A quick reading recommendation


I’ve been reading The Disordered Cosmos: A Journey Into Dark Matter, Spacetime, and Dreams Deferred, by Chanda Prescod-Weinstein. It’s the best science book I’ve read in a long while.

Part of it is a pop-science discussion of particle physics and cosmology, as they’re now understood. It may seem strange that the tiniest things and the biggest thing are such natural companion subjects. That is what seems to make sense, though. I’ve fallen out of touch with a lot of particle physics since my undergraduate days and it’s wonderful to have it discussed well. This sort of pop physics is for me a pleasant comfort read.

The other part of the book is more memoir, and discussion of the culture of science. This is all discomfort reading. It’s an important discomfort.

I discuss sometimes how mathematics is, pretensions aside, a culturally-determined thing. Usually this is in the context of how, for example, that we have questions about “perfect numbers” is plausibly an idiosyncrasy. I don’t talk much about the culture of working mathematicians. In large part this is because I’m not a working mathematician, and don’t have close contact with working mathematicians. And then even if I did — well, I’m a tall, skinny white guy. I could step into most any college’s mathematics or physics department, sit down in a seminar, and be accepted as belonging there. People will assume that if I say anything, it’s worth listening to.

Chanda Prescod-Weinstein, a Black Jewish agender woman, does not get similar consideration. This despite her much greater merit. And, like, I was aware that women have it harder than men. And Black people have it harder than white people. And that being open about any but heterosexual cisgender inclinations is making one’s own life harder. What I hadn’t paid attention to was how much harder, and how relentlessly harder. Most every chapter, including the comfortable easy ones talking about families of quarks and all, is several needed slaps to my complacent face.

Her focus is on science, particularly physics. It’s not as though mathematics is innocent of driving women out or ignoring them when it can’t. Or of treating Black people with similar hostility. Much of what’s wrong is passively accepting patterns of not thinking about whether mathematics is open to everyone who wants in. Prescod-Weinstein offers many thoughts and many difficult thoughts. They are worth listening to.

Reading the Comics, July 30, 2016: Learning Tools Edition


I thank Comic Strip Master Command for the steady pace of mathematically-themed comics this past week. It fit quite nicely with my schedule, which you might get hints about in weeks to come. Depends what I remember to write about. I did have to search a while for any unifying motif of this set. The idea of stuff you use to help learn turned up several times over, and that will do.

Steve Breen and Mike Thompson’s Grand Avenue threatened on the 24th to resume my least-liked part of reading comics for mathematics themes. This would be Grandma’s habit of forcing the kids to spend their last month of summer vacation doing arithmetic drills. I won’t say that computing numbers isn’t fun because I know what it’s like to work out how many seconds are in 50 years in your head. But that’s never what this sort of drill is about. The strip’s diverted from that subject, but it might come back to spoil the end of summer vacation. (I’m not positive what my least-liked part of the comics overall is. I suspect it might be the weird anti-participation-trophy bias comic strip writers have.)

Ryan North’s Dinosaur Comics reprint for the 25th is about the end of the universe. We’ve got several competing theories about how the universe is likely to turn out, several trillion years down the road. The difference between them is in the shape of space and how that shape is changing. I’ve mentioned sometimes the wonder of being able to tell something about a whole shape from local information, things we can tell without being far from a single point. The fate of the universe must be the greatest example of this. Considering how large the universe is and how little of it we will ever be able to send an instrument to, we measure the shape of space from a single point. And we can realistically project what will happen in unimaginably distant times. Admittedly, if we get it wrong, we’ll never know, which takes off some of the edge.

Dinosaur Comics reappears the 28th with some talk about number bases. It’s all fine and accurate enough, except for the suggestion that anyone would use base five for something other than explaining how bases work. I like learning about bases. When I was a kid this concept explained much to me about how our symbols for numbers work. It also helped appreciate that symbols are not these fixed or universal things. They’re our creations and ours to adapt for whatever reason we find convenient. In the past we’ve found bases as high as sixty to be convenient. (The division of angles into 360 degrees each of 60 minutes, each of those of 60 seconds, is an echo of that.) But when I was a kid doing alternate-base problems nobody knew what I was doing or why, except the mathematics teacher who said I might like the optional sections in the book. We only really need base ten, base two, and base sixteen, which might as well be base two written more compactly. The rest are toys, good for instruction and for fun. Sorry, base seven.

