johncarlosbaez

johncarlosbaez, 3 months ago (edited 3 months ago) to random

This curve is not an elliptic curve - because even though you can write it in as

y² = P(x)

with P a cubic polynomial, elliptic curves need to be smooth! We say this curve is 'singular', not smooth everywhere, because it crosses itself at one point, making a kind of X shape. Mathematicians call this point a 'node'. So this curve, which I'd rather write as

y² = x³ - x²

is called a 'nodal cubic'.

It's still fun to count the solutions of this equation in a finite field. Let's do it!

(1/n)

johncarlosbaez, 3 months ago (edited 3 months ago) to random

There's a bunch of bigshots on YouTube who pontificate about string theory, the mysteries of quantum mechanics, and other profound issues in physics. But you can't really learn much physics from most of them. It's just chat.

Angela Collier here is so much better! So much more humble - and so much more fun if you really care about physics. I actually learned something: how to estimate the distance of a pulsar!

When pulses of radio waves from a pulsar move through space, they get smeared out as they go, and you can use this to guess how far away the pulsar is. Why? Because waves of lower frequency move a bit slower. Why? Because they interact more with the ionized gas in the Milky Way.

But how much slower, and why? That's what she explains - and actually this part, how radio waves interact with ionized gas, is what will stick with me.

This is the first episode of a series she calls Coffee and The Problem:

"We have coffee and I solve a problem, and the idea is that it's like a cozy weekend morning and you pull out your notebook and you solve the problem right along with me. I will give you time to pause and solve it yourself if you want and compare your answer with mine if you want. That's the game! That's the fun."

This time she's solving a problem about estimating the distance of pulsar. The problem just hands you a formula. But she's good. She doesn't just use the formula, she shows how to derive it from more fundamental principles! And also, at the end, she raises the question I was worrying about all along: how reliable is this method in practice? So she's not blindly solving a problem: she's thinking about physics.

https://www.youtube.com/watch?v=iox8Z-NGGS8

johncarlosbaez, 3 months ago (edited 3 months ago) to random

I'm trying to learn some number theory - elliptic curves and their L-functions. It would help to do some calculations and look for patterns. But I'm no good at programming. Can anyone here write a little program in Sage, which I could modify? Even better, maybe some folks could join in and explore this together on Mathstodon.

I want to to count the solutions of some cubic equations with integer coefficients in two variables. I want to count the solutions in the field with pⁿ elements. If I choose the equation and a prime p, I want to see the number of solutions for n = 1, 2, ..., 10 (say).

Below I did this for

y² + y = x³ + x

Here the number of solutions in the field with 2ⁿ elements turns out to be

2ⁿ - (1+i)ⁿ - (1-i)ⁿ

In general for an elliptic curve and a prime p the number of solutions is

pⁿ - αⁿ - βⁿ

where α is some complex number and β is its complex conjugate. This is called Hasse's theorem, and I explained it starting here:

https://math.ucr.edu/home/baez/motives/8.html

But next I'd like to try

y² + y = x³

again with the prime p = 2. This one may follow a different pattern, because this curve has a "cusp" for p = 2.

[Narrator: or does it? Read the comments.]

Of course there must be people who understand all this stuff, but it's sort of fun to figure things out oneself... umm, at least if someone helps out a little. 😅

johncarlosbaez, 3 months ago (edited 3 months ago) to random

Here Ethan Siegel (@startswithabang) gives a clear explanation of the 'multiverse' and why a lot of physicists think it exists.

Actually 'multiverse' means multiple different things. Siegel focuses on just one — which is fine. But beware, there are others, and also lots of people who cluelessly confuse them.

Siegel focuses on the inflationary cosmology, why it could be correct, and how it could create many 'bubbles': regions of the universe that can't communicate with each other. I have my doubts about inflation, more than Siegel, but this scenario seems possible.

There's another meaning of the multiverse, which is that we can think of the universe as a quantum superposition of different approximately classical worlds in which different events occurred. This scenario seems almost unavoidable to me, and it's not in conflict with the first.

