iiinova — where your digital self keeps evolving

Reposted by Paul Fabel

Axel Gelfert @agelfert.bsky.social · 23h

I asked ChatGPT to create a cover for a course catalogue of a "university for people who have given up thinking due to AI over-reliance". To be fair, this is a pretty good take -- and more subtle than I expected... #aislop #criticalthinking #deskillling #chatGPT #philsky

ChatGPT-created fictitious cover of a course catalogue of a university for people who have give up thinking: android leaning back with a mug "AI does thinking for you" in front of him, and advertised degree programmes "BA in Artificial Thought" and "PhD in Epistemic Delegation".

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Paul Fabel @paulfabel.bsky.social · 29/08/2026

A man is reluctant to return a briefcase after buying some tent poles.

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Paul Fabel @paulfabel.bsky.social · 29/08/2026

An exhausted man is too tired to talk following a long stroll in the Texas back country.

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Paul Fabel @paulfabel.bsky.social · 29/08/2026

Entrepreneur tussles with management in northern California town after rejecting corporate buy out offer.

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Paul Fabel @paulfabel.bsky.social · 16/08/2026

Potential HW or test questions in math foundations. I give you SOME information about f:A-->B, e.g. A is infinite and f surjective. Then we can ask: Must B be infinite? Could B be infinite? Must be B finite? Could B be finite? Students can then justify with proofs and examples.

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Paul Fabel @paulfabel.bsky.social · 07/08/2026

Directions for the remote control.

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Paul Fabel @paulfabel.bsky.social · 29/07/2026

Thanks for the helpful explanations! Context refactored: Submonoids of products of walking arrows. If the monoid M={0,1} is the walking arrow (not a group), then the monoid M x M has order 4, but contains a submonoid T of order 3. T= {(0,0) , (1,0) , (1,1)}, which is not monogenic or product.

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Paul Fabel @paulfabel.bsky.social · 29/07/2026

Thanks for checking! Three endos {id,a,b } of the set {A,B,C} also helps bring this to life. Let a(x)=A. Let b(C)=B with fixed points A and B. No generative side effects since A is fixed by each of {id,a,b}. M={id,a,b} has the advertised properties, helped seen by the maps dynamics.

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Paul Fabel @paulfabel.bsky.social · 28/07/2026

Informative and fun to construct an abelian monoid of order 3 which is NOT monogenic (one generator). Since 3 is prime, this has no hope to factor as a product of monogenic monoids. It was NOT fun coaxing ChatGPT, free version, to find the example, a mixture of nonsense and turgid spoilers.

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Paul Fabel @paulfabel.bsky.social · 26/07/2026

Whether every quotient of every set admits a section is more or less equivalent to the axiom of choice, depending plausibly on your model of set theory. But a quotient of a 3 element abelian monoid might NOT embed in the starting monoid. {0,1,2,3...} mod {0,2,4...} is also a counterexample.

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Paul Fabel @paulfabel.bsky.social · 26/07/2026

The 3 element monoid with {1,f,ff} with ff not fff, and quotient {1,ff} , {f} shows 1) A quotient of a finite abelian monoid might NOT embed in the starting monoid. 2) The cancellative quotient of a finite abelian monoid might have cosets of different sizes. Useful to bring up in Foundations?

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Reposted by Paul Fabel

OrcGirlBoss 🐇🏹 @orc.bsky.social · 02/07/2026

I would like a physical copy of my game please.

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Reposted by Paul Fabel

Jennifer Ouellette @jenlucpiquant.bsky.social · 30/06/2026

Here’s Why Weather Forecasts Have Seemed So Inaccurate Lately. Meteorologists simply aren't able to gather as much data as they used to, and you're suffering for it. gizmodo.com/heres-why-we...

gizmodo.com

Here's Why Weather Forecasts Have Seemed So Inaccurate Lately

Meteorologists simply aren't able to gather as much data as they used to, and you're suffering for it.

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Paul Fabel @paulfabel.bsky.social · 07/06/2026

Given functions f,g :A-->A we can ask: 1) What is |<f>|? 2) Are f and g conjugate? 3) Are <f> and <g> isomorphic? Too easy? Warning: Questions about <f,g> are HARD, since `most' nonabelian groups have two generators.

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Paul Fabel @paulfabel.bsky.social · 07/06/2026

Teaching Foundations of Math in the Fall, but how to engage students in doable, compelling exercises, teachable for an exam, not totally trivial, and with long term healthy cognitive side effects? Monoids will work at lots of levels, starting with basic questions about f:A-->A with A finite.