Scott Meyer’s Basic Instructions rerun for the 27th is about everyone’s favorite bit of intransitivity. Rock-Paper-Scissors and its related games are all about systems in which any two results can be decisive but any three might not be. This prospect turns up whenever there are three or more possible outcomes. And it doesn’t require a system to be irrational or random. Chaos and counterintuitive results just happen when there’s three of a thing.

I remember, and possibly you remember too, learning of a computer system that can consistently beat humans at Rock-Paper-Scissors. It manages to do that by the oldest of game theory exploits, cheating. Its sensors look for the twitches suggesting what a person is going to throw and then it changes its throw to beat that. I don’t know what that’s supposed to prove since anyone who’s played a Sid Meier’s Civilization game knows that computers already know how to cheat.

Thom Bluemel’s Birdbrains, yes, you can be in my Reading The Comics post this week too. Don’t beg.

Bill Schorr’s The Grizzwells for the 28th is a resisted word problem joke. It doesn’t use the classic railroad or airplane forms, but it’s the same joke anyway.

'My drive to solve this [ a pie chart of the relationships between current, voltage, resistance, and power in an electrical circuit ] keeps getting interrupted by my desire to order a PIZZA'.
Benita Epstein’s Six Chix for the 29th of July, 2016. The pie chart’s valid, in case you need it, in which case you’re doing the mathematics of electric circuits. Current, voltage, resistance, and power all relate to one another in ways the chart makes clear once you know how to read it. Each of the quantities — I, V, R, or P — is equal to each of the expressions outside it. Pizza, meanwhile, is just a naturally funny word and thus appears in comic strips to amuse you.

Benita Epstein’s Six Chix for the 29th is probably familiar to the folks taking electronics. The chart is a compact map used as a mnemonic for the different relationships between the current (I), the voltage (V), the resistance (R), and the power (P) in a circuit. When I was a student we got this as two separate circles, one for current-voltage-resistance and one for power-current-voltage. Each was laid out like the T-and-O maps which pre-Renaissance Western Europe used to diagram the world. While I now see that as a convenient and useful tool, as a student, I was skeptical that it was any easier to use the mnemonic aid than it was to just remember “voltage equals current times resistance” and “power equals voltage times current”. I’ve always had an irrational suspicion of mnemonic devices. I’m trying to do better.

Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 30th is a return of the whiteboard full of symbols to represent deep thinking. The symbols don’t mean anything as equations, though that might be my limited perspective. And that also might represent the sketchy, shorthand way serious work is done. As an idea is sketched out weird bundles of symbols that don’t literally parse do appear. In a publishable paper this is all turned into neatly formatted and standard stuff. Or we introduce symbols with clear explanations of what they mean so that others can learn to read what we write. But for ourselves, in the heat of work, we’ll produce what looks like gibberish to others and that’s all right as long as we don’t forget what the gibberish means. Sometimes we do, but the gibberish typically helps us recapture a lost idea. (I offer the tale of a mathematician with pages of notes for a brilliant insight which she has to reconstruct from a lost memory to would-be short story writers looking for a Romantic hook.)

Original Problem! Expanding Galaxies and Rates of Change


Afiq Hatta here presents a nice little problem that mixes geometry and calculus. And it’s inspired by cosmology, to cover an extra subject.

The Arthur Christmas Problem


Since it’s the season for it I’d like to point new or new-wish readers to a couple of posts I did in 2012-13, based on the Aardman Animation film Arthur Christmas, which was just so very charming in every way. It also puts forth some good mathematical and mathematical-physics questions.

Opening the scene is “Could `Arthur Christmas’ Happen In Real Life?” which begins with a scene in the movie: Arthur and Grand-Santa are stranded on a Caribbean island while the reindeer and sleigh, without them, go flying off in a straight line. This raises the question of what is a straight line if you’re on the surface of something spherical like the Earth.

