Canadian Mathematical Society (CMS)/CAIMS Summer 2004 Meeting

Halifax, Nova Scotia 
June 13-15, 2004 	    

 
F. WILLIAM LAWVERE, University at Buffalo, 244 Mathematics Building
The map from Euler reals to Dedekind reals

    The rig of uppercuts in Q serves as value-space for metrics; call it the Dedekind reals for short (mapping a ring to it would only hit two-sided cuts, but that is a separate issue; if Q denotes the nonnegative rationals, then the term "arithmetic reals" may be justified, but for the issue addressed here, Q might as well be "the constant reals" coming from a lower topos). Euler affirmed that a real should be determined as a ratio between infinitesimals. Adopting a rational definition of "ratios", and conservatively interpreting the appropriate space of infinitesimals T as the representing object for the tangent-bundle functor, we call Euler reals the part R of the function-space TT that preserves the base point. (T is regarded as given as a reflection of physical experience, and R typically has a unique addition compatible with the obvious multiplication; if we define D as the part of R of square 0, the Kock-Lawvere axiom would affirm that there exist units of time, i.e., isomorphisms T® D, or equivalently certain non-unique semigroup structures on T itself (in contrast with the canonical multiplication on our R).) Philosophically, the Euler reals serve not only to parameterize motion but to provide a basis for the cause of motion; the cause operates at each single time. By contrast the Dedekind reals serve to measure by Q-approximations the result of motion; measuring, like a photograph, kills the particular motion. Thus the map from Euler reals to Dedekind reals, which is in urgent need of being understood in any smooth topos of interest, will therefore not be injective. Any given object in a smooth topos will induce a function presheaf on finite-dimensional varieties; since continuous functions are not usually smooth, it is unlikely that the Dedekind reals (even two-sided) will be included in R. An inclusion Q® R of constants is however to be expected, and forms one ingredient for constructing the map under discussion; the other ingredient is an ordering on R, inducing in the obvious way the map from R to parts of Q. Several treatments of SDG postulate this ordering, but it seems to always turn out that the ordering is not anti-symmetric and that closed intervals are closed under the addition of infinitesimals, manifesting the non-injectivity of the map. In some cases there are ways to construct the ordering "synthetically", i.e., by categorical operations, such as pizero (R) applied ultimately to the object T.