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Poincaré's doctrine of geometric conventionalism presents observers of space with a difficult quandary: how to know that space is Euclidean or non-Euclidean. He begins his discussion with some definitions and clarifications. He distinguishes between two types of phenomena, external and internal. External phenomena are involuntary and unaccompanied by muscular sensations, attributable to external objects. Internal phenomena are attributable to our own body and its movements and displacements. Further, changes in one category can be corrected by a correlative move in the other category. These displacements and their reciprocal influence on the other forms the basis of geometry. For this to be possible, Poincaré's space must be homogenous and isotropic.

These definitions serve simply as a base for Poincaré's doctrine of conventionalism. In his words, "Experiment tells us not what is the truest, but what is the most convenient geometry.". This is the conventionalist doctrine in a nutshell. With various thought-puzzles to illustrate his view, Poincaré makes the case for a virtually unknowable physical geometry. For instance, An experimental violation of the Euclidean internal angles theorem can be explained in at least 2 ways: If light travels in straight lines, then space must be non-Euclidean; If space is Euclidean, then light must bend. It is impossible as the subjective observer to know which of these mechanisms is at work, so we must simply choose one, and use it as if it were true. A Poincaré equation of geometry would look like this: Geometry of space + Behaviour of apparatus = Survey measurements. Sklar says: "The hypothesis about the actual geometrical structure of the physical world is an inference from the lawlike structure of our sensational experience. We do not perceive space directly as three-dimensional Euclidean; rather, we infer that this is its structure from the lawlike systematization that we find we can impose upon the true, immediate sensory data.".

Poincaré's conventionalist approach to the geometry of space seems to be a common sense approach also. It addresses the often bewildering idea of objectivity in observing events in science, and allows us a certain metaphysical certainty about our measurements of the most basic forms known- those of geometry. This is valuable to us not only for reasons of scientific self-assurance, but because it offers other possibilities of geometries, and allows us to imagine other ways of looking at the universe. In fact, Poincaré made a huge contribution to the field of chaos systems study; the unpredictability of chaotic systems arises due to their sensitivity to their initial conditions, such as their initial position and velocity. Two identical chaotic systems set in motion with slightly different initial conditions can quickly exhibit motions that are quite different. Poincaré concluded that he could not prove the solar system to be completely predictable. He was the first to state the defining feature of what later became known as chaos: "It may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible. ...". The ramifications of Poincaré's discovery were not fully appreciated by most scientists until computers allowed them to easily model and visualize chaotic systems. Before then, however, pioneering scientists and engineers at the National Aeronautics and Space Administration used Poincaré's work to send people and satellites into orbit.