Rose Horowitch, in The Atlantic, says something is deeply wrong with America’s schools:

For the past several years, America has been using its young people as lab rats in a sweeping, if not exactly thought-out, education experiment. Schools across the country have been lowering standards and removing penalties for failure. The results are coming into focus.

Part of the problem, according to Horowitch, is we’re giving away good grades for poor performance in high school classes. But those same signs of academic decay—low standards, reduced penalties—infect all levels of American schooling. The result? Rapidly eroding skills:

The decline started about a decade ago and sharply accelerated during the coronavirus pandemic. The average eighth grader’s math skills, which rose steadily from 1990 to 2013, are now a full school year behind where they were in 2013, according to the National Assessment of Educational Progress, the gold standard for tracking academic achievement.

But there’s a problem with Horowitch’s argument. Her piece is anchored by a fascinating report from UC San Diego about their surging remedial math program. She thinks this surge reflects a decline in basic math skills. But there’s no decline on California’s state tests, which instead show increases up until the pandemic (and slow recovery since).

This isn’t to say that Horowich is wrong—test scores have declined in California on NAEP. But when she points to a drop starting in 2013, that could only be in 8th Grade mathematics; 4th Grade math seems to have only dipped in California with the pandemic.

So—were standards only lowered for 8th Graders? And only for math? Because 8th Graders were looking pretty good at reading on NAEP in California until the pandemic.

My point isn’t to defend or focus on California. I’m also not saying that everything is fine. What I am saying is I find all of this very confusing. Some scores definitely started dropping around 2014, but not all of them. I don’t know how you can decide what’s going on in American schools based on these graphs. Whatever the cause of the decline is, it’s not obvious or simple—maybe it’s not even problematic.

It’s Mostly About the Weakest Students

Play around with NAEP scores and you’ll notice that the declines are concentrated in the weakest students. This seems to be the case also for TIMSS, an international assessment. On both the 4th and 8th grade exams, America’s lowest performing students peaked in 2011, while the strongest students continued improving through 2019.

So are American students “getting dumber,” as Matt Yglesias says?

First, that’s rude. But second—no, they aren’t! I mean, yes, even high fliers were impacted by the pandemic. Everyone was. But strong students were looking fine up until then. That cuts out some possible explanations: this isn’t about detracking, gifted education, grading in AP classes, or anything else that would primarily impact the strongest 10% of students.

So…is it test-based accountability? Economist Joshua Goodman thinks so. He points out that 2015 is when the Obama administration started issuing waivers that defanged No Child Left Behind. Up until then, high-stakes testing pushed schools to focus on lifting weaker students up.

Listen—I don’t know! I’d quibble that since NCLB was introduced in 2001, it can’t explain the decade of improvements leading up to that law. And because NCLB was focused on math and reading it can’t explain declines in civics or US history.1

But there’s another, more significant problem with this explanation…

It’s Not Just Kids

Chad Aldeman has been writing about declining test scores for years, and generally favors an explanation not unlike Goodman or Horowich’s. But it was from him I first learned that American scores on PIAAC, a test of workplace skills for adults of ages 16 to 65, have also been on the decline, arguably also peaking around 2014.

There were declines in every cohort, in both literacy and numeracy, even in the 55-65 group that has been out of school for over forty years.

Isn’t that…deeply weird? So it’s not just kids that have lost progress, but adults. How could schools possibly be responsible for that? And not all countries experienced a decline. So something is going on in America that impacted both students and adults beginning in 2014. What could it be?

It’s Not Just Phones, and it’s Not Just Here

Adults have phones. Teens have phones. Maybe it’s phones?

Goodman thinks this is part of the story. Horowich mentions this too. But as Aldeman points out, phones are everywhere and declines in TIMSS scores aren’t universal:

Smartphones and social media are global phenomena, and yet scores in Australia, England, Italy, Japan and Sweden have all risen over the last decade. A couple of other countries have seen some small declines (like Finland and Denmark), but no one has else seen declines like we’ve had here in the States.

(I’d also point out that 4th Graders typically don’t have phones, so they couldn’t explain declines in 4th Grade NAEP reading.)

But on the subject of international declines, and whether America is an outlier, there is another major international exam besides TIMSS—that’s PISA. And PISA scores have been down across the board since around 2014.

Weirdly, looking at this graph, America looks like it’s beating these trends in reading and science. How do you explain that? Why would countries participating in PISA have overall declines starting midway through the decade when TIMSS did not? Why would the US partly buck the trend on PISA, the test we traditionally do worse at, and then underperform the international trend on TIMSS, the exam we do much better on?

Now, there are differences between TIMSS and PISA. TIMSS is generally more of an achievement test with questions that closely resemble what students learn in school. PISA is designed to go a step further, requiring interpretation and application. Maybe that explains the different international trends…

But at this point, I want to throw my hands up—I just don’t know what’s going on!

So, what is it?

Here’s the situation: Americans are getting dumber...well, mostly not. But our lowest performing students seem to be losing ground. And, simultaneously, our adults. Some international tests show a similar decline happening in other countries. On other exams, America is on its own. What gives?

For a moment I had a theory that I liked—that it’s about the shift from paper to digital assessments. NAEP went digital in 2017. PISA went digital in 2015. TIMSS transitioned in 2019. Kids do a lot of lazy button pushing when they take these digital exams.

Maybe the issue is giving tests with the same devices we use to scroll YouTube?

This is definitely something to worry about (it’s a “mode effect”) but it seems as if the test-makers are on top of this. I guess I trust them not to mess up? Also, you’d figure that if there was a big problematic mode effect it would be more of a one-time hit. It probably wouldn’t cause a steady decline…right?

OK, another theory: is it just a coincidence that this tumult coincides with a huge influx of immigrants to the United States? Wouldn’t that change the adult workforce along with the school population? Demographic shift is the most accurate answer to so many educational questions—could it help us here?

I don’t know! This is all so speculative. I feel like I’m just making things up. And none of the explanations are completely satisfying.

Meanwhile the amount of confidence on display from our political writers is remarkable. They seem quite sure that American schools are in crisis, that students aren’t being taught, that there is some grand trick being played on the American public.

But the facts are confusing. So why the confidence?

