“I wish you hadn’t just told me not to touch it, because I don’t want to get into trouble and I didn’t even want to touch it, but your telling me not to makes me want to touch it!” my five-year-old exclaimed in frustration, apropos of something or other I’d asked him not to touch. Children are like that. Or, as the song “Never Say No” puts it: “Children, I guess, must get their own way the minute that you say no.”1
Adults are like that too. Being told what we can’t do takes us back to the time when we were powerless children, and sometimes we grownups respond to prohibitions in childish ways. Consider how many supposedly grown-up people have tantrums when they’re told they can’t enter a certain establishment unless they’re wearing a face mask! I sometimes wonder whether I’ve really matured as much as my change in station over the past half-century (from snotty pre-teen to tenured professor) would indicate; maybe I only seem more mature because, in my present life circumstances, fewer people tell me what I can’t do.
SQUARING THE CIRCLE
Among the adults who don’t like being told “You can’t do that” are many adults who enjoy math as a hobby, and the most common thing they’re told they can’t do is square the circle. Squaring the circle is the problem of constructing a square with the same area as a given circle, using only straightedge and compass in the classic Greek manner. (A straightedge is a ruler with all the markings removed, but by way of compensation, it can be as long as you need it to be. A compass is a tool for poking other kids in geometry class when the teacher’s back is turned.) Telling people “But it’s been proved that you can’t square the circle!” often proves to be an irresistible lure, and mathematicians regularly receive correspondence from strangers claiming to have found a solution.
David Richeson’s new book “Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity” (Princeton University Press, 2019) is devoted to the history of squaring the circle and three related problems: trisecting the angle, doubling the cube, and constructing (most) regular polygons. This well-written amply-illustrated book won’t fix the problem of amateur mathematicians insisting that they’ve solved one of the problems, because the people who most need to read the book either won’t read it or will leaf through it without understanding it. But many others will find it an enjoyable and informative read and a stunning illustration of the power of reason. (For a taste of Richeson’s writing, read his article “When Math Gets Impossibly Hard”.)
Modern circle-squarers have an illustrious forerunner in the person of the philosopher Thomas Hobbes, who believed to his dying day that he’d squared the circle. But the situation in Hobbes’ day was different: squaring the circle had not yet been proved impossible. So although each construction Hobbes proposed was wrong (as his combative correspondent the mathematician John Wallis was quick to point out), it was reasonable of Hobbes to hope that a workable construction existed and that perseverance would disclose it.2 Nowadays we know better, but it’s important to be precise about what it is that we know, because it’s easy to misunderstand what the claim of impossibility says and what it doesn’t say.
The proof of the impossibility of squaring the circle hinges on the subtle issue of precisely what sorts of geometrical constructions are permitted. We need to specify not just the tools that are to be used (straightedge and compass) but the way in which they are to be used.3 What nineteenth century mathematicians proved is that if one restricts oneself to using straightedges and compasses in certain specified ways, then certain geometric constructions are impossible. The “if … then …” nature of the claim is in keeping with the contingent character of pure mathematics, as deftly described by Clarence Wylie in the poem that concluded my first Mathematical Enchantments post.
The proofs of the four impossibility theorems Richeson discusses aren’t easy, but in philosophical essence they’re not that different from the claim that it’s impossible to find two even numbers whose sum is odd. If we agree on the meaning of “even” (an integer is even when it can be written as twice an integer), then two even numbers, say 2m and 2n, have sum 2m+2n, which (being equal to 2(m+n)) is again an even number. It would be foolish to object “But there are infinitely many even numbers to try, and you’ve only considered finitely many of them; how can you be sure someone cleverer than you won’t someday find two even numbers whose sum is odd?”
FROM GEOMETRY TO NUMBERS
How did mathematicians prove that squaring the circle is impossible? By turning it into a statement about operations on numbers.
