We study a contemporary need to complement analytic philosophy with pendular, synthetic approaches. We provide new definitions of the dyad analytics/synthetics and complete it with a natural third, horotics. Some historical trends to support a synthetic/horotic paradigm are studied: (i) Peirce’s ideas around his logic of continuity – non Cantorian continuum and existential graphs – emphasizing the importance of mathematical gestures, (ii) Gödel’s understanding of intuitionism as a synthetic counterpart of classical logic, along with a new horotic approach to his (...) work, (iii) Contemporary mathematical achievements (1950-2000), difficult to understand from analytical philosophy perspectives. Finally, we indicate some main features that a systematical synthetic/horotic reasoning should enforce, in order to fulfill its historic sequence. (shrink)

“Thought requires achievement for its own development, and without this development it is nothing. Thought must live and grow in incessant new and higher translations, or it proves itself not to be genuine thought.” – C. S. Peirce (CP 5.595) Introduction: Captivating Pictures and Liberating Gestures At the center of one of the most famous anecdotes involving a famous philosopher, we encounter what is commonly called in English a gesture, in fact, a Neapolitan gesture, though one made by a Tur...