On Sunday, November 10, 2019 at 6:17:10 AM UTC-6, Bruno Marchal wrote: > > > On 9 Nov 2019, at 02:22, Lawrence Crowell <[email protected] > <javascript:>> 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 > > The key word is useful. Infinitesimals are immensely useful in calculus and point-set topology. It provide a proof of the mean value theorem in calculus, which in higher dimension is Stokes' rule that in the language of forms lends itself to algebraic topology. Something that useful as I see it has some sort of ontology to it, even if it is in the abstract sense of mathematics.
LC > > 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 >> >> *Infinitesimals, Imaginaries, Ideals, and Fictions* >> https://arxiv.org/abs/1304.2137 >> >> *Leibniz vs Ishiguro: Closing a quarter-century of syncategoremania* >> 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] <javascript:>. > 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/aebf7dfe-6627-4c23-b0b5-9c2644e05fc1%40googlegroups.com.
