On Friday, August 17, 2018 at 11:53:54 AM UTC-7, Ross A. Finlayson wrote:
> On Thursday, August 16, 2018 at 2:41:04 PM UTC-7, Ross A. Finlayson wrote:
> > On Thursday, August 16, 2018 at 1:57:27 PM UTC-7, Ross A. Finlayson wrote:
> > > On Thursday, August 16, 2018 at 1:29:02 PM UTC-7, Ross A. Finlayson wrote:
> > > > On Thursday, August 16, 2018 at 12:55:35 PM UTC-7, Ross A. Finlayson wrote:
> > > > > On Thursday, August 16, 2018 at 9:25:31 AM UTC-7, FredJeffries wrote:
> > > > > > On Thursday, August 16, 2018 at 8:02:12 AM UTC-7, antonio.ma...@gmail.com wrote:
> > > > > > > Take, as an example, this limit
> > > > > > >
> > > > > > > (Dodero Baroncini Manfredi, Ghisetti Editions, 1980)
> > > > > > >
> > > > > > > lim(x->p) tan(x/2)^2 / log ((p-x)^2)
> > > > > > >
> > > > > > > easily transformed in...
> > > > > > >
> > > > > > > lim(x->0) cotan(x/2)^2 / 2log(x)
> > > > > > >
> > > > > > > I call "A" the "Infinity unit"
> > > > > > > and I call "n" the infinitesimal unit
> > > > > > > and I say that, every infinitesimal is lesser than any real
> > > > > > > and I say that, every infinity is greater than any real
> > > > > > >
> > > > > > > (0 < n < |x| < A)
> > > > > >
> > > > > > See Gordon Fisher's "The infinite and infinitesimal quantities of du Bois-Reymond and their reception"
> > > > > >
> > > > > > https://pdfs.semanticscholar.org/f8a7/ee5bb3e46f0bc3faa462857cc8e3ec8d9a28.pdf
> > > > >
> > > > > This kind of thing helps a lot
> > > > > to show the mutual necessities
> > > > > of completeness and rigor.
> > > > >
> > > > > There plainly are theories with
> > > > > infinities and infinitesimals
> > > > > (and even standard ones, and
> > > > > as even about contributing to
> > > > > the character of the real numbers
> > > > > and the standard real numbers),
> > > > > that's mathematics.
> > > >
> > > > Fisher's survey here is really
> > > > rather excellent and you might
> > > > compare it with Ehrlich's.
> > > >
> > > > It helps to identify that the
> > > > various schools about infinitesimals
> > > > that were modern in the era of Cantor
> > > > (du Bois-Reymond, Veronese, Stolz)
> > > > have quite a mathematical heritage
> > > > and relevant context for their consideration,
> > > > vis-a-vis today's hyper-reals or surreals
> > > > as only "conservative extensions" of the
> > > > formalism of Dedekind after Weierstrass
> > > > with Cantor, that there are others.
> > > >
> > > > This wasn't perhaps very widely or
> > > > well-known these decades here where
> > > > we discuss infinities and infinitesimals,
> > > > it helps a lot to improve the context
> > > > and helps thinkers in the acknowledgment
> > > > that these very issues which continue to
> > > > confound some today (because, they're
> > > > researches in mathematics and naturally
> > > > arise) at least already have many and
> > > > various schools of thought to so complement
> > > > what is the regular, standard usage of
> > > > infinity as about notions of infinitesimal
> > > > that is the infinitesimal analysis, the
> > > > integral calculus, the real analysis.
> > > >
> > > > Or, "Newton and Leibniz were quite right."
> > > >
> > > > These days to figure out how to address
> > > > what would be paradoxes and find again
> > > > (because, as mathematical objects they're
> > > > resistant to meddling and ignorance) some
> > > > of these "standard infinitesimals" (or as
> > > > I often discuss "iota-values"), it's at
> > > > least more reasonable for critical thinkers
> > > > (with very strong belief systems about
> > > > mathematics, and not just axiomatized
> > > > ignorance as simply-internally-consistent)
> > > > to be able to reflect on the conceptual
> > > > development of the theories of these
> > > > mathematical objects with a reasonably
> > > > enlightened view of history and not
> > > > just the received paradigm.