“America has been using its young people as lab rats in a sweeping, if not exactly thought-out, education experiment,” Horowitch writes, and would it be crazy to say that she sounds a little bit jealous?

The American school system is stubborn. It evolves slowly and only in complicated ways in response to the demands of outsiders. It’s hard to reach into classrooms. Maybe that’s the point here. Writers, advocates, policymakers—many of them love the idea that schools have been steered in the wrong direction. After all, that means they can be steered, and maybe you can even take back the wheel. Climb aboard, everyone! We’re headed back to the 2000s.

Abstractions are powerful tools. Given enough abstraction everything gets somewhat simple. Somewhat clear. Also somewhat wrong. Abstraction turns everything real, material, consequential into mostly nothing. The abstraction of “a relationship” hides all the love and care and desire it might entail. The abstraction of “the border” hides the violence its defense entails.

Given enough abstraction every monstrosity becomes just a kind of mental gymnastics. This is the domain of devil’s advocates kind of people form whom the real often political meaning of certain statements or movements has become fully replaced by a game of debate. Who cares about the position, let’s win!

I keep thinking about abstractions when looking at the state of the current “AI” bubble and with it the state of a lot of the global economy.

As most people reading this know, I am not a fan of “AI”. If I never have to hear or read or see the term “agentic” ever again that would still be too early. Not only do I despise the political project that is “AI”, the mental damage these systems do to us, the harm to the environment they create, etc. but the way the narrative has turned a narrowly useful technology (stochastic pattern generation and recognition using neural networks) into the hammer to demolish all established codes and practices that ensure some base level of quality: But hey, who wants their software be developed by a team of professionals who can actually understand, model and solve or mitigate security issues when you can also just use a slot machine?

I’ve found myself saying how much I’d love the “AI” bubble to pop if just to shut “AI” influencers up, if just to have some space to talk about real solutions to real problems again. But that only works in abstraction. Only works when nothing means too much, really.

For worse (there’s no better in this sentence) we have made “AI” the foundation of many core parts of our economy. Not “AI” systems – those don’t really work, or meaningfully increase productivity – but the belief in the (future) value of the handful of tech companies building these systems. The US would be in a recession without the data center buildout that tech is throwing all its savings at. A buildout that is not connected to any form of successful business yet. “AI” does not scale the way digital services usually scale but all we see currently is still the old “increase user numbers and hope a business plan will manifest itself” scheme. Maybe ChatGPT can give Sam Altman some form of strategy besides lying.

This has material consequences. When (not if) the bubble pops we will see a few things: The stock market will take a dive which will affect many people living in countries without pension systems who rely on the money they have invested in ETFs or stocks. I am not talking about some VC dude or millionaire losing a few millions or billions, just normal people who wanted to retire at some point. Companies who have bet on “AI” now no longer can claim that one “needs to get onboard” and ride the hype but will have to fix their budgets – quick. This will lead to squeezing employees even more, firing people to make the next quarter’s numbers look better. In the current political landscape the instability that right leaning tech oligarchs and their fans will have created will probably benefit the right. We’ll see a lot of blame going around (remember: “AI” can never fail us, we can only fail “AI”. If this thing crashes, we did not believe enough!) and cuts to social services and anything that gives people the feeling of living in a functioning society. Which only helps right wingers but well that’s Neoliberalism for you.

I love the abstraction of the “AI” bubble popping. But the very probably effects haunt me.

This shouldn’t be read as a “well, ‘AI’ is here to stay so let’s make the best of it” kind of thing. I neither think “AI” will automatically stay (again: see this) nor am I sedated enough to believe that political values and action have no meaning. While I think there are a few narrow use cases for machine learning, whatever is called “AI” today has very few redeeming qualities being built on extraction, violence, domination, colonialism and right wing, anti-labor politics. This is not a moment of capitulation but of reflection.

It’s always easy to cheer for a revolution. Shit is fucked up and bullshit and our institutions and structures don’t seem to be willing, capable or motivated to meaningfully move towards a better world so let’s fuck shit up! Burn down data centers. Get out some guillotines. The thing is: Revolutions mean that people get hurt. That ultimately people die. That doesn’t mean that revolutions are always wrong, but it means that the abstraction is again doing a lot of work hiding harm to people who just want to get through their workday to be home with their family.

My criticism of “AI” is about limiting injustice and suffering. The suffering of the communities who get data centers put in their midst drinking up all the water, taking the electricity while producing metric fucktons of emissions, e-waste and noise. The injustice done to kids who get to chat to an LLM instead of building meaningful relationships to teachers and mentors, who don’t get to figure out what they are good at cause everything they do looks worse than “AI” output so they use that instead of getting to be so much better than slop machines. The way that non-working stochastic parrots undermine labor power and therefore putting the livelihood of thousands of families at risk. My criticism comes from love and care for people, communities and societies. So my actions can’t abstract the effects of those away.

But what can be done? In a better world we’d see governments segmenting the toxic “AI” part of the economy off and insulating the actual economy against it. Slowly moving big public investments out of those companies (not the best word, they seem more like cults these days but we still call them companies). Preparing for that bubble to deflate without taking thousands of lives with it. But we see the opposite: Europe where I live keeps wanting to throw all the money they can find at more “AI”. Just do more “AI”, it will be so good, bro. Trust us, bro. Just 10 more billion. It’ll be super cool, bro. Governments treat hyped tech like a stoner threats hits of their bong.

So what can we do? That is the question. Moving our savings out of the stock market? Maybe – if you have some. Stopping to criticize “AI” systems and the narratives that push them? Never: We mustn’t prop up or protect those dangerous and harmful systems by inaction.

I don’t have all the answers (I rarely have any TBH) but I do think that we should be at least careful wit glorifying the bubble popping a bit. Sure it’s fun to predict when it will go down and how much money Softbank is gonna set on fire but I think that it is also our job as critics to make sure that the public understands that “the AI bubble popping” has material consequences for them. That joining a union might be a good idea right now. That getting smart and knowledgeable people on Works councils is more important than ever. That tech companies are not your friends or benevolent and that they’d sell your kids if it made the stock perform well.

