Search results for tag #math
I would suggest that folks who think using AI is great for mathematicians should think again. It seems as little as 10 minutes of use can be problematic. What else do we know that provides short-term gains at the expense of long-term loss?
Here, through a series of randomized controlled trials on human-AI interactions (N = 1,222), we provide causal evidence for two key consequences of AI assistance: reduced persistence and impairment of unassisted performance. Across a variety of tasks, including mathematical reasoning and reading comprehension, we find that although AI assistance improves performance in the short-term, people perform significantly worse without AI and are more likely to give up. Notably, these effects emerge after only brief interactions with AI (approximately 10 minutes). These findings are particularly concerning because persistence is foundational to skill acquisition and is one of the strongest predictors of long-term learning.From AI Assistance Reduces Persistence and Hurts Independent Performance, on arXiv https://arxiv.org/abs/2604.04721
#AI #GenAI #GenerativeAI #AgenticAI #AIAssistants #CognitiveImpairment #math #MathematicalReasoning #ReadingComprehension
🍕📐 Mathematicians used #geometry to solve the problem of dividing a circular shape into equal areas using off-center slices.
The #research explains how the curvature of a slice affects its structural integrity and its ability to hold toppings.
👉 https://www.scientificamerican.com/article/the-mathematically-correct-way-to-slice-a-pizza/
We knew, but the proof is nice.
"Apple just proved that AI models cannot do math. Not advanced math. Grade school math. The kind a 10-year-old solves"
The guess-the-next-words machines don’t actually understand anything.
https://nitter.poast.org/heynavtoor/status/2041243558833987600#m
I often see or hear that people don't do math(s). They treat it as if it is some kind of specialized information. I'm in a field where I use it frequently at work, and it was used extensively in post secondary education. Even outside of that I feel like it's pretty handy in many situations. Maybe it is ubiquitous access to pocket computers that helps people feel like that. I didn't grow up with them, so maybe that colors my perception.
So, purely for mental mathematics, deliberately leaving complexity aside, what is your frequency of use?
| I use math daily: | 10 |
| I use math often: | 2 |
| I use math sometimes: | 3 |
| I use math rarely: | 0 |
| I never use math: | 0 |
Zundamon's Theorem is like THE BEST math YouTube channel. And by best I mean silly.
The other day I had a fairly popular post talking about how mathematicians easily and often admit that they don't know things or don't understand things. Today at work a real-life example came up!
Original post linked to in the next toot since apparently I can't post a link and have an image at the same time ... wtf?!
I was helping a student with Calc I in my office. The question gave a function and asked for values of x where the tangent line was horizontal. The function is the first in the image.
This requires taking the derivative with the product rule. The result of this is the second in the image. Since the second term has a denominator (other than 1 of course) we need to combine the two terms so we can set the numerator to 0 and solve.
The result of this operation is the third in the image. Fractions are 0 when their numerators are 0, so the fourth line shows the equation to be solved.
The student got this far without any help but was unable to solve the equation. This is commonplace. After all, the hardest part of calculus is algebra. But I couldn't see how to solve it either, so I told the student this.
At this moment two of my colleagues were talking in the hall outside my office so I told the student I'd ask them about it. Neither knew how to solve it and told me as much. So I told the student, who was actually thrilled that none of us could solve it either.
So I asked Wolfram Alpha, which gave a solution using the Lambert W, aka the productlog function. I'm a combinatorial topologist -- I do graph theory of various kinds. I've heard of this function but otherwise know nothing at all about it. And I'm happy to admit it! Anyway, that's how mathematicians roll.
ETA: Of course this problem shouldn't have appeared in an introductory calculus text since no undergraduate at that level would be able to solve it, so its inclusion was a mistake of the author or the editor.
#Calculus #LambertW #HorizontalTangent #DifferentialCalculus #Math #Mathematics #Mathematicians
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Screenshot of five equations produced by LaTex. The TeX code follows, which is the best way I can think to describe this:
\documentclass{article}
\begin{document}
$$f(x)=x\ln(x+3)$$
$$f^{\prime}(x)=\ln(x+3) + \frac{x}{x+3}$$
$$f^{\prime}(x)=\frac{(x+3)\ln(x+3) + x}{x+3}$$
$$(x+3)\ln(x+3) + x =0$$
$$x=???$$
\end{document}
📺 https://peer.adalta.social/w/mjzW4Y2EoQBG3rxJRM8U5h
🔗 [🇩🇪🇺🇸🇫🇷](https://adalta.info/articles/prstn_who_116330456243696530_fr)
🔗 [ℹ️](https://numbword.com/")
Une énigme mathématique parfaite, révélant des schémas de résolution complexes.