“Returning To Arthur Christmas” was titled that because I’d left the subject for a couple weeks, as is my wont, and it gets a little bit more spoiler-y since the film seems to come down on the side of the reindeer moving on a path called a Great Circle. This forces us to ask another question: if the reindeer are moving around the Earth, are they moving with the Earth’s rotation, like an airplane does, or freely of it, like a satellite does?

“Arthur Christmas And The Least Common Multiple” starts by supposing that the reindeer are moving the way satellites do, independent of the Earth’s rotation, and on making some assumptions about the speed of the reindeer and the path they’re taking, works out how long Arthur and Grand-Santa would need to wait before the reindeer and sled are back if they’re lucky enough to be waiting on the equator.

“Six Minutes Off” shifts matters a little, by supposing that they’re not on the equator, which makes meeting up the reindeer a much nastier bit of timing. If they’re willing to wait long enough the reindeer will come as close as they want to their position, but the wait can be impractically long, for example, eight years, or over five thousand years, which would really slow down the movie.

And finally “Arthur Christmas and the End of Time” wraps up matters with a bit of heady speculation about recurrence: the way that a physical system can, if the proper conditions are met, come back either to its starting point or to a condition arbitrarily close to its starting point, if you wait long enough. This offers some dazzling ideas about the really, really long-term fate of the universe, which is always a heady thought. I hope you enjoy.

Reading the Comics, September 8, 2014: What Is The Problem Edition


Must be the start of school or something. In today’s roundup of mathematically-themed comics there are a couple of strips that I think touch on the question of defining just what the problem is: what are you trying to measure, what are you trying to calculate, what are the rules of this sort of calculation? That’s a lot of what’s really interesting about mathematics, which is how I’m able to say something about a rerun Archie comic. It’s not easy work but that’s why I get that big math-blogger paycheck.

Edison Lee works out the shape of the universe, and as ever in this sort of thing, he forgot to carry a number.
I’d have thought the universe to be at least three-dimensional.

John Hambrock’s The Brilliant Mind of Edison Lee (September 2) talks about the shape of the universe. Measuring the world, or the universe, is certainly one of the older influences on mathematical thought. From a handful of observations and some careful reasoning, for example, one can understand how large the Earth is, and how far away the Moon and the Sun must be, without going past the kinds of reasoning or calculations that a middle school student would probably be able to follow.

There is something deeper to consider about the shape of space, though: the geometry of the universe affects what things can happen in them, and can even be seen in the kinds of physics that happen. A famous, and astounding, result by the mathematical physicist Emmy Noether shows that symmetries in space correspond to conservation laws. That the universe is, apparently, rotationally symmetric — everything would look the same if the whole universe were picked up and rotated (say) 80 degrees along one axis — means that there is such a thing as the conservation of angular momentum. That the universe is time-symmetric — the universe would look the same if it had got started five hours later (please pretend that’s a statement that can have any coherent meaning) — means that energy is conserved. And so on. It may seem, superficially, like a cosmologist is engaged in some almost ancient-Greek-style abstract reasoning to wonder what shapes the universe could have and what it does, but (putting aside that it gets hard to divide mathematics, physics, and philosophy in this kind of field) we can imagine observable, testable consequences of the answer.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 5) tells a joke starting with “two perfectly rational perfectly informed individuals walk into a bar”, along the way to a joke about economists. The idea of “perfectly rational perfectly informed” people is part of the mathematical modeling that’s become a popular strain of economic thought in recent decades. It’s a model, and like many models, is properly speaking wrong, but it allows one to describe interesting behavior — in this case, how people will make decisions — without complications you either can’t handle or aren’t interested in. The joke goes on to the idea that one can assign costs and benefits to continuing in the joke. The idea that one can quantify preferences and pleasures and happiness I think of as being made concrete by Jeremy Bentham and the utilitarian philosophers, although trying to find ways to measure things has been a streak in Western thought for close to a thousand years now, and rather fruitfully so. But I wouldn’t have much to do with protagonists who can’t stay around through the whole joke either.

