There’s one math problem I find surprisingly deep in how it connects simple puzzles to profound ideas in logic, proof, and the rise of AI in mathematics.

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:

\( I_5(n) \)	\( I_3(n) \)	\( f(n) \)
\( 0 \)	\( 0 \)	\( 0 \)
\( 0 \)	\( 1 \)	\( 1 \)
\( 1 \)	\( 0 \)	\( 2 \)
\( 1 \)	\( 1 \)	\( 3 \)

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“.

One of the world’s premier security organizations has canceled the results of its annual leadership election after an official lost an encryption key needed to unlock results stored in a verifiable and privacy-preserving voting system.

The International Association of Cryptologic Research (IACR) said Friday that the votes were submitted and tallied using Helios, an open source voting system that uses peer-reviewed cryptography to cast and count votes in a verifiable, confidential, and privacy-preserving way. Helios encrypts each vote in a way that assures each ballot is secret. Other cryptography used by Helios allows each voter to confirm their ballot was counted fairly.

An “honest but unfortunate human mistake”

Per the association’s bylaws, three members of the election committee act as independent trustees. To prevent two of them from colluding to cook the results, each trustee holds a third of the cryptographic key material needed to decrypt results.

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a 23-year-old Cory Doctorow shows off Usenet, the Internet Underground Music Archive, Justin's Links from the Underground, and more to the CBC #

A reminder: Second Breakfast is on hiatus until the new year. Well, sort of, as here's a Second Breakfast email in your inbox. There are a couple of weeks' worth of news articles to link to, to be sure – the usual Friday stuff. But I've got a few other things to say, more in general, about the state of "AI" and education technology, about their utter indignity.

A few weeks ago, I attended a conference at Duquesne University – the 2025 Tech Ethics Symposium – where I spoke on a couple of panels about the future (of the "AI" industry, of higher education). It was a remarkable event – the first time in my 15+ years as an education writer (and 25+ years working in and adjacent to ed-tech) that I've been to a technology event where "No" was presented as a viable (indeed, perhaps even the moral) response to computing.

Typically, instead of encouraging statements (and subsequently, practices) of refusal, these events channel any and all resistance into a reframing, into acquiescence: these technologies might be dangerous and damaging – inextricable from (as we can see with "AI") fascism, eugenics, environmental destruction, war, imperialism, and growing wealth inequality – but somehow, somehow we can still bend them "for good." (????????)

Many people so desperately want to believe that they have a relationship with technology that is, at the very least, symbiotic. "Bicycle for the mind" cliches galore. But computing technologies, built by the most powerful companies in the world, run by the richest men in the world (the richest the world has ever seen) are all fundamentally committed to something else altogether: not to symbiosis but to extraction, exploitation, and domination.

This all plays out in "markets." And this plays out on bodies. It has already taken a massive toll on life, on work, on school, on knowledge. On human dignity.

Whose bodies? Whose autonomy? Whose dignity?

The House Oversight Committee's release of some of the documents from Jeffrey Epstein's estate should serve to remind us of this: the ways in which the elite move through the world with such thoughtlessness and with such impunity. That Larry Summers even still had a job teaching at Harvard – my god.

(Among the positions Summers has stepped down from this week: his role on the board of OpenAI.)

There is a real rot at the core of many of our institutions – and certainly at the core of those powerful players operating within and adjacent to them. "Artificial intelligence" emerges from this rot. It cannot be a bulwark against it. Among Epstein's friends (correspondents, visitors to his island) were luminaries in the field of "AI," including Marvin Minsky and Roger Schank. Luminaries in the field of technology and education.

Epstein, if you'll recall, was personally interested in all sorts of scientific research, but particularly in eugenics (he hoped, according to a 2019 NY Times article, to "seed the human race with his DNA") as well as in artificial intelligence. I feel like I'm losing my mind sometimes having to repeat this: "AI" and eugenics are inextricable.

Whatever belief that some folks have that they can "do good" with this technology, the technology itself bends in other directions (bends with a cc to jeevacation@gmail.com, I dare say). It bends towards the automation of everything. It bends towards indignity. It bends towards the end of education (certainly towards the end of the Department of Education), the reduction of school to "workforce prep," and the elimination altogether of a profession called "teaching" (in no small part because these folks have always viewed girls and women for other "uses" altogether).

Just a few more stories on ed-tech, "AI," and authoritarianism:

• "'Antivirus for libraries’: How a Texas startup is capitalizing on school book bans" – The San Antonio Express News on a company called Bookmarked that will identify "problematic material" for schools
• "'They Dragged Her Out in Front of the Kids': Inside Chicago’s ICE Raid at a Bilingual Daycare" – via Mother Jones
• 404 Media reports on how "DHS Is Deploying a Powerful Surveillance Tool at College Football Games"
• Speaking of school surveillance, a little bit of good news, for once – an update from Ian Linkletter on his legal battle with Proctorio
• "Stop AI in Malden Schools"

Stop it everywhere.

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Today's bird is the bearded vulture, which according to Wikipedia, is "the only known vertebrate whose diet consists of 70-90% bone." Although primarily a scavenger, the bearded vulture is also known to attack and kill live prey. You can decide what to make of this bird analogy.

Thanks for reading Second Breakfast. I really am taking a break from newslettering for the rest of the year. (Maybe...)