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# Incompleteness

ARTICLES

## On the Relevance of Negative Results

### 1. Scientific knowledge and critical insight.

The analysis of concepts, conducted on a comparative level if possible, as well as the (tentative) explanation of the philosophical project, should always accompany scientific work. In fact, critical reflections regarding existing theories are at the core of positive scientific constructions, because science is often constructed against the supposed tyranny and autonomy of « facts » which in reality are nothing but « small-scale theories ». Science is also often constructed by means of an audacious interpretation of « new » (and old) facts ; it progresses against the obvious and against common sense (le « bon sens ») ; it struggles against the illusions of immediate knowledge and must be capable of escaping from already established theoretical frameworks. For example, the very high level of mathematical technicity in the geometry of Ptolemaic epicycles constructed from clearly observable facts strongly perplexed numerous Renaissance thinkers such as Copernicus, Kepler and Galileo… : in order to account for the movements of the stars and for the « obvious » immobility of the earth, circles that were added to circles, centers of new circles, were established with and extraordinary geometrical finesse and gave way to uncountably many « publications » (of very high Impact Factor, at least till the middle of the XVII century). Yet they failed to convince the aforementioned revolutionary critical thinkers. And, as Bachelard rightly puts it, the construction of knowledge was then founded, as was Greek thought, upon an epistemological severance, which operates a separation with the previous ways of thinking.
But it is recent examples that interest us, where the critical view finds expression on a more punctual basis, by means of « negative results ». Let’s explain.
When Poincaré was working on the calculi of astronomers, on the dynamics of planets within their gravitational fields, he produced, by purely mathematical...