Paul Fabel
@paulfabel.bsky.social
Thinking and rethinking math.
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
Pass Line Craps EV should be 2(8/36)+
2[ (10/36)(5/11) + (8/36)(4/10) +(6/36)(3/9) ) =
98.58585858....%
Don't Pass: 2[3/36]+1[1/36]+
2[ (10/36)(6/11) + (8/36)(6/10) +(6/36)(6/9) ) =
98.636363....%
2[ (10/36)(5/11) + (8/36)(4/10) +(6/36)(3/9) ) =
98.58585858....%
Don't Pass: 2[3/36]+1[1/36]+
2[ (10/36)(6/11) + (8/36)(6/10) +(6/36)(6/9) ) =
98.636363....%
September 11, 2025 at 10:12 PM
Pass Line Craps EV should be 2(8/36)+
2[ (10/36)(5/11) + (8/36)(4/10) +(6/36)(3/9) ) =
98.58585858....%
Don't Pass: 2[3/36]+1[1/36]+
2[ (10/36)(6/11) + (8/36)(6/10) +(6/36)(6/9) ) =
98.636363....%
2[ (10/36)(5/11) + (8/36)(4/10) +(6/36)(3/9) ) =
98.58585858....%
Don't Pass: 2[3/36]+1[1/36]+
2[ (10/36)(6/11) + (8/36)(6/10) +(6/36)(6/9) ) =
98.636363....%
Reposted by Paul Fabel
I wrote about the Third Way memo and its list of forbidden words. Comparing the alleged desire for simple persuasive talk and the way those who want it never seem to arrive at the simplest most persuasive "talk" there is: principled action.
www.the-reframe.com/simple-talk-2/
www.the-reframe.com/simple-talk-2/
Simple Talk
Want to move people? Focus on principles, not tactics. An essay about finding a real third way, using the simplest and most persuasive talk there is.
www.the-reframe.com
September 7, 2025 at 5:51 PM
I wrote about the Third Way memo and its list of forbidden words. Comparing the alleged desire for simple persuasive talk and the way those who want it never seem to arrive at the simplest most persuasive "talk" there is: principled action.
www.the-reframe.com/simple-talk-2/
www.the-reframe.com/simple-talk-2/
I would recommend the help of a certain nuclear particle, if you were met with a surprise on this website today.
September 7, 2025 at 8:13 PM
I would recommend the help of a certain nuclear particle, if you were met with a surprise on this website today.
Reposted by Paul Fabel
better late than never!
by Eugenia Cheng
www.nytimes.com/2025/09/05/o...
(surely the ≈1k comments will be thoughtful and constructive...)
by Eugenia Cheng
www.nytimes.com/2025/09/05/o...
(surely the ≈1k comments will be thoughtful and constructive...)
Opinion | What I Got Wrong About D.E.I.
www.nytimes.com
September 5, 2025 at 11:46 PM
better late than never!
by Eugenia Cheng
www.nytimes.com/2025/09/05/o...
(surely the ≈1k comments will be thoughtful and constructive...)
by Eugenia Cheng
www.nytimes.com/2025/09/05/o...
(surely the ≈1k comments will be thoughtful and constructive...)
Please consider to read.
I wrote about the humility of expertise and the arrogance of ignorance, the profound laziness that attends believing that you can find knowledge by ignoring the accumulation of inherited human knowledge, and the ways lazy arrogance fuels narratives of fascism. www.the-reframe.com/your-ignoran...
Your Ignorance Doesn't Make You An Expert
Let's go chasing waterfalls.
www.the-reframe.com
September 1, 2025 at 4:40 AM
Please consider to read.
Math and computation and computational complexity?
Here's a must read from Scott Aaronson.
www.scottaaronson.com/papers/philo...
Here's a must read from Scott Aaronson.
www.scottaaronson.com/papers/philo...
www.scottaaronson.com
August 27, 2025 at 3:47 AM
Math and computation and computational complexity?
Here's a must read from Scott Aaronson.
www.scottaaronson.com/papers/philo...
Here's a must read from Scott Aaronson.
www.scottaaronson.com/papers/philo...
Finally watching Succession. Please no spoilers. Brian Cox is one of the all time greats.
August 27, 2025 at 3:43 AM
Finally watching Succession. Please no spoilers. Brian Cox is one of the all time greats.
'But a return to a more conversational, extemporaneous style will make higher education more interpersonal, more improvised and more idiosyncratic, restoring a sense of community to our institutions.'
Clay Shirky
Resonates. Heavy emphasis on idiosyncratic and improvised.
Clay Shirky
Resonates. Heavy emphasis on idiosyncratic and improvised.
August 27, 2025 at 3:22 AM
'But a return to a more conversational, extemporaneous style will make higher education more interpersonal, more improvised and more idiosyncratic, restoring a sense of community to our institutions.'
Clay Shirky
Resonates. Heavy emphasis on idiosyncratic and improvised.
Clay Shirky
Resonates. Heavy emphasis on idiosyncratic and improvised.
Reposted by Paul Fabel
When an insurance company is deciding whether to pay for your medical treatment, it generates a file, which should contain all records associated with your case, including documents explaining why the claim was denied.
You have a right to see this file.
(Published May 2023)
You have a right to see this file.
(Published May 2023)
Health Insurance Claim Denied? See What Insurers Said Behind the Scenes
Learn how to request your health insurance claim file, which can include details about what your insurer is saying about you and your case.
www.propublica.org
August 26, 2025 at 1:30 AM
When an insurance company is deciding whether to pay for your medical treatment, it generates a file, which should contain all records associated with your case, including documents explaining why the claim was denied.
You have a right to see this file.
(Published May 2023)
You have a right to see this file.
(Published May 2023)
Reposted by Paul Fabel
Not every story has two equal “sides.” Report the *truth*: that’s our job protected by the First Amendment.
Pritzker: To the members of the press who are assembled here today and listening across the country, I am asking for your courage to tell it like it is. This is not a time to pretend here that there are two sides to this story.
August 25, 2025 at 9:25 PM
Not every story has two equal “sides.” Report the *truth*: that’s our job protected by the First Amendment.
Reposted by Paul Fabel
The root of the problem is the idea of sacrificing some people for the betterment of other people.
Because sacrificing people *is* the problem.
Because sacrificing people *is* the problem.
Today I wrote about a general tendency to accept atrocious premises, and the need to reject fascism's false choices in order to find expansive and imaginative paths forward.
Breaking the premise, embracing the obstacles, pursing everything.
www.the-reframe.com/there-is-no-...
Breaking the premise, embracing the obstacles, pursing everything.
www.the-reframe.com/there-is-no-...
There Is No Trolley Problem
We actually aren't constrained into atrocious premises by thought exercises. Obstacles are a reason to fight, not an excuse to quit. Our friends and neighbors aren't distractions..
www.the-reframe.com
August 24, 2025 at 9:23 PM
The root of the problem is the idea of sacrificing some people for the betterment of other people.
Because sacrificing people *is* the problem.
Because sacrificing people *is* the problem.
Reposted by Paul Fabel
August 24, 2025 at 10:46 AM