On 05/14/2026 10:44 PM, Mikko wrote:
On 14/05/2026 18:30, olcott wrote:
On 5/14/2026 2:54 AM, Mikko wrote:
On 13/05/2026 18:28, olcott wrote:
On 5/13/2026 9:38 AM, Andr|- G. Isaak wrote:
On 2026-05-13 08:20, olcott wrote:
On 5/13/2026 6:56 AM, Andr|- G. Isaak wrote:
On 2026-05-13 05:18, olcott wrote:
On 5/12/2026 10:02 PM, Andr|- G. Isaak wrote:
On 2026-05-12 07:32, olcott wrote:
On 5/12/2026 2:05 AM, Mikko wrote:
On 11/05/2026 14:44, olcott wrote:
On 5/11/2026 2:24 AM, Mikko wrote:
On 10/05/2026 22:06, olcott wrote:
We don't have the knowledge that all undecidability is >>>>>>>>>>>>>> merely semantic
incoherence, and can't know because we already know that >>>>>>>>>>>>>> there is
undecidability that is not semantic incoherence. FOr >>>>>>>>>>>>>> example the
axiom system
reCx (1riax = x)
1.5 != 5 re| you are wrong and I am couinting the rest as >>>>>>>>>>>>> gibberish
Middle dot is a commonly used mathematical operator. In this >>>>>>>>>>>> context
where the purpose of the operation is not specified some >>>>>>>>>>>> other symbols
are often used instead, like rey or reO, or operands are just put >>>>>>>>>>>> side by
side with no operator between.
If commonly used mathematics is gibberish to you then we many >>>>>>>>>>>> safely
conclude that you have nothing useful to offer to the groups >>>>>>>>>>>> you
posted to.
reCx (xria1 = x)
reCxreCyreCz (xria(yriaz) = (xriay)riaz)
reCxreay (xriay = 1)
reCxreay (yriax = 1)
is useful for many purposes. But there are sentences like >>>>>>>>>>>>>>
reCxreay (xriay = yriax)
that are undecidable in that system. But there is notiong >>>>>>>>>>>>>> semantically
incoherent in that example or similar ones.
I'm curious to know how you would actually address Mikko's >>>>>>>>>> point here. He's pointed out the rather obvious error in your >>>>>>>>>> reading comprehension, but you've simply glossed over the
example.
In what sense is reCxreay (xriay = yriax) "semantically incoherent"? >>>>>>>>>>
Andr|-
Decimal point versus multiplication operator?
If it is a decimal point then it is incoherent.
Obviously its nor a decimal point. He gave you the actual axioms >>>>>>>> which define the dot operator, so there should be no dispute as >>>>>>>> to its meaning.
The point is that reCxreay (xriay = yriax) is not decidable from those >>>>>>>> axioms,yet it is clearly semantically coherent.
Andr|-
reCxreay (xriay = yriax) is proven true by reCxreCy (xriay = yriax) >>>>>>> which is proven true by the commutative property of multiplication. >>>>>>>
https://people.hsc.edu/faculty-staff/blins/classes/fall18/math105/ >>>>>>> Examples/AlgebraAxioms.pdf
But then you're introducing a new axiom which isn't part of the set >>>>>> of axioms which he introduced.
My thesis:
The entire body of knowledge that can be expressed
in language can be encoded as relations between finite strings.
Therefore, you're not dealing with the same theory.
*The entire body of knowledge that can be expressed in language*
Is the scope. That some knucklehead can fail to bother to define
that "cats are animals" so that his knucklehead formal system
does not know this IS OFF-TOPIC.
The entire body of knowledge includes that i-# = -1, j-# = -1, k-# = -1, >>>> and ijk = -1. From these we can infer that ij = k and ji = -k and
that ij rea ji, at least when we are talking about Hamilton's
quaternions.
i-| = i
i-# = -1
i-| = -i
The entire body of general knowledge expressed in
language includes the "atomic facts" about imaginary
numbers in a finite list and the semantic relations
between finite strings in a finite list such that
any combination of the above can be derived, no
longer a finite list.
Imaginary numbers themselves may be an incoherent notion.
In this case they would not be included in the body of
knowledge.
So a part of the body of knowledge is that multiplication is not
commutative except in restricted domains, so your above claim to
the contrary is not true.
Never seen "restricted" to mean "abelian".
The "restricted" usually refers to "restriction of comprehension",
or reductionist accounts making for things like "restricted relativity", vis-a-vis the "general" or "restricted" (or "special"),
though I have heard a joke about "abelian grapes", vis-a-vis,
abelian groups (those being groups that are commutative i.e.
that their operation is having the property of commutativity).
How about "adeles and etales" then, those are pretty simply
accounts of modularity about the integer moduli. Lots of
acounts of "algebra" are quite more direct when they're
"arithmetizations" besides "algebraizations".
Many people who've studied "algebra" never heard of "magmas".
("What's red and green and goes 90 mph?", is a question
that all school-children for decades before the Internet
would've heard.)
https://en.wikipedia.org/wiki/Adele_ring https://en.wikipedia.org/wiki/%C3%89tale https://en.wikipedia.org/wiki/Restricted_product
"... with these moduli problems which are "really" defined over
cyclotomic integer rings rather than over Z.
I'd suggest that by the time that adeles or etales are invoked,
that usually enough "almost-everywhere" has been invoked,
and that "almost-purity" is "somewhere-dirty".
Then accounts of integers themselves and the super-Archimedean,
e.g., having a point at infinity, and whether that's non-Archimedean,
speaks more to the character of "arithmetizations" than
"agebraizations", since most of the "algebraic geometry"
since "Bourbaki and Langlands" are not quite so much "geometers".
