Happy Pi Day!^{1} To celebrate I want to get away from software for a bit and talk about something special. You may have heard the story that the Indiana legislature tried to change the value of π, to something like 3 or 4 or 3.15 or something like that. This is usually shared as evidence that Indianans are dumb hicks, but I don't like leaving things at that. Why did they try to change π and what did they expect to happen?
I did the research, and now you get to hear the whole story. But to fully understand the context, I'll need to explain math.
I'll need to explain a lot of math.
The Western math tradition starts with the Greeks. They weren't the first civilization to do math, and many things attributed to them were discovered earlier by Babylonian, Egyptian, and Chinese mathematicians. But those discoveries come to us through Greek antiquity. They were also indisputably the best geometers of their time. The Greeks were especially interested in a class of problems known as "straight-edge and compass problems". Given an infinitely long edge with no markings and a circle-drawing compass, what shapes can you construct, and what can you get out of existing shapes?
You can do a lot more than you'd expect! For example, you can take an angle and "bisect it", or draw a line that splits it in exact halves. You can take a line and construct a 32-gon from it. You can take a square and construct another square with exactly twice the area.
The Greeks had several open questions, whether certain things were constructable or not. Just because they couldn't do it didn't mean it was impossible. Some problems were too difficult for the Greeks, but ultimately solvable, like constructing a 17-gon (in 1796). One of the longest standing problems was "squaring the circle": given a circle, can you construct a square with the same area?
Now we need to be very clear about what that means. It doesn't count if you can construct a square that's 0.00001% off. It doesn't count if you come up with a method that converges to exactness over infinite steps. It doesn't count if you use anything other than an unmarked straightedge and a compass. The Greeks already knew how to square the circle with a marked ruler or spiral. The question was if you could get the exact area with the minimum number of tools.
By the Renaissance the mathematical consensus was that squaring the circle was probably impossible, mostly because every approach people could think of had failed. In 1837, Pierre Wantzel put that conjecture on firmer ground. A circle with radius 1 has area π, so the square would need sides of length sqrt(π). Wantzel had proven you could only get this if π was algebraic, and mathematicians strongly suspected it was transcendental.
Algebraic? Transcendental? I'll steal Mark Dominus's explanation:
We will play a game. Suppose you have some number
x
. You start withx
and then you can add, subtract, multiply, or divide by any integer, except zero. You can also multiply byx
. You can do these things as many times as you want. If the total becomes zero, you win.For example, suppose
x
is2/3
. Multiply by 3, then subtract 2. The result is zero. You win!Suppose
x
iscuberoot(7)
. Multiply byx
, then byx
again, then subtract 7. You win!Suppose
x
issqrt(2) + sqrt(3)
. Here it's not easy to see how to win. But it turns out that if you multiply byx
, subtract 10, multiply byx
twice, and add 1, then you win. (This is not supposed to be obvious; you can try it with your calculator.)But if you start with
x=π
, you cannot win. There is no way to get fromπ
to 0 if you add, subtract, multiply, or divide by integers, or multiply byπ
, no matter how many steps you take. (This is also not supposed to be obvious. It is a very tricky thing!)[...] Numbers like which you can win are called algebraic. Numbers like π with which you can't win are called transcendental.
While "almost all" numbers are transcendental,^{2} proving a given number is transcendental is very difficult. To whit, we don't know if π + e
is. Still, in 1882 Ferdinand von Lindemann proved π was transcendental, meaning you couldn't get a side of length sqrt(π)
, meaning squaring the circle is impossible.
Again, I want to reiterate this isn't actually consequentual for real-world applications. Getting an approximation is easy, and if you absolutely must have an exact solution you can pull out a ruler.
Even though mathematicians proved that squaring the circle is impossible, there is one group of people who still enthusiastically try to find a way: cranks.
