Writing about medical science, such as it was 2,400 years ago, Hippocrates identified “knowledge” as belonging to those truly in the know and “opinion” as the province of the ignorant. Today, with so much of the world’s scientific knowledge just a few thumb taps away but competing for sunlight with prickly online opinions that multiply like thistles, it’s fair to ask why a magazine dedicated to illuminating science has a reader forum at all.

It wasn’t a total surprise when several prominent science publications moved years ago to curtail or eliminate comments on their websites, encouraging readers to release their precious thought spores into the social media wilderness instead. No doubt caution is warranted — a 2018 study by researchers at the Massachusetts Institute of Technology found that “lies spread faster than the truth.” The perils of social media echo chambers and conspiracy theory chat rooms are all too real.

But here’s why Quanta accepts (some) online comments. Simply put, we genuinely want to hear from you, our valued, well-informed, curious readers. At many news outlets, the comment magnets are articles that wade into the thorniest controversies, sparking flame wars from the tinder of entrenched political views. Here at Quanta, what counts as controversial might seem amusing in contrast. Our most-commented story of the year was about — you guessed it — the nature of time. And our second-most-commented story of 2020 was about gravity. (Both were written by ace physics reporter Natalie Wolchover.)

It’s not all controversy and debate, however. Some commenters write in because they’re captivated by the ideas in a piece and want to learn more; some write to thank the author for explaining tricky concepts like how Gödel’s proof works. Some offer creative solutions to Pradeep Mutalik’s brain-bending Insights puzzles. Some readers, like Sanman Yadav, ask earnest questions about hard-to-visualize ideas like the geometry of the universe, which we try to answer whenever we can. (We appreciate it when experts pop in to answer reader questions too.) And in May, after we published our very popular article about Lisa Piccirillo’s solution to the Conway knot problem, Diana Conway (the wife of John Conway, the famous mathematician for whom the problem was named and who sadly died of COVID-19 in April) wrote a congratulatory note to Piccirillo:

Bravo! I hope John didn’t offer you any money…. I used to joke with him about being on the hook for these prizes after he was gone. 🙂

We also welcome thoughtful critiques of our coverage, which can only serve to make us more aware of potential blind spots. And readers are quick to point it out when we slip up, helping us correct any factual errors that sneak past our editing and fact-checking defenses. Thank you!

Not all comments contribute to the conversation, unfortunately. That’s why we moderate them, and that’s why it can take a while for your posts to appear (we’re only able to moderate a couple of times each weekday during New York business hours). Some comments are rejected and do not appear at all. How do we make that call? Our commenting policy states that “Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation. Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected.” If your comment gets rejected, it’s for one of these reasons. Decrying the ills of censorship won’t get your comment approved. Sorry.

Thinking about our civil, constructive community of commenters, I’m reminded of prime1987, who left this generous note after visiting Quanta a few years ago:

… I was thinking how blissfully relaxing it is to visit a site whose raison d’être is pure knowledge and which forsakes all of today’s marketing cancer.

Then I read the comments, and realised there’s something still better to be found here. Because even the weakest comments — let’s charitably call ’em “least informed”; or even “most encumbered by erroneous conceptions” — attract either no reply at all, or at best, a patient, good-faith attempt to address the misunderstanding. …

Call me a jaded curmudgeon, but for me this makes Quanta an oasis of the internet — and a reason for some small remaining hope that our species isn’t irrevocably doomed to choke out of existence under a pile of waste while taking a selfie …

Hyperbole aside, I hope that our moderated comments section offers a safe haven for the free exchange of (on-topic) ideas about fundamental science and math. If you took the time to write one of the thousands of comments that we’ve approved over the past 12 months, thank you. This part of our “2020 in Review” series is dedicated to you.

Before launching into our editors’ picks for 2020, a caveat: There were far too many incisive questions, brilliant insights and entertaining remarks to include in this short roundup. Rather than wring our hands over which comments to crown the best of the year, I asked our editors to each pick one they personally found the most delightful — one they thought other readers might enjoy too. On today’s polarized internet, maybe that’s enough.

Here are their picks.

Michael Moyer, Deputy Editor

Rogue waves are sea ghosts, terrifying apparitions that arise without warning. And as with ghosts, many have doubted their existence. Data came only from sailors’ tales, for what those are worth. What physical mechanism could possibly raise a solitary wall of water out of calm seas? But then on New Year’s Day 1995, an oil platform in the North Sea made the first verified measurements of a rogue wave, and the theoretical race to explain the phenomenon was on. In time, two separate physical theories emerged to explain these giants, and researchers were split on what to believe. Then this year, mathematicians developed an overarching method of predicting when and where rogue waves might appear, as writer Charlie Wood described in “The Grand Unified Theory of Rogue Waves.” Such advance knowledge would be immensely useful for captains, as illustrated by the unnerving anecdote shared by reader Ron M:

A few years ago, taking an excursion on a catamaran from St. Martin to Tintamarre Island, we had a terrifying experience. The ocean was a little rough but no-one was concerned, we were all drinking away, having a blast. Suddenly the boat felt like it dropped, a lot. I had the feeling of my stomach lifting, same as when you descend in a plane or on a roller coaster. Then before us was a huge wall of water, then the captain yelled, “hold on.” The wave completely covered the entire boat, it felt like we were under water, then it was over, just as quick as it happened. The captain did a quick head count, and checked that everyone was OK. The only injury was the second mate who had slammed on the deck and shattered the plexiglass hatch with his back, he had some severe bruising and small cuts.

