Incompleteness irreducibility

Incompleteness (the fact that a model neglects some factors influencing the phenomenon it is modeling) has been proved all along all these works to be the major difficulty encountered by the classical approaches to modeling (logic and usual programming). This difficulty is the central justification for the adoption of Bayesian programming proposed as a way to take into account this incompleteness.

However, a legitimate question is: “Is this incompleteness reducible?” or stated in other terms: “Is it possible, by adding more and more variables to our model, to obtain a model with no hidden variables?”

If the studied phenomenon is not formal the answer to these questions is NO.

There are three fundamental arguments in favor of this negative answer:

• The very idea of exhaustivity is in contradiction to the concept of the model. A model is interesting if, and only if, it is (much) simpler than the studied phenomenon.
• The fundamental laws of physics are boundless in distance: any single particle is in interaction with all the others in the universe. Consequently, the entire universe should be necessary to describe completely any physical phenomenon even the simplest.
• “Chaotic” systems prove that it is impossible to provide a “simplified” model of some phenomenon, even some elementary ones, as it is necessary to know “exactly” their initial conditions to be able to predict their evolution. Henri Poincaré summarized this in beautiful words:

To find a better definition of hazard, we have to look for facts that we agree to qualify as fortuitous and for which probability calculus applies. We will then look for their common characteristics.

The first instance that we can choose in unstable equilibrium. If a cone rests on its point we know that it will fall down but we don’t know on which side. It seems that only hazard will decide. Would the cone be perfectly symmetric, would its axis be perfectly vertical, would there be absolutely no other force than gravity, it will not fall. But the slightest symmetry break will tilt it on one side or another and, as soon as it will be tilted, so little that it is, it will completely fall on that side. Even with a perfect symmetry, an infinitesimal juddering, a breath of air will tilt it of a few arc seconds and it will be sufficient to cause its fall and to determine the direction of this fall toward the initial inclination.

An infinitesimal cause that we overlook may determine a major effect that we cannot miss. We then say that this effect is due to hazard. Would we know exactly the laws of nature and the state of the universe at the initial instant, we could exactly predict the state of this same universe at the next moment. But, even with this perfect knowledge of the laws of nature, we have only an approximate knowledge of the initial state. If we can predict the next state with the same approximation, it’s all what we need, the phenomenon has been forecast, it is ruled by laws. However, it is not always the case, it may happen that slight differences in initial conditions generate huge ones in final phenomenon. The prediction becomes impossible and we are facing a fortuitous phenomenon.

Calcul des Probabilités, Henri Poincaré [1912]

In practice, however, it is not necessary to invoke these fundamental reasons to justify the irreducibility of incompleteness. A sensory motor system, either living or artificial, should evidently be able to take decisions with only a very partial knowledge of its interaction with its environment. Can we imagine that a bee has a complete model of its aerodynamic interaction with the environment to fly around without running into obstacles?

Original in French :

Pour trouver une meilleure définition du hasard, il nous faut examiner quelques-uns des faits qu’on s’accorde à regarder comme fortuits, et auxquels le calcul des probabilités paraît s’appliquer; nous rechercherons ensuite quels sont leurs caractères communs.

Le premier exemple que nous allons choisir est celui de l’équilibre instable; si un cône repose sur sa pointe, nous savons bien qu’il va tomber, mais nous ne savons pas de quel côté; il nous semble que le hasard seul va en décider. Si le cône était parfaitement symétrique, si son axe était parfaitement vertical, s’il n’était soumis à aucune autre force que la pesanteur, il ne tomberait pas du tout. Mais le moindre défaut de symétrie va le faire pencher légèrement d’un côté ou de l’autre, et dès qu’il penchera, si peu que ce soit, il tombera tout à fait de ce côté. Si même la symétrie est parfaite, une trépidation très légère, un souffle d’air pourra le faire incliner de quelques secondes d’arc; ce sera assez pour déterminer sa chute et même le sens de sa chute qui sera celui de l’inclinaison initiale.

Une cause très petite, qui nous échappe, détermine un effet considérable que nous ne pouvons pas ne pas voir, et alors nous disons que cet effet est dû au hasard. Si nous connaissions exactement les lois de la nature et la situation de l’univers à l’instant initial, nous pourrions prédire exactement la situation de ce même univers à un instant ultérieur. Mais, lors même que les lois naturelles n’auraient plus de secret pour nous, nous ne pourrions connaitre la situation qu’approximativement. Si cela nous permet de prévoir la situation ultérieure avec la même approximation, c’est tout ce qu’il nous faut, nous disons que le phénomène a été prévu, qu’il est régi par des lois; mais il n’en est pas toujours ainsi, il peut arriver que de petites différences dans les conditions initiales en engendrent de très grandes dans les phénomènes finaux; une petite erreur sur les premières produirait une erreur énorme sur les derniers. La prédiction devient impossible et nous avons le phénomène fortuit.

Calcul des Probabilités, Henri Poincaré [1912]