Then there's a more speculative version of the multiverse, where these events in the very early history of the universe include the universe settling down to have different approximate laws of physics. For example maybe it's a superposition of states where the Standard Model works the way we see, and states where there are other numbers of forces and other kinds of particles. This seems possible too, but I'd only get interested if someone 1) makes up a precise, consistent theory of physics where this happens, and 2) convinces me that this theory is likely to be correct. So far I don't think even 1) has happened.

An interesting thing about these multiverses is how little they matter. They might someday, but they're not something I spend much time on. There's so much math and physics that's more interesting and practical!

https://bigthink.com/starts-with-a-bang/scientists-think-multiverse-fiction/

johncarlosbaez, 3 months ago (edited 3 months ago) to random

Tired: Justin Bieber
Wired: Heinrich Biber

We tend to think of 20th-century classical music as more experimental than what came before, and that's largely true, but there have been some astounding explorations in music for a long time — like this 1673 piece by Heinrich Biber. This suite for string orchestra starts out straight, but at 1:45 it bursts into polytonality.

Biber was a court musician in Salzburg, and a famous violin virtuoso of his day. This piece is called "Battalia à 10", or roughly "Battalion for 10". It depicts an army preparing for war, getting drunk, marching, and fighting a battle - and it ends with a "lament for wounded musketeers". The second movement is a quodlibet, a type of 17th century drinking song in which people sing different folk songs simultaneously. Biber titled it “The lusty society of all types of humor”, and it mixes Slovak, Bohemian, Austrian and German tunes playing in different keys. He wrote on the score "here it is dissonant everywhere, for thus are drunkards accustomed to bellow with different songs.”

The melody in the third violin part here is the German folk song "Kraut und Rüben haben mich vertrieben" (“cabbages and turnips have driven me away”) — a melody which J .S. Bach later used in his Goldberg Variations of 1741. Were people eating too many cabbages and turnips at that time in Germany?

https://www.youtube.com/watch?v=dMVI7z5GYRU

johncarlosbaez, 3 months ago (edited 3 months ago) to random

I'm enjoying the "Nancy" comics here on Mastodon:

https://mathstodon.xyz/@nancycomics@mastodon.social

Most of them are not very funny - less funny than this one. Most leave me scratching my head wondering how this comic strip got into the newspapers and stayed there so long. But they have a kind of surreal charm.

johncarlosbaez, 3 months ago (edited 3 months ago) to random

GALOIS' LAST LETTER

Galois invented group theory - and in his last letter, written to a friend the night before he died in a duel, he had some interesting things to say about it.

"Möbius transformations" are functions

f(z) = (az + b)/(cz + d)

where z,a,b,c,d are complex numbers. But it's better to think of them as maps from the Riemann sphere to itself. The Riemann sphere is the complex plane together with a point called ∞, which lets us know what to do when we divide by zero.

Möbius transformations are precisely all the transformations of the sphere that preserve angles! That's why they're important, geometrically. We say they form a "group" because doing two angle-preserving transformations gives another one, and for any angle-preserving transformation there's another one that undoes it.

We can copy this story with other number systems replacing the complex numbers. And in his last letter, Galois considered the integers mod p where is a prime. He defined a group of Möbius transformations

f(z) = (az + b)/(cz + d)

where now z,b,c,d are integers mod p. This group acts on a finite version of the Riemann sphere that consists of the numbers 0,1,2,3,...,p-1 together with ∞.

This baby Riemann sphere has p+1 points. But Galois showed that the group of mod p Möbius transformations has a special property when p = 2, 3, 5, 7 or 11, but no higher prime! Namely, it also acts transitively on a set with p points. "Transitively" means these transformations can map any point to any other point.

For p = 5 our group is the symmetries of a dodecahedron, which acts on the set of 5 tetrahedra shown here!

(1/n)

johncarlosbaez, 3 months ago (edited 3 months ago) to random

WHAT MAKES PLANTS GO BAD?

Why do plants become carnivorous?

They do so only when it's very hard to get nutrients like nitrogen. They still get their energy from photosynthesis, and it costs a lot of energy for them to be carnivorous. So they only arise in unusual environments like acidic bogs that are nitrogen-poor but sunny. And some cease to be carnivorous in the winter, when there's less light.

