novel conceptual (and mathematical) structures which aim at a better understanding of the physical singularity of the living state of matter: the change of perspective on symmetries is at the core of our scientific proposal. Predictability, not even of the space of possibilities, is no longer at the center of knowledge construction. This construction aims at the understanding of the historical contingency of life (and, eventually, society), at the awareness of the role of our action in a totally unpredictable world, where we judge for the better, by making explicit the perspective (and values) that guide our acts.

*Giuseppe Longo has been a professor of mathematical logic and of Computer Science at the University of Pisa. He is currently Directeur de Recherche CNRS at the Ecole Normale Supérieure, Paris. He is editor-in-chief of *Mathematical Structures in Computer Science*. He recently extended his research interests and work to the epistemology of mathematics and theoretical biology.*