Cranks are people who have deep, unshakeable beliefs about things that look like scientific knowledge but are actually bugnuts crazy. For internet people, the most famous example may be time cube. Most cranks are more coherent in what they say, but they use the same kind of moon logic to justify their beliefs. There are cranks in every field; computer science cranks do things like refute the halting problem or prove P = NP or whatever the hell this masterpiece is.^{3}
There are a lot of mathematical cranks, and a lot of them obsess over squaring the circle and trisecting the angle. I think it's for a few reasons:
In A Budget of Paradoxes, Augustus De Morgan lists over thirty circle-squarers he corresponded with. Reading his accounts from 1872, it's amazing how similar his cranks are to ours. That's why I find crankery so incredibly fascinating. It's a contradiction of derangement and conformity. No matter what the field, no matter the era, no matter what the crank is saying, they follow the same script, they talk the same way. That's why you can often tell something is crankery without even reading it, because you see how it taps into the crank gestalt. It's the scientific version of outsider art.
Crankery comes from a lot of places. For Edward J Goodwin, tragically, it came from mental illness. He thought his proofs were divinely inspired, literally the word of God. And in 1888, God told him how to square the circle. You can read the full method here, but here's the gist: Take a 90° arc and draw a chord between the two edges. The ratio of the arc to the chord is 8:7, and the ratio of the chord to the base of the corresponding triangle is 10:7. Multiplying the two ratios gives the length of the base is 4/5ths that of the arc. Since this is a quarter of the circle, scaling by four gives us 16/5ths. From there the construction is trivial.
Followed that? I hope not! I spent three hours staring at the proof to figure out what the hell he was thinking. Let me tell you, forcing yourself to think like a crank is not easy. I now have his moon logic etched into my brain. I am now one step closer to becoming a crank myself. The things I do for you people.
The important thing for our story the ratios he used: 8:7 and 10:7. These are about 2% and 1% off from the actual values,^{4}, and using them gives you a value of π that's about 4% off (160/49). Goodwin proceeded to make two more mistakes that happened to cancel each other out in a way that gave him the slightly more accurate π=3.2.^{5} I should be clear that this wasn't his only mistake, just the only obvious computational mistake. It's not clear how he actually gets from redefining pi to squaring the circle. Assuming it worked, the sides of the constructed square would be about 1% longer than they should be, small enough to pass the sight-test.
Goodwin brings up the divergence between 3.2 and the "orthodox ratio" of 3.1416, and then claims that 3.1416 is wrong. This, to me, is the wonder and horror of the crankery. While their crankery is so out there, so empirically wrong, they're absolutely certain about what they believe. They don't have the metacognition to think it's possible they're wrong about something. It couldn't possibly be that he made a mistake somewhere, it's everybody else that screwed up. A lot of cranks have delusions of grandeur, comparing themselves to Galileo and Einstein. If I thought I knew the true secrets of the universe, I guess I would, too.
Now Goodwin had a method to literally do the impossible. How do you get the word out? By publishing it, of course! In 1894 he submitted his proof to the American Mathematical Monthly, which published it in their July issue.
How'd it possibly get in? Mostly due to luck. AMM had started just seven months earlier, and the editors weren't 100% sure what they wanted out of the publication. The high-level goal was to make mathematics more accessible and interesting to the general population, and they experimented with several ways of doing that. According to one historian, early on the printed "whatever was submitted (or at least as much as there was room for)". First they'd print the "high order" papers they received, then they printed everything else. Goodwin's proof was published under the "Queries and Information" section, which had no editorial insight whatsoever. ^{6}
But most people don't hear about the intricacies of editorial content policies: they hear "it got published" and believe "it got endorsed". This gave Goodwin the credibility he wanted, and further fueled his crank ambitions. Goodwin had already "copywrited" the proof in 1889, and now, believing he had the support of the mathematical world, wanted to make money off it. He reasoned a discovery of this degree would have to be taught in every school, which means every school would have to pay him to reproduce the proof. So in 1897 he approached the Indiana state legislature with a proposal: if they officially declared the proof correct, he'd let them use it free of charge, thereby saving a lot of money on royalties.
So yeah, Indiana's motivation was a budget crunch.
You can read the whole bill here.
