Math is a Sport
And the referees sometimes take too long to issue decisions.
So much of what goes on in contemporary discussion and early education is a simple appeal to some authority1 which, while often well-intentioned, seems to only support thinking in the act of rebellion. That’s not the spirit of educational philosophy, but preteens have little contact with the enormity of the history of progress of human understanding and thus are in a poor position to recognize society’s need for them to acquire skills and knowledge of their world at an early age, when their brains are relatively plastic.2 What society wants them to learn changes decade-by-decade and from place to place, but that they might have the freedom to pursue knowledge for themselves at some later date the minimum is frequently summarized as the three-Rs: reading, writing, and arithmetic. This last topic seems of a different character than the first two as sums and long division are certain rigid procedures with rules that must be memorized which lead inexorably to one correct result. And for much of a decade this activity is known simply as “math.” But it is not a good guide to the whole of mathematics.
Math, as an activity of human society, is a game.
In the past, it was a sporting event with duels and grudge matches.
So what is math? Math is clearly useful, but its utility in our society or even the sciences is not part of its definition. Arithmetic concerns itself with the written forms of numbers and how to manipulate them, but is not what mathematicians call “number theory.” Math is big and varied and beyond the capacity of any human to understand it all. So what’s math like at its very core. Eugenia Cheng writes: “Math uses logic and abstraction together. It uses logic to build rigorous arguments, and it uses abstraction to ensure that we are working in a world where logic can be made rigorous.”3 And that makes it a recreation where one invents worlds built of rules to see what wonders they hold and if they are doomed.
Does math use abstraction? Certainly we see this at every level, including early arithmetic education. When we write “1 + 2 = 3”, what do we mean by “3”? It is the symbols for the abstract thing which is “the same” between “three apples”, “three dogs”, “three rocks”, “three ideas”, “three wars”, “three clouds”, “three languages”, etc. The underlying abstract thing that lets us compare apples and oranges so that we may say we have the same number of apples as oranges, is called a counting number or positive integer and is the subject of number theory, while the choice of symbols we use to express and manipulate that number say “3” or “1293” is the subject of arithmetic. Number theory teaches us other ways to mean “3” such as “s(s(s(0)))“, “11₂” or “𝑝₂”4. It’s nearly the same type of abstraction as human language and to that end, Galileo wrote something like “Mathematics is the language with which God has written the universe“5 in that it is the natural language of the natural sciences which describe the behavior of the universe.
Does math use logic? Very much so. If math is viewed as a science, then logic is its laboratory where ideas are tested to see if they are admissible as mathematical thought. A realm of mathematics has to be consistent which means you cannot get it to lead to logical arguments where it would contradict itself. If your proposed rules of arithmetic allows you to conclude both that x = y and x ≠ y6, then it is a bad collection of rules. And that’s where the referees come in.
Math, as an activity of human society, is a game. In the past, it was a sporting event with duels7 and grudge matches. The assistant gatekeepers who help the editors determine if a work is published or not are literally called referees. Moreover, once a purported proof is published, people poke at it, sometimes for years, trying to find weak or broken spots in the arguments. In 1976, a proof of the four-color theorem was announced by Appel and Haken but with almost 2000 cases and computer calculation replacing human argument it was controversial for both flaws that could possibly derail the argument and that one nearly needed a computerized bookkeeping system just to be sure that one had read the 400+ page proof comprehensively. The proof program was replicated and the flaws shown to be both harmless and correctable, but the potential for hidden errors in the logic or program persisted until perhaps 2005 when the entire theory was implemented in a computer proof assistant which made sure all the rules of the game are strictly followed.8
Today, such computerized proof checkers are everywhere. Languages like Rust and Java seek to prove that code is safe before it executes. Tools like Coq, Lean, and Metamath allow building of shared libraries of math proofs with all their rules and assumptions spelled out both for the poor computer and the new student.
Computers are making slow inroads on finding original proofs and asking questions that are original and interesting topics for research, but mathematics remains very much a human endeavor, sometimes a solo recreation and sometimes a global team sport. Proven theorems of the past are being formalized for machines to check them and add their techniques in libraries. One of the things I hope to do is to take topics from these proof assistants and make them accessible to those that don’t yet follow their peculiar languages.
