On Thursday, July 13, 2023 at 11:51:10 PM UTC-7, Ross Finlayson wrote:
> On Tuesday, July 11, 2023 at 11:11:57 PM UTC-7, Ross Finlayson wrote:
> > On Sunday, July 9, 2023 at 9:56:52 AM UTC-7, Ross Finlayson wrote:
> > > Dana Scott fan club
> > >
> > > Been reading some more into Dana Scott. He has a pretty good intuition and is
> > > also a grandiose sort of hedge. Also he knows things and isn't wrong.
> > >
> > > Been reading a bit into the Habermas school or Frankfurt school.
> > >
> > > Cohen's "Equations from G-d" was a pretty good historical outline about
> > > Boole and de Morgan than about Russell about "pure mathematics" in the
> > > 19'th century, still though I believe in a stronger platonism and that there's
> > > a science of mathematics but its study is _of_ the real "pure mathematics".
> > >
> > > Was reading some Knuth the other day about combinatorics historically,
> > > quite a well-rounded guy.
> > Been reading Quine's "Set Theory" (and Quine's number theory and Quine's model theory, ...).
> >
> > I thought it was pretty good until he got up to real numbers and used the term "non-circularize
> > the argument" in an off-hand way. He started with a good discussion of class/set distinction
> > then put it aside and coat-tailed up past "higher-order equals". As a structuralist I don't much
> > agree except that "equals is first-order", so pretty much the usual coat-tails logician's coat-tailing
> > of "higher-order equals" comes across as "circularized". So, when Quine got to his real numbers
> > and was like "my rationals are reals instead of my reals are rationals" then there's a quibble about
> > least-upper-bound property, pretty much I was disappointed in him when he faked a quibble about
> > least-upper-bound property. Still, I'm only about half-way through so maybe there will be something
> > better to talk about later in it.
> >
> > Dana Scott's pretty good. He's like, "Oh you made an algebra? Here's a boolean lattice."
> >
> >
> > Reading the other days about Schwarz functions and their distributions and Heaviside's function
> > and hysteresis and ringing and Gibbs, from some late '90's papers from NASA, about doubling-spaces
> > and the non-standard and infinitesimals, I figure that it still makes pretty great sense the re-Vitali-ization
> > of measure theory ("after LUB, the other axiom, measure 1.0"), into doubling spaces and Ramsey theory,
> > figuring they'll need a foundations besides their applied.
> >
> > The stopping-derivative is kind of an interesting idea, I've been thinking about the identity dimension
> > and a bunch of great stuff that arrives from re-Vitali-ization and a deconstructive account of the
> > arithmetic and so on.
>
> Well I kept reading Quine's book on set theory, "Set Theory ...", and it's really pretty great
> and one of the better or the best overall books on set theory.
>
> He goes on to explain the various perspectives and approaches to the objects of set theory,
>
> elements have memberships (elt, set theory, Mengen),
> classes have members (contains, part theory, Unmenge),
>
> and explains various organizations of primary objects
>
> Frege and his numbers,
> von Neumann and about functions,
> Russell with types,
> Zermelo and well-foundings
>
> and about well-orderings and ordering theory.
>
> What I notice of it is as the various concerns of the concepts of the objects,
> circle about a common drain,
>
> set and part theory,
> ordering theory,
> number theory,
> function theory
>
> so this sits very well with my approaches to ubiquitous ordinals,
> topology and function theory making for geometry, that to make
> for a circularizing of the circularizing, has that pretty much I can mark
> the salient points in Quine that have where these approaches define
> each other in terms of each other, and suss out a unified approach to
> them-all.
>
> When it comes to coat-tails, here it's canonry, where fully I intend that
> it's one giant coat-tails. (And none.)
>
> For foundations, it's a foundations of logical objects, mathematical objects,
> all one theory.
>
>
> Yeah, I'm pretty happy I wrote an apologetics for modern mathematics and
> paleo-classical post-modern extra-standard ubiquitous ordinals in primary
> objects and ur-elements after all universal theory.
>
> Don't need another one, ....
>
> Quine shirt-sleeves quite a few good quotes on the topic.

Here's an example of a 2023 paper about continuous domains that references a Scott topology.

https://arxiv.org/abs/2301.09940

It sort of makes you wonder how such a, "countable continuous domain", could be, without tipping each other's carts.