📺 https://peer.adalta.social/w/6YAvi6rv1bkeHqpeDJTWRb
🔗 [🇩🇪🇺🇸🇫🇷](https://adalta.info/articles/prstn_who_116330456243696530_en)
🔗 [ℹ️](https://numbword.com/")
A Perfect Solution Reveals a Fundamental Flaw in the Game’s Design.
📺 https://peer.adalta.social/w/58mBZLC7KXoTXd8eFNADs2
🔗 [🇩🇪🇺🇸🇫🇷](https://adalta.info/articles/prstn_who_116330456243696530_de)
🔗 [ℹ️](https://numbword.com/")
Die Lösung: Eine ungewöhnliche Zahlenfolge und ihre sprachliche Entsprechung.
The futility of rushing too fast
Structurally it appear s very similar to the question off why buses often appear in threes’ and the stop go effect of red lights.
Having sat with the notion for about six months now, I think Jay's critique of the Church-Turing thesis has legs. I don't see clearly yet exactly where and how the limits of computation manifest in his own system(s), which of course they must. But I think he's correct that this thesis as it's colloquially presented (and taught to students, including me!) is misleading at best and false in a certain important sense. Apparently he is regularly called a crackpot for forwarding this critique even though it's straightforwardly demonstrated.
Waaldijk's book is more of a constructive mathematics exploration. In this it is closely related to computer science, but it's focused on traditionally mathematical notions like topological space. The latter is usually quite complicated, but Waaldijk shows that the core concept of compact space can be represented with finitely-branching trees, making these spaces amenable to computation. Since we imagine physics taking place in spaces that are topological (among other things) there's potentially an interesting bidirectional flow of ideas between computer science and physics.
Jay calls his central notion "natural trees". Waaldijk calls his central notion "natural spaces". In both cases I think the intended sense is "with minimal artifice".
✨🪲 Researchers are studying the mathematical models that allow thousands of #fireflies to blink in perfect unison.
By observing these bioluminescent #insects, scientists can better understand how decentralized systems coordinate complex behaviors without a single leader.
👉 https://arstechnica.com/science/2026/03/the-science-of-how-fireflies-stay-in-sync/
#biology #physics #nature #science #wildlife #entomology #math #bioluminesence
Astronomers have discovered 11 more moons around Saturn, bringing its total to 285--by far the most of any planet in the solar system.
The true number may be unknowable, if you count every ring particle as its own little moon.
https://minorplanetcenter.net/mpec/K26/K26F14.html #space #science #nature #math
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In the upper half of the illustration, the 21 largest Saturn satellites are represented full-scale with respect to each other and in relation to the parent planet (beige circular segment under the moons) and to the rings (in the background). The lower half shows the positions of several of the moons up to a distance of approximately seven Saturn radii from the planet's centre, which can be seen as an outline hemisphere having a radius of 71,500 kilometres at the left (shown in correct full scale).
"How Many Decimals of Pi Do We Really Need?" by #NASA Jet Propulsion Laboratory JPL - Short answer: for spacecraft navigation around the solar system (as far as humanity has launched anything so far) NASA determined 15 digits after the decimal point is sufficient. So for NASA, 𝝿 = 3.141592653589793 . That yields an error of about 1.5cm/0.5in at the extremities of the solar system. Good enough. https://www.jpl.nasa.gov/edu/news/how-many-decimals-of-pi-do-we-really-need/ #astronomy #space #math #PiDay
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article overview screenshot added because the page did not provide a social media thumbnail
How Many Decimals of Pi Do We Really Need?
Written by NASA/JPL EduOct. 24, 2022
[image: Pi to the 15th decimal is shown in a speckled starry band as a silhouetted face looks out over colorful concentric circles and black and white images of an atom, molecules, Earth, and the Voyager spacecraft.]