If we’ve got a line segment of length 1 (call it AB), we can use it as the radius of a circle whose area would be πr2 = π 12 = π. If we start doing straightedge and compass operations, we can construct new points all over the place, but if you study them closely you’ll find a numerical pattern governing all those points: the distance between any two of them is a number that can be derived from the number 1 using only the operations of addition, subtraction, multiplication, division, and square roots, or what’s called a constructible number.4 So if at the end of the construction there are four points WXYZ forming a square of side-length s, s will have to be a constructible number. On the other hand, if the square has area π (the area of the original circle), the side-length s will have to be the square root of π. (For more on the square root of π, see my earlier blog post.) So if there were a way to square the circle in the Greek manner, the square root of pi would have to be a constructible number.
Turning this around, if we knew that √π weren’t a constructible number, that is, if we knew that √π can’t be obtained from the number 1 using only +, −, ×, ÷, and √, we’d know that the circle can’t be squared in the Greek manner. In short, proving the impossibility of a geometric construction (squaring the circle) can be reduced to proving the impossibility of an arithmetic construction (constructing √π from 1 using only +, −, ×, ÷, and √). And that’s how mathematicians settled the ancient problem.
But how did mathematicians show that √π and π aren’t constructible? To answer this (at least in outline), I present and discuss a graded sequence of impossibilities.
#1. You can’t arrive at the number 99 by adding 2’s together (and performing no other operations), no matter how many 2’s you add.
#2. You can’t arrive at the number 1/3 by adding and subtracting finitely many fractions whose denominators are powers of ten, or if you prefer decimals to fractions, finitely many terminating decimals.
#3. You can’t arrive at the square root of 2 through finitely many operations of addition, subtraction, multiplication, and division starting from the number 1.
#4. You can’t arrive at the cube root of 2 through finitely many operations of addition, subtraction, multiplication, division, and extracting square roots starting from the number 1. (That is, the cube root of 2 is not constructible.)
#5. You can’t arrive at pi through finitely many operations of addition, subtraction, multiplication, division, and extracting square roots starting from the number 1.
The truth of #1 is pretty clear: if you draw a highly exclusive number line with only even integers marked on it, there’s a big hole where 99 belongs, and if you’re playing the adding-2’s game you can stop short of the hole or you can leap over it but you can’t arrive at it.
The truth of #2 is similar in spirit, but subtler. Imagine that our snooty number club has decided to extend membership not just to all the formerly-excluded odd integers but to the terminating decimals as well: 0.3, 0.33, and the like. But 1/3 isn’t equal to 0.3, or 0.33, or 0.333, etc. It isn’t equal to any terminating decimal (or to any number expressible as a sum or difference of terminating decimals, because the result is just another terminating decimals). You could say that the set of terminating decimals has a hole where 1/3 would be. The difference between the hole at 99 in scenario #1 and the hole at 1/3 in scenario #2 is that in scenario #2 the hole doesn’t have a zone of unreachability around it. Terminating decimals will get you as close to 1/3 as you like (.3, .33, .333, etc. to the left of 1/3, and .4, .34, .334, etc. to the right of 1/3), but if only finitely many digits are allowed, your terminus will be only be an approximation to 1/3, not the exact number 1/3.
Moving on to #3, notice first what you can build up from the number 1 by the permitted operations: counting numbers (1+1=2, 1+1+1=3, etc.), zero (1−1=0), negative numbers (0−1=−1, 0−2=−2, etc.), and fractions (2/3, etc.). You can imagine that the number-line club has decided to relax its membership requirements again and now admits 1/3 and all the other rational numbers. But since the square root of 2 is irrational4 you can’t arrive at the square root of 2 in this way. Some of the new numbers like 7/5 and 17/12 give something close to 2 when you square them, but none of them square to exactly 2.
