> On 9 Nov 2019, at 02:22, Lawrence Crowell <[email protected]> > wrote: > > We can think of infinitesimals as a manifestation of Gödel's theorem with > Peano number theory. There is nothing odd that is going to happen with this > number theory, but no matter how much we count we never reach "infinity." We > have then an issue of ω-consistency, and to completeness. To make this > complete we must then say there exists an element that has no successor. We > can now take this "supernatural number" and take the reciprocal of it within > the field of rationals or reals. This is in a way what infinitesimals are. > These are a way that Robinson numbers are constructed. These are as "real" in > a sense, just as imaginary numbers are. They are only pure fictions if one > stays strictly within the Peano number theory. They also have incredible > utility in that the whole topological set theory foundation for algebraic > geometry and topology is based on this.
Roughly thinking, I agree. It corroborates my feeling that first order logic is science, and second-order logic is philosophy. Useful philosophy, note, but useful fiction also. Bruno > > LC > > On Sunday, November 3, 2019 at 6:39:53 AM UTC-6, Philip Thrift wrote: > > Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, > And Their Foes From Berkeley To Russell And Beyond > https://arxiv.org/abs/1205.0174 <https://arxiv.org/abs/1205.0174> > > Infinitesimals, Imaginaries, Ideals, and Fictions > https://arxiv.org/abs/1304.2137 <https://arxiv.org/abs/1304.2137> > > Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania > https://arxiv.org/abs/1603.07209 <https://arxiv.org/abs/1603.07209> > > Leibniz frequently writes that his infinitesimals are useful fictions, and we > agree; but we shall show that it is best not to understand them as logical > fictions; instead, they are better understood as pure fictions. > > @philipthrift > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/bf376129-a933-4d79-9134-8568795df2a4%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/bf376129-a933-4d79-9134-8568795df2a4%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/1C3D928D-4AF3-4796-9B0A-E19367268DB3%40ulb.ac.be.