Marc Anderson’s Andertoons (September 6) was probably composed in the spirit of joking, but it does hit something that I understand baffles kids learning it every year: that subtracting a negative number does the same thing as adding a positive number. To be fair to kids who need a couple months to feel quite confident in what they’re doing, mathematicians needed a couple generations to get the hang of it too. We have now a pretty sound set of rules for how to work with negative numbers, that’s nice and logically tested and very successful at representing things we want to know, but there seems to be a strong intuition that says “subtracting a negative three” and “adding a positive three” might just be different somehow, and we won’t really know negative numbers until that sense of something being awry is resolved.

Andertoons pops up again the next day (September 7) with a completely different drawing of a chalkboard and this time a scientist and a rabbit standing in front of it. The rabbit’s shown to be able to do more than multiply and, indeed, the mathematics is correct. Cosines and sines have a rather famous link to exponentiation and to imaginary- and complex-valued numbers, and it can be useful to change an ordinary cosine or sine into this exponentiation of a complex-valued number. Why? Mostly, because exponentiation tends to be pretty nice, analytically: you can multiply and divide terms pretty easily, you can take derivatives and integrals almost effortlessly, and then if you need a cosine or a sine you can get that out at the end again. It’s a good trick to know how to do.

Jeff Harris’s Shortcuts children’s activity panel (September 9) is a page of stuff about “Geometry”, and it’s got some nice facts (some mathematical, some historical), and a fair bunch of puzzles about the field.

Morrie Turner’s Wee Pals (September 7, perhaps a rerun; Turner died several months ago, though I don’t know how far ahead of publication he was working) features a word problem in terms of jellybeans that underlines the danger of unwarranted assumptions in this sort of problem-phrasing.

Moose has trouble working out 15 percent of $8.95; Jughead explains why.
How far back is this rerun from if Moose got lunch for two for $8.95?

Craig Boldman and Henry Scarpelli’s Archie (September 8, rerun) goes back to one of arithmetic’s traditional comic strip applications, that of working out the tip. Poor Moose is driving himself crazy trying to work out 15 percent of $8.95, probably from a quiz-inspired fear that if he doesn’t get it correct to the penny he’s completely wrong. Being able to do a calculation precisely is useful, certainly, but he’s forgetting that in tis real-world application he gets some flexibility in what has to be calculated. He’d save some effort if he realized the tip for $8.95 is probably close enough to the tip for $9.00 that he could afford the difference, most obviously, and (if his budget allows) that he could just as well work out one-sixth the bill instead of fifteen percent, and give up that workload in exchange for sixteen cents.

Mark Parisi’s Off The Mark (September 8) is another entry into the world of anthropomorphized numbers, so you can probably imagine just what π has to say here.

In A Really Big Universe


I’d got to thinking idly about Olbers’ Paradox, the classic question of why the night sky is dark. It’s named for Heinrich Wilhelm Olbers, 1758-1840, who of course was not the first person to pose the problem nor to give a convincing answer to it, but, that’s the way naming rights go.

It doesn’t sound like much of a question at first, after all, it’s night. But if we suppose the universe is infinitely large and is infinitely old, then, along the path of any direction you look in the sky, day or night, there’ll be a star. The star may be very far away, so that it’s very faint; but it takes up much less of the sky from being so far away. The result is that the star’s intensity, as a function of how much of the sky it takes up, is no smaller. And there’ll be stars shining just as intensely in directions that are near to that first star. The sky in an infinitely large, infinitely old universe should be a wall of stars.

Oh, some stars will be dimmer than average, and some brighter, but that doesn’t matter much. We can suppose the average star is of average brightness and average size for reasons that are right there in the name of the thing; it makes the reasoning a little simpler and doesn’t change the result.

The reason there is darkness is that our universe is neither infinitely large nor infinitely old. There aren’t enough stars to fill the sky and there’s not enough time for the light from all of them to get to us.

But we can still imagine standing on a planet in such an Olbers Universe (to save myself writing “infinitely large and infinitely old” too many times), with enough vastness and enough time to have a night sky that looks like a shell of starlight, and that’s what I was pondering. What might we see if you looked at the sky, in these conditions?