Impossibility #4 is related to one of the problems Richeson writes about, namely the problem of doubling the cube. Just as squaring the circle can be reduced to the problem of determining whether π is constructible, doubling the cube can be reduced to the problem of determining whether the cube root of 2 is constructible. You can imagine that the number-line club has expanded its membership yet again, to allow the square root of 2 to join, along with lots of other numbers. Every positive number that’s a member is welcome to extend the invitation to its square root, and every new member is given that same license! But there are still holes in this more inclusive version of the number line, and it can be proved that one of those holes is where the cube root of 2 sits. If you just want to approximate the cube root of 2, there’s a nice way to do it with error that becomes as small as you like, but if you’re only allowed finitely many operations, there’s no way to hit that number on the nose.
Finally we get to impossibility #5. Is pi in the inclusive club of constructible numbers or isn’t it? In attempting to find the answer, we’re led to a disconcerting question: what is this π number anyway? I mean, even if we don’t know what the cube root of 2 is numerically, we know what numerical property it’s supposed to have: when you cube it, you’re supposed to get 2. But what property distinguishes the number that is π from the infinitely many numbers that aren’t π? We know the geometrical meaning of π as the ratio of a circumference to a diameter, but what do we know about the number π?
Here the story takes a long detour through calculus, and I’m not going to give the details. Read Richeson’s book (and the references he provides) if you want to know more!
Now we come upon an issue of public relations. It’s one thing to say “The sum of two even numbers is always even”, and another thing to say “You can’t find two even numbers whose sum isn’t even”. Even if they’re logically equivalent, they’re not psychologically equivalent: the latter assertion throws down a gauntlet.
I doubt anyone has ever spent much time trying to find two even integers whose sum is odd, because the reasons for the impossibility are pretty clear; but when the reasons are more intricate, as is the case for the four problems Richeson treats, and when the word “impossible” is used, people who relish a challenge are more prone to take the bait. And once a person has committed to the position that the mathematical establishment is wrong, it may be hard for them to back down with their pride intact.
Meanwhile, people who say that something is “impossible”, even when they’re in the right, may find themselves linked in people’s minds with all the fools who made the classic blunder of saying that such-and-such is impossible only to be proved wrong an embarrassingly short amount of time later. Yes, the most famous is “Heavier-than-air flying machines are impossible” (Lord Kelvin, 1895) but there are many others. Of course math is different from aeronautics and other sciences because proofs in mathematics have a kind of rigor not attainable in other fields, but not everybody understands that.
Labeling these results “theorems of impossibility” may not be the best look for the mathematical profession. I prefer to describe them as “theorems of necessity”, inasmuch as they assert that if you want to square the circle (say), then it’s necessary that you broaden the notion of construction that you’re allowing yourself. Come to think of it, the ancient Greek mathematicians didn’t limit themselves to what we moderns call “geometric constructions in the Greek style”; for instance, Archimedes, not limiting himself to the lines and circles that straightedge and compass afford, figured out a way to solve the angle-trisection problem using spirals.
I’m suggesting that theorems of impossibility should be recast in more positive form, so that the assertion becomes an invitation to creativity rather than a door slammed in one’s face. As a perverse exercise, I invite you to take some attractively positive result and recast it in a negative vein. The two assertions may be logically equivalent, but they can feel very different!5
I’m not saying that every proof of impossibility can be recast in a more positive form, but I think it’s worth trying. For instance, instead of saying “These properties of the real number system show that it’s impossible for the square of a number to be -1”, we can say “If we’re going to have a number system in which -1 has a square root, we’ll have to drop one of the following properties of the real number system.” Likewise, instead of saying “It’s impossible for the angles of a triangle to add up to less than 180 degrees”, we can say “If we want a geometry in which the angles of a triangle can add up to less than 180 degrees, then …” and then try to figure out how to finish the sentence.
The story of geometry isn’t finished, and you don’t need to understand the 19th century impossibility proofs to find new results in Euclidean geometry. Although the geometric topsoil has been pretty thoroughly turned by past prospectors, there’s still gold waiting to be found, and proofs of impossibility can steer you towards the gold by steering you away from the places where there isn’t any.