Best wishes,
Richard Penner
This has been my first Substack post. If you have a topic of passion, you might want to start your own long-form essay site or newsletter.
Appeal to Authority is a logical fallacy often used in place of a valid argument or genuine thinking about a subject. Although it’s a speedy way to convey actual facts, it lends the appearance of truth to false statements as well. It is anathema to mathematics.
In mathematics we do not appeal to authority, but rather you are responsible for what you believe.
— Richard Hamming, “Mathematics on a Distant Planet.” The American Mathematical Monthly 105, no. 7 (1998): 640–50. https://doi.org/10.2307/2589247.
Nevertheless, it would be unnecessarily wasteful of our effort to not trust any authorities because again of that vast pile of human knowledge thus far achieved. So we seek to distinguish good, reasonable, and unbiased sources from bad while leaving ourselves open to the possibility that even good sources can be wrong at times.
That’s plastic in the sense of exhibiting adaptability to change or variety in the environment. A plastic brain is one still actively adapting to new situations, picking up new skills, languages, and ways of thinking with relative alacrity. Sadly, this is something we lose over time, so society seeks to equip young people with skills early in life.
The Joy of Abstraction, Eugenia Cheng, 2023, Cambridge University Press, ISBN: 9781108477222, p. 24.
This is one of a number of unrelated cheeky references in book titles to The Joy of Sex (Alex Comfort, 1972), including The Joy of Sets (Keith Devlin, 1993) and The Joy of X (Steven Strogatz, 2013). But one would have to understand the nature of abstraction to explain the joke.
In Peano Arithmetic, 0 is the only postulated number, but every number has a successor, given by the s() function. If we defined 3 as the third number that follows 0, we have s(s(s(0))) = 3, and other rules of Peano Arithmetic allow us to prove s(0) + s(s(0)) = s(s(s(0))).
In binary, 11₂ = 1 × 2¹ + 1 × 2⁰ = 2 + 1 = 3.
𝑝₂ refers to the second prime number. We can define a bijection (a one-to-one correspondence) between the counting numbers and the primes, as:
That is 𝑝₁ = 2 and for larger n, 𝑝ₙ is the smallest counting number larger than 𝑝ₙ₋₁ such that nothing between 1 and it divides it.
This function is available in Mathematica as Prime[n] and it is the inverse of the prime-counting function π or, in Mathematica, PrimePi[n], so ∀n∈ℕ π(𝑝ₙ) = n.
Or actually, something more like:
La Filosofia è scritta in questo grandissimo libro, che continuamente ci stà aperto innanzi à gli occhi (io dico l’universo) ma non si può intendere se prima non s’impara à intender la lingua, e conoscere i caratteri nei quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi, & altre figure Geometriche, senza i quali mezi è impossibile à intenderne umanamente parola; senza questi è un aggirarsi vanamente per un’oscuro laberinto.
{Natural] philosophy is written in this huge book, which is continually open before our eyes (I mean the universe), but it cannot be understood if we do not first learn to understand the language and know the characters in which it is written. It is written in the mathematical language, and the characters are triangles, circles, & other geometric figures, without which means it is humanly impossible to understand a single word of it; without these it is a wandering in vain through a dark labyrinth.
Galileo Galilei, Il Saggiatore (1623), p. 25. w/Google Translation.
Thanks to Michael Molinsky for researching this.
Normally, I would shorten these two contradictory statements to just 1 = 0, but that is a valid statement in arithmetic modulo 1, where 1/2 + 2/3 ≡ 1/6 mod 1 and 1 ≡ 0 mod 1.
Yes, really! See Cubic Equations and Mathematical Duels by Enrico Degiuli or Fabor Toscano and Arturo Sangalli (tr.). The Secret Formula: How a Mathematical Duel Inflamed Renaissance Italy and Uncovered the Cubic Equation. (2020) Princeton University Press, ISBN: 9780691183671.
For a human-sized explanation, see Georges Gonthier, “Formal Proof — The Four-Color Theorem,” Notices of the AMS 55, no. 11, (2008), 1382–93