On 05/15/2026 07:27 AM, Ross Finlayson wrote:
On 05/14/2026 10:44 PM, Mikko wrote:
On 14/05/2026 18:30, olcott wrote:
On 5/14/2026 2:54 AM, Mikko wrote:
On 13/05/2026 18:28, olcott wrote:
On 5/13/2026 9:38 AM, Andr|- G. Isaak wrote:
On 2026-05-13 08:20, olcott wrote:
On 5/13/2026 6:56 AM, Andr|- G. Isaak wrote:
On 2026-05-13 05:18, olcott wrote:
On 5/12/2026 10:02 PM, Andr|- G. Isaak wrote:
On 2026-05-12 07:32, olcott wrote:
On 5/12/2026 2:05 AM, Mikko wrote:
On 11/05/2026 14:44, olcott wrote:
On 5/11/2026 2:24 AM, Mikko wrote:
On 10/05/2026 22:06, olcott wrote:
We don't have the knowledge that all undecidability is >>>>>>>>>>>>>>> merely semantic
incoherence, and can't know because we already know that >>>>>>>>>>>>>>> there is
undecidability that is not semantic incoherence. FOr >>>>>>>>>>>>>>> example the
axiom system
reCx (1riax = x)
1.5 != 5 re| you are wrong and I am couinting the rest as >>>>>>>>>>>>>> gibberish
Middle dot is a commonly used mathematical operator. In this >>>>>>>>>>>>> context
where the purpose of the operation is not specified some >>>>>>>>>>>>> other symbols
are often used instead, like rey or reO, or operands are just put >>>>>>>>>>>>> side by
side with no operator between.
If commonly used mathematics is gibberish to you then we many >>>>>>>>>>>>> safely
conclude that you have nothing useful to offer to the groups >>>>>>>>>>>>> you
posted to.
reCx (xria1 = x)
reCxreCyreCz (xria(yriaz) = (xriay)riaz)
reCxreay (xriay = 1)
reCxreay (yriax = 1)
is useful for many purposes. But there are sentences like >>>>>>>>>>>>>>>
reCxreay (xriay = yriax)
that are undecidable in that system. But there is notiong >>>>>>>>>>>>>>> semantically
incoherent in that example or similar ones.
I'm curious to know how you would actually address Mikko's >>>>>>>>>>> point here. He's pointed out the rather obvious error in your >>>>>>>>>>> reading comprehension, but you've simply glossed over the >>>>>>>>>>> example.
In what sense is reCxreay (xriay = yriax) "semantically incoherent"?
Andr|-
Decimal point versus multiplication operator?
If it is a decimal point then it is incoherent.
Obviously its nor a decimal point. He gave you the actual axioms >>>>>>>>> which define the dot operator, so there should be no dispute as >>>>>>>>> to its meaning.
The point is that reCxreay (xriay = yriax) is not decidable from those
axioms,yet it is clearly semantically coherent.
Andr|-
reCxreay (xriay = yriax) is proven true by reCxreCy (xriay = yriax) >>>>>>>> which is proven true by the commutative property of multiplication. >>>>>>>>
https://people.hsc.edu/faculty-staff/blins/classes/fall18/math105/ >>>>>>>> Examples/AlgebraAxioms.pdf
But then you're introducing a new axiom which isn't part of the set >>>>>>> of axioms which he introduced.
My thesis:
The entire body of knowledge that can be expressed
in language can be encoded as relations between finite strings.
Therefore, you're not dealing with the same theory.
*The entire body of knowledge that can be expressed in language*
Is the scope. That some knucklehead can fail to bother to define
that "cats are animals" so that his knucklehead formal system
does not know this IS OFF-TOPIC.
The entire body of knowledge includes that i-# = -1, j-# = -1, k-# = -1, >>>>> and ijk = -1. From these we can infer that ij = k and ji = -k and
that ij rea ji, at least when we are talking about Hamilton's
quaternions.
i-| = i
i-# = -1
i-| = -i
The entire body of general knowledge expressed in
language includes the "atomic facts" about imaginary
numbers in a finite list and the semantic relations
between finite strings in a finite list such that
any combination of the above can be derived, no
longer a finite list.
Imaginary numbers themselves may be an incoherent notion.
In this case they would not be included in the body of
knowledge.
So a part of the body of knowledge is that multiplication is not
commutative except in restricted domains, so your above claim to
the contrary is not true.
Never seen "restricted" to mean "abelian".
The "restricted" usually refers to "restriction of comprehension",
or reductionist accounts making for things like "restricted relativity",
vis-a-vis the "general" or "restricted" (or "special"),
though I have heard a joke about "abelian grapes", vis-a-vis,
abelian groups (those being groups that are commutative i.e.
that their operation is having the property of commutativity).
How about "adeles and etales" then, those are pretty simply
accounts of modularity about the integer moduli. Lots of
acounts of "algebra" are quite more direct when they're
"arithmetizations" besides "algebraizations".
Many people who've studied "algebra" never heard of "magmas".
("What's red and green and goes 90 mph?", is a question
that all school-children for decades before the Internet
would've heard.)
https://en.wikipedia.org/wiki/Adele_ring
https://en.wikipedia.org/wiki/%C3%89tale
https://en.wikipedia.org/wiki/Restricted_product
"... with these moduli problems which are "really" defined over
cyclotomic integer rings rather than over Z.
I'd suggest that by the time that adeles or etales are invoked,
that usually enough "almost-everywhere" has been invoked,
and that "almost-purity" is "somewhere-dirty".
Then accounts of integers themselves and the super-Archimedean,
e.g., having a point at infinity, and whether that's non-Archimedean,
speaks more to the character of "arithmetizations" than
"agebraizations", since most of the "algebraic geometry"
since "Bourbaki and Langlands" are not quite so much "geometers".
