hudebnik

I went to grad school to study math and computer science, but I happened to be at a school (UCSD) with a top-notch experimental-music department, so I took a year's worth of electronic music and a year's worth of computer music. In those courses, there was an occasional discussion of alternative scales: rather than the 12 equal intervals that make up the modern Western-culture octave, what if you used a different number, or if you used unequal intervals, or if you didn't base things on an octave at all? In particular, somebody pointed out that a 19-step equal-tempered scale gives pretty nice-sounding intervals; why don't we use that?

"Nice-sounding", or "pure", is shorthand for "small-integer frequency ratios", which the human ear seems to like. A unison is two exactly-equal frequencies; if you play two notes that are almost in tune, say 440 Hz and 441 Hz, you can hear a "wow-wow-wow" at the frequency of the difference, in this case 1 beat per second; guitar players routinely use this to confirm that two strings are really in tune with one another.
A pure octave is a 2:1 frequency ratio, e.g. 440 Hz and 880 Hz. A pure fifth is ideally a 3:2 frequency ratio, a pure fourth a 4:3 frequency ratio, a pure major third a 5:4 frequency ratio, and a pure minor third a 6:5 frequency ratio.

Pythagorean tuning, which as its name suggests is thousands of years old, is based on fifths and octaves: go up a pure fifth, then another pure fifth, and you're a little over an octave from where you started, so go down a pure octave, and repeat. Unfortunately, if you tune everything by pure fifths and octaves, you end up almost but not quite where you started: 12 pure fifths is a frequency ratio of 129.746, while 7 pure octaves takes you back down by a ratio of 128, leaving you at a frequency 1.36% (the "Pythagorean comma") higher than where you started. This is enough to be audible, even to the un-trained musician: all the intervals sound beautifully in tune except this one real clunker. Which means when you tune an instrument, you have to decide in advance which intervals you want to sound good, and which one can be horribly out of tune; in practice, you decide what keys you're most likely to play in, and pick an interval that's unlikely to show up in those keys.

Another problem with Pythagorean tuning is that getting all the fifths right gives you terrible thirds: a Pythagorean major third is a ratio of 1.266, over a percent "wider" than the ideal 1.25, while a Pythagorean minor third is 1.185, over a percent "narrower" than the ideal 1.2.

People have tried various ways to solve the problem: split the comma in thirds, so there are a bunch of beautiful intervals and three less-bad ones ("third-comma meantone"), or split it in fourths ("quarter-comma meantone"), or split it in sixths ("sixth-comma meantone"), or split it in twelfths so all the intervals are equally slightly out of tune. That's "equal temperament", the way most pianos are tuned today. In equal temperament, a fifth is a frequency ratio of 1.4983, about 0.1% narrower than a Pythagorean fifth; a fourth is likewise about 0.1% wider than a Pythagorean fourth; a major third is 1.26, almost a percent too wide; and a minor third is 1.19, almost a percent too narrow.

In the 15th-17th centuries there was a different approach, called "just tuning": rather than insisting on fifths über alles, maybe we should aim for getting thirds in tune. Combining a pure major third (5:4) and a pure minor third (6:5) actually gives you a 3:2 pure fifth, so you haven't lost that after all, and the thirds sound much better than in Pythagorean tuning. But tuning the thirds still doesn't help you get all the way around the scale and end up back where you started; some intervals will be beautifully in tune, while others are horrible, and you just try to place the horrible ones in out-of-the-way corners of the keys you usually play in.

By the way, modern singers, and players of un-fretted stringed instruments like violins, are taught to use just intonation, adjusting vocal pitch or finger position slightly so that each third, fourth, or fifth is in tune when you get to it, even though it may not be in tune with where the piece started. Since most musical pieces don't modulate very far from their original key, you can usually get away with this and end up in the same key you started in, but theoretically if you played on the violin a long piece that modulated all the way around the circle of fifths, you'd end up off by a Pythagorean comma from where you started; in particular, the notes you played on open strings at the beginning would have to be fretted at the end.

I was in an early-music workshop on Friday, and the teacher mentioned the "Berkeley manuscript", a 1370's-vintage music theory text currently owned by UC Berkeley. We had a copy (in facing-page translation) at home, so I've been reading through it in the last few days. In the first section the author describes the hexachord (or deducciones -- the word "hexachord" was invented retroactively by 19th-century musicologists) system, how the C, G, and F hexachords overlap, how and why one switches from one hexachord to another, how intervals that should be a whole tone in the gamut can be narrowed to a semitone by explicitly marking one note "mi" or "fa" (what we would now call "sharp" or "flat"), etc.

