1. The body and its structure Euclid and Dynamic Geometry: an end is always a beginning

2. Plan for today: 1.From local to global organization of knowledge 1.1. Deduction as a tool: “given p obtain q” 1.2. Euclidean discourse is bout Aristotelian abstraction: The physical space as the object of mathematical ex- ploration. 1.3. Lessons; from local to global = from physical to symbolic space (or: from Empiricism to Platonism) 1.4. Is the nature of space captured by our species’ vision? The fifth postulate: A hisdden belief and a desired goal. 1.5. Twenty centuries looking for the same…

3. Saccheri (1667-1733). The systemic disease A purely logical quest…to establish a metaphysical belief.

4. KANT( 1724-1804): The shaping of thinking a.What we think takes the shape of our (categorical) glass… b. We were born Euclidian creatures…we are prisoners of our eyes…THIS IMPLIES SPACE HAS TO BE SEEN AS EUCLIDIAN BY NATURE. Has anybody seen Piaget?

5. GAUSS (1777-1855) /LOBACHEVSKY(1792-1856) The nature of space cannot be explored away form empirical research. In other words: what we see is NOT necessarily what IS Our biology is not destiny (but almost) The global structure of space cannot be deduced from the local nature of it: The whole is more than the sum of its pieces

6. Geometric space is…semiotic space What does this mean? 1st answer: it is a representational space 2nd answer: it is a symbolic space; The agent embodies her activity 3rd conclusion: Mathematics is a model of how humans think