Buried in "Bourbaki and Langlands" are many results "old-wrapped-as-new"
and the account since Bourbaki of muddying "inequality"
with "strict inequality" when "algebraic geometry" had a great
fracturing between the algebraists and geometers, then has that
the "function theory" and "topology" of "algebraic geometry"
is often neither, for accounts like Lefschetz as more from
the "geometer's" side of "algebraic geometry".
So, after "arithmetic and analysis and algebra", then the
"function theory and topology" are often all over themselves
in "algebra" and not so much "arithmetic and analysis".
Here a "continuous topology" is its own initial and final topology,
and "functions" include the Cartesian and some examples after
geometry and number theory the "non-Cartesian", making, for example, for where "countable continuous domains" have their proper models
then that it's a result in the Cantorian that some functions are non-Cartesian, or "un-Cartesian", so that besides being constructively
the countable continuous domains, they don't contradict the
un-countability.
The "algebraic geometry" then "differential geometry" are often
quite all over themselves in their own particular sub-fields
and with their own particular accounts of language, and
not-quite-everywhere "everywhere", any almost-everywhere.
I.e., "algebraic geometers" aren't exactly geometers,
and "differential geometers" aren't exactly analysts.
"Strictly", ....
On 05/15/2026 07:46 AM, Ross Finlayson wrote:
On 05/15/2026 07:27 AM, Ross Finlayson wrote:
On 05/14/2026 10:44 PM, Mikko wrote:
On 14/05/2026 18:30, olcott wrote:
On 5/14/2026 2:54 AM, Mikko wrote:
On 13/05/2026 18:28, olcott wrote:
On 5/13/2026 9:38 AM, Andr|- G. Isaak wrote:
On 2026-05-13 08:20, olcott wrote:
On 5/13/2026 6:56 AM, Andr|- G. Isaak wrote:
On 2026-05-13 05:18, olcott wrote:
On 5/12/2026 10:02 PM, Andr|- G. Isaak wrote:
On 2026-05-12 07:32, olcott wrote:
On 5/12/2026 2:05 AM, Mikko wrote:
On 11/05/2026 14:44, olcott wrote:
On 5/11/2026 2:24 AM, Mikko wrote:
On 10/05/2026 22:06, olcott wrote:
We don't have the knowledge that all undecidability is >>>>>>>>>>>>>>>> merely semantic
incoherence, and can't know because we already know that >>>>>>>>>>>>>>>> there is
undecidability that is not semantic incoherence. FOr >>>>>>>>>>>>>>>> example the
axiom system
reCx (1riax = x)
1.5 != 5 re| you are wrong and I am couinting the rest as >>>>>>>>>>>>>>> gibberish
Middle dot is a commonly used mathematical operator. In this >>>>>>>>>>>>>> context
where the purpose of the operation is not specified some >>>>>>>>>>>>>> other symbols
are often used instead, like rey or reO, or operands are just put
side by
side with no operator between.
If commonly used mathematics is gibberish to you then we many >>>>>>>>>>>>>> safely
conclude that you have nothing useful to offer to the groups >>>>>>>>>>>>>> you
posted to.
reCx (xria1 = x)
reCxreCyreCz (xria(yriaz) = (xriay)riaz)
reCxreay (xriay = 1)
reCxreay (yriax = 1)
is useful for many purposes. But there are sentences like >>>>>>>>>>>>>>>>
reCxreay (xriay = yriax)
that are undecidable in that system. But there is notiong >>>>>>>>>>>>>>>> semantically
incoherent in that example or similar ones.
I'm curious to know how you would actually address Mikko's >>>>>>>>>>>> point here. He's pointed out the rather obvious error in your >>>>>>>>>>>> reading comprehension, but you've simply glossed over the >>>>>>>>>>>> example.
In what sense is reCxreay (xriay = yriax) "semantically incoherent"?
Andr|-
Decimal point versus multiplication operator?
If it is a decimal point then it is incoherent.
Obviously its nor a decimal point. He gave you the actual axioms >>>>>>>>>> which define the dot operator, so there should be no dispute as >>>>>>>>>> to its meaning.
The point is that reCxreay (xriay = yriax) is not decidable from those
axioms,yet it is clearly semantically coherent.
Andr|-
reCxreay (xriay = yriax) is proven true by reCxreCy (xriay = yriax) >>>>>>>>> which is proven true by the commutative property of
multiplication.
https://people.hsc.edu/faculty-staff/blins/classes/fall18/math105/ >>>>>>>>> Examples/AlgebraAxioms.pdf
But then you're introducing a new axiom which isn't part of the set >>>>>>>> of axioms which he introduced.
My thesis:
The entire body of knowledge that can be expressed
in language can be encoded as relations between finite strings.
Therefore, you're not dealing with the same theory.
*The entire body of knowledge that can be expressed in language* >>>>>>> Is the scope. That some knucklehead can fail to bother to define >>>>>>> that "cats are animals" so that his knucklehead formal system
does not know this IS OFF-TOPIC.
The entire body of knowledge includes that i-# = -1, j-# = -1, k-# = -1, >>>>>> and ijk = -1. From these we can infer that ij = k and ji = -k and
that ij rea ji, at least when we are talking about Hamilton's
quaternions.
i-| = i
i-# = -1
i-| = -i
The entire body of general knowledge expressed in
language includes the "atomic facts" about imaginary
numbers in a finite list and the semantic relations
between finite strings in a finite list such that
any combination of the above can be derived, no
longer a finite list.
Imaginary numbers themselves may be an incoherent notion.