But in the fifth section of the treatise (which I haven't gotten to yet; this is what the teacher and the editor say) the author says a mi-fa semitone really shouldn't be half a tone, but two-thirds of a tone (a "major semitone"; the remaining third is a "minor semitone"). If each of the five whole tones in an octave is three units, and each of the two semitones is two units, that makes... 19!

So how well does this actually work, using modern mathematical techniques (roots and exponents)?

• A major third, ut-mi or fa-la, is two whole tones, or 6/19 of an octave, which is a ratio of 1.24469, less than half a percent off from a just major third at 1.25 -- about equally bad as a third-comma-meantone major third, although narrow where that was wide.


• A minor third, re-fa or mi-sol, is a whole tone and a major semitone, or 5/19 of an octave, which is a ratio of 1/2001, less than a hundredth of a percent off from a just minor third.


• A fourth, ut-fa or re-sol or mi-la, is two whole tones and a major semitone, 8/19 of an octave, a ratio of 1.3389, about half a percent off from a Pythagorean fourth.


• Similarly, a fifth, ut-sol or re-la, is three whole tones and a major semitone, or 11/19 of an octave, a ratio of 1.49376, again about half a percent off. (This had to be true, since the perfect fourth and perfect fifth are one another's inverses: if one was half a percent off, the other would have to be likewise.)

So this system has reasonably-good fifths, fourths, and major thirds, and very good minor thirds, and it has the conceptual advantage of equal temperament that you don't need to decide which key you want to sound best: any piece of music can be transposed into any other key and its intervals will sound the same as they did before.

If you wanted to build a keyboard in this system, it would probably have the usual seven white keys (A, B, ... G), with two black keys between A and B, C and D, D and E, F and G, and G and A, and one black key between B and C and between E and F.

Yesterday's Supreme Court decision in Allen v. Milligan, somewhat surprisingly, doesn't further eviscerate the Voting Rights Act of 1965 (as amended in 1982): by a 6-3 margin, the Court agreed with a Federal appeals court that the Alabama map of Congressional districts wrongly denied its 27% black population a fair voice in electing Representatives. I haven't read the details of the appeals-court or Supreme court decision, but the case appears to involve the usual "packing and cracking": putting most of the state's black voters in one district, which almost always elects a black Democrat, so that the remaining six districts have a white Republican majority and almost always elect white Republicans.

Justice Clarence Thomas, in dissent, complains that the Court cannot force a state to use race as a criterion in drawing district boundaries; racism is racism, even in the interest of a minority. Of course, his criticism on this point is one-sided and hypocritical: he seldom objects when a state uses race as a criterion in favor of a white majority, as long as they're not so foolish as to put it in writing. But what if the state didn't use race as a criterion in favor of a white majority? Would he have a legitimate point?

What would a truly race-blind redistricting system look like? It could use not geography, but something with no correlation with race or political party, such as your Social Security number or your birthday (as a number from 1-366). If your Social Security number, divided by 7, has a remainder of 1, you're in district 1; a remainder of 2, district 2; and so on. Absolutely and utterly race-blind, and disastrous for minority representation, because every district in the state would be 27% black; considering the importance of race in Alabama politics, it would be rare to ever see a black representative elected at all.

In Justice Thomas's dream world, the end of overt racism by government would lead to race becoming irrelevant to politics, people would win elections on individual merit, unaffected by race, and in the long run the percentage of black elected officials might resemble the percentage of blacks in the population. But in the real world, that could be a very long run indeed, and it's not clear that Alabama society would actually move in that direction at all; in the interim, black Alabamans would have even less representation than they have now.

The fundamental problem actually isn't about race; it applies to race, political party, sexual orientation and identification, and any other characteristic (call it X) that voters see as politically important. Single-seat, winner-take-all elections amplify majorities. If 60% of voters have characteristic X, an X-blind electoral map will generally give at least 60%, but usually far more than 60%, of the seats to candidates with characteristic X.