Nonetheless, trisectors and circle-squarers and the like aren’t going to go away, and that’s a good thing. I propose that mathematical crankery ought to be encouraged among certain people as a way of channeling their latent susceptibility to outlandish beliefs; crazy notions about circles and circumferences are less harmful to society than crazy notions about pederasts and pizza parlors.6 Nor am I limiting my advocacy of mathematics-as-pacifier to the pacification of mathematical amateurs. I wish that one-time professional mathematician Ted “Unabomber” Kaczynski, instead of trying to change the world by killing people, had gone the way of Michael Atiyah, a truly great mathematician who at the end of his life came to mistakenly believe he’d proved the (still-unproved) Riemann Hypothesis. Atiyah’s delusions made him a happy man in his final days and hurt no one. Not all delusions are so harmless.
Thanks to Sandi Gubin, Joe Malkevitch, David Richeson, Evan Romer, and Stan Wagon.
#1. The song is from the musical “The Fantasticks”, which in turn is based on the 19th century play “Les Romanesques” by Edmond Rostand; both demonstrate the way prohibitions can backfire. A much earlier example comes in the Book of Genesis. Are we sure that Adam and Eve would have even noticed the Tree of Knowledge of Good and Evil if God hadn’t pointed it out to them, saying “Now whatever else you eat, don’t eat that“?
#2. For more on the Hobbes-Wallis conversation, see Martin Gardner’s essay “The Transcendental Number Pi”, chapter 8 in Gardner’s “New Mathematical Diversions”. Especially memorable is a quote from Hobbes that Gardner includes: “All you have said is error and railing; that is, stinking wind, such as a jade lets fly when he is too hard girt upon a full belly.” Hobbes was the more colorful writer, but Wallis was right on the math. Neither convinced the other. The correspondence has lessons for the age of Twitter.
#3. The last proviso matters because if you’re allowed to mark your straightedge, then the angle trisection problem ceases to be impossible and has a solution known to the ancient Greeks.
#4. Of course this assertion requires proof! I’m glad this is a blog and not a textbook, so I get to leave things there.
#5. As an example, I’ll “negativize” the sexy Banach-Tarski paradox. It’s sometimes couched in the form “You can divide a solid ball into a finite number of pieces and then reassemble those pieces to form two solid balls of the exact same size as the original” (though when it’s stated this way, I want to shout “No I can’t and neither can you!”). One could phrase this attention-grabbing positive claim in a negative fashion, asserting the nonexistence of a notion of “volume” that’s invariant under rotation and additive under finite dissection. This is equivalent to the usual statement (although the equivalence is trickier than you might think). But would this be a good way to sell it? I don’t think so.
#6. A friend who read an early draft of this essay suggested that the set of people who do kooky mathematics may be mostly disjoint from the set of people who engage in kooky politics (leaving aside Dr. Shiva Ayyadurai’s recent dabbling in election numerology). I asked David Richeson about the people who write back to him to share their trisections etc., and he said: “As for your question from earlier today—yes, I’ve definitely seen an uptick in crankish emails. But I have not been inundated. They do have some interesting variety. Some of them come by mail instead of email—with photocopied pages of complex geometric drawings. Some of them (quite a few of them) are written by people with advanced degrees. Some are degrees like psychology or medicine. Some are engineers. A lot of the people say that they’ve been working on these problems for years. I don’t know that I’ve received any who have explicitly said that my book is wrong. Rather, they just want to share their discoveries. They think I’ll be pleased to have learned that these problems are not, in fact, impossible. I have not gotten any like Woody Dudley writes about—by people who don’t want to share their solutions because they could be moneymaking ideas.” Here Richeson is referring to the books “Mathematical Cranks” and “The Trisectors” by Underwood Dudley.