"In the infinitary logic", ....

It's funny if you search for "countable continuous domain" nothing shows up, but "modern foundations" "set theory"
"countable continuous domain" sort of arrives here.

On Sunday, November 19, 2023 at 6:44:08 PM UTC-8, Ross Finlayson wrote:
> On Thursday, July 13, 2023 at 11:51:10 PM UTC-7, Ross Finlayson wrote:
> > On Tuesday, July 11, 2023 at 11:11:57 PM UTC-7, Ross Finlayson wrote:
> > > On Sunday, July 9, 2023 at 9:56:52 AM UTC-7, Ross Finlayson wrote:
> > > > Dana Scott fan club
> > > >
> > > > Been reading some more into Dana Scott. He has a pretty good intuition and is
> > > > also a grandiose sort of hedge. Also he knows things and isn't wrong.
> > > >
> > > > Been reading a bit into the Habermas school or Frankfurt school.
> > > >
> > > > Cohen's "Equations from G-d" was a pretty good historical outline about
> > > > Boole and de Morgan than about Russell about "pure mathematics" in the
> > > > 19'th century, still though I believe in a stronger platonism and that there's
> > > > a science of mathematics but its study is _of_ the real "pure mathematics".
> > > >
> > > > Was reading some Knuth the other day about combinatorics historically,
> > > > quite a well-rounded guy.
> > > Been reading Quine's "Set Theory" (and Quine's number theory and Quine's model theory, ...).
> > >
> > > I thought it was pretty good until he got up to real numbers and used the term "non-circularize
> > > the argument" in an off-hand way. He started with a good discussion of class/set distinction
> > > then put it aside and coat-tailed up past "higher-order equals". As a structuralist I don't much
> > > agree except that "equals is first-order", so pretty much the usual coat-tails logician's coat-tailing
> > > of "higher-order equals" comes across as "circularized". So, when Quine got to his real numbers
> > > and was like "my rationals are reals instead of my reals are rationals" then there's a quibble about
> > > least-upper-bound property, pretty much I was disappointed in him when he faked a quibble about
> > > least-upper-bound property. Still, I'm only about half-way through so maybe there will be something
> > > better to talk about later in it.
> > >
> > > Dana Scott's pretty good. He's like, "Oh you made an algebra? Here's a boolean lattice."
> > >
> > >
> > > Reading the other days about Schwarz functions and their distributions and Heaviside's function
> > > and hysteresis and ringing and Gibbs, from some late '90's papers from NASA, about doubling-spaces
> > > and the non-standard and infinitesimals, I figure that it still makes pretty great sense the re-Vitali-ization
> > > of measure theory ("after LUB, the other axiom, measure 1.0"), into doubling spaces and Ramsey theory,
> > > figuring they'll need a foundations besides their applied.
> > >
> > > The stopping-derivative is kind of an interesting idea, I've been thinking about the identity dimension
> > > and a bunch of great stuff that arrives from re-Vitali-ization and a deconstructive account of the
> > > arithmetic and so on.
> >
> > Well I kept reading Quine's book on set theory, "Set Theory ...", and it's really pretty great
> > and one of the better or the best overall books on set theory.
> >
> > He goes on to explain the various perspectives and approaches to the objects of set theory,
> >
> > elements have memberships (elt, set theory, Mengen),
> > classes have members (contains, part theory, Unmenge),
> >
> > and explains various organizations of primary objects
> >
> > Frege and his numbers,
> > von Neumann and about functions,
> > Russell with types,
> > Zermelo and well-foundings
> >
> > and about well-orderings and ordering theory.
> >
> > What I notice of it is as the various concerns of the concepts of the objects,
> > circle about a common drain,
> >
> > set and part theory,
> > ordering theory,
> > number theory,
> > function theory
> >
> > so this sits very well with my approaches to ubiquitous ordinals,
> > topology and function theory making for geometry, that to make
> > for a circularizing of the circularizing, has that pretty much I can mark
> > the salient points in Quine that have where these approaches define
> > each other in terms of each other, and suss out a unified approach to
> > them-all.
> >
> > When it comes to coat-tails, here it's canonry, where fully I intend that
> > it's one giant coat-tails. (And none.)
> >
> > For foundations, it's a foundations of logical objects, mathematical objects,
> > all one theory.
> >
> >
> > Yeah, I'm pretty happy I wrote an apologetics for modern mathematics and
> > paleo-classical post-modern extra-standard ubiquitous ordinals in primary
> > objects and ur-elements after all universal theory.
> >
> > Don't need another one, ....
> >
> > Quine shirt-sleeves quite a few good quotes on the topic.
>
>
> Here's an example of a 2023 paper about continuous domains that references a Scott topology.
>
> https://arxiv.org/abs/2301.09940
>
> It sort of makes you wonder how such a, "countable continuous domain", could be, without tipping each other's carts.
>
> "In the infinitary logic", ....
>
> It's funny if you search for "countable continuous domain" nothing shows up, but "modern foundations" "set theory"
> "countable continuous domain" sort of arrives here.

