Chronicles of Harry

So, last summer, I learned from HK about the horeshoe crabs in New York City. She had -- I think, I forget the exact story -- been out exploring the distant parts of the city, and had come across Broad Channel, and she saw the horseshoe crabs on the beach there, and she gathered some people to go see them again. So she and I and some other friends went out to see them.

See, around May and June, it's horsehsoe crab mating season. I had no idea that horseshoe crabs even lived in the waters around here, but they do. And during the mating season they come up into the shallows to mate. But, you can't see them at just any time. My understanding is, they come out mostly when the tide is highest. You want to go at high tide, but not just at high tide, you want to go at high tide around a spring tide.

Anyway, last year's trip out to Broad Channel and we saw lots of horseshoe crabs. Although, at first, we made the mistake of heading north to the wildlife refuge, rather than south to the beach. The wildlife refuge did not have horseshoe crabs. What did have was lots and lots of mosquitoes. We all got really bitten before deciding to turn back and head to the beach instead.

I wanted to try the same thing again this year -- but maybe we didn't need to go all the way out to Broad Channel? The city has a webpage listing where horseshoe crabs can be seen. None of these locations are all that convenient, but Kaiser Park, in Coney Island, seemed like the most overall accessible.

So a few friends and I tried it out. Unfortunately, we made a bit of an error in timing. Yes, we went near a spring tide... but I hadn't checked the detailed tide charts, and it was actually near low tide when we went. Oops. We ended up mostly only seeing dead horseshoe crabs -- although we did find one live one, buried in the sand. Once the tide came in, it came out, and we got to see it swimming around! Still, just one is a bit disappointing. We did see a really cool bird which I have since learned was a yellow-crowned night heron!

We (well, not the same group) tried again two weeks later, same location. In this case we knew it would be around low tide, but, well, Kate insisted it be then (and then didn't show up...). The result was basically the same. A bunch of dead horseshoe crabs, only one living one. Well -- I spotted one living one buried in the sand before the others arrived, but once they showed up and the tide had come in a bit, I couldn't find it anymore. I guess it swam away? We did not see any herons this time, which was a bit disappointing, as Kippy (a birder) had come hoping to see one.

We did see lots of fiddler crabs! They're difficult to see, because in addition to being small, you can only see them from some distance -- get anywhere near them and they'll scuttle back into their holes (their holes covered that section of the beach). But move away and they'll come out and wave their claws up and down!

Anyway, I decided to make one final attempt this past Sunday. This time, the tides should be right. (And for whatever reason, this spring tide was projected to be especially high, higher than the other recent ones. I wonder why?) HK suggested we try Calvert Vaux Park, across the creek from Coney Island rather than on it, although, like Kate, she then didn't show up! But this time it worked -- we did find horseshoe crabs! And also an egret, and a fiddler crab that wasn't so good at hiding, and also lots of cats? By the entrance near 27th Ave I encountered 3 cats. Huh.

The park seems to consist of 3 peninsulas (do all 3 count as the same park? I think the south one is separate), but the three seem pretty split off from one another -- I think they all have separate entrances. We didn't try the northern peninsula because we just kind of missed it, but it looks like there's a beach there we could have tried. We did try the cove that lies between the northern and central peninsulas, since there was a beach there, but it was a dump, and also had no horseshoe crabs. (It also required taking some paths through the brush that were not the easiest.)

Ultimately the place we found horseshoe crabs was one of the beaches on the south side of the central peninsula. This is basically where we found all the animals, except the cats! There weren't as many as at Broad Channel last year -- possibly still an issue of timing, we were a bit after high tide and a few days before the highest tides -- but still a good number. One of them was tagged! So, yes, I reported it afterward.

We didn't try the southern peninsula (which I think is actually not part of the park). We were kind of tired from the heat at that point, and I'm not sure that the southern peninsula has any beaches, or how to get to them if so.

June is ending now so I think that'll be it for this year. But I'll go look for them again next year, I figure, and that time I'll be more considerate of just when high tide is...

(Meanwhile, the goats arrive on July 12th!)

So, here's something that had bugged me for a long time. Jacobsthal multiplication is associative, but the proof of this is unsatisfying. Normally the way you prove an operation is associative is that you show that a multiary version of it makes sense; you don't just literally show that a(bc)=(ab)c, you show that abc makes sense on its own as a ternary thing (or with even more operands) and that a(bc) and (ab)c are both equal to this.

