> On 10 Nov 2019, at 20:09, Lawrence Crowell <[email protected]> > 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] >> <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.
Which infinitesimals? The informal one by Newton or Leibniz? Their recovering in non-standard analysis? Of in synthetic (category based) geometry? Personally, despite I am logician, I don’t really believe in non standard analysis. I find the Cauchy sequences more useful, and directly understandable (the “new” infinitesimal requires an appendix in either mathematical logic or in category theory). > 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. Abstract topology is enough here, in the Kolmogorov topological abstract spaces. You don’t need formal infinitesimal to have a mean value theorem in calculus. I guess you are OK with this. > Something that useful as I see it has some sort of ontology to it, even if it > is in the abstract sense of mathematics. Like physics, when we assume mechanism, it exists in the phenomenological sense, which is the case of all interesting thing. But to solve the mind-body problem, we need to be clear on the ontology, and with mechanism, the natural numbers (accompanied by their usual + and * laws) or anything Turing equivalent is enough, and cannot be extended, without making the phenomenology exploding (full of “white rabbits”). Bruno > > 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 <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] <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] > <mailto:[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 > > <https://groups.google.com/d/msgid/everything-list/aebf7dfe-6627-4c23-b0b5-9c2644e05fc1%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/E71C25AE-2DB6-4C0F-BCE6-8BC2A7D978C8%40ulb.ac.be.