The “AI” bubble will deflate. But as cathartic as a “POP” might feel right now, we need to build the structures that ensure that this event doesn’t put a lot of harm on people who had nothing to do with it. Let Marc Andreessen or Satya Nadella or Sundar Pitchai and all those tech bros lose their money, make sure that after the storm nobody forgets that these people gambled with our lives and societies to make number go up. But we can’t just focus on holding those men accountable – as righteous and good that might feel (and holy cow do I want to see many of those people put on trial). We need to start building lifeboats and barriers protecting our peers, neighbors, families and communities.

We are all we have. Only solidarity will get us through this.

Paul McCartney is releasing a new track. It’s his first new song in five years—so that’s a big deal. But there’s something even more significant about this 2 minute 45 second release.

The song is silent. It’s a totally blank track—except for a bit of hiss and background noise.

What’s going on? Has Paul McCartney run out of melodies at age 83? Is he nurturing his inner John Cage. Did he simply forget to turn on the mic?

No, none of the above.

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Macca is releasing this track as a protest against AI.

His new ‘music’ is part of an album entitled Is This What We Want? It’s already available on digital platforms, and is now coming out on vinyl. All proceeds will go to the non-profit organization Help Musicians.

“The album consists of recordings of empty studios and performance spaces,” according to the website. In addition to McCartney, more than a thousand musicians are participating, including:

Kate Bush, Annie Lennox, Damon Albarn, Billy Ocean, Ed O’Brien, Dan Smith, The Clash, Mystery Jets, Jamiroquai, Imogen Heap, Yusuf / Cat Stevens, Riz Ahmed, Tori Amos, Hans Zimmer, James MacMillan, Max Richter, John Rutter, The Kanneh-Masons, The King’s Singers, The Sixteen, Roderick Williams, Sarah Connolly, Nicky Spence, Ian Bostridge, and many more.

I keep hearing that protest music is dead—and has been losing momentum since the Vietnam War. But there’s now a new war, and it’s stirring up creators in every artistic idiom.

They are fighting for their livelihoods and IP rights. And, so far, it’s been a losing battle.

You can see the new battle lines across the entire creative landscape.

Vince Gilligan, one of the most brilliant minds in TV, admits that he “hates AI.” He calls it the “world’s most expensive plagiarism machine.” For his new show Pluribus, he has added this disclaimer to the credits:

This show was made by humans.

AI represents the exact opposite of creativity, Gilligan warns. It steals the work of others. So any attempt to legitimize it as a creative tool is built on lies. A bank robber might just as well pretend to be a financier. Or an art forger claim to be Picasso.

Filmmakers are reaching the same conclusion.

Oscar-winning director Guillermo del Toro says he would “rather die” than use AI in his movies. You might even view his latest film Frankenstein as a pointed attack on technology gone wild. He describes Dr. Frankenstein as

similar in some ways to the tech bros. He’s kind of blind, creating something without considering the consequences.

But here’s where things start to get really creepy. The headline above comes from Variety, a leading voice for the entertainment industry. But the new publisher for Variety is a huge fan of AI—and sees it as essential to the future of the periodical.

It’s worth noting that this publisher started her career at Variety by selling ads, not writing. And that gives you a clear sense of the people on the other side of the battle field.

The people who have built careers on their creativity are now mobilizing. But the overseers who prioritize finance and profits will fight them at every turn. You might think that these two parties need each other—but that’s not how the bosses see it.

They love AI because it will reduce their dependence on human artists—who are often stubborn difficult people. Even worse, great artists are expensive people, so the suits inside the boardroom dream of replacing them with servile bots.

Very few of the bosses will say this openly. They can’t afford to stir up their creative workers—not yet. It’s too early and AI tech isn’t robust enough to replace all those folks in the cast and crew. But if you don’t think this is the plan, you don’t know how the people in those expensive boardroom chairs think and act.

Just take a look at the new AI “talent studio” Xicoa. A few weeks ago, it introduced an AI actress named Tilly Norwood. She’s a sweet brunette who looks like the girl next door—provided that you live inside a simulation.

The creative community was disgusted by this. But movie studios and agents reached out to the company, eager to explore ways of working together. In the aftermath, Xicoa announced that it is developing another 40 AI-generated actors.

According to one inside source, all the studios and major film companies are looking at AI projects. But everything is top secret, wrapped up in non-disclosure agreements—so we can only guess at the details. But the threat is clearly escalating at a rapid pace.

We see the exact same thing in music. Big records labels complained about AI—until they got a cut of the action.

• So Warner Bros had a strong copyright infringement case against an AI music company—but then decided to reach a settlement.

• Universal Music did the exact same thing.

• Sony is also cutting AI music deals.

I believe that the music industry could put AI companies out of business—the robbery of IP is so severe that this could be Napster all over again. Flesh-and-blood musicians would be protected, and real creativity could flourish.

But the bosses don’t want that. They will sell out the musicians—just so long as they make some money in the transaction.

And the exact same thing is happening in publishing.

Let me repeat: AI companies could be stopped simply by prosecuting them for violating copyrights. Why isn’t this happening?

Who wants to hear a bot sing of love it has never experienced? Who wants a painting made by something with no eyes to see?

The answer is simple—and sad.

Instead of protecting artist rights, the big companies in the culture sphere are seeking collaboration and quick settlements. Creators absolutely need to understand this. It’s not clear that they can trust their own labels or publishers—or maybe not even their own lawyers.

This is the new culture war.

And it’s very different from the old culture war—which was a dim reflection of politics. This new battle is happening inside the culture world itself, and threatens to cut off artists from their own longstanding partners and support systems.

This new culture war will only escalate. The stakes are too high, and artists can’t afford to stay on the sidelines. But they face heavy odds, with the richest people on the planet opposed to their efforts.

How will this battle get decided? It really comes down to the audience. If they prefer AI slop, we will witness the total degradation of arts and entertainment.

I’d like to think that people are too smart to fall for this crude simulation of human creative expression. Who wants to hear a bot sing of love it has never experienced? Who wants a nature poem from a digital construct that exists outside of nature? Who wants a painting made by something with no eyes to see?

Will the public find this charming. Or even plausible? Maybe a few twelve year olds and fools, but not serious people. That’s my hunch.

In any event, we will soon find out.