But for Jacobsthal multiplication, the only proof I knew of its associativity was to show that a×(b×c) was equal to (a×b)×c; I wasn't able to find an interpretation of a×b×c on its own.

Now, some years ago, Paolo Lipparini found an order-theoretic interpretation of the Jacobsthal product, but it wasn't one that led to an immediate proof of associativity. But based on an observation of Isa Vialard, I think I finally have one! It's a little awkward, but of course it is; it's Jacobsthal multiplication, after all.

Let's say A and B are WPOs, and let's consider the lexicographic product A·B (note: B here is the coordinate that's compared first). Then o(A)o(B) ≤ o(A·B) ≤ o(A)×o(B).

But, actually, we can say more. Suppose B has k maximal elements, so that o(B) has finite part greater than or equal to k. Then, Vialard noted, we have o(A·B) = o(A)·(o(B)-k) + o(B)×k.

In particular, if o(B) has finite part equal to its number of maximal elements, then o(A·B)=o(A)×o(B). (In fact, as long as A has multiple distinct powers of ω in its Cantor normal form, the converse also holds.)

Btw, the finite part of o(B) is equal to the number of elements of B that have only finitely many elements above them. We could call such elements "almost maximal". So B satisfies this condition iff every almost maximal element is maximal.

Anyway, let's call a WPO that satisfies this condition "flat-topped". Note that flat-topped WPOs are closed under disjoint union, under Cartesian product, and, yes, under lexicographic product! So we can interpret α×β as the type of A·B for any flat-topped A,B with o(A)=α and o(B)=β; and we can do similarly with α×β×γ, etc. So this does it! We've interpreted multiary × and proved associativity nicely. Hooray!

EDIT 5/5: Actually, on writing to Lipparini about this, he pointed out to me that another observation of Vialard provides an even easier answer, namely, without needing the flat-topped condition, one has w(A·B)=w(A)×w(B), where w is width. I hadn't noticed this because, well, I've never really paid a lot of attention to widths, tbh.

Why didn't they call it the Super Switch??

There's a question that I occasionally see asked on the internet -- what's the big deal about Eliezer Yudkowsky? Oh, he suggested that artificial intelligence might be dangerous? That's not original! People have been talking about robot uprisings for as long as they've been talking about robots. Why is this Yudkowsky guy getting credit for it?

And I think this question is interesting for what it reveals about how much things have changed since, oh, 20 years ago, let's say. Yes, people have been talking about robot uprisings for as long as they've been talking about robots. But the question you have to ask is, who was saying artificial intelligence might be dangerous? Because back then, the general assumption among... technical people, science fans, transhumanists, etc -- the general assumption among that crowd was that of course artificial intelligence would be a good thing, just like technology generally is a good thing; and that if you thought otherwise, if you thought actually it might be dangerous, you were a backwards luddite. (This is roughly the view expressed now by the "e/acc" crowd.) Moreover, the arguments for it being dangerous were largely based on anthropomorphization. Yes, there were people talking about the dangers of artificial intelligence -- but the sort of person who might go work on artificial intelligence, or become any sort of expert on it, would never listen to that sort of person, and with good reason; they generally weren't worth listening to. They really were often anti-technology luddites, and their arguments were generally pretty bad.

(Frankly, I'm not even the best person to be relating this -- someone older, who had an adult's view of all this, would be better. I was prompted to finally write this post due to a conversation with an acquaintance today who occasionally attends OBNYC, but who also attends New York skeptics' meetings, and mentioned how the latter is a very different, substantially older crowd. I like to point out sometimes that a lot of Yudkowsky's writing is fairly continuous with older writings on rationality by people like Feynman, Sagan, etc; but this is only somewhat true, as evidenced by this split!)

So the significance of Eliezer Yudkowsky -- I mean, certainly not the only significance, likely not the main signifiance, but the significance for the purposes of this question -- isn't that he proposed that artificial intelligence could be dangerous; it's that he A. convinced of the risks of artificial intelligence the sort of people who were inclined to dismiss such, and B. did so by pointing out that the arguments for AI being safe are in fact based on anthropomorphization and rebutting these arguments in detail.

