Paradox, Natural Mathematics, Relativity and Twentieth-Century Ideas

Here we examine the influence of the new set theory historiography - in particular the works of Garciadiego, Grattan-Guinness and Ferreiros - on our understanding of major twentieth-century ideas, including those of Einstein, Kimura, Sraffa and Godel. Conclusions regarding the nature of set theory have been upended by this new work, and the mathematical approach of readers will be challenged. Therefore, it is recommended that readers familiarize themselves with the new historiography - particularly Garciadiego - before venturing further. In particular, we identify, for the first time, the way Einstein used practical geometry - what today is known as constructivism - in his formulation of the relativity of simultaneity. Not quite knowing what they were looking for, physicists have nevertheless indirectly made this the work of physics since Einstein promulgated the relativity of simultaneity in 1905. Thus, a century-long search comes to an end.We also lay the groundwork for answering the question, where the constructivist intervention is made in the arguments of Sraffa, Kimura and Godel? This question also arises with respect to the Pythagorean theorem