While world record holders may have memorized more than 70,000 digits of pi, a JPL engineer explains why you really only need a tiny fraction of that for most calculations – even at NASA.
[...]
Weekend project: my kid needed a tool to visualize math concepts, so I coded one. The power of vibe coding. Actually the power of professional vibe coding 😊 #vibecoding #math #professionalVibeCoding
They teach you the "English system" on antenna calculations in the US.
468 / Frequency = half wave dipole in feet
Then multiple by 12 for inches, divide by two for each side blah blah blah
In meters, 142.5 / Frequency = half wave dipole... but it corresponds to the band names.
In meters, for a 10 meter dipole at 28.500... you need an antenna that is 5.000 meters (half wave of 10 meters). 🙄 That's WAY EASIER.
20 meters at 14.250Mhz? 14.250/142.5 = 10 meter half wave dipole.
A lab mate shared this write up of Don Knuth using LLMs to solve a math problem: https://www-cs-faculty.stanford.edu/~knuth/papers/claude-cycles.pdf
It's clear that using Claude did help them arrive at some new understanding here, which is wonderful. I'm happy for them.
However, I'm upset by how much they personify Claude and attribute the solution to "him."
From this narrative, it's clear that the humans were very actively involved from beginning to end. Claude was a helpful tool, but it did not solve this problem on its own. What role did it actually play? How was it like or unlike a human collaborator on this problem?
It did generate a crucial insight, but where did that come from? Was it plagiarized from some unknown source? Did it "just emerge" from text completion and interpolation in latent space? Do we need some other explanation for Claude's apparent creativity?
These folks don't care. They just wanted a solution, which they attribute to Claude, and leave it at that. I think that's a serious problem.
#math #economy #cantaloupes #whatTheHell
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Charles M Schulz PEANUTS cartoon of Sally at school looking at a sheet of paper saying "ONLY IN MATH PROBLEMS CAN YOU BUY 60 CANTALOUPES AND NO ONE ASKS WHAT THE HELL IS WRONG WITH YOU"
Ooof, this smells of a sign/math error somewhere, ouch.
"...the software that should have pointed Lunar Trailblazer’s solar panels toward the Sun instead pointed them 180 degrees away from the Sun...."
Hands down the coolest website I've seen recently!
People calculated the longest (human) lines of sight for any point on earth!
https://map.alltheviews.world/longest/-127.77009398744215_50.17687825913197
#Maps #Mountains #Science #Math #Hiking #Dreams
Is it just me, or isn't it incredibly weird that all the USB powerbanks and lots of mobile device batteries use the unit mAh ... and then values like 20 000 or 5 000.
20 000mAh is the same as 20Ah.
#jouRapNal #book18 day 75
20260226Thu
#3goodThings #couch #volunteer #french #threeGoodThings
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(illustration of a couch)
iL is picking up Yt after volunteering and tells me i may have to couch surf or be homeless
Az sent me a Tik Tok where my name was sung
Ma is back to hs with a makeup quiz & some french verb conjugations
One of my greatest learning laments were my failed attempts at Calculus in high school and post college. By the time I got to Calculus in high school I was so burnt out, and had a terrible math teacher the year prior that one of my favorite academic interests had suffered a grievous wound.
I attempted a college Calculus course after I attained my Bachelor's degree. Leading up that attempt I took trigonometry course by correspondence and a pre-calculus course both of which I excelled at.
Got into the Calculus course then life started lifing and I was unable to complete it.
All this to say, I still want to know Calculus, even if I never use it outside of learning it.
#music #math #algebra #earworms
'Behind many great melodies, researchers found something surprisingly powerful: symmetry. Their work shows that advanced algebra can reveal deep musical patterns that are not always obvious by ear or even on a written score.'
https://uwaterloo.ca/news/media/secret-math-behind-catchy-melodies
Mathstodon.xyz is a Mastodon server for people who love maths, and includes LaTeX rendering in the web interface. Maths chat is especially welcome, but any topic of conversation following the code of conduct is OK.
This server has a post size of up to 1729 characters.
You can find out more at https://mathstodon.xyz/about or contact the admin account @christianp
#FeaturedServer #Mathematics #Maths #Math #Mastodon #Fediverse #FreeFediverse
Took Algebra at a local community college and had to work really hard to get an 'A-'. Credit did not transfer to my university, so had to take algebra again (I think I got a 'B' but the teacher was much worse than the one at the community college.