You can try to fix this with X-based redistricting, ensuring that only about 60% of the districts have pro-X majorities, but that has several problems. First, it leads to weirdly shaped, unnatural districts, such as the one that gave "gerrymandering" its name. Second, it requires government to explicitly consider X, and to discriminate on the basis of X, in its official proceedings, and that feels wrong, even if saying so puts me in agreement with Clarence Thomas (which also feels wrong). Third, it intentionally involves creating districts in which the general election is a foregone conclusion, and all the real action happens in primaries, which are dominated by more-extreme members of the respective parties. And fourth, what if political decisions depend not only on X, but on unrelated characteristic Y? Do we need to draw districts representing every possible combination of X and Y values?

Some democracies have tried to resolve this using proportional representation by party: if Party Z gets 30% of the votes, Party Z gets roughly 30% of the seats in a legislature (or one house of a legislature, or something). Unfortunately, this (a) fails to recognize any politically-important characteristic that isn't (yet) associated with a political party, and (b) gives political parties an official status in the (state or Federal) Constitution, whereas I would prefer to burn political parties to the ground.

A better answer is a voting system that tends to produce proportional representation (in a state's Congressional delegation or its state legislature), not only by race or party but by whatever characteristic voters feel is important. This cannot happen as long as each district elects a single winner independently; you need either a way for votes in one district to affect outcomes in another, or (more plausibly) multi-seat districts. And you need a way for each voter to express a preference, not only for a single candidate, but for a whole class of candidates who have characteristics that voter considers important (race, party, sexual orientation/identity, musical talent, business acumen, etc; the voter doesn't need to say or even consciously know what the characteristics are). That could in principle mean voters rank the candidates from first to last, or assign each candidate an independent numeric score from 1 to 10, or check off all the candidates they "approve" of, but you definitely need more than a single name per voter.

There are a lot of voting systems out there, each designed to solve a different problem and placing different priorities on the desirable properties of a voting system. I know of one designed specifically to achieve proportional representation, without pre-judging what characteristics voters will find important: Single Transferrable Vote, which is reminiscent of Instant Runoff but applied to multi-seat districts. Each voter ranks all the candidates in the district, and the system first looks at all the first-place votes. Any candidate with at least a threshold number of first-place votes is elected, and the ballots that listed that candidate in first place are removed from the pool, those voters having had their say. (There are variants involving redistributing the "surplus votes" over the threshold.) Of the remaining candidates, some with especially small numbers of first-place votes are eliminated, and we look at the second-place candidates on the ballots that chose them first. This continues in multiple rounds until all the seats are filled.

The system has many of the same problems as IRV: it pays more attention to whom you vote for than whom you vote against, thus favoring divisive candidates over consensus candidates; it's sensitive to "butterfly effects", in that a handful of votes changing hands between minor candidates can have a cascading effect on major candidates; and it's non-parallelizable, in that you can't do any substantial part of the counting work at the precinct level, you have to do everything globally. And it's even more complicated, difficult to explain, and difficult to justify than IRV. Among other things, there's no single obviously-right way to choose the thresholds for "elected" and "eliminated", nor any single obviously-right way to decide which surplus votes to redistribute. But it does a very good job of producing proportional representation, if there are enough seats per district (ideally, a single district for the whole state).

Is there a way to achieve something like proportional representation, but with a simpler, easier-to-understand, easier-to-justify voting system? I'm a big fan of the Borda count for single-seat elections; maybe that could be adapted? In the Borda system, if you vote for 7 candidates, your first choice gets 7 points, your second 6 points, and so on. So if there are 7 seats to fill, it seems natural to give them to the candidates with the top 7 point totals. The result won't be as nearly proportional as STV would produce, but you would expect a coherent, disciplined minority party to be able to elect a few top candidates over lower-ranked candidates of the majority party. But I don't think that works. I'm still working through the algebra, but it looks as though a majority party can fill all the seats, no matter what minority parties do, simply by telling its voters to rank the majority-party candidates randomly and ahead of all the other-party candidates. Furthermore, the majority party's advantage increases with the addition of minor, long-shot candidates -- not because they take votes away from either "real" party, but because they decrease the point ratio between a party's top-ranked and last-ranked candidates, and whenever this ratio is less than the ratio in party allegiance among voters, the majority party wins.

You could perhaps fix the above problem by doing a Borda count in which the number of candidates you can vote for (and thus the maximum number of points you give any one of them) is substantially less than the number of seats to fill. But that defeats one of the appeals of the Borda system: the fact that you can meaningfully downvote a candidate by ranking it last, and downvotes matter just as much as upvotes. I think Borda just doesn't work for multi-seat elections.