But it seems that people just coming to this conversation now often don't realize all this! They don't think there's anything usual about technical people, transhumanists, etc., considering AI to be dangerous -- they think of it as continuous with earlier arguments about robot uprisings, rather than considering those earlier arguments as something such people would have dismissed.

Somewhat similarly -- and this is something Yudkowsky himself has often remarked upon -- you get people who've never bothered to actually read Yudkowsky calling him an anti-progress luddite, which is of course not remotely correct. Indeed, the fact that he was not a luddite, and had credibility as a transhumanist, lent credibility to his eventual turn against artificial intelligence, made it influential rather than something reflexively dismissed! The thing (well, one of the things, but the relevant thing) that was unusual about Yudkowsky (but, thanks to him, is not unusual anymore) was his stance of being pro-technology and pro-progress but against the development of artificial intelligence specifically. But some people who comment on things based only on impressions assume that it still must be the case that anyone opposed to artificial intelligence must be a luddite. This is basically the opposite way one might miss Yudkowsky's significance; instead of failing to realize that the past was not like the present, one could fail to realize that the present is no longer like the past. (Well, OK, that's not quite right, because these people aren't so much assuming that the present is like the past, as they are assuming that the present is like the obvious thing you expect. It isn't!)

(Things are also different from 15 years ago in that it seems like transhumanism as it used to exist seems to be much reduced, because discussion of artificial intelligence has largely subsumed it all! Not entirely, but to a pretty good extent.)

Anyway, yeah, some context for those confused about such things...

-Harry

Yes solutions have been up for a while but I've been busy! Anyway, Mystery Hunt this year!

I once again went up to Boston for Mystery Hunt this year. Still on Plant.

The Hunt had an interesting structure this year. The first round had five different metapuzzles -- four ordinary metas and one supermeta, although I believe it was designed to be solvable without solving all of the other metas first -- and solving each one of the five unlocked a different subsequent round. Each of these later rounds had their own metapuzzle or metapuzzles, and solving the metapuzzles for all rounds unlocked the runaround. We made it to all five of the later rounds, but didn't solve any of their metas.

The other way the structure of the hunt was unusual was that, rather than solves unlocking specific puzzles, instead they separately "discovered" puzzles and gave you keys. You see, when you discovered a puzzle, it would still be locked -- they'd give you a name and a minimal description that they considered spoiler-free -- and you had to spend a key to unlock it. The idea was that this would allow teams to focus their efforts on the sort of puzzle that they prefer. I think this was a positive effect, but there was also a negative one. You know how clues and free solves require strong team leadership to make sure they get used? I think keys have a similar sort of thing going on. Without strong leadership a team might just not spend their keys and not unlock enough puzzles!

Also, one of the subsequent rounds this year was a fish round, and, I have mixed feelings about that. I mean, it's nice to have some easy puzzles, but they're so long, and it's kind of exhausting to solve so many puzzles and still not unlock the meta. If you want to have a fish round, maybe break it up into several rounds, rather than making it one long one?

The other reason I'm maybe a bit cool on fish rounds is that if there's always easy puzzles available, it becomes less appealing to sit down with the harder ones and stare at them and try lots of things. It's distracting! At least it's not like last year where, AIUI, the fish round barely contributed to progression... here at least it was a necessary part of making it to the runaround. But, yeah, if you want to have easier puzzles, maybe just mix them in with the other puzzles as an occasional "here have an easy one" rather than sticking them all in one round to make sure one's always available?

I'm actually a little unsure I helped all that much this year! I mean I definitely contributed to the team but I feel like I was often so distracted... well, I guess I'll go over the individual puzzles and see!

(Also: Li-Mei was on the writing team this year! Hi Li-Mei! :) )
( Cut for spoilers )
OK and I think that's all I have to say about this year!

Next year I think I might go to the JMM because it'll be in Washington DC, so I'll probably stay home for Mystery Hunt and solve remotely. But we'll see...

-Harry

Let's start with the less surprising one: A formula for primes (thanks to Jeff Lagarias for bringing this to my attention).

Everybody know there's no "formula for the prime numbers", right? Most mathematicians would I think view the question of a "formula for primes" as a bit silly -- the obsession with "closed-form formulas" is for high-schoolers, and what constitutes a "closed-form formula" isn't well-defined anyway, outside of very specific contexts. Most mathematical functions don't have formulas; high-schoolers always ask about a formula for this, a formula for that; you have to just tell them, look, stop worrying about a "formula" for it, it is what it's defined to be and that's it, OK? Math isn't about formulas.