In this case they would not be included in the body of
knowledge.
So a part of the body of knowledge is that multiplication is not
commutative except in restricted domains, so your above claim to
the contrary is not true.
Never seen "restricted" to mean "abelian".
The "restricted" usually refers to "restriction of comprehension",
or reductionist accounts making for things like "restricted relativity", >>> vis-a-vis the "general" or "restricted" (or "special"),
though I have heard a joke about "abelian grapes", vis-a-vis,
abelian groups (those being groups that are commutative i.e.
that their operation is having the property of commutativity).
How about "adeles and etales" then, those are pretty simply
accounts of modularity about the integer moduli. Lots of
acounts of "algebra" are quite more direct when they're
"arithmetizations" besides "algebraizations".
Many people who've studied "algebra" never heard of "magmas".
("What's red and green and goes 90 mph?", is a question
that all school-children for decades before the Internet
would've heard.)
https://en.wikipedia.org/wiki/Adele_ring
https://en.wikipedia.org/wiki/%C3%89tale
https://en.wikipedia.org/wiki/Restricted_product
"... with these moduli problems which are "really" defined over
cyclotomic integer rings rather than over Z.
I'd suggest that by the time that adeles or etales are invoked,
that usually enough "almost-everywhere" has been invoked,
and that "almost-purity" is "somewhere-dirty".
Then accounts of integers themselves and the super-Archimedean,
e.g., having a point at infinity, and whether that's non-Archimedean,
speaks more to the character of "arithmetizations" than
"agebraizations", since most of the "algebraic geometry"
since "Bourbaki and Langlands" are not quite so much "geometers".
Buried in "Bourbaki and Langlands" are many results "old-wrapped-as-new"
and the account since Bourbaki of muddying "inequality"
with "strict inequality" when "algebraic geometry" had a great
fracturing between the algebraists and geometers, then has that
the "function theory" and "topology" of "algebraic geometry"
is often neither, for accounts like Lefschetz as more from
the "geometer's" side of "algebraic geometry".
So, after "arithmetic and analysis and algebra", then the
"function theory and topology" are often all over themselves
in "algebra" and not so much "arithmetic and analysis".
Here a "continuous topology" is its own initial and final topology,
and "functions" include the Cartesian and some examples after
geometry and number theory the "non-Cartesian", making, for example, for
where "countable continuous domains" have their proper models
then that it's a result in the Cantorian that some functions are
non-Cartesian, or "un-Cartesian", so that besides being constructively
the countable continuous domains, they don't contradict the
un-countability.
The "algebraic geometry" then "differential geometry" are often
quite all over themselves in their own particular sub-fields
and with their own particular accounts of language, and
not-quite-everywhere "everywhere", any almost-everywhere.
I.e., "algebraic geometers" aren't exactly geometers,
and "differential geometers" aren't exactly analysts.
"Strictly", ....
https://www.jmilne.org/math/apocrypha.html :
"Finally, a story to keep in mind the next time you ask a totally stupid question at a major lecture. During a Bourbaki seminar on the status of
the classification problem for simple finite groups, the speaker
mentioned that it was not known whether a simple group (the monster)
existed of a certain order. "Could there be more than one simple group
of that order?" asked Weil from the audience. "Yes, there could" replied
the speaker. "Well, could there be infinitely many?" asked Weil."
The "K-theory" or "representation" theory is often hiding essentially "mis-representation" or truncations/approximations then to get the
neat algebraic properties then for what arithmetic provides about
geometry. "Almost-purity?", ..., actually dirty.
On 05/15/2026 07:54 AM, Ross Finlayson wrote:
On 05/15/2026 07:46 AM, Ross Finlayson wrote:
On 05/15/2026 07:27 AM, Ross Finlayson wrote:
On 05/14/2026 10:44 PM, Mikko wrote:
On 14/05/2026 18:30, olcott wrote:
On 5/14/2026 2:54 AM, Mikko wrote:
On 13/05/2026 18:28, olcott wrote:
On 5/13/2026 9:38 AM, Andr|- G. Isaak wrote:
On 2026-05-13 08:20, olcott wrote:
On 5/13/2026 6:56 AM, Andr|- G. Isaak wrote:
On 2026-05-13 05:18, olcott wrote:
On 5/12/2026 10:02 PM, Andr|- G. Isaak wrote:
On 2026-05-12 07:32, olcott wrote:
On 5/12/2026 2:05 AM, Mikko wrote:
On 11/05/2026 14:44, olcott wrote:
On 5/11/2026 2:24 AM, Mikko wrote:
On 10/05/2026 22:06, olcott wrote:
We don't have the knowledge that all undecidability is >>>>>>>>>>>>>>>>> merely semantic
incoherence, and can't know because we already know that >>>>>>>>>>>>>>>>> there is
undecidability that is not semantic incoherence. FOr >>>>>>>>>>>>>>>>> example the
axiom system
reCx (1riax = x)
1.5 != 5 re| you are wrong and I am couinting the rest as >>>>>>>>>>>>>>>> gibberish
Middle dot is a commonly used mathematical operator. In this >>>>>>>>>>>>>>> context
where the purpose of the operation is not specified some >>>>>>>>>>>>>>> other symbols
are often used instead, like rey or reO, or operands are just >>>>>>>>>>>>>>> put
side by
side with no operator between.
If commonly used mathematics is gibberish to you then we >>>>>>>>>>>>>>> many
safely
conclude that you have nothing useful to offer to the groups >>>>>>>>>>>>>>> you
posted to.
reCx (xria1 = x)
reCxreCyreCz (xria(yriaz) = (xriay)riaz)
reCxreay (xriay = 1)
reCxreay (yriax = 1)
is useful for many purposes. But there are sentences like >>>>>>>>>>>>>>>>>
reCxreay (xriay = yriax)
that are undecidable in that system. But there is notiong >>>>>>>>>>>>>>>>> semantically
incoherent in that example or similar ones.
