Infinitesimals and the continuum (Q1908685)

The author is interested in expanding upon concepts relative to synthetic differential geometry (SDG), a system that requires for the infinitesimal concepts expounded intuitionistic logic. He claims that SDG follows the intuitive infinitesimal ideas of many of the great scientific thinkers of past ages. He argues that a model for SDG contains explicit entities that represent continuous infinitesimals, that they can be specifically determined and that SDG gives a ``remarkable conceptional simplification'' of differential geometry. In the last section, he contrasts the continuous infinitesimals with those discovered by Robinson and appears to claim that the Robinson infinitesimals are somehow or other indistinguishable and less useful. Remarks: I respectfully submit that the implications given in this paper are false. It has been well established that the Robinson Nonstandard Analysis and infinitesimals and infinite numbers follow the exact intuitive ideas of Newton and Leibniz. Further, classical logic is still the foundation of modern science. The remarkable Robinson infinitesimal and infinite numbers are not investigated internal to the model but are investigated in the external world. Their properties are ``perfectly'' detectable, to use the author's words relative to SDG, when properly viewed from this external world. Finally, as established by this reviewer and others, the Robinson approach also gives a ``remarkable simplification'' of differential geometry as well as almost all of standard mathematics.