Fizz Buzz is a counting game that has become oddly popular in the world of computer programming as a simple test of basic programming skills. The rules of the game are straightforward. Players say the numbers aloud in order beginning with one. Whenever a number is divisible by 3, they say 'Fizz' instead. If it is divisible by 5, they say 'Buzz'. If it is divisible by both 3 and 5, the player says both 'Fizz' and 'Buzz'. Here is a typical Python program that prints this sequence:

for n in range(1, 101):
    if n % 15 == 0:
        print('FizzBuzz')
    elif n % 3 == 0:
        print('Fizz')
    elif n % 5 == 0:
        print('Buzz')
    else:
        print(n)

Here is the output: fizz-buzz.txt. Can we make the program more complicated? The words 'Fizz', 'Buzz' and 'FizzBuzz' repeat in a periodic manner throughout the sequence. What else is periodic? Trigonometric functions! Perhaps we can use trigonometric functions to encode all four rules of the sequence in a single closed-form expression. That is what we are going to explore in this article. By the end, we will obtain a discrete Fourier series that can take any integer \( n \) and select the corresponding text to be printed.

Contents

• Definitions
  • Symbol Functions
  • Fizz Buzz Sequence
• Indicator Functions
• Complex Exponentials
• Cosines
• Discrete Fourier Transform
• Conclusion

Definitions

Before going any further, we establish a precise mathematical definition for the Fizz Buzz sequence. We begin by introducing a few functions that will help us define the Fizz Buzz sequence later.

Symbol Functions

We define a set of four functions \( \{ s_0, s_1, s_2, s_3 \} \) for integers \( n \) by: \begin{align*} s_0(n) &= n, \\ s_1(n) &= \mathtt{Fizz}, \\ s_2(n) &= \mathtt{Buzz}, \\ s_3(n) &= \mathtt{FizzBuzz}. \end{align*} We call these the symbol functions because they produce every term that appears in the Fizz Buzz sequence. The symbol function \( s_0 \) returns \( n \) itself. The functions \( s_1, \) \( s_2 \) and \( s_3 \) are constant functions that always return the literal words \( \mathtt{Fizz}, \) \( \mathtt{Buzz} \) and \( \mathtt{FizzBuzz} \) respectively, no matter what the value of \( n \) is.

Fizz Buzz Sequence

Now we can define the Fizz Buzz sequence as the sequence \[ (s_{f(n)}(n))_{n = 1}^{\infty} \] where \[ f(n) = \begin{cases} 1 & \text{if } 3 \mid n \text{ and } 5 \nmid n, \\ 2 & \text{if } 3 \nmid n \text{ and } 5 \mid n, \\ 3 & \text{if } 3 \mid n \text{ and } 5 \mid n, \\ 0 & \text{otherwise}. \end{cases} \] The notation \( m \mid n \) means that the integer \( m \) divides the integer \( n, \) i.e. \( n \) is a multiple of \( m. \) Equivalently, there exists an integer \( c \) such that \( n = cm . \) Similarly, \( m \nmid n \) means that \( m \) does not divide \( n, \) i.e. \( n \) is not a multiple of \( m. \) With the above definitions in place, we can expand the first few terms of the sequence explicitly as follows: \begin{align*} (s_{f(n)}(n))_{n = 1}^{\infty} &= (s_{f(1)}(1), \; s_{f(2)}(2), \; s_{f(3)}(3), \; s_{f(4)}(4), \; s_{f(5)}(5), \; s_{f(6)}(6), \; s_{f(7)}(7), \; \dots) \\ &= (s_0(1), \; s_0(2), \; s_1(3), \; s_0(4), s_2(5), \; s_1(6), \; s_0(7), \; \dots) \\ &= (1, \; 2, \; \mathtt{Fizz}, \; 4, \; \mathtt{Buzz}, \; \mathtt{Fizz}, \; 7, \; \dots). \end{align*} Note how the function \( f(n) \) produces an index \( i \) which we then use to select the symbol function \( s_i(n) \) to produce the \( n \)th term of the sequence. We therefore call \( f(n) \) the index function.

Indicator Functions

Here is the index function \( f(n) \) from the previous section with its cases and conditions rearranged to make it easier to spot interesting patterns: \[ f(n) = \begin{cases} 0 & \text{if } 5 \nmid n \text{ and } 3 \nmid n, \\ 1 & \text{if } 5 \nmid n \text{ and } 3 \mid n, \\ 2 & \text{if } 5 \mid n \text{ and } 3 \nmid n, \\ 3 & \text{if } 5 \mid n \text{ and } 3 \mid n. \end{cases} \] This function helps us to select another function \( s_{f(n)}(n) \) which in turn determines the \( n \)th term of the Fizz Buzz sequence. Our goal now is to replace this piecewise formula with a single closed-form expression. To do so, we first define indicator functions \( I_m(n) \) as follows: \[ I_m(n) = \begin{cases} 1 & \text{if } m \mid n, \\ 0 & \text{if } m \nmid n. \end{cases} \] The formula for \( f(n) \) can now be written as: \[ f(n) = \begin{cases} 0 & \text{if } I_5(n) = 0 \text{ and } I_3(n) = 0, \\ 1 & \text{if } I_5(n) = 0 \text{ and } I_3(n) = 1, \\ 2 & \text{if } I_5(n) = 1 \text{ and } I_3(n) = 0, \\ 3 & \text{if } I_5(n) = 1 \text{ and } I_3(n) = 1. \end{cases} \] Do you see a pattern? Here is the same function written as a table:

Do you see it now? If we treat the values in the first two columns as binary digits and the values in the third column as decimal numbers, then in each row the first two columns give the binary representation of the number in the third column. For example, \( 3_{10} = 11_2 \) and indeed in the last row of the table, we see the bits \( 1 \) and \( 1 \) in the first two columns and the number \( 3 \) in the last column. In other words, writing the binary digits \( I_5(n) \) and \( I_3(n) \) side by side gives us the binary representation of \( f(n). \) Therefore \[ f(n) = 2 \, I_5(n) + I_3(n). \] We can now write a small program to demonstrate this formula:

for n in range(1, 101):
    s = [n, 'Fizz', 'Buzz', 'FizzBuzz']
    i = (n % 3 == 0) + 2 * (n % 5 == 0)
    print(s[i])

We can make it even shorter at the cost of some clarity:

for n in range(1, 101):
    print([n, 'Fizz', 'Buzz', 'FizzBuzz'][(n % 3 == 0) + 2 * (n % 5 == 0)])

What we have obtained so far is pretty good. While there is no universal definition of a closed-form expression, I think most people would agree that the indicator functions as defined above are simple enough to be permitted in a closed-form expression.