Now, one of my sons is learning trigonometry by taking a C programming course from Pikuma... and he's interested in it.
https://pikuma.com/courses/learn-3d-computer-graphics-programming
#math
RT: https://poa.st/objects/60e46c4f-82ce-4554-bd0a-585b162786d6
Happy birthday to #mathematician Johann Peter Gustav Lejeune Dirichlet (13 February 1805 – 5 May 1859)! My lino block print illustrates the famous mathematical tool known as the pigeonhole principle, which states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. You can imagine a bunch, call it m, pigeonholes and n pigeons, where n > m; 🧵
https://minouette.etsy.com/listing/1297260728
#linocut #printmaking #sciart #mathart #mastoArt #math #histsci
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My linocut print of the Pigeonhole Principle shows Dirichlet in blue, a bearded man with heavy brow, wearing a suit with arms crossed, looking a bit cross. Behind him is a bank of pigeonholes in a gradient of blue to gold. Each one contains one or two pigeons in black on grey with turquoise and violet markings. The pigeonholes may extend infinitely in all directions - the shelves appear to go on beyond the 4 x 7 array of pigeonholes shown.
What is a number? When you think of the concept, what do think of? Answer in the poll!
In your conception, is this kind of number fundamental, or do you feel that it relies on something more fundamental? If so, is that simply a standard construction (like the rationals from the integers) or something different?
#math #maths #mathematics
| A natural number: | 0 |
| An integer: | 0 |
| A rational number: | 0 |
| A computable number: | 0 |
| A real number: | 0 |
| An ordinal or cardinal number: | 0 |
| Something else?: | 0 |
Mathematicians discover new ways to make round shapes
https://www.scientificamerican.com/article/mathematicians-discover-new-ways-to-make-round-shapes/
Wow, if you are a fan of watches--and you want to win the geek of the year award--get yourself an Arithmo Slide Rule watch by Juvenia. I just saw one appraised on Antiques Roadshow, and it was not in very good condition, for $7000-8000! I've never worn a watch, but if I had one of these, I just might wear it! So cool!
Photo via: https://fabsuisse.com/juvenia-arithmo-calculating-slide-rule-steel-1950s/
#Geek #Math #Mathematics #Watch #Vintage #Antique #Science #Engineering
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A color photo of a round vintage wrist watch. The interesting thing about it is the outside dial is a slide rule!
Know why cats are good at math? It is because they study and practice in their daily lives.
#math #caturday #caturdayeveryday
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a cat studying a math book with a chapter in fractions
🌎📐 Working at the Naval Proving Ground in #Virginia, Dr. Gladys West used complex algorithms to map the #Earth’s irregular "geoid" shape.
Her mathematical foundation, combined with #Einstein’s relativity, is what allows the #GPS on your phone to be accurate within centimeters today.
👉 https://bigthink.com/starts-with-a-bang/gladys-west-einstein-gps/
#math #stem #blackhistory #physics #history #technology #innovation
Vulcan | The Planet That Didn't Exist
https://www.youtube.com/watch?v=iJyweEcpsGc
Re-watching one of my favorite #youtube #documentaries. It is a nice #historical approach to how #sicence is constantly working to improve and challenge it self.
#relativity #astronomy #documentary #mathematics #physics #math #stem #history #historyofscience
> Dr. Gladys West, Mathematician Whose Work Made GPS Possible, Dies at 95
https://thezebra.org/2026/01/18/dr-gladys-west-mathematician-whose-work-made-gps-possible-dies-at-95/
#RIP #history #math #GPS #science #BlackHistory #BlackMastodon
How can drugs get more expensive? He already brought down their prices "600%, 700%, 1000%".
The 500k-ton typo: Why data center copper math doesn't add up
#HackerNews #dataCenterCopper #typo #math #investing #news #technology
Mesopotamian pottery from 8,000 years ago may indicate mathematical thinking. I was told there would be no math
I need some #math / #machinelearning / #AI / #physics people to confirm for me that the topology of latent space shows non-Euclidean characteristics. This is not for a technical project; I'm trying to understand just how well cultural theorists are using their mathy metaphors. Thanks in advance!
If you were writing a bignum library in C what would you call it?
