Paul Fabel
@paulfabel.bsky.social
Thinking and rethinking math.
For cancellative abelian monoids, a nontrivial quotient map can have trivial kernel.
Map the free monoid F(C) over Cantor space C onto the additive non-negative reals, so that the right-shift map of F(C) represents halving.
Knowledge of the kernel of a quotient map is NOT one stop shopping.
Map the free monoid F(C) over Cantor space C onto the additive non-negative reals, so that the right-shift map of F(C) represents halving.
Knowledge of the kernel of a quotient map is NOT one stop shopping.
February 4, 2026 at 3:05 PM
For cancellative abelian monoids, a nontrivial quotient map can have trivial kernel.
Map the free monoid F(C) over Cantor space C onto the additive non-negative reals, so that the right-shift map of F(C) represents halving.
Knowledge of the kernel of a quotient map is NOT one stop shopping.
Map the free monoid F(C) over Cantor space C onto the additive non-negative reals, so that the right-shift map of F(C) represents halving.
Knowledge of the kernel of a quotient map is NOT one stop shopping.
Reposted by Paul Fabel
AI is getting good at math, but USING it still feels like looking up the answers in the back of the textbook.
Useful as a sanity check but generally hopeless as one stop shopping for understanding.
Exploring, developing tactics, finding examples and counterexamples, is not replaceable by AI.
Useful as a sanity check but generally hopeless as one stop shopping for understanding.
Exploring, developing tactics, finding examples and counterexamples, is not replaceable by AI.
January 31, 2026 at 8:46 PM
AI is getting good at math, but USING it still feels like looking up the answers in the back of the textbook.
Useful as a sanity check but generally hopeless as one stop shopping for understanding.
Exploring, developing tactics, finding examples and counterexamples, is not replaceable by AI.
Useful as a sanity check but generally hopeless as one stop shopping for understanding.
Exploring, developing tactics, finding examples and counterexamples, is not replaceable by AI.
Unlike groups, point preimages need NOT all have the same size under monoid epimorphisms.
There is an M_3-->M_2 counterexample with trivial kernel.
Let AAA=A and BB=B, in the domain and codomain.
Map A to B and id to id.
There is an M_3-->M_2 counterexample with trivial kernel.
Let AAA=A and BB=B, in the domain and codomain.
Map A to B and id to id.
January 31, 2026 at 7:48 PM
Unlike groups, point preimages need NOT all have the same size under monoid epimorphisms.
There is an M_3-->M_2 counterexample with trivial kernel.
Let AAA=A and BB=B, in the domain and codomain.
Map A to B and id to id.
There is an M_3-->M_2 counterexample with trivial kernel.
Let AAA=A and BB=B, in the domain and codomain.
Map A to B and id to id.
Want to localize your favorite monoid M?
Define all arbitrary quotients of M.
Then show that the quotients of M form a thin category with all joins and meets.
Mod M by the universal group kernel, so that the quotient M1 is cancellative.
Now Mod M1 x M1 by the diagonal.
Define all arbitrary quotients of M.
Then show that the quotients of M form a thin category with all joins and meets.
Mod M by the universal group kernel, so that the quotient M1 is cancellative.
Now Mod M1 x M1 by the diagonal.
January 31, 2026 at 6:09 PM
Want to localize your favorite monoid M?
Define all arbitrary quotients of M.
Then show that the quotients of M form a thin category with all joins and meets.
Mod M by the universal group kernel, so that the quotient M1 is cancellative.
Now Mod M1 x M1 by the diagonal.
Define all arbitrary quotients of M.
Then show that the quotients of M form a thin category with all joins and meets.
Mod M by the universal group kernel, so that the quotient M1 is cancellative.
Now Mod M1 x M1 by the diagonal.
The finite discrete space F is orderable but most automorphisms h:F-->F are NOT order preserving.
Same holds if we replace F with Cantor space C.