Complex Exponentials

In the previous section, we obtained the formula \[ f(n) = I_3(n) + 2 \, I_5(n) \] which we then used as an index to look up the text to be printed. We also argued that this is a pretty good closed-form expression already.

However, in the interest of making things more complicated, we must ask ourselves: What if we are not allowed to use the indicator functions? What if we must adhere to the commonly accepted meaning of a closed-form expression which allows only finite combinations of basic operations such as addition, subtraction, multiplication, division, integer exponents and roots with integer index as well as functions such as exponentials, logarithms and trigonometric functions. It turns out that the above formula can be rewritten using only addition, multiplication, division and the cosine function. Let us begin the translation. Consider the sum \[ S_m(n) = \sum_{k = 0}^{m - 1} e^{2 \pi i k n / m}, \] where \( i \) is the imaginary unit and \( n \) and \( m \) are integers. This is a geometric series in the complex plane with ratio \( r = e^{2 \pi i n / m}. \) If \( n \) is a multiple of \( m , \) then \( n = cm \) for some integer \( c \) and we get \[ r = e^{2 \pi i n / m} = e^{2 \pi i c} = 1. \] Therefore, when \( n \) is a multiple of \( m, \) we get \[ S_m(n) = \sum_{k = 0}^{m - 1} e^{2 \pi i k n / m} = \sum_{k = 0}^{m - 1} 1^k = m. \] If \( n \) is not a multiple of \( m, \) then \( r \ne 1 \) and the geometric series becomes \[ S_m(n) = \frac{r^m - 1}{r - 1} = \frac{e^{2 \pi i n} - 1}{e^{2 \pi i n / m} - 1} = 0. \] Therefore, \[ S_m(n) = \begin{cases} m & \text{if } m \mid n, \\ 0 & \text{if } m \nmid n. \end{cases} \] Dividing both sides by \( m, \) we get \[ \frac{S_m(n)}{m} = \begin{cases} 1 & \text{if } m \mid n, \\ 0 & \text{if } m \nmid n. \end{cases} \] But the right-hand side is \( I_m(n). \) Therefore \[ I_m(n) = \frac{S_m(n)}{m} = \frac{1}{m} \sum_{k = 0}^{m - 1} e^{2 \pi i k n / m}. \]

Cosines

We begin with Euler's formula \[ e^{i x} = \cos x + i \sin x \] where \( x \) is a real number. From this formula, we get \[ e^{i x} + e^{-i x} = 2 \cos x. \] Therefore \begin{align*} I_3(n) &= \frac{1}{3} \sum_{k = 0}^2 e^{2 \pi i k n / 3} \\ &= \frac{1}{3} \left( 1 + e^{2 \pi i n / 3} + e^{4 \pi i n / 3} \right) \\ &= \frac{1}{3} \left( 1 + e^{2 \pi i n / 3} + e^{-2 \pi i n / 3} \right) \\ &= \frac{1}{3} + \frac{2}{3} \cos \left( \frac{2 \pi n}{3} \right). \end{align*} The third equality above follows from the fact that \( e^{4 \pi i n / 3} = e^{6 \pi i n / 3} e^{-2 \pi i n / 3} = e^{2 \pi i n} e^{-2 \pi i n/3} = e^{-2 \pi i n / 3}. \)

The function above is defined for integer values of \( n \) but we can extend its formula to real \( x \) and plot it to observe its shape between integers. As expected, the function takes the value \( 1 \) whenever \( x \) is an integer multiple of \( 3 \) and \( 0 \) whenever \( x \) is an integer not divisible by \( 3. \)

Similarly, \begin{align*} I_5(n) &= \frac{1}{5} \sum_{k = 0}^4 e^{2 \pi i k n / 5} \\ &= \frac{1}{5} \left( 1 + e^{2 \pi i n / 5} + e^{4 \pi i n / 5} + e^{6 \pi i n / 5} + e^{8 \pi i n / 5} \right) \\ &= \frac{1}{5} \left( 1 + e^{2 \pi i n / 5} + e^{4 \pi i n / 5} + e^{-4 \pi i n / 5} + e^{-2 \pi i n / 5} \right) \\ &= \frac{1}{5} + \frac{2}{5} \cos \left( \frac{2 \pi n}{5} \right) + \frac{2}{5} \cos \left( \frac{4 \pi n}{5} \right). \end{align*} Extending this expression to real values of \( x \) allows us to plot its shape as well. Once again, the function takes the value \( 1 \) at integer multiples of \( 5 \) and \( 0 \) at integers not divisible by \( 5. \)

Recall that we expressed \( f(n) \) as \[ f(n) = I_3(n) + 2 \, I_5(n). \] Substituting these trigonometric expressions yields \[ f(n) = \frac{1}{3} + \frac{2}{3} \cos \left( \frac{2 \pi n}{3} \right) + 2 \cdot \left( \frac{1}{5} + \frac{2}{5} \cos \left( \frac{2 \pi n}{5} \right) + \frac{2}{5} \cos \left( \frac{4 \pi n}{5} \right) \right). \] A straightforward simplification gives \[ f(n) = \frac{11}{15} + \frac{2}{3} \cos \left( \frac{2 \pi n}{3} \right) + \frac{4}{5} \cos \left( \frac{2 \pi n}{5} \right) + \frac{4}{5} \cos \left( \frac{4 \pi n}{5} \right). \] We can extend this expression to real \( x \) and plot it as well. The resulting curve takes the values \( 0, 1, 2 \) and \( 3 \) at integer points, as desired.

