The map from Euler reals to Dedekind reals is not injective (1) The rig of uppercuts in Q serves as value-space for metrics; call it the Dedekind reals for short. (Mapping a ring to the Dedekind reals would only hit two-sided cuts, but that is a separate issue. If Q denotes the nonnegative rationals, then the term "arithmetic reals" would be justified, but for the issue addressed here, Q might as well be "the constant reals" coming from a lower topos). (2) 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 T of infinitesimals as the representing object for the tangent-bundle functor, I call Euler reals the part R of the function-space T^T that preserves the base point. (T is regarded as given as a reflection of physical experience, so not every topos has one. 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 require 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)). (3) Philosophically, the Euler reals serve not only to parameterize motion but also to provide a means to express a cause of motion; the cause operates at each single time, as is reflected in the fact that T has a unique point. By contrast, the Dedekind reals serve to measure, by Q-approximations, the changes resulting from motion between pairs of times. Measuring, like photographing, kills the particular motion; thus the map from Euler reals to Dedekind reals, should not be expected to be injective. That map needs to be understood in any smooth topos of interest,. (a) Of course, measuring can still derive information, perhaps even enough information, about the causes of motion too: we can pass to another moving quantity, e.g. velocity via a speedometer, and then measure that via rational approximation. There is an analogy with algebraic topology: pizero is a very crude measure of a space, it would seem, but as Sammy liked to point out, if you apply an appropriate geometrical endofunctor first, then pizero can deliver lots of useful information.) (b) Any given object in a smooth topos will induce a function presheaf on finite-dimensional varieties; since continuous functions are not usually smooth functions, it is unlikely that the Dedekind reals (even two-sided) will be included in R.) (4) However, an inclusion Q->R of constants is to be expected; it 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 such an ordering, but it always seems to turn out that the ordering is not anti-symmetric (in particular that any closed interval is closed under the addition of infinitesimals), illustrating the non-injectivity of the map. (In some cases there seem to be ways to construct the ordering "synthetically", i.e., by categorical operations, such as pizero(Aut(T)), applied ultimately to the object T.) I hope this will suggest some clarification of the questions raised by Thomas and Andrej. Bill Lawvere