There's a continuum of plants ranging from noncarnivorous to so-called "protocarnivorous" plants to fully carnivorous plants. Protocarnivorous plants trap and kill insects or other animals but don't have special enzymes to digest their prey. Some of these evolve to become carnivorous. But don't think of evolution as goal-directed: depending on changes in their environment, some evolve away from being carnivorous. It's mainly a matter of how easy it is to get nitrogen.

There's an interesting conflict in this game. Plants have many clever ways of trapping insects and forcing the insects to pollinate them. This is a perfect first step in becoming protocarnivorous. But this sets up a battle of competing forces: will evolution optimize the trap for pollination or for killing insects and feeding the plant?

This is called the 'pollinator-prey conflict'. Many carnivorous plants try to have it both ways! And some are good at releasing insects that pollinate the plant, while killing others.

The game gets more complicated, too. One species of tree frog in Borneo specializes in eating insects caught in pitcher plants. And there's an ant that dives into pitcher plants to eat insects caught there - and also lubricates the top of the plant to make it easier for insects to fall in!

johncarlosbaez, 3 months ago (edited 3 months ago) to random

Dallol is in Ethiopia near the Red Sea. It's the hottest place year-round on the planet, 125 meters below sea level, and it's geologically active. Intrusions of magma into ground water have repeatedly caused massive explosions, so the place is pocked with craters.

It's a highly dynamic place: active hot springs go inactive and new springs emerge from day to day, and their colors change from white to green, lime, yellow, gold, orange, red, purple and ochre.

These springs discharge hot brine that is more acidic than anywhere else on Earth: some other hot springs have a pH of less than 2, but here the pH is less than zero! The brine here is also anoxic: instead of oxygen, it contains carbon dioxide, hydrogen sulfide, nitrogen and sulfur dioxide. On top of that it's 10 times more salty than seawater, and it contains deadly concentrations of magnesium.

Can life survive here? In 2019 a team of scientists concluded that while the nearby salt plains are teeming with salt-loving microorganisms, there is no life in Dallol's brine ponds.

But then, that same year, another team found utra-small structures entombed in the mineral deposits - and identified them as very ancient organisms adapted to hypersaline environments! They're called Nanohaloarchaea.

More details and pictures:

https://en.wikipedia.org/wiki/Dallol_(hydrothermal_system)

There's also a ghost town nearby, left over from a potash mine. Check out my next post for a photo of this truly uninviting place!

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johncarlosbaez, 3 months ago (edited 3 months ago) to random

The CEO of Exxon just said "we've waited too long" to tackle climate change. He blames "society" and "activists".

Yes, Exxon. Yes, Exxon. Yes, Exxon: the corporation that for decades has been spending millions to slow progress on climate change, despite its own research showing the problem was urgent.

He said:

"We've waited too long to open the aperture on the solution sets in terms of what we need, as a society, to start reducing emissions.... Frankly, society, and the activist — the dominant voice in this discussion — has tried to exclude the industry that has the most capacity and the highest potential for helping with some of the technologies."

What is this — some sort of dark, twisted joke?

The interview is here:

https://web.archive.org/web/20240228015207/https://fortune.com/2024/02/27/exxon-ceo-darren-woods-interview-pay-the-price-for-net-zero/

https://www.salon.com/2024/02/29/exxon-ceo-to-world-climate-isnt-our-fault-now-pay-the-price/

johncarlosbaez, 3 months ago (edited 3 months ago) to random

The number 30 is pretty cool. Check out the numbers relatively prime to it and smaller than it:

1, 7, 11, 13, 17, 19, 23, 29

Except for 1 they're all prime! 30 is the largest number with this property.

It's not that hard to see why. Suppose we had a bigger number with this property. It needs to be divisible by 2, 3, and 5, or else some product of two of these primes would be relatively prime to it and smaller than it, but not prime. So the next option is 60. But that doesn't work since 7² is smaller than 60 and relatively prime to it but not prime. So our number needs to be divisible by 7 too, so the next option is 210. But this doesn't work since 11² is relatively prime to it and smaller than it, but not prime. And so on - it's a losing battle.