I'm curious to know how you would actually address Mikko's >>>>>>>>>>>>> point here. He's pointed out the rather obvious error in your >>>>>>>>>>>>> reading comprehension, but you've simply glossed over the >>>>>>>>>>>>> example.
In what sense is reCxreay (xriay = yriax) "semantically incoherent"?
Andr|-
Decimal point versus multiplication operator?
If it is a decimal point then it is incoherent.
Obviously its nor a decimal point. He gave you the actual axioms >>>>>>>>>>> which define the dot operator, so there should be no dispute as >>>>>>>>>>> to its meaning.
The point is that reCxreay (xriay = yriax) is not decidable from those
axioms,yet it is clearly semantically coherent.
Andr|-
reCxreay (xriay = yriax) is proven true by reCxreCy (xriay = yriax) >>>>>>>>>> which is proven true by the commutative property of
multiplication.
https://people.hsc.edu/faculty-staff/blins/classes/fall18/math105/ >>>>>>>>>>
Examples/AlgebraAxioms.pdf
But then you're introducing a new axiom which isn't part of the >>>>>>>>> set
of axioms which he introduced.
My thesis:
The entire body of knowledge that can be expressed
in language can be encoded as relations between finite strings. >>>>>>>>
Therefore, you're not dealing with the same theory.
*The entire body of knowledge that can be expressed in language* >>>>>>>> Is the scope. That some knucklehead can fail to bother to define >>>>>>>> that "cats are animals" so that his knucklehead formal system
does not know this IS OFF-TOPIC.
The entire body of knowledge includes that i-# = -1, j-# = -1, k-# = >>>>>>> -1,
and ijk = -1. From these we can infer that ij = k and ji = -k and >>>>>>> that ij rea ji, at least when we are talking about Hamilton's
quaternions.
i-| = i
i-# = -1
i-| = -i
The entire body of general knowledge expressed in
language includes the "atomic facts" about imaginary
numbers in a finite list and the semantic relations
between finite strings in a finite list such that
any combination of the above can be derived, no
longer a finite list.
Imaginary numbers themselves may be an incoherent notion.
In this case they would not be included in the body of
knowledge.
So a part of the body of knowledge is that multiplication is not
commutative except in restricted domains, so your above claim to
the contrary is not true.
Never seen "restricted" to mean "abelian".
The "restricted" usually refers to "restriction of comprehension",
or reductionist accounts making for things like "restricted
relativity",
vis-a-vis the "general" or "restricted" (or "special"),
though I have heard a joke about "abelian grapes", vis-a-vis,
abelian groups (those being groups that are commutative i.e.
that their operation is having the property of commutativity).
How about "adeles and etales" then, those are pretty simply
accounts of modularity about the integer moduli. Lots of
acounts of "algebra" are quite more direct when they're
"arithmetizations" besides "algebraizations".
Many people who've studied "algebra" never heard of "magmas".
("What's red and green and goes 90 mph?", is a question
that all school-children for decades before the Internet
would've heard.)
https://en.wikipedia.org/wiki/Adele_ring
https://en.wikipedia.org/wiki/%C3%89tale
https://en.wikipedia.org/wiki/Restricted_product
"... with these moduli problems which are "really" defined over
cyclotomic integer rings rather than over Z.
I'd suggest that by the time that adeles or etales are invoked,
that usually enough "almost-everywhere" has been invoked,
and that "almost-purity" is "somewhere-dirty".
Then accounts of integers themselves and the super-Archimedean,
e.g., having a point at infinity, and whether that's non-Archimedean,
speaks more to the character of "arithmetizations" than
"agebraizations", since most of the "algebraic geometry"
since "Bourbaki and Langlands" are not quite so much "geometers".
Buried in "Bourbaki and Langlands" are many results "old-wrapped-as-new" >>> and the account since Bourbaki of muddying "inequality"
with "strict inequality" when "algebraic geometry" had a great
fracturing between the algebraists and geometers, then has that
the "function theory" and "topology" of "algebraic geometry"
is often neither, for accounts like Lefschetz as more from
the "geometer's" side of "algebraic geometry".
So, after "arithmetic and analysis and algebra", then the
"function theory and topology" are often all over themselves
in "algebra" and not so much "arithmetic and analysis".
Here a "continuous topology" is its own initial and final topology,
and "functions" include the Cartesian and some examples after
geometry and number theory the "non-Cartesian", making, for example, for >>> where "countable continuous domains" have their proper models
then that it's a result in the Cantorian that some functions are
non-Cartesian, or "un-Cartesian", so that besides being constructively
the countable continuous domains, they don't contradict the
un-countability.
The "algebraic geometry" then "differential geometry" are often
quite all over themselves in their own particular sub-fields
and with their own particular accounts of language, and
not-quite-everywhere "everywhere", any almost-everywhere.
I.e., "algebraic geometers" aren't exactly geometers,
and "differential geometers" aren't exactly analysts.
"Strictly", ....
https://www.jmilne.org/math/apocrypha.html :
"Finally, a story to keep in mind the next time you ask a totally stupid
question at a major lecture. During a Bourbaki seminar on the status of
the classification problem for simple finite groups, the speaker
mentioned that it was not known whether a simple group (the monster)
existed of a certain order. "Could there be more than one simple group
of that order?" asked Weil from the audience. "Yes, there could" replied
the speaker. "Well, could there be infinitely many?" asked Weil."
