Comparative Historical Studies (1): Viète’s Analytic Art in \(\lambda \) -Philosophy

In his Ars Analytica (The Analytic Art) François Viète introduces a powerful notation that allowed him to analyze geometrical problems in algebraic terms. Viète’s zetetics is the first step of his algebraic analysis; in it the given geometrical problem is transformed into the corresponding algebraic problem. Similarly, the field of \(\lambda \) -zetetics proposed here is the first step of any kind of transformative analysis. In the \(\lambda \) -zetetics step, the given algorithmic problem (or problem of philosophical analysis) is transformed into the corresponding \(\lambda \) -definability or \(\lambda \) -encoding problem. In both cases is the transformed problem, not the original one, which is solved. (Several examples from both Viète’s and Church’s works are discussed in the chapter.) Moreover, in Viète there is an assumed equivalence between solving a geometrical problem (involving curves) and solving its corresponding algebraic problem (involving roots of equations). \(\lambda \) -definability in Church is an explicitly established equivalence between solving an algorithmic problem (involving a numeric function) and solving its corresponding \(\lambda \) -definability problem (involving \(\lambda \) -terms on Church numerals). Finally, Viète mentions the convenience of his symbolic logistics over Diophantus’ numerical logistics. In a partially similar way, Turing (1937) and Kleene (1981) mention the replacement of Turing machines by the more convenient \(\lambda \) -definitions.