This is a quasi-serious question.
The working name is “Rx” for internal reasons. I’ll stick with that if I can’t get something better.
String Theory Inspires a Brilliant, Baffling New Math Proof
Years ago, an audacious Fields medalist outlined a sweeping program that, he claimed, could be used to resolve a major problem in algebraic geometry. Other mathematicians had their doubts. Now he says he has a proof.
https://www.quantamagazine.org/string-theory-inspires-a-brilliant-baffling-new-math-proof-20251212/
One thing I like about this book is its approach to eigenvalues and eigenvectors. Most linear algebra books present eigenvalues as roots of the "characteristic polynomial", which is built from the "determinant", which in turn has some formula defining it. These objects are rarely motivated geometrically, and so you're left with limited understanding of just what an eigenvalue is or why linear transformations on finite-dimensional vector spaces must have them. Axler avoids determinants till Chapter 9 of the book, focusing instead on linear operators. The fact that operators must have eigenvalues pops out of the observation that iterating an operator on a given non-zero starting vector results in a set of vectors that must eventually become linearly dependent. This fact also leads to the development of the characteristic polynomial; you can then come at the determinant from this, more geometric, perspective.
Here's one. If you're given a function, you can treat argmax of that function as a set-valued function varying over all subsets of its domain, returning a subset--the argmaxima let's call them--of each subset. argmax x∈S f(x) is a subset of S, for any S that is a subset of the function f's domain. Another way to think of this is that argmax induces a 2-way partitioning of any such input set S into those elements that are in the argmax, and those that are not.
Now imagine you have some way of splitting any subset of some given set into two pieces, one piece containing the "preferred" elements and the other piece the rest, separating the chaff from the wheat if you will. It turns out that in a large variety of cases, given only a partitioning scheme like this, you can find a function for which the partitioning is argmax of that function. In fact you can say more: you can find a function whose codomain is (a subset of) some n-dimensional Euclidean space. You might have to relax the definition of argmax slightly (but not fatally) to make this work, but you frequently can (1). It's not obvious this should be true, because the partitioning scheme you started with could be anything at all (as long as it's deterministic--that bit's important). That's one thing that's interesting about this observation.
Another, deeper reason this is interesting (to me) is that it connects two concepts that superficially look different, one being "local" and the other "global". This notion of partitioning subsets into preferred/not preferred pieces is sometimes called a "solution concept"; the notion shows up in game theory, but is more general than that. You can think of it as a local way of identifying what's good: if you have a solution concept, then given a set of things, you're able to say which are good, regardless of the status of other things you can't see (because they're not in the set you're considering). On the other hand, the notion of argmax of a function is global in nature: the function is globally defined, over its entire domain, and the argmax of it tells you the (arg)maxima over the entire domain.
In evolutionary computation and artificial life, which is where I'm coming from, such a function is often called an "objective" (or "multiobjective") function, sometimes a "fitness" function. One of the provocative conclusions of what I've said above for these fields is that as soon as you have a deterministic way of discerning "good" from "bad" stuff--aka a solution concept--you automatically have globally-defined objectives. They might be unintelligible, difficult to find, or not very interesting or useful for whatever you're doing, but they are there nevertheless: the math says so. The reason this is provocative is that every few years in the evolutionary computation or artificial life literature there pops up some new variation of "fitnessless" or "objective-free" algorithms that claim to find good stuff of one sort of another without the need to define objective function(s), and/or without the need to explicitly climb them (2). The result I'm alluding to here strongly suggests that this way of thinking lacks a certain incisiveness: if your algorithm has a deterministic solution concept, and the algorithm is finding good stuff according to that solution concept, then it absolutely is ascending objectives. It's just that you've chosen to ignore them (3).
Anyway, returning to our friend argmax, it looks like it has a kind of inverse: given only the "behavior" of argmax of a function f over a set of subsets, you're often able to derive a function g that would lead to that same behavior. In general g will not be the same as f, but it will be a sibling of sorts. In other words there's an adjoint functor or something of that flavor hiding here! This is almost surely not a novel observation, but I can say that in all my years of math and computer science classes I never learned this. Maybe I slept through that lecture!