> > >
> > > "When he turns to the infinitely small, DU BOIS-REYMOND
> > > says: "The statement that the quantity of division points
> > > on the unit segment is infinitely large, generates with
> > > logical necessity a belief in the infinitely small [same,
> > > 71]. His argument in favor of this is rather breathtaking:
> > > "For if we consider what we presented above as to the correct
> > > concept of quantity, that points on a length do not follow
> > > one another without distance, so cannot adjoin one another,
> > > but are always separated by segments, so that points alone
> > > can never form a segment, then also the infinitely many
> > > points are separated by infinitely many segments, and so
> > > finite, that is, finite in number of these segments can
> > > be contained in the unit segment [perhaps he means the
> > > unit segment cannot contain only a finite number of these
> > > segments], because by the arbitrariness of the unit length,
> > > any segment, however, small, must have the same character
> > > as the unit segment, so that infinitely many division
> > > points must again be present on it [same, 72]. All one
> > > sentence! And this is the complete argument. For he now
> > > concludes: "It therefore results that the unit segment
> > > decomposes into infinitely many subsegments of which
> > > none is finite. Thus the infinitely small actually exists."
> > > [same, 72] "
> > >
> > >
> > > This isn't unfamiliar, that there are infinitesimals
> > > and in the real numbers (and as of the line).
> > >
> > > That's accessible to reason in terms of "trust".
> >
> >
> > "MESCHKOWSKI has nevertheless argued that CANTOR'S
> > argument is not invalid since he was assuming a kind
> > of "continuity" [MESCHKOWSI(~1967, 120]. (Of course,
> > one now knows that DEDEKIND completeness or "continuity"
> > implies the Archimedean property, hence the non-existence
> > of infinitesimals.) However, there are kinds of continuity
> > for linear continua (e.g., the kind introduced by VERONESE)
> > which do not imply the Archimedean property. CANTOR does
> > not specify what kind of continuity he has in mind, and
> > indeed his only reference to continuity in this argument
> > is a single use of the adjective "stetiges" in a very
> > general way [CANTOR 1887 in 1932, 407; MESCHKOWSKI 1967,
> > 118, 120]"
> >
> >
> > Here it's generally known that I frame continuity
> > at least in terms of:
> > line continuity,
> > field continuity,
> > signal continuity.
> >
> >
> > "Then, in a subsequent article commenting on the
> > one by VIVANTI, RODOLFO BETTAZZI remarks that even
> > if the gap in CANTOR'S alleged proof were filled,
> > it would not demonstrate the non-existence of infinitesimals,
> > but would only "prove that Cantor's numbers are not
> > sufficient to exhaust the idea of magnitude [grandezza:
> > quantity?]" [BETTAZZI 1891, 179]."
> >
> >
> > This here is sometimes "Eudoxus/Dedekind/Cauchy is insufficient."
> > (Or rather as I put it.)
> >
> > Seems I've found a bunch of compatible idealogues.
> >
> >
> > "... indeed, confusions between kinds of infinitesimals
> > were common at the time (and still are). "
> >
> >
> > Fisher's survey and history is really quite relevant
> > to those who would understand more about what thought
> > goes into numbers. It reminds of Berkeley's treatise
> > that is so often referenced as about "ghosts of departed
> > quantities", but that really, upon further reading,
> > much like the "hobgoblins of minds" is usually taken
> > out of context, is an ironic defense.
> >
> > Is there such a rich chronology about Veronese?
> > This is where Borel and Pincherle and Hausdorff
> > followed from du Bois-Reymond, I'm much interested
> > in what is "Veronese's definition of continuity".
> >
> >
> >
> > "... PEANO'S real purpose is not to describe a system
> > which would be of use in analysis, but to speak about
> > actual versus potential infinities and infinitesimals.
> > He concludes: "Thus a new category of actual or constant
> > infinitesimal entities is presented. And one sees that
> > the difference between the actual or constant infinites,
> > and potential or variable ones, a question which has so
> > much impassioned the mathematical philosophers, is a question
> > of pure language. The function 1/x, for x tending to infinity,
> > is a variable infinitesimal; its end is a constant infinitesimal"
> > [PEANO 1910 in 1957, 362]. Thus PEANO contradicts his contention
> > of 1892, following CANTOR, that constant infinitesimals are impossible. "
> >
> >
> >
> > Ah, progress.
> Cf.
>
> P. Cantu: "The role of epistemological models
> in Veronese's and Battazi's theory of magnitudes",
> https://philarchive.org/archive/CANTRO-2v1
>
> J.L. Bell: "The Continuous and the Infinitesimals
> in Mathematics and Philosophy",
> https://books.google.com/books?id=Eq8ZualfMRkC&lpg=PA196&ots=Njn5GLsC3m&dq=Veronese%20continuum&pg=PA196#v=onepage&q&f=false
>
> "As a geometer Veronese naturally took an
> essentially geometric view of the continuum."
>
>
>
> This isn't unfamiliar to me as about these
> various definitions of continuity, and also
> as about the polydimensional points, and
> about a spiral space-building curve as a
> natural continuum (and as of an ur-analytic
> geometry).