It is fun to talk about it and laugh now, but when facing that wall of water, I had an ugly, grim feeling.

John Rennie, Deputy Editor

What makes a forest more than a bunch of trees, or an ant colony more than a lot of ants? For that matter, what makes each of us a human individual and not just 35 trillion agglomerated cells? The problem of defining what an individual is crops up repeatedly in biological theory; as one scientist told Jordana Cepelewicz in her thought-provoking article on this subject from July, biology is “a science of individuality.” That article elicited passionate responses from readers, but the first one was from Jon Richfield, a frequent and enthusiastic contributor to our comment section:

I love, love, love this. I have been thinking along similar lines for some decades, in fact from the late seventies when Dawkins’ description of species and their relation to information, and Patterson’s common gene pool definition of species started me … on the concept of “entity,” something it took me years to come to terms with. Salter’s “individuals” did not to my mind make much sense, least of all in biology.

… A species in an ecological structure is rather like an eddy in a rocky stream bed, or a cloud-cap over a mountain, created by the wind passing over; it adapts to the circumstances, it has to pass through its energy and substance, and when the rock, water, wind, or mountain changes or goes away, it must adapt or go extinct. …

Bill Andrews, Senior Editor

My favorite comment is a bit of a cheat, because it’s actually a pair of comments, a perennial question and a thoughtful answer. They appeared in response to Kevin Hartnett’s article about a breakthrough in determining the existence of theoretical rectangles, though they could just as easily have been a response to any sufficiently specific development in pure mathematics:

River Creature
As a non-mathematician person, what are the practical applications of this proof? The math is interesting to see but for me the application/uses are what I can understand.

Zemyla Cenh
In math, more than any other discipline, no one knows the practical applications of a problem until they’re found. 100 years ago, mathematicians studied prime numbers and elliptic curves for their beauty. Now, they’re used to cryptographically secure our bank transactions. Group theory is used to explain results in everything from quantum physics to Rubik’s cubes.

Someone reading this article may just go, “Aha! This will help me solve a problem in underwater basket weaving!” and change millions of lives. We won’t know until it happens.

Natalie Wolchover, Senior Writer/Editor

The subtitle of my article on Kurt Gödel’s infamous 1931 incompleteness theorems states that the theorems “destroyed the search for a mathematical theory of everything.” But Rachael Alvir, a doctoral student in mathematics specializing in Gödel’s work, questioned that widespread conception of the theorems, for the interesting reason that Gödel himself continued to believe in the knowability of all mathematical truth:

… Too many of Godel’s papers to cite here for this, but my favorite quote is below. It talks about what one cannot conclude from the results of his 1931 paper <3

You all know Hilbert’s famous words that every mathematician is convinced that for any precisely formulated mathematical question a unique answer can be found … The answer [to this given by the 1931 incompleteness theorems] may have two different meanings: (1) it may mean that the problem in its original formulation has a negative answer, or (2) it may mean that through the transition from evidence to formalism something was lost. It is easily seen that the second is the case, since the number-theoretic questions which are undecidable in a given formalism are always decidable by evident inferences not expressible in the given formalism. … So the belief in the decidability of every mathematical question is not shaken by this result.
[From Collected Works, Volume 3]

In fact, nothing Godel ever proved can be used to say he “destroyed the search for a mathematical theory of everything,” and he was actually after such a “mathematical theory of everything” all his life! … So, if Godel’s theorems from 1931 didn’t show what the [sub]title of this article claims, then what did? Clearly the [sub]title of the article is a widespread belief; I simply think myself that it is a widespread misconception (which I don’t blame the author for, given that it is widespread even among trusted sources).

Of course it is actually Cohen’s and Godel’s proofs showing the independence of CH [the continuum hypothesis] which makes us think that maybe there is no single mathematical system or set of self-evident axioms which is complete and consistent, since CH is a known undecidable question for which a method of obtaining its solution is unknown. Godel’s famous paper on CH says something in agreement with this.

That does not mean, of course, that it has no solution. It simply means that it is not known yet if there is a way of obtaining that solution mathematically. … It could go either way whether or not CH has a mathematical solution, or whether or not all mathematical problems have mathematical solutions. Perhaps the most important part of that picture, though, is the fact that the question is open. …