#ComputerScience #math #argmax #SolutionConcepts #CoevolutionaryAlgorithms #CooptimizationAlgorithms #optimization #EvolutionaryComputation #EvolutionaryAlgorithms #GeneticAlgorithms #ArtificialLife #InformativeDimensions
(1) If you're familiar with my work on this stuff then the succinct statement is: partial order decomposition of the weak preference order induced by the solution concept, when possible, yields an embedding of weak preference into ℝ^n for some finite natural number n; the desired function can be read off from this (the proofs about when the solution concept coincides with argmax of this function have some subtleties but aren't especially deep or hard). I skipped this detail, but there's also a "more local" version of this observation, where the domain of applicability of weak preference is itself restricted to a subset, and the objectives found are restricted to that subdomain rather than fully global.
(2) The latest iteration of "open-endedness" has this quality; other variants include "novelty search" and "complexification".
(3) Which is fair of course--maybe these mystery objectives legitimately don't matter to whatever you're trying to accomplish. But in the interest of making progress at the level of ideas, I think it's important to be precise about one's commitments and premises, and to be aware of what constitutes an impossible premise.
There’s a fine line between a numerator and a denominator.
Only a fraction of people will find this funny.
Say you have a notion of "context", and a way of ordering these so that some contexts are larger, more expansive than, or "above" others. And let's say in each context, there is a set of things that are identifiable as "best". I'm being vague because you can instantiate this basic idea pretty broadly. For instance, maybe the contexts are states of information in a search algorithm and "best" refers to the possible solutions that seem best in each state of information; as you search, you change (increase) your state of information, and might change your might about which possible solutions are the best one. As another example, the contexts could be possible worlds and "best" refers to which propositions are true in each possible world; as you progress from one possible world to the next, you might change your mind about what propositions are true.
Anyway, with that simple setup you can associate to each thing the set of all contexts in which it appears best. This set could be empty or could be very large or anything in between. Then the lower order shows up as a weak preference relationship among all the things: one thing is lower preference than another if, for each context in which it appears best, there's a larger or equal context in which the other thing seems best. Put differently, any time you think the first thing is best, there's a way to increase your context such that the other thing appears best. This is exactly the lower order between the sets of contexts in which each thing seems best. If the set of contexts in which one thing seems best is higher up the lower order (😝) than the set of contexts in which the other seems best, then the former thing is weakly preferred to the latter.
The intuition in a search setting is that contexts are states of information, a kind of compendium of what you've learned so far in your search. If x and y are possible solutions, and for every context (state of information) in which you think x is the best there is always a bigger context--i.e., with more information--in which you think y is best instead, you ought to prefer y to x. The rationale is that any time you think x is best there's a way to learn a little more and change your mind to think y is best instead, which justifies preferring y to x.
Applied to modal logic, this notion corresponds to validity: if in every possible world where the proposition p is true there is an accessible world in which proposition q is true, then "p implies possibly q" is true in every world (valid).
The appearance of "possibly" is suggestive I think, and concords with this being a weak preference. "Necessarily" would be a strong preference, but I'd expect (in the sense of demand) a search process follow such a preference directly.
#math #ComputerScience #search #CoevolutionaryAlgorithm #SolutionConcept #ModalLogic
#SlowScience #math #CoevolutionaryAlgorithm #SolutionConcept
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Picture of a blackboard with some math scribbled on it. Below in the foreground there's a computer screen with some of the math typeset. I'm happy to describe to math to anyone who's interested but I'm not going to attempt to put a more detailed description here.
But the American Ornithological Society is making an effort with respect to bird names, and working through the controversies: https://americanornithology.org/english-bird-names/aos-pilot-project-to-change-harmful-english-common-bird-names/
and I think all of science and math can and should follow their lead. The world doesn't need "McCown's longspur" (McCown being a Confederate general complicit in genocide), and we don't need, for example, anything named after people like Gentzen either if you ask me: "In April 1939 Gentzen swore the oath of loyalty to Adolf Hitler as part of his academic appointment"; "Under a contract from the SS, Gentzen worked for the V-2 project" (from https://en.wikipedia.org/wiki/Gerhard_Gentzen)
Science results and math theorems should not be named after people, and we should undertake to rename any that currently are. We should prioritize renaming results or theorems named after white men and other privileged categories of people, with special attention to cases where a privileged person accepted or was assigned credit for work a less-privileged person did.