Ten years ago, or so, somebody gave me a copy of Neal Stephenson's novel
Anathem, as I'd previously read and enjoyed
Snow Crash and
Cryptonomicon.
Snow Crash, as I recall (I don't see it on the shelf now), was a normal-sized paperback novel, while
Cryptonomicon was a (let's see...) 910-page hardback (not counting technical appendix) involving half a dozen story lines in different time periods, all somewhat absurdist-surreal in a Catch-22 sort of way.
Anathem promised to resemble the latter, at least in size (890 pages, not counting several appendices).
I dove into
Anathem, read several chapters, and got bogged down. Came back to it a few months later, read a few more chapters, and got bogged down. Came back to it a year later, read a few more chapters, and got bogged down. Came back to it a year later, starting from the beginning to get a running start, and got bogged down. The issue seemed to be that Stephenson had spent an enormous amount of effort on world-building and forgotten about plot: nothing ever happened. And there was a remarkable amount of philosophical argument.
For whatever reason, last month I came back to it, started from the beginning to get a running start, and read a few pages whenever I was waiting for a long compile during my work day. And actually
finished it this time. In fact, a great deal happens in the last, say, 200 pages, which feel sorta like an ordinary SF novel in their own right, but nothing that happens in those last 200 pages would make sense without the world-building and philosophical argument.
So let's see what I can summarize without excessive spoilerization. The book is set in a world quite similar to ours, and from the first page you can tell that people speak and think in a Romance-based language that's not quite any of the ones we know. It's an old culture: it has written history going back at least 5000 years, at a level of detail comparable to what we have for the past 500 years.
And it's a culture in which academics have consciously (and not entirely voluntarily) divorced themselves from popular culture, living sequestered in what are effectively monasteries. To maintain their perspective and avoid being polluted and confused by transient fads, many of the (co-ed) monastic/academics live under a rule that they can only leave the monastery, or even
hear news from outside the monastery, once a year. To maintain even more perspective, other monastics live under a stricter ten-year rule: no information from the outside can get in except at ten-year intervals. An even more elite group live under a hundred-year rule, and the really hard-core ones live under a thousand-year rule; they've only opened their gates twice since the monastic rules were set up 3000 years ago. (The monastics aren't celibate, but they're sterile due to their diet; an unanswered mystery, which bothers the protagonist from time to time, is how the Hundreders and Thousanders stay populated.) There are even rumors of a sect living under a 10,000-year rule, which has another 7,000 years to go before they'll open their gates.
So some of the book explores the resulting sense of detachment. For example, the monastics and the outside world experience linguistic drift, so the longer they've been apart, the less they can understand one another. The monastics of different sects can mostly understand one another because they're all trained from the same books in Classical Orth, the analogue of Classical Latin. And when they
do observe the outside world, it's all "the city had gone through cycles of growth and shrinkage over the millennia, and was currently in a shrinking phase, as shown by such-and-such characteristics typical of shrinking cities," or "the current governmental institution was dominated by a religious belief system with properties X and Y, but not Z," all properties identified in other religious belief systems over the millennia. The monastics have observed how the outside world thinks about them, and have categorized such beliefs into a dozen or more named Iconographies, each of which recurs every few hundred years (
e.g. the monastics are a bunch of mostly-harmless mental patients, the monastics are secretly developing technology that will bring about a Golden Age when they finally share it with the rest of us, the monastics are a bunch of malevolent magicians scheming to take over the world, the monastics are a bunch of malevolent magicians who have already taken over the world but are keeping it secret,
etc.) In short, the monastics view the current outside world as anthropologists might view an isolated tribe, comparing and contrasting it with other isolated tribes while trying to avoid getting thrown in a stew-pot.
Another thread of the book starts with a philosophical question which (in our world) is called Mathematical Realism. Mathematicians studying a topic like (say) prime numbers generally feel as though the concept of prime numbers was
inevitable -- that
any culture, or even any
species, with sufficient cognitive powers would eventually come up with the same concept, and prove the same theorems about them. Indeed, a great deal of mathematics has this feel of inevitability to it: we're not
inventing new things, but
discovering things that in some sense were already out there waiting to be discovered; when we prove a theorem, we now know not only that it's true, but that it
couldn't have been false in any conceivable world. A mathematics developed by small furry creature from the planet Zrax would have isomorphic conceptual structures to our own, differing only in names. But where is "out there"? Is it a world of Platonic ideals, made up of isosceles triangles and automorphism groups in the same way our world is made up of photons, electrons, protons, etc.? And how do we
know this stuff is inevitable, as opposed to being artifacts of our own culture or historical accident?