Well, uh, it turns out that one of the premises above is wrong. And in fact this information isn't new -- the article I linked cites a result of Mazzanti from 2002 (!) that any Kalmár elementary function can be expressed in terms of addition, multiplication, exponentiation, monus (minus, but saturating if you try to go below 0), and floored division.

That's a huge class of functions -- implying that, contrary to the point of view expressed above, most number theoretic functions like divisor count, sum of divisors, etc, do have "formulas"! And yes, that includes the n'th prime function. The paper I linked isn't proving the existence of such a formula, which follows from the Mazzanti result; rather it's just trying to optimize it a bit.

Now all this assumes you're OK with monus and floored division. But:
1. I think very little of the power here is coming from monus; I think pretty much every instance of monus used to construct p_n (and lots of these other number-theoretic functions) could just be minus instead, I don't think they ever really perform a subtraction without knowing that the difference is nonnegative.
2. If you're not OK with monus, but you are OK with max or min, well, then monus is just max{n-m,0}
3. If you're not OK with that, well, we're allowing exponentiation, so you can write |n-m| as √((n-m)²) and then say it's (|n-m|+(n-m))/2 (this is a little questionable in my book but hey it would satisfy a high schooler :P )
4. As for floored division, unlike monus it really is quite essential to making this work -- evidently floored division is way more powerful than I ever realized! But like, would a high schooler be OK with floors in their formula? I'm pretty sure the answer is yes. :P

So, wow, that sure overturned some things I thought! But wait, here's something crazier:
"Any function I can actually write down is measurable, right?"

Like, every mathematician knows, that if you can explicitly write down a function, you don't really have to worry about whether it's measurable right? Not only do all the things one might normally do to make a function preserve measurability, but it's actually impossible to constructively prove the existence of a non-measurable set (and therefore function). So if you can write down a function explicitly, you can't prove it's non-measurable.

Somehow it never occurred to me before there's a loophole there: No, it can't be provably non-measurable... but maybe it's not provably measurable. And that's exactly what this paper claims to do -- it writes down an explicit function whose measurability is independent of ZFC. I haven't yet taken a good look at this one to understand how it does it, but I should really sit down sometime and do that.

But, even having looked inside a bit, I was still confused -- how could it be the case that it wouldn't be provably measurable, given that the definition wasn't anything too crazy, and all the common ways of making functions preserve measurability? I think the answer here is that while the infimum or supremum of a countable collection of measurable functions is again measurable, there isn't any principle that says that if f:X×Y→R is measurable, then so is x↦inf_y∈Y f(x,y). Huh! I would never have guessed that that operation need not (provably) preserve measurability. Crazy!

Sometimes, the right formula can do a lot!

So here's a question Shaked asked a bit ago. He said, say you've got a finite poset P; let's associate a number to P that is the smallest number of pairs x<y you need to specify for this to generate the order via transitive closure. Given a poset on n elements, what's the largest this number can be?

My initial thought was that (at least when n was a power of 2) maybe the cube would maximize it. This turned out to be pretty far from correct. Like, very approximately, this gets you n(log n)². What I didn't notice at the time is that Shaked's number actually has a much simpler description, because the minimum set of pairs you need to specify is precisely the order's covering relation! So the question is just, what poset on n elements maximizes the size of the covering relation, or equivalently the number of edges in the Hasse diagram?

Well, you can get ⌊n²/4⌋ by just making two tiers, divided up as evenly as possible, and declaring everything in the top tier to be greater than everything in the bottom tier (with different elements in the same tier incomparable). Is this the best possible? Yes! A Hasse diagram has to be triangle-free (can't have transitive triangles because it's a covering relation, can't have cyclic triangles because that wouldn't yield a partial order), and Turán's theorem or Mantel's theorem says this is the (unique) triangle-free graph on n vertices with the maximum number of edges. There you go! The uniqueness of the undirected graph doesn't quite translate to uniqueness of the poset, though, because you could instead have three tiers, with the size of the middle tier equalling (or almost equalling) the combined sizes of the other two -- that's a different poset, but it gets you the same undirected graph. It's not too hard to see that this is the only way to vary things, though.