I'm not too clear on how it got passed the house. Some people speculate that the house of representatives were math-illiterate and didn't realize that "ratio of the diameter and circumference is as five-fourths to four" was talking about π, but it seems at first they knew this was bull! According to the Telegraph:
Gast of Bloomington, a Democrat, moved, amid great laughter, that the bill be referred to the Finance Committee, as it has made itself responsible for the solving of great problems, and since it has the time to do the job. Another representative arose and said that he believed the Committee on Swamplands was the appropriate place for successful wrestling with the problems. Midst general cheerfulness the Speaker then referred the "Squaring of the Circle" to the Committee on Swamplands where, in the swamp, the bill will find a deserved grave.
But then the Committee on Canals ("swamplands") bounced it to the Committee on Education, which came back with a wholehearted endorsement by the state superintendent, so the bill got passed 67-0. Why the about-face? My best guess is that nobody, ironically enough, wanted to look dumb. Most people aren't familiar with crankery and can't easily distinguish between nonsense and knowledge that only sounds like nonsense because you're not a subject expert. Do you really want to be the one arguing with Einstein? This was at least enough to cow the Indianapolis Journal:
The average editor will not gain much by trying to make fun of a discovery that has been indorsed by the American Mathematical Journal, approved by the professors of the National Astronomical Observatory at Washington, including Professor Hall, who discovered the moons of Mars; declared absolutely perfect by professors at Ann Arbor and Johns Hopkins Universities, and copyrighted as original in seven countries of Europe.^{7} The average editor is hardly well enough versed in high mathematics to attempt to down such an array of authorities as that.
Maybe it cowed the HoR too? It seems like a good enough explanation to me, but it's just speculation.
Regardless of why, it did pass, and the Senate was going to pass it too. But it happened that a mathematician from Perdue Purdue was in the Capitol that week, and some senators asked his opinion, and he explained why the proof was nonsense, and the bill was tabled indefinitely. I think this supports my hypothesis. The crank could pass for an expert and get people to second-guess themselves. But when an actual recognized expert gets involved then the spell is broken.
Goodwin died five years later at the age of 77. Many of the people he knew still believed in him. From his obituary:
As years went on and he saw the child of his genius still unreceived by the scientific world, he came broken with disappointment, although he never lost hope and trusted that before his end came he would see the world awakened to the greatness of his plan and taste for a moment the sweetness of success. He was doomed to disappointment, and in the peaceful confines of village life the tragedy of a fruitless ambition was enacted.
What's the lesson here? Most people chalk it up to "Politicians/Indianans/Americans are dumb". I find it more fascinating as a lesson about crankery. It's a failure mode of our epistemic norms, absolute confidence crossed with absolute wrongness. And it's not something most people are prepared for. Goodwin was, by all accounts, a charming and well-spoken man. He acted like someone worth taking seriously, so people did, and almost broke geometry on his behalf.
But aside from his near-success, Goodwin was just like any other crank. He thought the same way, he wrote the same way, he acted the same way. I sometimes get emails with that same crank energy. In the past hundred years it hasn't changed one bit.
That's all I have on the bill, next week we'll get back to proving the moon is fake. Happy π day everyone!
To stave off the obvious comments: yes, I know about τ, and yes, I agree it's a better choice for the circle constant. But nobody ever tried to change τ by legislative fiat, now did they? ↩
Sketch of the proof: the reals are uncountable. Every algebraic number is the root of some polynomial. Polynomials can be represented as tuples of coefficients, ie x² - 3
→ (1, 0, -3). The set of all finite lists is countable, which means the algebraics are too. Remove the countable algebraics from the uncountable reals and you're left with the transcendentals. ↩
No serious watch it it's glorious ↩
π/2 and sqrt(2), respectively. ↩
1) He multiplied by 7/10 instead of 10/7. 2) He multiplied by 4 (circumference = 4 quadrant arcs) but didn't divide by 2 (diameter = 2 radii). ↩
There were also a lot of flamewars in the Queries and Information section. ↩
Goodwin claimed he talked to Asaph Hall on this, and everybody took him at his word. As for why this man in particular, it might be because Hall was the first person to publicly carry out the Buffon's Needle experiment, a means of determining π via random sampling. I can imagine Goodwin hearing about that and thinking that Hall was trying to find the true value of pi, another aπkores. ↩
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