Furthermore, those numbers I listed form a palindrome:

30 - 1 = 29, 30 - 7 = 23, 30 - 11 = 19, 30 - 13 = 17,
30 - 17 = 13, 30 - 19 = 11, 30 - 23 = 17, 30 - 29 = 1

This stuff is not a meaningless coincidence: it comes from 30 being the "Coxeter number" of the group E8. Notice there are 8 numbers on our list - that actually comes from the "8" in E8. If you add 1 to each of these numbers you get the so-called "magic numbers" for E8:

2, 8, 12, 14, 18, 20, 24, 30

Why are they magic? For example, if you take any maximal torus in E8, its normalizer mod its centralizer is a finite group with

2 × 8 × 12 × 14 × 18 × 20 × 24 × 30 = 696,729,600

elements. And if you double these magic numbers, subtract one from each and sum them up:

3 + 15 + 23 + 27 + 35 + 39 + 47 + 59 = 248

you get the dimension of the group E8.

Wacky stuff! For explanations go to the bottom of this page:

https://math.ucr.edu/home/baez/octonions/integers/integers_5.html

johncarlosbaez, 3 months ago (edited 3 months ago) to random

If you have a line of pendulums each attracting its neighbors, you can form waves of swinging pendulums that move down the line. These can act like particles!

More precisely, when the pendulums form a twist like a 360° twist in a ribbon, you get either a "kink" or an "antikink" depending on whether the twist is clockwise or counterclockwise. Here you see a "kink" and an "antikink" collide and move through each other. The total twist - the number of kinks minus the number of antikinks - is always conserved. At the moment of collision you can see that the total twist is zero in this example.

All this is described by the "sine-Gordon equation". Mathematicians and physicists have studied the hell out of this equation - it turns out to have a lot of depth to it.

I got this animation from Kanehisa Takasaki's webpage:

https://www2.yukawa.kyoto-u.ac.jp/~kanehisa.takasaki/soliton-lab/gallery/solitons/sg-e.html

and he has some more there. Can you guess what happens when two kinks collide?

(1/n)

johncarlosbaez, 3 months ago (edited 3 months ago) to random

The Law of Excluded Middle says that for any statement P, "P or not P" is true.

𝗜𝘀 𝘁𝗵𝗶𝘀 𝗹𝗮𝘄 𝘁𝗿𝘂𝗲? In classical logic it is. But in intuitionistic logic it's not.

So, in intuitionistic logic we can ask what's the 𝙥𝙧𝙤𝙗𝙖𝙗𝙞𝙡𝙞𝙩𝙮 that a randomly chosen statement obeys the Law of Excluded Middle. And the answer is "at most 2/3 - or else your logic is classical".

This is a very nice new result by Benjamin Bumpus and Zoltan Kocsis:

https://bmbumpus.com/2024/02/27/degree-of-classicality/

Of course they had to make this more precise before proving it. Just as classical logic is described by Boolean algebras, intuitionistic logic is described by something a bit more general, called Heyting algebras. They proved that in a finite Heyting algebra, if more than 2/3 of the statements obey the Law of Excluded Middle, then it must be a Boolean algebra!

johncarlosbaez, 3 months ago (edited 3 months ago) to random

Hey! Please sign this petition - it could make a real difference:

https://www.change.org/p/urge-duke-university-to-reconsider-closing-their-herbarium

Duke University want to close its herbarium, a massive collection of 825,000 plant specimens. The Dean of Natural Sciences gives this reason: "we are a university with limited resources". But Duke has an endowment of $11.6 billion - and this is a key resource for studying biodiversity!

“There are no herbariums that could absorb something like this,” said Kathleen Pryer, the director of the herbarium. “I’m very concerned that it will end up in a warehouse somewhere and become forgotten.”

14,306 people have signed the petition to stop this. Can you help bring it up to 15,000?

Or if you want to read more first:

https://web.archive.org/web/20240224094530/https://www.nytimes.com/2024/02/21/science/duke-herbarium.html

I'll quote a bit:

"Herbariums have been a mainstay of biology for centuries. Botanists return from expeditions with dried leaves, flowers, stems and seeds, which are then stored for posterity. Some specimens have even been the basis for naming new species.

But herbariums are also valuable because they include plants collected over long stretches of time, helping scientists track the impact of humans on the environment. Some collections have shown that plants have shifted their ranges as the planet has warmed, for example.