When Shaked initially posed this problem -- before we realized what the invariant he defined actually was -- I said that it reminded me vaguely of order dimension. In reality I don't think it's all that similar, but that was what I thought at first. This raises a question: What's the highest order dimension you can make with a poset on n points? This question has been studied, it turns out; the answer is ⌊n/2⌋ for n≥4. (This is violated for n=2 and n=3, since dimension 2 is obviously possible for both -- if the maximum were 1, every order would be total!) When n is even, this bound is attained by putting n/2 points in a top tier, n/2 corresponding points in a bottom tier, and having everything in the bottom tier be less than everything in the top tier except its corresponding point above. So, kind of similar to the maximizer to Shaked's problem! Moreover, for even n≥8 this is also unique, and in the case n=6 it's pretty close to unique (there's only two alternatives, or only one if we don't count duals as separate). (The n=4 case isn't at all unique, as anything non-total will do.) For the odd case you can just do this for n-1 and then toss in an additional point that does whatever you want (so this case won't be at all unique).

I don't know, not a lot to say about this but I thought it was neat.

We're down to just 178 books remaining now. :P

So here's something unusual that happened recently. I got an email from some guy who wanted to ask me about something that had been added to the ordinal arithmetic Wikipedia page. He says, hey, someone added to the article this alternate recursion for natural multiplication, but didn't include a citation for it... you seem to be a person who knows about ordinal arithmetic; is this alternate definition correct? If so, do you know a citation for it?

I took a look, and, huh, this alternate definition was one I'd never seen before! So, if it was correct, I certainly didn't know a cite for it!

But, was it correct? The claim was that α⊗β was the supremum of all (α'⊗β)⊕β for α'<α together with all (α⊗β')⊕α for β'<β.

This recursion actually works for (α,β)<(ω2,ω2), but it isn't correct; at (ω2,ω2), it fails. The natural product of ω2 and ω2 is of course ω²4, but applying this recursion instead yields ω²3. So I wrote back and said all this and the guy removed it from the article (I didn't feel like getting involved myself).

But, huh! What an odd thing to happen. I wonder who added it -- OK, I guess I can go check that myself if I really care. But I wonder where it came from? Was it just a bit of incorrect original research? Like this person came up with what they thought was an alternate definition, just stuck it on Wikipedia like you're not supposed to do, and then it turned out not to be correct? Probably! But it would have been quite interesting had it actually turned out to be copied from somewhere...

There's a few other notable things here. First off, I'm flattered that someone thought I was the person to ask about ordinal arithmetic! I mean, I can't disagree that I'm a good person to ask, I've spent quite a bit of time examining ordinal arithmetic, not just the ordinary operations and the natural operations and some intermediate operations (that's the paper that caught his attention), but also weirder ones... have I never discussed [the operation that, having invented, I have named] Nim-Jacobsthal multiplication here?? Huh, I'll have to write about sometime. (No idea when I'd ever actually write that up for publication... it's somewhere near the bottom of the pile...) Still, I'm not who I would expect someone to go to first for such a thing as the semi-outsider that I am, even if I am a good choice!

The other thing is that I didn't notice this change myself. This page is actually on my watchlist, you see; the thing is that I haven't actually regularly checked my Wikipedia watchlist in years. And, wow, it looks like I haven't written about this here either.

Basically, some years ago, I got into a stupid edit war over Allan Lichtman's "Keys to the White House". I say it was a "stupid" edit war because it wasn't over the validity of the system or anything (which is obviously terrible!) -- it was purely about phrasing. At some point I just got too worked up about it and couldn't continue further, and, well, it led to me just not really checking my watchlist anymore.

Recently, though, I went and finally removed that page from my watchlist (after reading an article about how the "The Keys to the White House" isn't just bad, it's actually dishonest -- Lichtman goes back and changes things after the fact so that he appears to be always right) so I can start checking it again without encountering that. So far, well, I haven't actually done that. But maybe I will!

(And maybe I'll write sometime here about Nim-Jacobsthal multiplication, but if I do, you'll see why it's near the bottom of the pile publication-wise. :P But I guess that means a blog post is more suited to it...)

This was my entry in this year's ACX book review contest. Since it's not one of the finalists, I'll repost it here. Apologies for the uncharacteristic use of fancy Unicode characters -- I originally wrote this in a Google Doc as per the contest rules and I don't care to re-edit it. :P (Also, yeah, the "book review" format is why there's the "I recommend this book" section at the end as I try to wrap up... not something I'd normally include!)