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Reposted by Paul Fabel

Maggie Wiggin @maggie162.bsky.social · 01/06/2026

My god, Octavia Butler, just miss ONCE

Page from a book saying:

From EARTHSEED: THE BOOKS OF THE LIVING
Choose your Leaders with wisdom and forethought.
To be led by a coward is to be controlled
by all that the coward fears.
To be led by a fool is to be led
by the opportunists who control the fool.
To be led by a thief is to offer up your most precious treasures to be stolen.
To be led by a liar
is to ask to be told lies.
To be led by a tyrant is to sell yourself and those you love into slavery.
t

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Paul Fabel @paulfabel.bsky.social · 03/05/2026

Partial answer: 2-adics yields one stop shopping when upgrading guesses solving x^2=y mod q. The group of solutions to x^(2^m)=1 mod q embeds in a 2-Prufer group A=(C_2_infty), valid for all primes q>2. But Endo(A) is THE ring of 2-adics, we halve in A, and canonically transport back to Z_q.

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Paul Fabel @paulfabel.bsky.social · 20/04/2026

Yep! Couple a beers but no AI involved. BTW same result and proof holds if we assume all spaces are compact and weakly Hausdorff. Non-trivial example: Alexandroff compactifcation X of your favorite T_2 space Y which is NOT locally compact , e.g. Y is rationals with order topology.

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Paul Fabel @paulfabel.bsky.social · 20/04/2026

Catchy rhyme, Say it in time. Useful fact, Closed is compact. Problem is hacked, Push forward and back.

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Paul Fabel @paulfabel.bsky.social · 08/04/2026

Yeah, I'm waffling a bit too. If the starting monoid is cancellative, abelian, totally ordered, and if all nonempty's have a greatest lower bound, for example if M is the natural numbers, then x-->[0,x] is the way to go, mapping nonzero x-->[0,x) is NOT a homomorphism, even if we map 0 to {0}.

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Paul Fabel @paulfabel.bsky.social · 07/04/2026

But we can in fact get best of both of worlds employing your ideas! Map 0 to {0} and nonzero x to [0,x). The submonoid K is the bounded sets in 2^M of the form {0} union any bounded union of half open intervals [0,x). K is isomorphic to additive non-negs, and M-->K is a monoid monomorphism.

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Paul Fabel @paulfabel.bsky.social · 07/04/2026

Yes! Thanks! Indeed one stop shopping as colimit, the additive naturals or integers, indexed over naturals, with monomorphic bonding maps x--> N(n) x with N(n) = (2 3 5 ...p(n))^n. (The less elegant inclusions of fractions is useful as a sanity check too).

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Paul Fabel @paulfabel.bsky.social · 07/04/2026

But don't we get additive rationals as colimit of infinite cyclic groups A1 -->A2-->A3... with inclusion bonding maps, A_n the rational fractions m/[(2 3 5 ...p_n)^n] ? (m is an integer and p_n the nth prime)

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Paul Fabel @paulfabel.bsky.social · 07/04/2026

Arghh! Thanks! Replace with the additive dyadic rationals, the colimit of the infinite cyclic group Z under the doubling map x-->x+x, indexed over the naturals Z-->Z-->Z...

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Paul Fabel @paulfabel.bsky.social · 07/04/2026

A monoids versus semigroups categorical tussle, MUST identities map to identities under maps which preserve multiplication? Good ways to construct additive reals avoiding Dedekind cuts and Cauchy sequences in short supply. In this case, (at most) 2-1 quotients are compatible with the algebra.

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Paul Fabel @paulfabel.bsky.social · 07/04/2026

Yes! But the open ray [0,0) can be uninhabited? So if we want x-->[0,x] to be a monomorphism, M--> 2^M, (adding subsets A and B of M, to get A+B in the codomain) we should to KEEP the closed intervals, so that 0 maps to [0,0]. i.e. the formula x-->[0,x) breaks when x=0, and M is skeletal.

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Paul Fabel @paulfabel.bsky.social · 29/03/2026

Can the right divisibility preorder on a noncommutative monoid be total? (a<=b if ac=b for some c). Yes, according to Eliahu Levy ( 1947-2023). arxiv.org/pdf/2006.00886

arxiv.org

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Reposted by Paul Fabel

Fietser @americanfietser.bsky.social · 27/03/2026

We need a vehicle weight tax. If you can justify buying an $80k, 3,000 kg Wagoneer to haul one bag of groceries, taxpayers shouldn't be subsidizing your pavement wear and tear plus the added danger to our streets. If you can afford the payments, you can afford to stop asking society to subsidize it.