The collections have become even more useful as technology has advanced. With improved DNA sequencing, researchers have begun to extract genetic material from dried plant specimens, tackling old scientific questions such as the origin of the world’s crops.

Botanists are far from finished documenting the diversity of plants. And every year, they identify new species that need to be stored because many are already threatened with extinction."

johncarlosbaez, 3 months ago (edited 3 months ago) to random

Galileo put a lot of work into explaining why we don't feel the motion of the Earth. I just learned that similarly, early scientists had to explain why we don't hear the music of the spheres!

Aristotle writes:

"Some thinkers suppose that the motion of bodies of that size must produce a noise, since on our earth the motion of bodies far inferior in size and in speed of movement has that effect. Also, when the sun and the moon, they say, and all the stars, so great in number and in size, are moving with so rapid a motion, how should they not produce a sound immensely great? Starting from this argument and from the observation that their speeds, as measured by their distances, are in the same ratios as musical concordances, they assert that the sound given forth by the circular movement of the stars is a harmony. Since, however, it appears unaccountable that we should not hear this music, they explain this by saying that the sound is in our ears from the very moment of birth and is thus indistinguishable from its contrary silence, since sound and silence are discriminated by mutual contrast. What happens to men, then, is just what happens to coppersmiths, who are so accustomed to the noise of the smithy that it makes no difference to them."

Aristotle didn't believe in the music of the spheres. But much later, Kepler was fascinated by it. Here's a picture from his book 𝘏𝘢𝘳𝘮𝘰𝘯𝘺 𝘰𝘧 𝘵𝘩𝘦 𝘞𝘰𝘳𝘭𝘥𝘴.

Now we know each planet makes gravitational waves with a frequency equal to its year - very faint, and a very deep bass. But as neutron stars spiral into each other, in the last second their gravitational waves make a powerful 'chirp' with frequencies that soar over an octave above middle C!

johncarlosbaez, 3 months ago to random

I hope Trump wakes up every morning and looks at this clock:

https://payuptrump.com/

johncarlosbaez, 3 months ago to random

Last weekend in Munich, Christopher A. Wray, the F.B.I. director, said that hacking operations from China were now directed against the United States at “a scale greater than we’d seen before.” And at a recent congressional hearing, Mr. Wray said China’s hacking program was larger than that of “every major nation combined.”

“In fact, if you took every single one of the F.B.I.’s cyberagents and intelligence analysts and focused them exclusively on the China threat, China’s hackers would still outnumber F.B.I. cyberpersonnel by at least 50 to one,” he said.

[from the New York Times]

johncarlosbaez, 3 months ago (edited 3 months ago) to random

At first glance it's amazing that one of the great British composers of the 1400s largely sank from view until his works were rediscovered in 1850.

But the reason is not hard to find. When the Puritans took over England, they burned not only witches and heretics, but also books — and music! They hated the complex polyphonic choral music of the Catholics.

So, in the history of British music, between the great polyphonists Robert Fayrfax (1465–1521) and John Taverner (1490–1545), there was a kind of gap — a silence — until the Peterhouse Partbooks were rediscovered.

These were an extensive collection of musical manuscripts, handwritten by a single scribe between 1539 and 1541. Most of them got lost somehow and were found only in the 1850s. Others were found even later, in 1926! They were hidden behind a panel in a library — probably hidden from Puritans.

The 1850 batch contains wonderful compositions by Nicholas Ludford (~1485–1557). One music scholar has called him "one of the last unsung geniuses of Tudor polyphony". Another wrote:

"it is more a matter of astonishment that such mastery should be displayed by a composer of whom virtually nothing was known until modern times."

Ludford's work was first recorded only in 1993, and much of the Peterhouse Partbooks have been recorded only more recently. A Boston group called Blue Heron released a 5-CD set, starting in 2010 and ending in 2017. It's magnificent!

Here is the Sanctus from Nicholas Ludford's "Missa Regnum mundi". It has long, sleek lines of harmony; you can lose yourself trying to follow all the parts.

https://www.youtube.com/watch?v=qLrgI6-3I-s

johncarlosbaez, 3 months ago (edited 3 months ago) to random

How deep is your bass?

