On Sunday, November 10, 2019 at 1:09:41 PM UTC-6, Lawrence Crowell wrote: > > 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]> >> 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 > >
It is interesting that infinitesimal calculus [ https://www.math.wisc.edu/~keisler/foundations.pdf ] is still a "backbench" calculus - not mattering so much in science, at least in terms of education. Maybe that's a problem with science. @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]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/047d9a57-4a8f-41c4-909f-d6085e95ab05%40googlegroups.com.
