AXIOMATIC CONSTRUCTION OF THE TANGENT BUNDLE IN SYNTHETIC DIFFERENTIAL GEOMETRY - Sebastian, Ian
Synthetic differential geometry (SDG) is a reformulation of differential geometry based on an axiom that is synthetic in nature, in the sense that it relates to geometric objects and the relations between those objects, such that a rigorous calculus that incorporates nilpotent infinitesimals can be done. As a basic assumption, we assume the existence of a basic continuum object R which obeys a principle we call the Microaffineness Principle, which states that every curve can be seen as a composition of microstraight lines, which is somewhat in line with the classical understanding of the continuum, an ideal object on which true continuous variation is possible. To be exact, we assume that R obeys the Kock-Lawvere axiom, which states that all functions from the domain D = {d ? R : d2 = 0} to R can be uniquely described as an affine function. Having been allowed to work with infinitesimal elements explicitly, we are now able to carry on rigorous constructions of geometric objects in a more general manner that better accomodates the geometric intuition we might naturally have of the object, through arguing with infinitesimal elements. However, to work with this principle, we have to sacrifice the universality of the Law of Excluded Middle, a principle of logic which states that P ?¬P always holds for all propositions P. A significant consequence of this is the loss of our ability to use non-constructive arguments of proof, particularly the non-constructive version of the reductio ad absurdum argument. In this Thesis, we will examine the benefits we gain from this sacrifice.