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The Globe: How the Earth Became Round, by James Hannam

We all know that the world is round; people have known this for thousands of years. Why, even the ancient Greeks knew the world was round; Eratosthenes famously computed its circumference. But – naïvely – it looks flat. Presumably, people didn’t always know the world was round, and at some point they figured it out. Who figured it out? How did the idea spread? Who knew, when? “The Globe: How the Earth Became Round” gives us an answer to this question, and it’s a disorienting one.

The Greeks didn’t always know the world was round; they had to figure it out. But they did figure it out. So, given how important astronomy was to so many ancient civilizations, you might reasonably guess that other advanced civilizations of the time – the Persians, the Indians – figured it out too.

James Hannam tells us that no, they didn’t. Sure, they found out – but they all learned it, directly or indirectly, from the Greeks. According to Hannam, the idea that the world is round was discovered precisely once. (OK, maybe not by one person, but only by one civilization.)

In the case of, say, Persia, this perhaps isn’t too surprising; they had a lot of contact with Greece, so it makes sense that they’d learn it by transmission before they ever solved the problem themselves. But what about China? They had plenty of time to figure it out for themselves before Greek learning on the matter was brought to them. Well, two millennia later, when the Europeans showed up to bring the news… even their best astronomers still had no idea.

The world is round?

Now, there’s three things we might mean when we say “the world is round” – the Earth is circular; the Earth is spherical; or the universe is spherical. (Or the universe is circular, but that doesn’t seem to have been ever much believed; or, I suppose, that the universe is a 3-sphere, but that way of thinking wasn’t available to the ancients. These days we’d say that the universe is overall flat!)

Obviously, it’s the middle one of these that we care about here, because it’s the one that’s true, and the one that took work to determine. The idea that the Earth is circular was commonplace among cultures that believed the Earth to be flat – and so was the idea that the universe is spherical. Sorting all this out in the sources can be a bit tricky – these can all be described as “the world is round”, and not only in English, so statements are frequently ambiguous!

But it is possible to sort out a history with enough reading, and this book attempts to present that history; a history of who thought it was round when, and of many of the ways in which people thought it was flat.

The ancient Greeks were the ones who first figured out the globe. There’s not really one discoverer among the Greeks, but Hannam gives the credit to Aristotle for properly putting the pieces together; saying not only that the Earth is round but also how we can know this, and, importantly, stating explicitly the idea that on a round Earth, “down” is not a fixed direction in space, but rather towards the Earth’s center, thereby explaining why the oceans don’t drain off and such (a point that would confuse flat-earthers throughout the ages).

From Greece the idea of the Globe spread to Rome, to India, to Persia; to the Christians, the Muslims, the Jews; to Europe; and eventually to China. The book has chapters (sometimes more than one) on all of these, as well as some others I left out.

The book talks about opposition to the idea of the globe as well as support for it. Like, prior to the ninth century, Jews were largely skeptical of the idea of a round Earth because of the idea’s Greek origin. And yes, the book does discuss modern flat-earthism at the end.

But if you want a full rundown, go read the book. I’m not here to recap how the idea from ancient Greece outward; I’m here to discuss what in that history is particularly surprising, and the implications.

The Chinese case gets worse

So – until the Jesuits started showing up in the 1500s, hoping to trade superior knowledge of astronomy for permission to preach freely, Chinese astronomers were not aware that the earth was round.

Now, I say “Chinese astronomers”, but I should be clearer about what I mean by that. Because back in the late 13th century, when Khublai Khan conquered China, he set up a second astronomy bureau in his capital, one that practiced Islamic astronomy, as needed for computing the Islamic calendar. The Muslims, of course, had long known that the Earth is round; they’d learned it from the Greeks, and also from the Indians who’d learned it from the Greeks.

Which means, yes, that there were people in China who knew that the Earth was round. But the Chinese astronomers and the Islamic astronomers – both set up in the Chinese capital – apparently didn’t talk to each other, because 300 years later the Chinese astronomers still held to a flat Earth!

An unsettling counterfactual

All this raises the question: what if the ancient Greeks hadn’t figured it out?

It’s easy to say that if they hadn’t learned it from the Greeks first, India and Persia would have figured it out themselves. Or the Muslims, later. They all had perfectly capable astronomers, after all. But… so did China. And they didn’t.