If you're interested in some of Scott's examples of topologies and
topologies via logic, you might appreciate Vickers' "Topology via Logic".

The usual open topology of course and zero being rational gets into
why otherwise for example rationals and irrationals would be indistinguishable
except for their countability. "Topology via Logic" introduces others.

This Hofmann-Mislove theorem also sort of provides a complement/alternative,
to something like Dedekind completeness, if you don't not look the other way.

Then, one might aver that this leads to contradictions unless there's some way
to model various sorts of continuous domains, like line/field/signal reals.

On Monday, November 20, 2023 at 2:03:03 PM UTC-8, Ross Finlayson wrote:
> On Sunday, November 19, 2023 at 6:44:08 PM UTC-8, Ross Finlayson wrote:
> > On Thursday, July 13, 2023 at 11:51:10 PM UTC-7, Ross Finlayson wrote:
> > > On Tuesday, July 11, 2023 at 11:11:57 PM UTC-7, Ross Finlayson wrote:
> > > > On Sunday, July 9, 2023 at 9:56:52 AM UTC-7, Ross Finlayson wrote:
> > > > > Dana Scott fan club
> > > > >
> > > > > Been reading some more into Dana Scott. He has a pretty good intuition and is
> > > > > also a grandiose sort of hedge. Also he knows things and isn't wrong.
> > > > >
> > > > > Been reading a bit into the Habermas school or Frankfurt school.
> > > > >
> > > > > Cohen's "Equations from G-d" was a pretty good historical outline about
> > > > > Boole and de Morgan than about Russell about "pure mathematics" in the
> > > > > 19'th century, still though I believe in a stronger platonism and that there's
> > > > > a science of mathematics but its study is _of_ the real "pure mathematics".
> > > > >
> > > > > Was reading some Knuth the other day about combinatorics historically,
> > > > > quite a well-rounded guy.
> > > > Been reading Quine's "Set Theory" (and Quine's number theory and Quine's model theory, ...).
> > > >
> > > > I thought it was pretty good until he got up to real numbers and used the term "non-circularize
> > > > the argument" in an off-hand way. He started with a good discussion of class/set distinction
> > > > then put it aside and coat-tailed up past "higher-order equals". As a structuralist I don't much
> > > > agree except that "equals is first-order", so pretty much the usual coat-tails logician's coat-tailing
> > > > of "higher-order equals" comes across as "circularized". So, when Quine got to his real numbers
> > > > and was like "my rationals are reals instead of my reals are rationals" then there's a quibble about
> > > > least-upper-bound property, pretty much I was disappointed in him when he faked a quibble about
> > > > least-upper-bound property. Still, I'm only about half-way through so maybe there will be something
> > > > better to talk about later in it.
> > > >
> > > > Dana Scott's pretty good. He's like, "Oh you made an algebra? Here's a boolean lattice."
> > > >
> > > >
> > > > Reading the other days about Schwarz functions and their distributions and Heaviside's function
> > > > and hysteresis and ringing and Gibbs, from some late '90's papers from NASA, about doubling-spaces
> > > > and the non-standard and infinitesimals, I figure that it still makes pretty great sense the re-Vitali-ization
> > > > of measure theory ("after LUB, the other axiom, measure 1.0"), into doubling spaces and Ramsey theory,
> > > > figuring they'll need a foundations besides their applied.
> > > >
> > > > The stopping-derivative is kind of an interesting idea, I've been thinking about the identity dimension
> > > > and a bunch of great stuff that arrives from re-Vitali-ization and a deconstructive account of the
> > > > arithmetic and so on.
> > >
> > > Well I kept reading Quine's book on set theory, "Set Theory ...", and it's really pretty great
> > > and one of the better or the best overall books on set theory.
> > >
> > > He goes on to explain the various perspectives and approaches to the objects of set theory,
> > >
> > > elements have memberships (elt, set theory, Mengen),
> > > classes have members (contains, part theory, Unmenge),
> > >
> > > and explains various organizations of primary objects
> > >
> > > Frege and his numbers,
> > > von Neumann and about functions,
> > > Russell with types,
> > > Zermelo and well-foundings
> > >
> > > and about well-orderings and ordering theory.
> > >
> > > What I notice of it is as the various concerns of the concepts of the objects,
> > > circle about a common drain,
> > >
> > > set and part theory,
> > > ordering theory,
> > > number theory,
> > > function theory
> > >
> > > so this sits very well with my approaches to ubiquitous ordinals,
> > > topology and function theory making for geometry, that to make
> > > for a circularizing of the circularizing, has that pretty much I can mark
> > > the salient points in Quine that have where these approaches define
> > > each other in terms of each other, and suss out a unified approach to
> > > them-all.
> > >
> > > When it comes to coat-tails, here it's canonry, where fully I intend that
> > > it's one giant coat-tails. (And none.)
> > >
> > > For foundations, it's a foundations of logical objects, mathematical objects,
> > > all one theory.
> > >
> > >
> > > Yeah, I'm pretty happy I wrote an apologetics for modern mathematics and
> > > paleo-classical post-modern extra-standard ubiquitous ordinals in primary
> > > objects and ur-elements after all universal theory.
> > >
> > > Don't need another one, ....
> > >
> > > Quine shirt-sleeves quite a few good quotes on the topic.
> >
> >
> > Here's an example of a 2023 paper about continuous domains that references a Scott topology.
> >
> > https://arxiv.org/abs/2301.09940
> >
> > It sort of makes you wonder how such a, "countable continuous domain", could be, without tipping each other's carts.
> >
> > "In the infinitary logic", ....
> >
> > It's funny if you search for "countable continuous domain" nothing shows up, but "modern foundations" "set theory"
> > "countable continuous domain" sort of arrives here.
> If you're interested in some of Scott's examples of topologies and
> topologies via logic, you might appreciate Vickers' "Topology via Logic".
>
> The usual open topology of course and zero being rational gets into
> why otherwise for example rationals and irrationals would be indistinguishable
> except for their countability. "Topology via Logic" introduces others.
>
> This Hofmann-Mislove theorem also sort of provides a complement/alternative,
> to something like Dedekind completeness, if you don't not look the other way.
>
> Then, one might aver that this leads to contradictions unless there's some way
> to model various sorts of continuous domains, like line/field/signal reals.

