Jerry Fodor (2003) sees Hume’s Treatise as the foundational document of cognitive science, though he concedes that “Descartes got there first.” However, Hume’s “Cartesianism” is an ambiguous inheritance since Hume’s representational account (and Fodor’s) is closer to Malebranche’s version than Descartes’ own. Descartes shared the ‘pragmatism’ and ‘direct realism’ of Arnauld and later Reid – the...

A. Medina-Santiago Center for Research, Development and Technological Innovation, University of Science and Technology Descartes, Chiapas, México Email: [email protected] E. V. Toledo-Nuñez Master of Technology in Computer Science, University of Science and Technology Descartes, Chiapas, México Email: [email protected] R. Soto-Alvarez Master of Technology in Computer Science...

The Descartes circle theorem states that if four circles are mutually tangent in the plane, with disjoint interiors, then their curvatures (or “bends”) bi = 1 ri satisfy the relation (b1 + b2 + b3 + b4) 2 = 2(b1 + b 2 2 + b 2 3 + b 2 4). We show that similar relations hold involving the centers of the four circles in such a configuration, coordinatized as complex numbers, yielding a complex Des...

We present a short proof of Descartes Circle Theorem on the “curvaturecenters” of four mutually tangent circles. Key to the proof is associating an octahedral configuration of spheres to four mutually tangent circles. We also prove an analogue for spheres. It can be traced back to at least Descartes that four mutually tangent circles have curvatures (reciprocals of radii) satisfying the relatio...

In this work, we formally proved Descartes Rule of Signs, which relates the number of positive real roots of a polynomial with the number of sign changes in its coefficient list. Our proof follows the simple inductive proof given by Arthan [1], which was also used by John Harrison in his HOL Light formalisation. We proved most of the lemmas for arbitrary linearly-ordered integrity domains (e.g....