The "K-theory" or "representation" theory is often hiding essentially
"mis-representation" or truncations/approximations then to get the
neat algebraic properties then for what arithmetic provides about
geometry. "Almost-purity?", ..., actually dirty.
https://publish.uwo.ca/~jbell/
https://www.researchgate.net/publication/344371879_Reflections_on_Bourbaki%27s_Notion_of_Structure_and_Categories
"Here Bourbaki/Dieudonn|- uses the phrase "tower of Babel" with its usual connotation of "place of confusion"."
"Thus the tower of Babel might be seen, not as representing the
jumble of separate practices that Bourbaki deplores, but rather as the
unity that they wished to impose on mathematics. In that case,
Bourbaki's |elements would, ironically perhaps, amount precisely to the attempt to build a mathematical "tower of Babel". The
fact that Bourbaki failed - as is well-known - to complete his grandiose project as originally conceived was not, as in Genesis, the result of
God sowing linguistic confusion - the Bourbaki members, after all, still spoke a common mathematical language. Rather, BourbakirCOs project was
simply too ambitious to be brought to completion. Nevertheless, the |el|-ments, unlike the tower of Babel, remains a
magnificent plinth."
"Bourbaki identifies three basic types of mathematical structure -
structures m|?res" - or "mother structures". These are algebraic, order,
and topological structures, which can be summed up as the "three C's" : Combination, Comparison and Continuity."
These quotes of Bell, who also from his web-page has a variety of
essays on continuity, are insightful, and here it can be very plainly
obvious that a usual account of "ZFC + Martin's Axiom a.k.a. the
illative or univalency: the strength of ZFC and two large cardinals",
has that
well-foundedness and well-ordering and well-dispersion
are as of three sorts "rulialities/regularities" that make for
"combination, comparison, and continuity",
then usually that those are to be reversed in order.
Bourbaki/Dieudonne wrote (as quoted by Bell): "To set up the axiomatic
theory of a given structure amounts to the deduction of the logical consequences of the axioms of the structure, excluding every other
hypothesis on the elements under consideration (in particular, every hypothesis as to their own nature)."
Bell: "Accordingly, the departure from the scene of the concept of set
opened the way for Bourbaki to maintain that the unity of mathematics
stems, not from the set concept, but from the concept of structure."
Then, here those are as of structural realists' heno-theories,
not nominalist fictionalists' accounts of the not-thereof.
Bell: "In any case, as already remarked, Bourbaki's general concept of structure, organized into species, plays only a very minor role in his
actual development of mathematics. By and large, only specific kinds of structure are discussed: e.g., topological spaces, algebraic structures,
and combinations of the two such as topological groups. In
practice, the role of structure in general is played by the defining
axioms of the various species of structures."
It doesn't take a very generous or biased reading of Bell to have him throwing quite a bit of shade on Bourbaki as alike a troll who espouses
a universal for completion then immediately recidivates.
As to why "category theory" and "model theory" need definitions of
"faithful" the functors and witnesses the models, the structures in the theory, is because they haven't been.
Bell's great paper "Reflections on Bourbaki's Notion of Structure and Categories" is rather a quite fantastic survey and introduction, and
while it has any number of mathematical formulas each of which will
halve the popular audience, it as well has quite a number of diagrams
which speak directly to graphical learners.
Bell: "In 1940 Stone established what amounts to the duality between the category of compact Hausdorff spaces and the category of real
C*-algebras - commutative rings equipped rings with an order structure
and a norm naturally possessed by rings of bounded real-valued intensive quantities."
Accounts like Naimark's "Normed Rings" is quite a different and more conscientious approach, than just "equipping" the things, and forgetting
the differences.
Bell: "In the transition from Bourbaki's account of mathematics to its category-theoretic formulation, what is the fate of the "mother
structures"? It is remarkable that, given Bourbaki`s distaste for logic, their mother structures came to play a key role in establishing the connection between category theory and logic."
Bell closes the paper: "By contrast, category theorists - those, at
least, who are sensitive to such issues (and those constitute the
majority) - regard set theory as a kind of ladder leading from pure discreteness to the depiction of the mathematical landscape in terms of
pure Form. Categorists are no different from artists in finding the
landscape (or its depiction, at least) more interesting than the ladder, which should, following Wittgenstein's advice, be jettisoned after ascent."
There's that damn Wittgenstein again: claiming that it's
inter-subjective that nobody else is, and stealing the last parachute.
On 05/15/2026 08:34 AM, Ross Finlayson wrote:
On 05/15/2026 07:54 AM, Ross Finlayson wrote:
On 05/15/2026 07:46 AM, Ross Finlayson wrote:
On 05/15/2026 07:27 AM, Ross Finlayson wrote:
On 05/14/2026 10:44 PM, Mikko wrote:
On 14/05/2026 18:30, olcott wrote:
On 5/14/2026 2:54 AM, Mikko wrote:
On 13/05/2026 18:28, olcott wrote:
On 5/13/2026 9:38 AM, Andr|- G. Isaak wrote:
On 2026-05-13 08:20, olcott wrote:
On 5/13/2026 6:56 AM, Andr|- G. Isaak wrote:
On 2026-05-13 05:18, olcott wrote:
On 5/12/2026 10:02 PM, Andr|- G. Isaak wrote:
On 2026-05-12 07:32, olcott wrote:
On 5/12/2026 2:05 AM, Mikko wrote:
On 11/05/2026 14:44, olcott wrote:
On 5/11/2026 2:24 AM, Mikko wrote:
On 10/05/2026 22:06, olcott wrote:
We don't have the knowledge that all undecidability is >>>>>>>>>>>>>>>>>> merely semantic
incoherence, and can't know because we already know that >>>>>>>>>>>>>>>>>> there is
undecidability that is not semantic incoherence. FOr >>>>>>>>>>>>>>>>>> example the
axiom system
reCx (1riax = x)
1.5 != 5 re| you are wrong and I am couinting the rest as >>>>>>>>>>>>>>>>> gibberish
Middle dot is a commonly used mathematical operator. In >>>>>>>>>>>>>>>> this
context
where the purpose of the operation is not specified some >>>>>>>>>>>>>>>> other symbols
are often used instead, like rey or reO, or operands are just >>>>>>>>>>>>>>>> put
side by
side with no operator between.