What about Europe? In the very worst case, Europe would have to have figured it out by the time of Kepler at the latest, surely? I mean, one might say that the idea of a round Earth is a prerequisite to the ideas of Copernicus, Brahe, and Kepler – and I imagine not having it would have slowed things down some. But given that the scientific community of Europe (and not just a few isolated figures) were discovering heliocentrism and related ideas at this time, this to me suggests that there was a general capability there, that certainly they would have figured out the globe as part of the astronomical revolution regardless.

But whether they actually would have is impossible to know. And of course – the astronomical revolution is still a pretty late time to figure all this out! That’s almost as late as the idea spreading to China. Would they have figured it out earlier? One would think so, but again, the example of China suggests perhaps not.

Perhaps Europeans would have figured it out from all the long-distance sailing they were doing? That’s before the Copernican revolution, but only by about a century or so. (And even if sailors had known, how long would the idea have taken to spread to astronomers?) But other people, including the Chinese, were already doing long-distance sailing by then, including the use of stellar navigation, and Zheng He’s voyages recorded plenty of evidence for the globe, and yet China didn’t figure it out; was European long-distance sailing different enough that it would have mattered?

I’ve gotten pretty used to the idea of multiple discovery being the norm, of ideas coming up when their time is ripe. New things are figured out because the groundwork has been laid, the measuring instruments have been improved, new evidence has been coming in, and so forth. The astronomical revolution in Europe is a fine example – invent the telescope and the rest seems inevitable.

The idea I’ve most often seen as an example of the opposite is general relativity; people often say that if it hadn't been for Einstein, it wouldn’t have been figured out for another decade or longer. (Which is kind of odd, actually, given that Einstein wasn't the only one at the time developing a geometric theory of gravity. I would have thought that the idea of a geometrical theory of gravity, rather than the specific field equation, would have been the hard part, but I suppose not?)

I'm used to considering the idea of the globe as so basic that of course it must have been multiply discovered; after all, the Greeks knew it! And indeed, within Greece – among that particular milieu – it does seem to have been something of a gradual (if not multiple) discovery. Certainly it’s nowhere near as non-obvious as General Relativity! But perhaps we should regard it to be pretty high on the scale of non-obviousness, perhaps approaching that of heliocentrism? That doesn’t sound right to me. And yet, the lack of an Aristotle delayed the Chinese discovery of the globe not by a mere decade, but rather two millenia. What was so special about ancient Greece, that the time was ripe there but not elsewhere?

More ancient Greek speculation

It really does seem that ancient Greece had more speculation than other places about non-flat shapes for the Earth. Everywhere had cosmological speculation, but in most places all of this elaborated on the idea of a flat Earth rather than replacing it.

For instance, the very first person to suggest the idea of a spherical Earth may have been the Greek philosopher Philolaus; his surviving work doesn’t explicitly state this but seems to suggest it. The notable thing here is that Philolaus is not a geocentrist; he has the Earth, and the other planets, orbiting the “universal fire” or “universal hearth”. So… he’s a heliocentrist two millenia early? Nope! The Sun is distinct from the universal fire, and also orbits it. Huh.

Meanwhile, about a hundred years before Philolaus, a Greek philosopher named Archelaus correctly reasoned that the Earth can’t be flat due to the difference in sunrise and sunset times as one travels East and West… but his conclusion was that the Earth was shaped like a concave bowl, which gets backwards how those times should vary. Seems he had the basic idea, but failed to think it through! Still, ultimately it was ancient Greece that was thinking about this and ancient Greece that got it.

Where does the Sun go at night?

There is, as I mentioned earlier, a fair amount of discussion of just what sorts of flat-earth cosmologies various ancient peoples believed in. For instance, lots of civilizations thought that the Sun goes underneath the Earth at night, but others said, how can it go under the Earth, it’s solid underneath there! The Talmud, for instance, says that at night the Sun travels from West to East not underneath the Earth, but rather around its northern edge. I would suggest reading the book for more of this nature.

But once again it’s the case of China that’s eyebrow-raising. Let’s consider – one of the classical pieces of evidence that the Earth is round is that as you travel North and South, you see different stars; some stars become hidden behind the Earth, which is only possible because of its curvature.