Left Coopers’ Lake at 1:40 PM yesterday. First stop, pizza in Grove City. Stopped for the night near Delaware Water Gap, which on the way out had been a bit over four hours from home (really bad traffic). Had breakfast at an apparent local favorite spot, then drove home, arriving a little before noon (no significant traffic this time). Backed the trailer into the garage, unloaded the car, unloaded the trailer, re-loaded the trailer with empty chests and things that live there year-round, walked to the farm market for sweet corn, zucchini, etc. D. started a couple of loads of laundry. Retrieved hounds from boarding kennel half a mile away. Took rapid-read Covid tests: both negative.

I managed to get all the way through Pennsic without sunburn, but driving home (even though I was in the driver's seat most of the way, and therefore on the north side of the car) sunburned my hands to the "itchy" level.

At some point during Pennsic, D. picked up a class handout somebody had left behind, on Ptolemaic mathematics, so I read that when I needed a break from unloading and unpacking. Some of it is basic stuff about converting between decimal and sexagesimal, some is standard 9th-grade geometry involving similar triangles, and then they get into spherical geometry, which is much trickier, particularly when trigonometry hasn’t been invented yet. The workhorse tool seems to be something called Menelaus’s Theorem, which specifies a proportion among several arcs in a figure formed by the intersection of four great circles… except it isn’t really a proportion among the arcs, but rather among their chords. And even that isn’t quite right: it’s actually the chords of the double angles of each of these arcs. How did anybody come up with that? I guess the “double angles” part is a transformation between an angle whose vertex is on the opposite side of the sphere and one whose vertex is the center of the sphere, and the “chords” part allows us to work with nice simple planar triangles. Anyway, one can use this theorem to calculate the angle between equator and ecliptic at various times of year (at the equinoxes it’s zero), and thence the positions of sun, moon, and planets.

Then I picked up a book that’s been sitting on the coffee table since Christmas: Seb Falk’s The Light Ages: the Surprising Story of Medieval Science. And a good deal of the first chapter is about the same sort of astronomy: if you live in one place for years, you can observe the northernmost and southernmost places the sun rises and sets, observe the lengths of noonday shadows at various times of the year, and thence derive things like your latitude and the Earth’s axial tilt, which go into the aforementioned Ptolemaic calculations. Of course, everything is much easier if you assume the Earth is a sphere, and its orbit and those of the moon and all the planets are circles. I wonder how much more difficult it would be on a planet with a significantly eccentric orbit. Among other things, the warmest and coldest parts of the year might not correlate closely with the highest and lowest noonday suns, which would change the societal motivation for doing astronomy. And the obvious existence of a non-perfect circle in Creation would have interesting theological implications.

Last Monday we gave a concert in the performing arts tent, entitled “An Evening at the Salle des Ardents: Medieval Smoky Jazz”. All circa-1400 music. There’s a recording, which I need to retrieve from the recorder and edit on the desktop machine before posting it for public consumption. The opening number was a little chaotic, so I may substitute in a recording of it from a rehearsal, but most of the pieces went well and I think are suitable for posting. [Edit: see here. There are a number of wince-worthy moments in the concert, but a bunch of good moments too.]

To do before next Pennsic:

• Buy and/or build a new tent.
  

  Roof panels are largely sewn together, but that's where they've been for several years; no significant progress in Covidtime. Then need to cut and sew together wall pieces, and build or commission poles, and make a bunch of rope attachment points and stake loops, and so on. shalmestere points out that we're much more likely to have a new tent by next Pennsic if we order it from a tentsmith.



• Finish building new "birdcage" music stand.
  

  Parts are mostly cut out, but they need to be stuck together, and we need an upright and a foot.



• Rebuild cooler chest.
  

  The ends are fine, but the lid and sides are permanently warped, so it never actually closes, which defeats its purpose of hiding a styrofoam cooler; in addition, one of the lid's reinforcing battens which had been loose for years finally fell off completely this Pennsic. Also want to attach handles to the ends, which would make it enormously easier to carry.



• Lose enough belly to fit into my red linen pourpoint.
  

  shalmestere has a photo of me in it from eleven years ago today, but when I tried it on at Pennsic, it was nowhere near closing.