However each order preserving h:C-->C quotients to
h^:[0,1]-->[0,1] preserving the `dyadic rationals'
And vice versa, each h^ lifts to an h.
Same holds if we replace F with Cantor space C.
However each order preserving h:C-->C quotients to
h^:[0,1]-->[0,1] preserving the `dyadic rationals'
And vice versa, each h^ lifts to an h.
January 26, 2026 at 6:29 PM
The finite discrete space F is orderable but most automorphisms h:F-->F are NOT order preserving.
Same holds if we replace F with Cantor space C.
However each order preserving h:C-->C quotients to
h^:[0,1]-->[0,1] preserving the `dyadic rationals'
And vice versa, each h^ lifts to an h.
Same holds if we replace F with Cantor space C.
However each order preserving h:C-->C quotients to
h^:[0,1]-->[0,1] preserving the `dyadic rationals'
And vice versa, each h^ lifts to an h.
Reposted by Paul Fabel
Explore this gift article from The New York Times. You can read it for free without a subscription. www.nytimes.com/2026/01/24/o...
Opinion | State Terror Has Arrived
www.nytimes.com
January 25, 2026 at 12:17 PM
Explore this gift article from The New York Times. You can read it for free without a subscription. www.nytimes.com/2026/01/24/o...
Reposted by Paul Fabel
I wrote about how fucked up the past year has been as a person who's covered the far right for over a decade and who lives in DC. The guardrails are gone, and everyone knows it.
Everything those of us on this beat warned about happened, and then some. postsfromunderground.ghost.io/the-not-so-g...
Everything those of us on this beat warned about happened, and then some. postsfromunderground.ghost.io/the-not-so-g...
the not-so "golden age," one year on
Expo, a Swedish antiracist magazine that I contribute to, asked me to write a personal reflection how researching and reporting on the far right has felt during the first year of Donald Trump's return...
postsfromunderground.ghost.io
January 22, 2026 at 10:44 PM
I wrote about how fucked up the past year has been as a person who's covered the far right for over a decade and who lives in DC. The guardrails are gone, and everyone knows it.
Everything those of us on this beat warned about happened, and then some. postsfromunderground.ghost.io/the-not-so-g...
Everything those of us on this beat warned about happened, and then some. postsfromunderground.ghost.io/the-not-so-g...
Reposted by Paul Fabel
New article! On life, death, and the rampage of ICE.
sarahkendzior.substack.com/p/a-shining-...
sarahkendzior.substack.com/p/a-shining-...
A Shining Mausoleum on a Hill
Life, death, and the rampage of ICE.
sarahkendzior.substack.com
January 15, 2026 at 4:37 PM
New article! On life, death, and the rampage of ICE.
sarahkendzior.substack.com/p/a-shining-...
sarahkendzior.substack.com/p/a-shining-...
Reposted by Paul Fabel
Testing our sanity
January 15, 2026 at 9:24 PM
Testing our sanity
Reposted by Paul Fabel
Reposted by Paul Fabel
In a talk called "A reintroduction to proofs"
emilyriehl.github.io/files/reintr...
I've speculated about teaching an undergraduate level introduction to proofs course but using dependent type theory as the implicit formal system in place of set theory and first order logic.
emilyriehl.github.io/files/reintr...
I've speculated about teaching an undergraduate level introduction to proofs course but using dependent type theory as the implicit formal system in place of set theory and first order logic.
emilyriehl.github.io
December 29, 2025 at 9:09 PM
In a talk called "A reintroduction to proofs"
emilyriehl.github.io/files/reintr...
I've speculated about teaching an undergraduate level introduction to proofs course but using dependent type theory as the implicit formal system in place of set theory and first order logic.
emilyriehl.github.io/files/reintr...
I've speculated about teaching an undergraduate level introduction to proofs course but using dependent type theory as the implicit formal system in place of set theory and first order logic.
Some things are MORE EQUAL than others??
YouTube video by Sheafification of G
www.youtube.com
December 21, 2025 at 4:33 AM
Strong Floor, No Ceiling, Can't Leak!