If commonly used mathematics is gibberish to you then we >>>>>>>>>>>>>>>> many
safely
conclude that you have nothing useful to offer to the >>>>>>>>>>>>>>>> groups
you
posted to.
reCx (xria1 = x)
reCxreCyreCz (xria(yriaz) = (xriay)riaz) >>>>>>>>>>>>>>>>>> reCxreay (xriay = 1)
reCxreay (yriax = 1)
is useful for many purposes. But there are sentences like >>>>>>>>>>>>>>>>>>
reCxreay (xriay = yriax)
that are undecidable in that system. But there is notiong >>>>>>>>>>>>>>>>>> semantically
incoherent in that example or similar ones.
I'm curious to know how you would actually address Mikko's >>>>>>>>>>>>>> point here. He's pointed out the rather obvious error in your >>>>>>>>>>>>>> reading comprehension, but you've simply glossed over the >>>>>>>>>>>>>> example.
In what sense is reCxreay (xriay = yriax) "semantically incoherent"?
Andr|-
Decimal point versus multiplication operator?
If it is a decimal point then it is incoherent.
Obviously its nor a decimal point. He gave you the actual >>>>>>>>>>>> axioms
which define the dot operator, so there should be no dispute as >>>>>>>>>>>> to its meaning.
The point is that reCxreay (xriay = yriax) is not decidable from those
axioms,yet it is clearly semantically coherent.
Andr|-
reCxreay (xriay = yriax) is proven true by reCxreCy (xriay = yriax) >>>>>>>>>>> which is proven true by the commutative property of
multiplication.
https://people.hsc.edu/faculty-staff/blins/classes/fall18/math105/ >>>>>>>>>>>
Examples/AlgebraAxioms.pdf
But then you're introducing a new axiom which isn't part of the >>>>>>>>>> set
of axioms which he introduced.
My thesis:
The entire body of knowledge that can be expressed
in language can be encoded as relations between finite strings. >>>>>>>>>
Therefore, you're not dealing with the same theory.
*The entire body of knowledge that can be expressed in language* >>>>>>>>> Is the scope. That some knucklehead can fail to bother to define >>>>>>>>> that "cats are animals" so that his knucklehead formal system >>>>>>>>> does not know this IS OFF-TOPIC.
The entire body of knowledge includes that i-# = -1, j-# = -1, k-# = >>>>>>>> -1,
and ijk = -1. From these we can infer that ij = k and ji = -k and >>>>>>>> that ij rea ji, at least when we are talking about Hamilton's
quaternions.
i-| = i
i-# = -1
i-| = -i
The entire body of general knowledge expressed in
language includes the "atomic facts" about imaginary
numbers in a finite list and the semantic relations
between finite strings in a finite list such that
any combination of the above can be derived, no
longer a finite list.
Imaginary numbers themselves may be an incoherent notion.
In this case they would not be included in the body of
knowledge.
So a part of the body of knowledge is that multiplication is not
commutative except in restricted domains, so your above claim to
the contrary is not true.
Never seen "restricted" to mean "abelian".
The "restricted" usually refers to "restriction of comprehension",
or reductionist accounts making for things like "restricted
relativity",
vis-a-vis the "general" or "restricted" (or "special"),
though I have heard a joke about "abelian grapes", vis-a-vis,
abelian groups (those being groups that are commutative i.e.
that their operation is having the property of commutativity).
How about "adeles and etales" then, those are pretty simply
accounts of modularity about the integer moduli. Lots of
acounts of "algebra" are quite more direct when they're
"arithmetizations" besides "algebraizations".
Many people who've studied "algebra" never heard of "magmas".
("What's red and green and goes 90 mph?", is a question
that all school-children for decades before the Internet
would've heard.)
https://en.wikipedia.org/wiki/Adele_ring
https://en.wikipedia.org/wiki/%C3%89tale
https://en.wikipedia.org/wiki/Restricted_product
"... with these moduli problems which are "really" defined over
cyclotomic integer rings rather than over Z.
I'd suggest that by the time that adeles or etales are invoked,
that usually enough "almost-everywhere" has been invoked,
and that "almost-purity" is "somewhere-dirty".
Then accounts of integers themselves and the super-Archimedean,
e.g., having a point at infinity, and whether that's non-Archimedean, >>>>> speaks more to the character of "arithmetizations" than
"agebraizations", since most of the "algebraic geometry"
since "Bourbaki and Langlands" are not quite so much "geometers".
Buried in "Bourbaki and Langlands" are many results
"old-wrapped-as-new"
and the account since Bourbaki of muddying "inequality"
with "strict inequality" when "algebraic geometry" had a great
fracturing between the algebraists and geometers, then has that
the "function theory" and "topology" of "algebraic geometry"
is often neither, for accounts like Lefschetz as more from
the "geometer's" side of "algebraic geometry".
So, after "arithmetic and analysis and algebra", then the
"function theory and topology" are often all over themselves
in "algebra" and not so much "arithmetic and analysis".