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Paul Fabel @paulfabel.bsky.social · 24/03/2026

Thanks! I am having my come to semi-groups moment.

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Paul Fabel @paulfabel.bsky.social · 24/03/2026

Thanks Michael, Indeed, I should have used preorder instead, I was conflating, (incorrectly?) preorder with nonstrict poset. but preorder relation is the way to go x<=x, plus transitivity.

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Paul Fabel @paulfabel.bsky.social · 24/03/2026

Thanks! Indeed the relation is transitive and that's about it. I am allowing for nonstrict posets, e.g. if the starting monoid is a group we have x<=y universally. (I will delete, edit, or clarify if I've posted obvious nonsense).

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Paul Fabel @paulfabel.bsky.social · 24/03/2026

If the abeliain monoid (M,*) determines a total order on (M,<), then the bounded rays comprise a totally ordered and left complete submonoid of 2^M. Now take the cancellative quotient, and we get the non-neg additive reals, if M is the non-neg dyadic rationals. All this is `easy'.

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Paul Fabel @paulfabel.bsky.social · 24/03/2026

f:M-->G is trivial if there is k in M so that km=mk=k for all m in M, if M is a monoid and G is a group, and f is a homomorphism.

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Paul Fabel @paulfabel.bsky.social · 22/03/2026

Ignoring order, the additive rationals are discrete, a colimit of cyclic groups. But if we restrict to the monoid of non-neg rationals, algebra gives us x<y iff x+z=y, we get order topology. Induced order on monoid of bounded rays, almost the non-neg reals. Mod out by [0,x)=[0,x] when x rational.

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Paul Fabel @paulfabel.bsky.social · 22/03/2026

Submonoid category. Exercises and examples. From S={a,b} we can create M(S), the monoid of functions f:M-->M under composition. From M(S) we can create the category C(S) of submonoids of M(S). |C(S)|=5 in this case. We can then ask, given x and y in C, are x and y isomorphic? |H(x,y)|=?

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Paul Fabel @paulfabel.bsky.social · 22/03/2026

NEVER meaning this relation yields the trivial order on a group, unlikely useful since the starting data is inhabited.

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Paul Fabel @paulfabel.bsky.social · 21/03/2026

Monoids are essential building blocks of groups. We sometimes get a useful partial order on a monoid declaring x<=y if zx=y has a solution. Sometimes this internal algebra creates a total order, e.g. on the natural numbers. But this NEVER happens in a group, since x <= y holds for all x and y.

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Reposted by Paul Fabel

Leah McElrath @leahmcelrath.bsky.social · 06/03/2026

Malignant narcissists experience increased paranoia, denial, and rage as death nears. They often externalize these experiences through destruction and abuse of others in an effort to maintain a sense of control. I suspect all of these wars are in part related to Trump’s awareness of his mortality.

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Reposted by Paul Fabel

A.R. Moxon @juliusgoat.bsky.social · 28/04/2024

Seven lessons learned from a quarter century in a war-oriented society. It's 2001—the year the movies promised we'd make contact with aliens—and the United States has rather recently been attacked by terrorists who flew passenger planes into buildings. www.the-reframe.com/war-or-nothi...

the-reframe.com

War or Nothing

A tale across the years; seven lessons learned from a quarter-century in a war-oriented society, where the greatest crime is any opposition to killing.

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Paul Fabel @paulfabel.bsky.social · 10/02/2026

Yes, despite pragmatic rarity of checking off conditions. Metrizability demands monoid of additive non-negative reals. Origin stories of latter highly nontrivial, done carefully. First class examples of limits, colimits, and monoid quotients emerge, done with care. Reusable tools writ large.

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Reposted by Paul Fabel

A.R. Moxon @juliusgoat.bsky.social · 08/02/2026

Today I wrote about the racist accommodation to the racist response to Bad Bunny's Super Bowl show; the moral hazard of extending good faith to those who act in bad faith; and how alienating white bigots is actually part of the solution, not a part of the problem. www.the-reframe.com/hating-the-g...

the-reframe.com

Hating the Game

The cooperation game, the murder game, and acting in good faith with people you know are acting in bad faith.

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Paul Fabel @paulfabel.bsky.social · 04/02/2026

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.

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Paul Fabel @paulfabel.bsky.social · 31/01/2026

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.

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Paul Fabel @paulfabel.bsky.social · 31/01/2026

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.

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Paul Fabel @paulfabel.bsky.social · 29/01/2026

Substituting x=7 certifies or refutes a random trig identity, armed with a scientific calculator, working in radians.

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