Geometries of Nature, Living Systems and Human Cognition

Mathematics stems out from our ways of making the world intelligible by its peculiar conceptual stability and unity; we invented it and used it to single out key regularities of space and language.  This is exemplified and summarised below in references to the main foundational approaches to Mathematics, as proposed in the last 150 years.  Its unity is also stressed: in this paper, Mathematics is viewed as a "three dimensional manifold" grounded on logic, formalisms and invariants of space; we will appreciate by this both its autonomous generative nature and its effectiveness.  But effectiveness is also due to the fact that we re-construct the world by Mathematics: we organise knowledge of space and language by Mathematics, and give meaning by it to their structuring.  But, what is "meaning", for us living and historical beings?  What does "mathematical intuition" refer to?  We will try to propose an understanding of these crucial aspects of the mathematical praxis, often disregarded as "magic" or as beyond any scientific analysis.  Finally, some limits of the remarkable, but reasonable effectiveness of Mathematics will be sketched, in particular in reference to its applications in Biology and in human cognition.