• Make more braes and shirts


• Repair existing braes and shirts


• Make more points and aiglets

To do in the next week or two:

• Repair the drinking jug I knocked off the top of the piano while putting away krummhorns from Pennsic ✓


• Wash and put away clothes and dishes from Pennsic ✓


• Put away stuff that was left lying around from Pennsic packing ✓


• Replenish groceries ✓


• Cook normal food ✓


• Edit and post concert audio ✓


• Hem, and attach a linen collar-lining to, the gown that shalmestere made me two weeks ago from one of her old dresses


• Add eyelets to the tailed hose shalmestere made me a few weeks ago


• Finish building the fleece harp-case we started several months ago


• Contact harp-maker to say "yes, we're home from vacation now; feel free to ship the double-strung harp we commissioned two years ago" ✓


• Get Rid Of Stuff
  

  I'd like to set us the challenge that every day we throw/give away one item that's been in our possession, but unused, for years. I figure after a month or two of that, we might notice a difference.

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.

So last night we were both in a mood to watch something not-particularly-challenging, and on a whim Turned On the Cable Box (which happens every few months in our house) and started flipping through the hundreds of channels to see if anything looked appealing. We landed on a series of back-to-back episodes of "Futurama", which we usually see only when we're staying in a motel, which hasn't happened in almost a year because covid.

One episode stands out in my mind because the resolution to the plot problems rests on a theorem of permutation theory. The professor has built a mind-swapping device, and two characters swap minds, then have second thoughts and want to swap back, but the way the device works, once a pair of minds have been swapped, that exact pair can never be swapped again. But all is not lost: they can swap through a third party: (a,b) = (a,c)(c,b)(a,c). No, wait: that repeats the (a,c) pair; maybe a fourth party? Before they can explore that possibility, the third party runs off in mid-sequence, and various other swaps happen for other motivations... eventually (IIRC) eight different characters are all in the wrong bodies, and would all like to get back home. Is this even possible?

In the penultimate scene, two black NBA players (why? because it's Futurama, of course) show up, do a bunch of figuring on a blackboard (which they still have in Futurama), and announce that no matter how screwed-up the permutation of minds and bodies, it's possible to get back to the original state with at most two "extra players". Which they do, in a dizzying montage of swaps that leaves everybody in the right body, and they all live happily ever after.

OK, so the theorem is "for any sequence of swaps on a set of n objects, there's an additional sequence of swaps (on the original set plus at most two additional objects) that inverts the original sequence, without the same swap ever appearing more than once in the combined sequence." Why is this true, and why two additional objects -- one would have expected either one or infinitely many? And how does the number of swaps grow? I guess it can't grow more than quadratically in n because there are only C(n+2,2) = (n+2)(n+1)/2 possible swaps on n+2 objects without repeating.

The n=2 case from above: (a,b) can be inverted by (a,c)(b,d)(a,d)(b,c) without violating the no-repeats rule. And it really can't be done with only one additional "player", as one can see by exhaustive search: there are only 3 possible swaps on {a,b,c}, and neither (a,c), (b,c), (a,c)(b,c), nor (b,c)(a,c) is equivalent to (a,b). (Side question: how big is this "exhaustive search"? There are only C(n+1,2) = n(n+1)/2 possible swaps, but then you have to look at all permutations of at most this many swaps... how many permutations are there of at most m objects, where m=C(n+1,2)? Let F(m) = 1 + m + m(m-1) + m(m-1)(m-2) + ... + m!; then F(0)=1, F(1)=2, F(2)=5, F(3)=16, F(4)=65, F(5)=306.... nothing obvious except that it grows at least as fast as m!. I should know this, or be able to solve it; I've taken combinatorics, thirty years ago. I'll get back to this.)

It'll be difficult to do the underlying theorem by induction because the no-repeats rule imposes a global constraint that doesn't lend itself to self-reduction. But maybe it's simpler than that: just grab the last swap in the sequence that needs to be reversed, label it (a,b), label the "extra players" c and d, do (a,c)(b,d)(a,d)(b,c) as above, and iterate. We know that nothing involving c or d was in the original sequence, so that doesn't violate the no-repeats rule, but it's not obvious that nothing involving a or b has already been used in the "undoing" sequence. We know that (a,b) doesn't appear twice in the original sequence, and if that's the only occurrence of either a or b in the original sequence, the above works, but it's likely that (a,*) and (b,*) each appears more than once in the original sequence, which could use up our allotted (a,c), (a,d), (b,c), and (b,d) swaps. I'll get back to this.