Now we can write our Python program as follows:

from math import cos, pi
for n in range(1, 101):
    s = [n, 'Fizz', 'Buzz', 'FizzBuzz']
    i = round(11 / 15 + (2 / 3) * cos(2 * pi * n / 3)
                      + (4 / 5) * cos(2 * pi * n / 5)
                      + (4 / 5) * cos(4 * pi * n / 5))
    print(s[i])

Discrete Fourier Transform

The keen-eyed might notice that the expression we obtained for \( f(n) \) is a finite Fourier series. This is not surprising, since the output of a Fizz Buzz programme depends only on \( n \bmod 15 . \) Any function on a finite cyclic group can be written exactly as a finite Fourier expansion. In this section, we recover the same function \( f(n) \) using the discrete Fourier transform. It is worth mentioning here that the calculations presented here are quite tedious to do by hand. Nevertheless, this section offers a glimpse of how such coefficients are calculated. By the end, we will arrive at exactly the same \( f(n) \) as before. There is nothing new to discover here. We simply obtain the result by a different, more direct and noticeably tedious method. If this doesn't sound interesting to you, you may safely skip this section.

We know that \( f(n) \) is a periodic function with period \( 15. \) To apply the discrete Fourier transform, we look at one complete period using the indices \( n = 1, 2, \dots, 15. \) Over this period, the values are: \begin{array}{c|ccccccccccccccc} n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline f(n) & 0 & 0 & 1 & 0 & 2 & 1 & 0 & 0 & 1 & 2 & 0 & 1 & 0 & 0 & 3 \end{array} The discrete Fourier transform gives us the constants defined by: \[ c_k = \frac{1}{15} \sum_{n = 1}^{15} f(n) e^{-2 \pi i k n / 15}, \] for \( k = 1, 2, \dots, 15. \) These constants reconstruct \( f(n) \) via the inverse transform: \[ f(n) = \sum_{k = 1}^{15} c_k e^{2 \pi i k n / 15} \] where \( n \in \mathbb{Z}. \) Let us compute the first constant: \[ c_1 = \frac{1}{15}\left( \begin{aligned} e^{-2 \pi i \cdot 3/15} &+ 2 e^{-2 \pi i \cdot 5/15} + e^{-2 \pi i \cdot 6/15} + e^{-2 \pi i \cdot 9/15} \\ &+ 2 e^{-2 \pi i \cdot 10/15} + e^{-2 \pi i \cdot 12/15} + 3 e^{-2 \pi i \cdot 15/15} \end{aligned} \right) = 0. \] Similarly, \[ c_2 = \frac{1}{15}\left( \begin{aligned} e^{-2 \pi i \cdot 6/15} &+ 2 e^{-2 \pi i \cdot 10/15} + e^{-2 \pi i \cdot 12/15} + e^{-2 \pi i \cdot 18/15} \\ &+ 2 e^{-2 \pi i \cdot 20/15} + e^{-2 \pi i \cdot 24/15} + 3 e^{-2 \pi i \cdot 30/15} \end{aligned} \right) = 0. \] and \[ c_3 = \frac{1}{15}\left( \begin{aligned} e^{-2 \pi i \cdot 9/15} &+ 2 e^{-2 \pi i \cdot 15/15} + e^{-2 \pi i \cdot 18/15} + e^{-2 \pi i \cdot 27/15} \\ &+ 2 e^{-2 \pi i \cdot 30/15} + e^{-2 \pi i \cdot 36/15} + 3 e^{-2 \pi i \cdot 45/15} \end{aligned} \right) = \frac{2}{5}. \] Continuing in this manner, we find all the constants: \begin{align*} c_1 &= c_2 = c_4 = c_7 = c_8 = c_{11} = c_{13} = c_{14} = 0, \\ c_3 &= c_6 = c_9 = c_{12} = \frac{2}{5}, \\ c_5 &= c_{10} = \frac{1}{3}, \\ c_{15} &= \frac{11}{15}. \end{align*} Using the inverse transform, we get \begin{align*} f(n) &= \sum_{k = 1}^{15} c_k \, e^{2 \pi i k n / 15} \\ &= \frac{11}{15} + \frac{2}{5} \left( e^{2 \pi i \cdot 3n/15} + e^{2 \pi i \cdot 6n/15} + e^{2 \pi i \cdot 9n/15} + e^{2 \pi i \cdot 12n/15} \right) \\ &\phantom{=} \phantom{\frac{11}{15}} + \frac{1}{3} \left( e^{2 \pi i \cdot 5n/15} + e^{2 \pi i \cdot 10n/15} \right) \\ &= \frac{11}{15} + \frac{2}{5} \left( e^{6 \pi i n/15} + e^{12 \pi i n/15} + e^{18 \pi i n/15} + e^{24 \pi i n/15} \right) \\ &\phantom{=} \phantom{\frac{11}{15}} + \frac{1}{3} \left( e^{10 \pi i n/15} + e^{20 \pi i n/15} \right) \\ &= \frac{11}{15} + \frac{2}{5} \left( e^{6 \pi i n/15} + e^{12 \pi i n/15} + e^{-12 \pi i n/15} + e^{-6 \pi i n/15} \right) \\ &\phantom{=} \phantom{\frac{11}{15}} + \frac{1}{3} \left( e^{10 \pi i n/15} + e^{-10 \pi i n/15} \right) \\ &= \frac{11}{15} + \frac{2}{5} \left( 2 \cos \left( \frac{6 \pi n}{15} \right) + 2 \cos \left( \frac{12 \pi n}{15} \right) \right) \\ &\phantom{=} \phantom{\frac{11}{15}} + \frac{1}{3} \left( 2 \cos \left( \frac{10 \pi n}{15} \right) \right) \\ &= \frac{11}{15} + \frac{4}{5} \cos \left( \frac{2 \pi n}{5} \right) + \frac{4}{5} \cos \left( \frac{4 \pi n}{5} \right) + \frac{2}{3} \cos \left( \frac{2 \pi n}{3} \right). \end{align*} This gives us exactly the same function \( f(n) \) we obtained earlier. The difference is only in how we arrived there. Working out Fourier coefficients by hand is slow and mechanical. In practice these sums are almost always computed automatically by numerical software or computer algebra systems. Still, this exercise shows that our humble Fizz Buzz index function can be expressed precisely using the machinery of Fourier analysis.