Dibs.
Dibs.
December 2, 2025 at 5:40 AM
Strong Floor, No Ceiling, Can't Leak!
Dibs.
Dibs.
Why is Sarah Kendzior blocked?
Hopefully this post will be moot, not mute.
Hopefully this post will be moot, not mute.
November 12, 2025 at 4:48 AM
Why is Sarah Kendzior blocked?
Hopefully this post will be moot, not mute.
Hopefully this post will be moot, not mute.
Suppose the free group G on countably many generators is a sequential Hausdorff topological group, refining the usual product topology so that sequentially closed sets are closed.
What ensures some planar continuum X has pi_1(X) homeomorphic to G, with the natural quotient topology?
What ensures some planar continuum X has pi_1(X) homeomorphic to G, with the natural quotient topology?
November 10, 2025 at 10:09 PM
Suppose the free group G on countably many generators is a sequential Hausdorff topological group, refining the usual product topology so that sequentially closed sets are closed.
What ensures some planar continuum X has pi_1(X) homeomorphic to G, with the natural quotient topology?
What ensures some planar continuum X has pi_1(X) homeomorphic to G, with the natural quotient topology?
Tread carefully in compactly generated free abelian topological groups.
Closed subgroups need not be compactly generated.
If A is the group of all bounded sequences of integers, so that uniformly bounded null sequences converge to 0, let B denote the subgroup with b in B iff n divides b_n.
Closed subgroups need not be compactly generated.
If A is the group of all bounded sequences of integers, so that uniformly bounded null sequences converge to 0, let B denote the subgroup with b in B iff n divides b_n.
November 7, 2025 at 4:55 PM
Tread carefully in compactly generated free abelian topological groups.
Closed subgroups need not be compactly generated.
If A is the group of all bounded sequences of integers, so that uniformly bounded null sequences converge to 0, let B denote the subgroup with b in B iff n divides b_n.
Closed subgroups need not be compactly generated.
If A is the group of all bounded sequences of integers, so that uniformly bounded null sequences converge to 0, let B denote the subgroup with b in B iff n divides b_n.
Hausdorff quotients of locally compact complete separable metric spaces need not be 1st countable.
Start with X= [0,1] x {1,2,3,....}
Now glue together (0,n) and (0,m).
The quotient space is not 1st countable, despite being locally connected and uniquely arcwise connected.
Start with X= [0,1] x {1,2,3,....}
Now glue together (0,n) and (0,m).
The quotient space is not 1st countable, despite being locally connected and uniquely arcwise connected.
November 1, 2025 at 8:00 PM
Hausdorff quotients of locally compact complete separable metric spaces need not be 1st countable.
Start with X= [0,1] x {1,2,3,....}
Now glue together (0,n) and (0,m).
The quotient space is not 1st countable, despite being locally connected and uniquely arcwise connected.
Start with X= [0,1] x {1,2,3,....}
Now glue together (0,n) and (0,m).
The quotient space is not 1st countable, despite being locally connected and uniquely arcwise connected.
Reposted by Paul Fabel
In the near future economy, I bet the skill of “can type a prompt into ChatGPT and repeat whatever it says” will be less in demand than “understands things enough to catch when ChatGPT makes an error or hallucinates.”
Wow. Just wow.
"Students pay premium prices for information that AI now delivers instantly and for free. A business student can ask ChatGPT to explain supply chain optimization or generate market analysis in seconds. The traditional lecture-and-test model faces its Blockbuster moment."
"Students pay premium prices for information that AI now delivers instantly and for free. A business student can ask ChatGPT to explain supply chain optimization or generate market analysis in seconds. The traditional lecture-and-test model faces its Blockbuster moment."
When Knowledge is Free, What are Professors For?
Higher Education Must Stop Competing with AI on Information and Start Teaching What Machines Can’t Do
www.forbes.com
October 16, 2025 at 5:28 PM
In the near future economy, I bet the skill of “can type a prompt into ChatGPT and repeat whatever it says” will be less in demand than “understands things enough to catch when ChatGPT makes an error or hallucinates.”