I was sitting in the front row at some kind of math colloquium, where a woman was describing something called "abstract pairs" of matrices. (Even in the dream, I thought this was a lousy name, as there was nothing abstract about them.) The defining property had to do with the sums of various 2x2 sub-matrices, and she demonstrated how, if this property held, any 2x2 submatrix that appeared in one quadrant of the matrices must also appear in the other three quadrants, or something like that. I asked a question to clarify: "so, if I understand this, abstract pairs are really rare, but if you've got one, you only need to know what's in one quadrant and that tells you everything?" The speaker replied "No, not quite," but before she could explain why not, a woman in the back row had a question of her own, which she introduced by saying "I was the one who replied to your paper in the Bratislava Proceedings with 'I love you!'" And I don't remember what her question was. And then I woke up.

Why I'm having dreams about a math colloquium in the first place, much less one about this particular class of matrices, I don't know....

shalmestere brought home an advance reader's copy of The Math Myth (to be published in March 2016), thinking I might be interested. The main point of the book is that American schools, from middle school through college, require students to take a lot of math courses that they will never use, that do not make them better thinkers, that do not increase their earning ability or their competence as citizens of a democracy, but that do serve to filter out and discourage a lot of kids from reaching their potential in non-mathematical fields.

Which is almost certainly true. I've actually had occasion to use a little bit of calculus on the job (in doing analysis of algorithms), and in answering idle-curiosity physics problems, and I've used trigonometry to design tents, and I've used both trigonometry and linear algebra to write graphics programs, but 99.99% of the U.S. population will never need to do any of those things, either on the job or in private life. Most people need to be able to do arithmetic (with the aid of a calculator, but they need enough of a feel for numbers that they can tell whether the answers are at all plausible), and read a graph, and it would be nice if they knew that correlation isn't causation, and what statistical significance means. As a logician, I'd like it if ordinary citizens knew that "not all cats are grey" is equivalent to "at least one cat is not grey", and that "if that's a duck, then I'm Henry Ford" is not equivalent to "if that's not a duck, then I'm not Henry Ford".

Anyway, the author documents vast numbers of students whose only academic problem is an inability to pass such irrelevant math classes (middle school, high school, or college), but who are denied the opportunity to study Shakespeare or Swahili or spot-welding. He points out that most of the doomsaying about an imminent shortage of STEM-qualified workers comes from employers who have a strong vested interest in creating an overabundance of such workers. And he likes to illustrate things with sample test questions.

Here's a question he likes:

A rectangular-shaped fuel tank measures 27-1/2 inches in length, 3/4 of a foot in width, and 8-1/4 inches in depth. How many gallons will the tank contain? (231 cubic inches = 1 gallon)

(a) 7.366 gallons
(b) 8.839 gallons
(c) 170,156 gallons

He likes this because it tests "did you read the question carefully?" -- specifically, did you convert 3/4 of a foot into 9 inches -- and do you know what needs to be multiplied and what divided? He says if you failed to notice the "feet", you would get the incorrect answer (a) (in fact, he's misplaced a decimal point: you would get .7366 gallons, which you should also be able to rule out through common sense because a tank that big has got to hold more than a gallon!)

Here's a question he doesn't like:

Two charges (+q and -q) each with mass 9.11 x 10³¹ kg, are place 0.5 m apart and the gravitational force (F_g) and electric force (F_e) are measured. If the ratio F_g/F_e is 1.12 x 10^-77, what is the new ratio if the distance between the charges is halved?

(a) 2.24 x 10^-77
(b) 1.12 x 10^-77
(c) 5.6 x 10^-78
(d) 2.8 x 10^-78

I have to confess I do like this question, because it doesn't require doing any arithmetic at all, only remembering that both gravitational and electrical forces follow an inverse-square law. Although if you have two masses of almost 10³² kg half a meter apart from one another, they're both black holes and you have bigger things to worry about than measuring the forces between them.

However, I don't see a lot of benefit in asking this question on an MCAT (which is where it allegedly came from). Yes, identifying the relevant and irrelevant features of a problem is important to a doctor, but physics isn't.

For a certain board game, two dice are thrown to determine the number of spaces to move. One player throws the two dice and the same number comes up on each of the dice. What is the probability that the sum of the two numbers is 9?
(a) 0
(b) 1/6
(c) 2/9
(d) 1/2
(e) 1/3

Again, he doesn't like this question, and I do: it requires no arithmetic, no probability, no combinatorics, only the ability to see past the irrelevant stuff to what matters.

Is this what they call a "trick" question? One that requires common-sense reasoning, not just the application of a memorized procedure? If so, I'm all for them.

Anyway, I've only read a quarter of the book; we'll see what else he has to say.