Conclusion

To summarise, we have defined the Fizz Buzz sequence as \[ (s_{f(n)}(n))_{n = 1}^{\infty} \] where \[ f(n) = \frac{11}{15} + \frac{2}{3} \cos \left( \frac{2 \pi n}{3} \right) + \frac{4}{5} \cos \left( \frac{2 \pi n}{5} \right) + \frac{4}{5} \cos \left( \frac{4 \pi n}{5} \right). \] and \( s_0(n) = n, \) \( s_1(n) = \mathtt{Fizz}, \) \( s_2(n) = \mathtt{Buzz} \) and \( s_3(n) = \mathtt{FizzBuzz}. \) A Python program to print the Fizz Buzz sequence based on this definition was presented earlier. That program can be written more succinctly as follows:

from math import cos, pi
for n in range(1, 101):
    print([n, 'Fizz', 'Buzz', 'FizzBuzz'][round(11 / 15 + (2 / 3) * cos(2 * pi * n / 3) + (4 / 5) * (cos(2 * pi * n / 5) + cos(4 * pi * n / 5)))])

We can also wrap this up nicely in a shell one-liner, in case you want to share it with your friends and family and surprise them:

python3 -c 'from math import cos, pi; [print([n, "Fizz", "Buzz", "FizzBuzz"][round(11/15 + (2/3) * cos(2*pi*n/3) + (4/5) * (cos(2*pi*n/5) + cos(4*pi*n/5)))]) for n in range(1, 101)]'

We have taken a simple counting game and turned it into a trigonometric construction: a finite Fourier series with a constant term \( 11/15 \) and three cosine terms with coefficients \( 2/3, \) \( 4/5 \) and \( 4/5. \) None of this makes Fizz Buzz any easier, of course, but it does show that every \( \mathtt{Fizz} \) and \( \mathtt{Buzz} \) now owes its existence to a particular set of Fourier coefficients. We began with the modest goal of making this simple problem more complicated. I think it is safe to say that we did not fall short.

Read on website | #python | #programming | #mathematics | #puzzle

A few weeks ago I was minding my own business, peacefully reading a well-written and informative article about artificial intelligence, when I was ambushed by a passage in the article that aroused my pique. That’s one of the pitfalls of knowing too much about a topic a journalist is discussing; journalists often make mistakes that most readers wouldn’t notice but that raise the hackles or at least the blood pressure of those in the know.

The article in question appeared in The New Yorker. The author, Stephen Witt, was writing about the way that your typical Large Language Model, starting from a blank slate, or rather a slate full of random scribbles, is able to learn about the world, or rather the virtual world called the internet. Throughout the training process, billions of numbers called weights get repeatedly updated so as to steadily improve the model’s performance. Picture a tiny chip with electrons racing around in etched channels, and slowly zoom out: there are many such chips in each server node and many such nodes in each rack, with racks organized in rows, many rows per hall, many halls per building, many buildings per campus. It’s a sort of computer-age version of Borges’ Library of Babel. And the weight-update process that all these countless circuits are carrying out depends heavily on an operation known as matrix multiplication.

Witt explained this clearly and accurately, right up to the point where his essay took a very odd turn.

HAMMERING NAILS

Here’s what Witt went on to say about matrix multiplication:

“‘Beauty is the first test: there is no permanent place in the world for ugly mathematics,’ the mathematician G. H. Hardy wrote, in 1940. But matrix multiplication, to which our civilization is now devoting so many of its marginal resources, has all the elegance of a man hammering a nail into a board. It is possessed of neither beauty nor symmetry: in fact, in matrix multiplication, a times b is not the same as b times a.”

The last sentence struck me as a bizarre non sequitur, somewhat akin to saying “Number addition has neither beauty nor symmetry, because when you write two numbers backwards, their new sum isn’t just their original sum written backwards; for instance, 17 plus 34 is 51, but 71 plus 43 isn’t 15.”

The next day I sent the following letter to the magazine:

“I appreciate Stephen Witt shining a spotlight on matrices, which deserve more attention today than ever before: they play important roles in ecology, economics, physics, and now artificial intelligence (“Information Overload”, November 3). But Witt errs in bringing Hardy’s famous quote (“there is no permanent place in the world for ugly mathematics”) into his story. Matrix algebra is the language of symmetry and transformation, and the fact that a followed by b differs from b followed by a is no surprise; to expect the two transformations to coincide is to seek symmetry in the wrong place — like judging a dog’s beauty by whether its tail resembles its head. With its two-thousand-year-old roots in China, matrix algebra has secured a permanent place in mathematics, and it passes the beauty test with flying colors. In fact, matrices are commonplace in number theory, the branch of pure mathematics Hardy loved most.”

Confining my reply to 150 words required some finesse. Notice for instance that the opening sentence does double duty: it leavens my many words of negative criticism with a few words of praise, and it stresses the importance of the topic, preëmptively¹ rebutting editors who might be inclined to dismiss my correction as too arcane to merit publication.

I haven’t heard back from the editors, and I don’t expect to. Regardless, Witt’s misunderstanding deserves a more thorough response than 150 words can provide. Let’s see what I can do with 1500 words and a few pictures.

THE GEOMETRY OF TRANSFORMATIONS

As a static object, matrices are “just” rectangular arrays of numbers, but that doesn’t capture what they’re really about. If I had to express the essence of matrices in a single word, that word would be “transformation”.

One example of a transformation is the operation f that takes an image in the plane and flips it from left to right, as if in a vertical mirror.

Another example is the operation g that that takes an image in the plane and reflects it across a diagonal line that goes from lower left to upper right.

The key thing to notice here is that the effect of f followed by g is different from the effect of g followed by f. To see why, write a capital R on one side of a square piece of paper–preferably using a dark marker and/or translucent paper, so that you can still see the R even when the paper has been flipped over–and apply f followed by g; you’ll get the original R rotated by 90 degrees clockwise. But if instead, starting from that original R, you were to apply g followed by f, you’d get the original R rotated by 90 degrees counterclockwise.

Same two operations, different outcomes! Symbolically we write g ◦ f ≠ f ◦ g, where g ◦ f means “First do f, then do g” and f ◦ g means “First do g, then f”.² The symbol ◦ denotes the meta-operation (operation-on-operations) called composition.