Rappers are put to shame by the 19th-century physicist Hermann von Helmholtz, who tried to find the point at which a note is so low it stops sounding like a note and you start feeling individual vibrations!

How? Using 32-foot tall organ pipes! Anyone who claims their measly little car speakers give a deep bass has gotta be kidding.

The biggest organ pipes are 64 feet tall - for example the Contra-Trombone in the Sydney Town Hall Grand Organ. I wonder what that feels like!

Helmholtz wrote:

"The 16-foot C of the organ, with 88 vibrations in a second, certainly gives a tolerably continuous sensation of drone, but does not allow us to give it a definite position in the musical scale. We almost begin to observe the separate pulses of air, notwithstanding the regular form of the motion. In the upper haIf of the 32-foot octave, the perception of the separate pulses becomes still clearer, and the continuous part of the sensation, which may be compared with a sensation of tone, continually weaker, and in the lower half of the 32-foot octave we can scarcely be said to hear anything but the individual pulses, or if anything else is really heard, it can only be weak upper partial tones, from which the musical tones of stopped pipes are not quite free."

From his great book On the Sensations of Tone as a Physiological Basis for the Theory of music, free here:

https://archive.org/details/onsensationsofto00helmrich

johncarlosbaez, 3 months ago (edited 3 months ago) to random

Yesterday I showed you a bunch of cute pictures of tuning systems, like this one.

On the outside you see a 'circle of fifths' with numbers saying how sharp or flat each fifth is. Bigger positive numbers mean more sharp, and I make them brighter green. Bigger negative numbers mean more flat, and I make them brighter red.

Inside there's a 'star of thirds' saying how sharp or flat each major third is. The same rules apply.

Basically, bright colors sound bad. Black is best. You'd like all the numbers to be zero and all the arrows to be black. But that's impossible! It's like lumps in a carpet that's too big for your floor: if you make parts of it nice, other parts get lumpy.

So what's allowed?

(1/n)

johncarlosbaez, 3 months ago (edited 3 months ago) to random

In the most popular tuning system today, each of the fifths is slightly flat - 1/12 of a comma flat, to be precise. That's why you see -1/12 on each arrow in the circle here.

That's not so bad. What's bad is that each major third is 2/3 of a comma sharp! That's why you see 2/3 on the arrows in the star inside the circle.

There's no way to get everything perfect. But different tuning systems handle this problem differently. I've just figured out a new way of drawing tuning systems, to show this.

Next, check out one of the oldest systems: Pythagorean tuning!

(1/n)

johncarlosbaez, 3 months ago to random

Today I got some questions about logic, like:

what do provability, decidability, consistency, and completeness mean?
how do we work with equality in set theory?

I don't know articles that explain topics like this to non-mathematicians, clearly and crisply, without becoming overlong or heavy with notation. Do you?

I looked around and found this "Introduction to First-Order Logic":

https://builds.openlogicproject.org/content/first-order-logic/introduction/introduction.pdf

but the very first sentence is

"You are probably familiar with first-order logic from your first introduction to formal logic."

which is basically a way of saying "fuck you - if you don't know this stuff already I won't explain it to you".

As a student I liked Boolos and Jeffrey's book "Computability and Logic":

http://alcom.ee.ntu.edu.tw/system/privatezone/uploads/Logic/20090928151927_George_S._Boolos,_John_P._Burgess,_Richard_C.

but that's more like a course than what I'm thinking of here: a collection of essays that explain different topics in plain English.

I also liked Hofstadter's "Gödel, Escher, Bach", but that's a massive quirky elaborate tale, not a simple clear explanation.

Wikipedia articles are packed with information but they aren't self-contained, clearly written essays. Articles in the Stanford Encyclopedia of Philosophy are better in some ways, but they often "show off" by including more advanced material.

Sigh....

johncarlosbaez, 3 months ago (edited 3 months ago) to random

How can we save the wonderful wildness of nature?

In 2017, Douglas Eger founded Intrinsic with the goal of creating "natural asset companies" (NACs). The idea is that a landowner works with investors to create a NAC that licenses the rights to the ecosystem services the land produces. If the company is listed on an exchange, a public offering of shares would provide the landowner with a revenue stream.