Reposted by Paul Fabel
The 400 richest Americans are now worth a record $6.6 trillion.
The entire bottom 50% of America is worth just $4.2 trillion.
Read that back.
When 400 people control more wealth than half a country’s population, we have a very serious problem.
The entire bottom 50% of America is worth just $4.2 trillion.
Read that back.
When 400 people control more wealth than half a country’s population, we have a very serious problem.
October 12, 2025 at 11:30 PM
The 400 richest Americans are now worth a record $6.6 trillion.
The entire bottom 50% of America is worth just $4.2 trillion.
Read that back.
When 400 people control more wealth than half a country’s population, we have a very serious problem.
The entire bottom 50% of America is worth just $4.2 trillion.
Read that back.
When 400 people control more wealth than half a country’s population, we have a very serious problem.
ONE of the tricks/ideas lying behind proofs of quadratic reciprocity, is the `half the units' trick.
Suppose A is an abelian group.
Suppose U is a ` not both u and -u subset', if u is in U then -u is NOT in U.
Then vU is ALSO a ` not both u and -u' subset, for each automorphism v:A-->A.
Suppose A is an abelian group.
Suppose U is a ` not both u and -u subset', if u is in U then -u is NOT in U.
Then vU is ALSO a ` not both u and -u' subset, for each automorphism v:A-->A.
October 8, 2025 at 3:07 PM
ONE of the tricks/ideas lying behind proofs of quadratic reciprocity, is the `half the units' trick.
Suppose A is an abelian group.
Suppose U is a ` not both u and -u subset', if u is in U then -u is NOT in U.
Then vU is ALSO a ` not both u and -u' subset, for each automorphism v:A-->A.
Suppose A is an abelian group.
Suppose U is a ` not both u and -u subset', if u is in U then -u is NOT in U.
Then vU is ALSO a ` not both u and -u' subset, for each automorphism v:A-->A.
AI has a WAYS to go before it can do the basics reliably.
47^(73)= 932 mod 1009 , we can do this quickly with a cheap calculator, via repeated doubling, since 73=64+8+1.
But BOTH chatgpt and google AI cannot stick the landing on the question:
What is 47^(73) mod 1009?
47^(73)= 932 mod 1009 , we can do this quickly with a cheap calculator, via repeated doubling, since 73=64+8+1.
But BOTH chatgpt and google AI cannot stick the landing on the question:
What is 47^(73) mod 1009?
September 22, 2025 at 8:17 PM
AI has a WAYS to go before it can do the basics reliably.
47^(73)= 932 mod 1009 , we can do this quickly with a cheap calculator, via repeated doubling, since 73=64+8+1.
But BOTH chatgpt and google AI cannot stick the landing on the question:
What is 47^(73) mod 1009?
47^(73)= 932 mod 1009 , we can do this quickly with a cheap calculator, via repeated doubling, since 73=64+8+1.
But BOTH chatgpt and google AI cannot stick the landing on the question:
What is 47^(73) mod 1009?
Reposted by Paul Fabel
This is either engineered to create mass chaos in the cruelest possible fashion, or is a consequence of complete incompetence. I suspect both, though by different people in the chain of command leading to this executive order.
September 20, 2025 at 6:52 PM
This is either engineered to create mass chaos in the cruelest possible fashion, or is a consequence of complete incompetence. I suspect both, though by different people in the chain of command leading to this executive order.
Reposted by Paul Fabel
You're invited to sign this open letter
New people are signing our open letter almost everyday, and it is so heartening to see. Especially good to see familiar names sign.
openletter.earth/open-letter-...
openletter.earth/open-letter-...
Open Letter: Stop the Uncritical Adoption of AI Technologies in Academia
openletter.earth
September 19, 2025 at 7:17 AM
You're invited to sign this open letter