Here a "continuous topology" is its own initial and final topology,
and "functions" include the Cartesian and some examples after
geometry and number theory the "non-Cartesian", making, for example,
for
where "countable continuous domains" have their proper models
then that it's a result in the Cantorian that some functions are
non-Cartesian, or "un-Cartesian", so that besides being constructively >>>> the countable continuous domains, they don't contradict the
un-countability.
The "algebraic geometry" then "differential geometry" are often
quite all over themselves in their own particular sub-fields
and with their own particular accounts of language, and
not-quite-everywhere "everywhere", any almost-everywhere.
I.e., "algebraic geometers" aren't exactly geometers,
and "differential geometers" aren't exactly analysts.
"Strictly", ....
https://www.jmilne.org/math/apocrypha.html :
"Finally, a story to keep in mind the next time you ask a totally stupid >>> question at a major lecture. During a Bourbaki seminar on the status of
the classification problem for simple finite groups, the speaker
mentioned that it was not known whether a simple group (the monster)
existed of a certain order. "Could there be more than one simple group
of that order?" asked Weil from the audience. "Yes, there could" replied >>> the speaker. "Well, could there be infinitely many?" asked Weil."
The "K-theory" or "representation" theory is often hiding essentially
"mis-representation" or truncations/approximations then to get the
neat algebraic properties then for what arithmetic provides about
geometry. "Almost-purity?", ..., actually dirty.
https://publish.uwo.ca/~jbell/
https://www.researchgate.net/publication/344371879_Reflections_on_Bourbaki%27s_Notion_of_Structure_and_Categories
"Here Bourbaki/Dieudonn|- uses the phrase "tower of Babel" with its usual
connotation of "place of confusion"."
"Thus the tower of Babel might be seen, not as representing the
jumble of separate practices that Bourbaki deplores, but rather as the
unity that they wished to impose on mathematics. In that case,
Bourbaki's |elements would, ironically perhaps, amount precisely to the
attempt to build a mathematical "tower of Babel". The
fact that Bourbaki failed - as is well-known - to complete his grandiose
project as originally conceived was not, as in Genesis, the result of
God sowing linguistic confusion - the Bourbaki members, after all, still
spoke a common mathematical language. Rather, BourbakirCOs project was
simply too ambitious to be brought to completion. Nevertheless, the
|el|-ments, unlike the tower of Babel, remains a
magnificent plinth."
"Bourbaki identifies three basic types of mathematical structure -
structures m|?res" - or "mother structures". These are algebraic, order,
and topological structures, which can be summed up as the "three C's" :
Combination, Comparison and Continuity."
These quotes of Bell, who also from his web-page has a variety of
essays on continuity, are insightful, and here it can be very plainly
obvious that a usual account of "ZFC + Martin's Axiom a.k.a. the
illative or univalency: the strength of ZFC and two large cardinals",
has that
well-foundedness and well-ordering and well-dispersion
are as of three sorts "rulialities/regularities" that make for
"combination, comparison, and continuity",
then usually that those are to be reversed in order.
Bourbaki/Dieudonne wrote (as quoted by Bell): "To set up the axiomatic
theory of a given structure amounts to the deduction of the logical
consequences of the axioms of the structure, excluding every other
hypothesis on the elements under consideration (in particular, every
hypothesis as to their own nature)."
Bell: "Accordingly, the departure from the scene of the concept of set
opened the way for Bourbaki to maintain that the unity of mathematics
stems, not from the set concept, but from the concept of structure."
Then, here those are as of structural realists' heno-theories,
not nominalist fictionalists' accounts of the not-thereof.
Bell: "In any case, as already remarked, Bourbaki's general concept of
structure, organized into species, plays only a very minor role in his
actual development of mathematics. By and large, only specific kinds of
structure are discussed: e.g., topological spaces, algebraic structures,
and combinations of the two such as topological groups. In
practice, the role of structure in general is played by the defining
axioms of the various species of structures."
It doesn't take a very generous or biased reading of Bell to have him
throwing quite a bit of shade on Bourbaki as alike a troll who espouses
a universal for completion then immediately recidivates.
As to why "category theory" and "model theory" need definitions of
"faithful" the functors and witnesses the models, the structures in the
theory, is because they haven't been.
Bell's great paper "Reflections on Bourbaki's Notion of Structure and
Categories" is rather a quite fantastic survey and introduction, and
while it has any number of mathematical formulas each of which will
halve the popular audience, it as well has quite a number of diagrams
which speak directly to graphical learners.
Bell: "In 1940 Stone established what amounts to the duality between the
category of compact Hausdorff spaces and the category of real
C*-algebras - commutative rings equipped rings with an order structure
and a norm naturally possessed by rings of bounded real-valued intensive
quantities."
Accounts like Naimark's "Normed Rings" is quite a different and more
conscientious approach, than just "equipping" the things, and forgetting
the differences.
Bell: "In the transition from Bourbaki's account of mathematics to its
category-theoretic formulation, what is the fate of the "mother
structures"? It is remarkable that, given Bourbaki`s distaste for logic,
their mother structures came to play a key role in establishing the
connection between category theory and logic."
Bell closes the paper: "By contrast, category theorists - those, at
least, who are sensitive to such issues (and those constitute the
majority) - regard set theory as a kind of ladder leading from pure
discreteness to the depiction of the mathematical landscape in terms of
pure Form. Categorists are no different from artists in finding the
landscape (or its depiction, at least) more interesting than the ladder,
which should, following Wittgenstein's advice, be jettisoned after
ascent."
There's that damn Wittgenstein again: claiming that it's
inter-subjective that nobody else is, and stealing the last parachute.
"Damn Wittgenstein": https://old.reddit.com/r/PhilosophyMemes/comments/10a8hen/damn_wittgenstein/
https://old.reddit.com/r/PhilosophyMemes/
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