Anybody with a mathematics degree has probably sat a topology course.

Then, usually the usual open topology is very well explored, there are others.

The book "Counterexamples in Topology" is pretty great, ..., there are others.

See, one reason I'm a fan of Dana Scott is that he knows enough about the
entire edifice of the usual "modern mathematics" to arrive at why it's reasonable
that there are alternative derivations with alternative conclusions, that, thusly
as mathematics, would need to see a quite necessary revision of what's otherwise,
"foundations", for it all to sit together without contradicting itself.

On 11/19/2023 06:44 PM, Ross Finlayson wrote:
>
> "In the infinitary logic", ....
>
> It's funny if you search for "countable continuous domain" nothing shows up, but "modern foundations" "set theory"
> "countable continuous domain" sort of arrives here.
>
>

Or, it was so that day.

The other day I heard that the circle and
box notations are kind of coming from linguistics,
this amused me and speaks to the confluence of
formal languages and formal languages.

Quite a fan of relevance logic.

I think Scott should be more firm, he's contributed some great things
for modern logic, "classical relevance logic" if everything has to be
"classical" still, but he doesn't really point that out to the
"classical quasi-modal logic" people directly, figuring that people
will eventually pick his up because of thoroughnessand and correctness,
it's still involved putting them together, not just having "won"
but making a "win-win".

I.e., I think Scott should be more firm like "these collected
results for example of mine really make the quasi-modal not modal".