Sound impractical? Here's what happened. The Rockefeller Foundation kicked in about $1.7 million to fund the effort. In 2021, Intrinsic announced its plan to list NACs on the New York Stock Exchange. They filed an application with the Securities Exchange Commission (SEC).

Then the American Stewards of Liberty, a Texas group that campaigns against conservation measures, picked up on the plan. Through both grass-roots organizing and high-level lobbying, they argued that NACs were a Trojan horse for foreign governments and “global elites” to lock up large swaths of rural America, prevent mining in parks, etc. The SEC started to get tons of criticism.

A group of 25 Republican attorneys general called it illegal and part of a “radical climate agenda.” On Jan. 11 this year, the Republican chairman of the House Natural Resources Committee sent a letter demanding a slew of documents relating to the proposal. Less than a week later, the SEC gave up on allowing NACs.

This makes me think the idea actually might have been practical! Otherwise why work so hard to kill it?

The above is paraphrased from a NY Times article. You can read the whole thing for free here:

https://web.archive.org/web/20240218132637/https://www.nytimes.com/2024/02/18/business/economy/natural-assets.html

johncarlosbaez, 3 months ago to random

So much of what we're suffering in the US - and elsewhere - is due to the capture of income by the rich. To quote:

"How big is this elephant? A staggering $50 trillion. That is how much the upward redistribution of income has cost American workers over the past several decades.

This is not some back-of-the-napkin approximation. According to a groundbreaking new working paper by Carter C. Price and Kathryn Edwards of the RAND Corporation, had the more equitable income distributions of the three decades following World War II (1945 through 1974) merely held steady, the aggregate annual income of Americans earning below the 90th percentile would have been $2.5 trillion higher in the year 2018 alone. That is an amount equal to nearly 12 percent of GDP—enough to more than double median income—enough to pay every single working American in the bottom nine deciles an additional $1,144 a month. Every month. Every single year.

Price and Edwards calculate that the cumulative tab for our four-decade-long experiment in radical inequality had grown to over $47 trillion from 1975 through 2018. At a recent pace of about $2.5 trillion a year, that number we estimate crossed the $50 trillion mark by early 2020. That’s $50 trillion that would have gone into the paychecks of working Americans had inequality held constant—$50 trillion that would have built a far larger and more prosperous economy—$50 trillion that would have enabled the vast majority of Americans to enter this pandemic far more healthy, resilient, and financially secure."

https://time.com/5888024/50-trillion-income-inequality-america/

johncarlosbaez, 4 months ago (edited 4 months ago) to random

Someone very sweet, a total stranger, wrote:

"I wanted to send you a short thank you. Early in my college career, I found your article on how to learn math and physics. As a child I experienced educational neglect and knew very little about math or science, or even how to study it! I was lost before I even started."

"I have lived by your quote, 𝗴𝗲𝘁 𝗶𝗻𝘁𝗼 𝘁𝗵𝗲 𝗵𝗮𝗯𝗶𝘁 𝗼𝗳 𝗺𝗮𝗸𝗶𝗻𝗴 𝗶𝘁 𝗰𝗹𝗲𝗮𝗿 𝘄𝗵𝗲𝘁𝗵𝗲𝗿 𝘆𝗼𝘂 𝗸𝗻𝗼𝘄 𝘀𝗼𝗺𝗲𝘁𝗵𝗶𝗻𝗴 𝗳𝗼𝗿 𝘀𝘂𝗿𝗲 𝗼𝗿 𝗮𝗿𝗲 𝗷𝘂𝘀𝘁 𝗴𝘂𝗲𝘀𝘀𝗶𝗻𝗴, but couldn't remember where I had read it! I recently found it in my journal from my 1st week of college! Most amazingly, this approach works for every single subject!"

It's interesting to see someone who firmly latched onto that principle and profited from it. I know a bunch of math grad students who are really good in other ways but still don't impose that discipline. They trip up all the time. It's great to have intuitions that go beyond what you can prove, but it's bad to mistake those for certainty.

My article on how to learn math and physics is here:

https://math.ucr.edu/home/baez/books.html

I should update it. For example, my advice on courses still seems good, but there are a lot more online courses now.