The fact that the order in which transformations are applied can affect the outcome shouldn’t surprise you. After all, when you’re composing a salad, if you forget to pour on salad dressing until after you’ve topped the base salad with grated cheese, your guests will have a different dining experience than if you’d remembered to pour on the dressing first. Likewise, when you’re composing a melody, a C-sharp followed by a D is different from a D followed by a C-sharp. And as long as mathematicians used the word “composition” rather than “multiplication”, nobody found it paradoxical that in many contexts, order matters.

THE ALGEBRA OF MATRICES

If we use the usual x, y coordinates in the plane, the geometric operation f can be understood as the numerical operation that sends the pair (x, y) to the pair (−x, y), which we can represented via the 2-by-2 array

where more generally the array

stands for the transformation that sends the pair (x, y) to the pair (ax+by, cx+dy). This kind of array is called a matrix, and when we want to compose two operations like f and g together, all we have to do is combine the associated matrices under the rule that says that the matrix

composed with the matrix

equals the matrix

For more about where this formula comes from, see my Mathematical Enchantments essay “What Is A Matrix?”.

There’s nothing special about 2-by-2 matrices; you could compose two 3-by-3 matrices, or even two 1000-by-1000 matrices. Going in the other direction (smaller instead of bigger), if you look at 1-by-1 matrices, the composition of

and

is just

so ordinary number-multiplication arises as a special case of matrix composition; turning this around, we can see matrix-composition as a sort of generalized multiplication. So it was natural for mid-19th-century mathematicians to start using words like “multiply” and “product” instead of words like “compose” and “composition”, at roughly the same time they stopped talking about “substitutions” and “tableaux” and started to use the word “matrices”.

In importing the centuries-old symbolism for number multiplication into the new science of linear algebra, the 19th century algebraists were saying “Matrices behave kind of like numbers,” with the proviso “except when they don’t”. Witt is right when he says that when A and B are matrices, A times B is not always equal to B times A. Where he’s wrong is in asserting that is a blemish on linear algebra. Many mathematicians regard linear algebra as one of the most elegant sub-disciplines of mathematics ever devised, and it often serves as a role model for the kind of sleekness that a new mathematical discipline should strive to achieve. If you dislike matrix multiplication because AB isn’t always equal to BA, it’s because you haven’t yet learned what matrix multiplication is good for in math, physics, and many other subjects. It’s ironic that Witt invokes the notion of symmetry to disparage matrix multiplication, since matrix theory and an allied discipline called group theory are the tools mathematicians use in fleshing out our intuitive ideas about symmetry that arise in art and science.

So how did an intelligent person like Witt go so far astray?

PROOFS VS CALCULATIONS

I’m guessing that part of Witt’s confusion arises from the fact that actually multiplying matrices of numbers to get a matrix of bigger numbers can be very tedious, and tedium is psychologically adjacent to distaste and a perception of ugliness. But the tedium of matrix multiplication is tied up with its symmetry (whose existence Witt mistakenly denies). When you multiply two n-by-n matrices A and B in the straightforward way, you have to compute n² numbers in the same unvarying fashion, and each of those n² numbers is the sum of n terms, and each of those n terms is the product of an element of A and an element of B in a simple way. It’s only human to get bored and inattentive and then make mistakes because the process is so repetitive. We tend to think of symmetry and beauty as synonyms, but sometimes excessive symmetry breeds ennui; repetition in excess can be repellent. Picture the Library of Babel and the existential dread the image summons.

G. H. Hardy, whose famous remark Witt quotes, was in the business of proving theorems, and he favored conceptual proofs over calculational ones. If you showed him a proof of a theorem in which the linchpin of your argument was a 5-page verification that a certain matrix product had a particular value, he’d say you didn’t really understand your own theorem; he’d assert that you should find a more conceptual argument and then consign your brute-force proof to the trash. But Hardy’s aversion to brute force was specific to the domain of mathematical proof, which is far removed from math that calculates optimal pricing for annuities or computes the wind-shear on an airplane wing or fine-tunes the weights used by an AI. Furthermore, Hardy’s objection to your proof would focus on the length of the calculation, and not on whether the calculation involved matrices. If you showed him a proof that used 5 turgid pages of pre-19th-century calculation that never mentioned matrices once, he’d still say “Your proof is a piece of temporary mathematics; it convinces the reader that your theorem is true without truly explaining why the theorem is true.”

If you forced me at gunpoint to multiply two 5-by-5 matrices together, I’d be extremely unhappy, and not just because you were threatening my life; the task would be inherently unpleasant. But the same would be true if you asked me to add together a hundred random two-digit numbers. It’s not that matrix-multiplication or number-addition is ugly; it’s that such repetitive tasks are the diametrical opposite of the kind of conceptual thinking that Hardy loved and I love too. Any kind of mathematical content can be made stultifying when it’s stripped of its meaning and reduced to mindless toil. But that casts no shade on the underlying concepts. When we outsource number-addition or matrix-multiplication to a computer, we rightfully delegate the soul-crushing part of our labor to circuitry that has no soul. If we could peer into the innards of the circuits doing all those matrix multiplications, we would indeed see a nightmarish, Borgesian landscape, with billions of nails being hammered into billions of boards, over and over again. But please don’t confuse that labor with mathematics.

Join the discussion of this essay over at Hacker News!

This essay is related to chapter 10 (“Out of the Womb”) of a book I’m writing, tentatively called “What Can Numbers Be?: The Further, Stranger Adventures of Plus and Times”. If you think this sounds interesting and want to help me make the book better, check out http://jamespropp.org/readers.pdf. And as always, feel free to submit comments on this essay at the Mathematical Enchantments WordPress site!

ENDNOTES

#1. Note the New Yorker-ish diaresis in “preëmptively”: as long as I’m being critical, I might as well be diacritical.

#2. I know this convention may seem backwards on first acquaintance, but this is how ◦ is defined. Blame the people who first started writing things like “log x” and “cos x“, with the x coming after the name of the operation. This led to the notation f(x) for the result of applying the function f to the number x. Then the symbol for the result of applying g to the result of applying f to x is g(f(x)); even though f is performed first, “f” appears to the right of “g“. From there, it became natural to write the function that sends x to g(f(x)) as “g ◦ f“.