On Monday, November 11, 2019 at 3:28:05 AM UTC-6, Bruno Marchal wrote: > > > On 10 Nov 2019, at 20:09, Lawrence Crowell <[email protected] > <javascript:>> 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. > > > Which infinitesimals? The informal one by Newton or Leibniz? Their > recovering in non-standard analysis? > Of in synthetic (category based) geometry? > > If one is sticking to a more formal approach then Leibniz Really Weierstrass is the guy who got this straight.
> 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). > > These things are not about belief or nonbelief. They are formal models, and as I see it one works with any particular model if it is useful. > > > 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. > > > The MVT relies upon calculus f'(c) = (f(a) - f(b))/(a - b) or the integral form ∫f(x)dx = f(c)(a - b) for b to a limits in integral. So infinitesimals are there at least implicitly. When it comes to point set topology I prefer to get past that as quickly as possible and get to cohomology, homotopy or cobordism. > > 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 > > I don't have thoughts on the mind-body problem. I have no particular theory about consciousness or anything related. LC -- 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/6ce67a53-c135-46e3-9dc1-9937acf530e4%40googlegroups.com.
