Comparative Historical Studies (2): A Brief Analysis of Descartes’ Correlation-Hypothesis and Church’s Thesis

According to E. Grosholz, initially, Descartes’ correlation-hypothesis (concerning geometry and algebra) can be described as an incompletely justified and corregible conjecture. I claim that the genesis of Church’s thesis can be described in the same way and the testimonies of Rosser and Kleene offer some evidence of it. Descartes, developing Viète’s ideas, reduced geometrical problems to a standard or canonical form. Similarly, in the \(\lambda \) -calculus any algorithmic problem is also reduced to a canonical form: \(\lambda \) -terms. Now, Descartes’ and Church’s hypotheses were not completely justified by either of the fields independently, because their scope and limits were not yet clear back then. In fact, Kleene emphasized in his testimonial that it was not clear at the time how profound the correlation held, nor how extensive it was. Moreover, neither Descartes nor Church noticed that due to the tentative nature of each correlation-hypotheses, a continuous process of verification (or refutation) of each one was required (but Post did realize that at the time). Finally, Rosser emphasizes the fact that neither of them (Church, Kleene, and Rosser himself) was sure about Church’s thesis during 1933 even though the evidence in favor of it was growing very fast.