This doesn't happen only in number theory or group theory. In theoretical computer science, people point out that the problem classes Rec [recursive], r.e. [recursively enumerable], and P [polynomial-time-computable] can each be defined in at least half a dozen simple, elegant, apparently unrelated ways that all turn out to be exactly equivalent, and conclude from this robustness that they're "natural" or "real" classes. Discoveries, not inventions. Which makes people more motivated to study them: it feels more rewarding to study something that "really exists" than to study a historical contingency that might have worked out any of a hundred slightly different ways instead.
An alternative view could be called Mathematical Formalism: this dispenses with the question of whether these mathematical concepts are "real", and says "we've agreed to manipulate symbols according to such-and-such rules; what results do we get by following those rules? What if we changed the rules slightly; what results would we get then?" This approach is summarized in the Bertrand Russell quote
"Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." We "don't know what we are talking about" because there
is no "about": we're just shuffling symbols according to arbitrary rules. And "we don't know whether it's true" because a statement's "truth" is a property not of the symbols but of a particular world (while in a different world, the same statement isn't true).
In our world, mostly starting in the 20th century, logicians have made serious progress reconciling these views. We can study a set of axioms and their implications under certain syntactical proof rules (we call this "proof theory"), and (following Tarski
et al) we can also talk about concrete worlds that "model" those axioms and proof rules in the sense that, when you interpret each formal symbol as a particular object in or assertion about the world, the axioms all turn out to be true and the proof rules preserve truth ("model theory"). Logicians explored the difference between formal provability and semantic validity, showing that in certain simple systems they can be made to coincide perfectly ("Completeness"), while for complex systems of arithmetic they can't (to oversimplify Gödel's famous Incompleteness Theorems). For example, one of the tasks my thesis advisor set me (and at which I never entirely succeeded) was to take a particular result that a colleague had proven using model theory, and see if I could reach the same or a similar conclusion using proof theory instead. There's a branch of logic called "modal logic" which talks about not whether things
are true but whether they
could be or
must be true; logicians for the past few centuries had dismissed this as meaningless mysticism, but it was given a precise proof system in the 1930's, and a precise semantics in the 1960's; see
this overview article.
In the world of
Anathem, the realist and formalist schools have been feuding for thousands of years, the formalists accusing the realists of quasi-religious mysticism and the realists accusing the formalists of pointless mental masturbation. The realists hypothesize that yes, there
really is a world (or "causal domain") of Platonic ideals from which we get all our mathematical ideas, which means that information somehow flows from that abstract world to our concrete one. But once you've postulated two worlds with a unidirectional information flow, a natural question (to any mathematician) is "why only two?" Could there be
several "concrete" worlds, independent of one another but all getting their ideas from the same abstract world? Could there be
several abstract worlds, independent of one another but all contributing ideas to our world? Could there be an even-more-abstract world from which the Platonic world gets its "ideas"? Could there be an even-more-concrete world that gets its "ideas" from ours? Could there be an arbitrary DAG (directed acyclic graph) of worlds connected by information flow? And why does it have to be acyclic: could there be information flow in both directions? (Although if there's information flow in both directions between two worlds, they've effectively collapsed into one causal domain.) The formalists, of course, say this is all mystical nonsense. Until they meet visitors from one of these other worlds... but that's getting into spoiler territory.
A seemingly-unrelated philosophical thread in the book is about epistemology within a single concrete world. When you see a frying pan, or a friend, how do you know it's "the same" frying pan or friend as it was yesterday, or as it was when you looked at it from the other side? There's an argument from simplicity -- Occam's Razor (or "Gardan's Steelyard" in the world of
Anathem) -- that it's simpler to hypothesize one persistent object than to hypothesize distinct objects that coincidentally have a lot of observable properties in common. But those observable properties aren't identical; they vary, but only in certain
predictable ways. If I move clockwise around the frying pan, I expect to stop seeing what was on its right side, and see additional things adjacent to what was on its left side, and things fall down but not up, and people get older but not younger, and so on. All this requires building a complex model of how the physical world works, which implies a lot of conclusions about "what
would happen
if ...": if I looked at the frying pan from the other side, if the table the pan is on weren't present, if my friend were wearing different clothes,
etc. In other words, you can't make sense of even a single concrete world without a lot of counterfactual reasoning -- or in other words, reasoning about worlds that are extremely similar to this one, but differ in one particular way. The realists of
Anathem further hypothesize that this ability of the human mind is implemented using quantum computation: that rather than one classical brain exploring all these counterfactual worlds classically, it actually explores
all of them at once in a superposition of quantum states, or in parallel universes, or something. And we're back to talking about multiple worlds -- except that now the information flow among them (
via quantum interference), while limited, is symmetrical. Each version of your brain directly perceives the universe that it's in, as well as picking up some quantum interference from versions of itself in other nearby universes, and uses this to form counterfactual models and thus to reason about its own world.
So now that I've
finally finished the book, I have some things to think about. And some logic to reread.