Chinese astronomers were aware of this phenomenon, but they had a different explanation. Their explanation, which they also used to explain where the Sun goes at night was – and I swear I am not making this up – that the universe had a draw distance. If you get too far from something, its light simply can’t reach you. The book doesn’t use the term “draw distance”, but that’s basically what it’s describing!

Notice, by the way, how that explanation requires the Sun and stars to be quite close! “The Gnomon of the Zhou”, from around the year 20 CE, gave the distance to the heavenly canopy as 43,000 kilometers, and the “draw distance” as 90,000 kilometers. Which doesn’t seem to make sense offhand, because 90,000 is more than twice 43,000, so that would mean the Sun would be always visible. Hm, well, “The Gnomon of the Zhou” isn’t a very consistent text in the first place, and may not be reflective of what later astronomers thought.

It’s possible that I’m portraying things uncharitably here, and that actually Chinese astronomers said that the light becomes continuously attenuated over distance, only eventually fading out entirely, but the book sure makes it sound like a draw distance. Certainly “The Gnomon of the Zhou” does, although, again, that may not be the best text to rely on.

Either way this explanation is baffling, because when the Sun sets it clearly vanishes a bit at a time, rather than uniformly fading out (with the attenuation explanation) or vanishing suddenly (with the draw distance explanation). Apparently the common people generally said that the Sun becomes hidden behind a mountain at night – a fairly common belief among flat-Earth-believing cultures (who often further identified this mountain with the mountain believed to exist at the world’s center), and to my mind a more sensible one – but Chinese astronomers said otherwise.

I wish the book had gone into more detail on the question of how well this explanation worked for explaining the vanishing stars upon North/South travel; how well does this explanation actually match the data? How big is the anomaly, and were people worried about it? Did Chinese scholars of optics have opinions on this mechanism? Unfortunately, the book answers none of these questions.

For that matter, which way is North?

“The Gnomon of the Zhou” contains more surprising assertions I wish the book had discussed further. I mentioned above that most civilizations that believed in a flat earth believed that the world was a circle; in China, however, it was believed to be a square. Now if you believe in a flat Earth this raises the question of, where is the center of the world, something many civilizations had opinions about. In China, the center was generally believed to be the capital city, although which capital city varied (earlier Dengfeng, later Kaifeng).

“The Gnomon of the Zhou”, however, has it different. It says that the center of the world is at the North pole. And that’s a statement that I really have to wonder what it means.

As mentioned, the Chinese believed that the Earth was a square; moreover, it seemed they considered it to be an “axis-aligned” square, with sides running North-South and sides running “East-West”.

If the “North pole” means the northernmost point, then already we’ve got a problem; such a square Earth has a northern edge, not a northernmost point. But even if we ignore that, the northernmost point sure as hell isn’t at the center!

I really have to wonder how whoever wrote that was interpreting the ideas of “North”, “South”, “East”, and “West”. My general understanding was that in China at that time these were understood as fixed directions in space, with the axis-aligned picture I mentioned earlier. For the North pole to be at the center, that would instead suggest a view where North is not a fixed direction, but rather “towards the center” – much as how on a round Earth, “down” is not a fixed direction but rather towards the center. But if the authors were thinking this way, it doesn’t seem to have led them to the idea of a round Earth. Which isn’t surprising; that’s quite a leap.

This point, of what “The Gnomon of the Zhou” meant by this claim, is something the book really needed to discuss more, I’d say. I don’t know how possible it is to answer this question, especially as that particular work just isn’t all that consistent in the first place; unfortunately “The Globe” never mentions the question at all.

I recommend this book

I’d recommend reading “The Globe”. I’ve covered what I consider to be the highlights, but the book discusses a lot more. It does have some notable omissions as mentioned above. Also, a bit of a warning about the prose – when it goes off its main topic, it sometimes reads like a high-schooler’s essay (although never in the endnotes, so I guess this was a deliberate stylistic choice?). This mostly goes away once you get past the first few chapters, though, and the main topic comes more into focus.

Of course really the main thing I’ve taken from this book is that unsettling counterfactual that it never explicitly asks – what if the ancient Greeks hadn’t discovered the Earth to be round? How long would it have taken India, Persia, Europe, to all get it? This question is for sure going to haunt me. I guess you don’t really need to go read the book for it to haunt you too! If you want your speculation on the matter to be a bit more informed, though, probably this is a good place to start.