Well, light, obviously; we can imagine the sky looking as bright as the sun, but in all directions above the horizon. The sun takes up a very tiny piece of the sky — it’s about as wide across as your thumb, held at arm’s length, and try it if you don’t believe me (better, try it with the Moon, which is about the same size as the Sun and easier to look at) — so, multiply that brightness by the difference between your thumb and the sky and imagine the investment in sunglasses this requires.

It’s worse than that, though. Yes, in any direction you look there’ll be a star, but if you imagine going on in that direction there’ll be another star, eventually. And another one past that, and another past that yet. And the light — the energy — of those stars shining doesn’t disappear because there’s a star between it and the viewer. The heat will just go into warming up the stars in its path and get radiated through.

This is why interstellar dust, or planets, or other non-radiating bodies doesn’t answer why the sky could be dark in a vast enough universe. Anything that gets enough heat put into it will start to glow and start to shine from that light. The stars will slow down the waves of heat from the stars behind them, but given enough time, it will get through, and in an infinitely old universe, there is enough time.

The conclusion, then, is that our planet in an Olbers Universe would get an infinite amount of heat pouring onto it, at all times. It’s hard to see how life could possibly exist in the circumstance; water would boil away — rock would boil away — and the planet just would evaporate into dust.

Things get worse, though: it’s not just our planet that would get boiled away like this, but as far as I can tell, the stars too. Each star would be getting an infinite amount of heat pouring into it. It seems to me this requires the matter making up the stars to get so hot it would boil away, just as the atmosphere and water and surface of the imagined planet would, until the star — until all stars — disintegrate. At this point I have to think of the great super-science space-opera writers of the early 20th century, listening to the description of a wave of heat that boils away a star, and sniffing, “Amateurs. Come back when you can boil a galaxy instead”. Well, the galaxy would boil too, for the same reasons.

Even once the stars have managed to destroy themselves, though, the remaining atoms would still have a temperature, and would still radiate faint light. And that faint light, multiplied by the infinitely many atoms and all the time they have, would still accumulate to an infinitely great heat. I don’t know how hot you have to get to boil a proton into nothingness — or a quark — but if there is any temperature that does it, it’d be able to.

So the result, I had to conclude, is that an infinitely large, infinitely old universe could exist only if it didn’t have anything in it, or at least if it had nothing that wasn’t at absolute zero in it. This seems like a pretty dismal result and left me looking pretty moody for a while, even I was sure that EE “Doc” Smith would smile at me for working out the heat-death of quarks.

Of course, there’s no reason that a universe has to, or even should, be pleasing to imagine. And there is a little thread of hope for life, or at least existence, in a Olbers Universe.

All the destruction-of-everything comes about from the infinitely large number of stars, or other radiating bodies, in the universe. If there’s only finitely much matter in the universe, then, their total energy doesn’t have to add up to the point of self-destruction. This means giving up an assumption that was slipped into my Olbers Universe without anyone noticing: the idea that it’s about uniformly distributed. If you compare any two volumes of equal size, from any time, they have about the same number of stars in them. This is known in cosmology as “isotropy”.

Our universe seems to have this isotropy. Oh, there are spots where you can find many stars (like the center of a galaxy) and spots where there are few (like, the space in-between galaxies), but the galaxies themselves seem to be pretty uniformly distributed.

But an imagined universe doesn’t have to have this property. If we suppose an Olbers Universe without then we can have stars and planets and maybe even life. It could even have many times the mass, the number of stars and everything, that our universe has, spread across something much bigger than our universe. But it does mean that this infinitely large, infinitely old universe will have all its matter clumped together into some section, and nearly all the space — in a universe with an incredible amount of space — will be empty.

I suppose that’s better than a universe with nothing at all, but somehow only a little better. Even though it could be a universe with more stars and more space occupied than our universe has, that infinitely vast emptiness still haunts me.

(I’d like to note, by the way, that all this universe-building and reasoning hasn’t required any equations or anything like that. One could argue this has diverted from mathematics and cosmology into philosophy, and I wouldn’t dispute that, but can imagine philosophers might.)