I’m sorry not to be able to offer more about his mathematical work. If someone knows of a mathematics-history podcast with a similar goal, please leave a comment. I’d love to know and to share with other people.
I continue to share things I’ve heard, rather than created. Peter Adamson’s podcast The History Of Philosophy Without Any Gaps this week had an episode about Nicholas of Cusa. There’s another episode on him scheduled for two weeks from now.
Nicholas is one of those many polymaths of the not-quite-modern era. Someone who worked in philosophy, theology, astronomy, mathematics, with a side in calendar reform. He’s noteworthy in mathematics and theology and philosophy for trying to understand the infinite and the infinitesimal. Adamson’s podcast — about a half-hour — focuses on the philosophical and theological sides of things. But the mathematics can’t help creeping in, with questions like, how can you tell the difference between a straight line and the edge of a circle with infinitely large diameter? Or between a circle and a regular polygon with infinitely many sides?
The BBC’s In Our Time program, and podcast, did a 50-minute chat about the longitude problem. That’s the question of how to find one’s position, east or west of some reference point. It’s an iconic story of pop science and, I’ll admit, I’d think anyone likely to read my blog already knows the rough outline of the story. But you never know what people don’t know. And even if you do know, it’s often enjoyable to hear the story told a different way.
The mathematics content of the longitude problem is real, although it’s not discussed more than in passing during the chat. The core insight Western mapmakers used is that the difference between local (sun) time and a reference point’s time tells you how far east or west you are of that reference point. So then the question becomes how you know what your reference point’s time is.
This story, as it’s often told in pop science treatments, tends to focus on the brilliant clockmaker John Harrison, and the podcast does a fair bit of this. Harrison spent his life building a series of ever-more-precise clocks. These could keep London time on ships sailing around the world. (Or at least to the Caribbean, where the most profitable, slavery-driven, British interests were.) But he also spent decades fighting with the authorities he expected to reward him for his work. It makes for an almost classic narrative of lone genius versus the establishment.
But, and I’m glad the podcast discussion comes around to this, the reality more ambiguous than this. (Actual history is always more ambiguous than whatever you think.) Part of the goal of the goal of the British (and other powers) was finding a practical way for any ship to find longitude. Granted Harrison could build an advanced, ingenious clock more accurate than anyone else could. Could he build the hundreds, or thousands, of those clocks that British shipping needed? Could anyone?
And the competing methods for finding longitude were based on astronomy and calculation. The moment when, say, the Moon passes in front of Jupiter is the same for everyone on Earth. (At least for the accuracy needed here.) It can, in principle, be forecast years, even decades ahead of time. So why not print up books listing astronomical events for the next five years and the formulas to turn observations into longitudes? Books are easy to print. You already train your navigators in astronomy so that they can find latitude. (This by how far above the horizon the pole star, or the sun, or another identifiable feature is.) And, incidentally, you gain a way of computing longitude that you don’t lose if your clock breaks. I appreciated having some of that perspective shown.
(The problem of longitude on land gets briefly addressed. The same principles that work at sea work on land. And land offers some secondary checks. For an unmentioned example there’s triangulation. It’s a great process, and a compelling use of trigonometry. I may do a piece about that myself sometime.)
Also a thing I somehow did not realize: British English pronounces “longitude” with a hard G sound. Huh.
I have another mathematics-themed podcast to share. It’s again from the BBC’s In Our Time, a 50-minute program in which three experts discuss a topic. Here they came back around to mathematics and physics. And along the way chemistry and mensuration. The topic here was Pierre-Simon Laplace, who’s one of those people whose name you learn well as a mathematics or physics major. He doesn’t quite reach the levels of Euler — who does? — but he’s up there.
Laplace might be best known for his work in celestial mechanics. He (independently of Immanuel Kant) developed the nebular hypothesis, that the solar system formed from the contraction of a great cloud of dust. We today accept a modified version of this. And for studying the question of whether the solar system is stable. That is, whether the perturbations every planet has on one another average out to nothing, or to something catastrophic. And studying probability, which has more to do with these questions than one might imagine. And then there’s general mechanics, and differential equations, and if that weren’t enough, his role in establishing the Metric system. This and more gets discussion.
This is not the whole of her work, though my understanding is she’d be worth noticing even if it were. Part of the greatness of the translation was putting Newton’s mathematics — which he had done as geometric demonstrations — into the calculus of the day. The experts on In Our Time’s podcast argue that she did a good bit of work advancing the state of calculus in doing this. She’d also done a good bit of work on the problem of colliding bodies.
A major controversy was, in modern terms, whether momentum and kinetic energy are different things and, if they are different, which one collisions preserve. Châtelet worked on experiments — inspired by ideas of Gottfried Wilhelm Liebniz — to show kinetic energy was its own thing and was the important part of collisions. We today understand both momentum and energy are conserved, but we have the advantage of her work and the people influenced by her work to draw on.
She’s also renowned for a paper about the nature and propagation of fire, submitted anonymously for the Académie des Sciences’s 1737 Grand Prix. It didn’t win — Leonhard Euler’s did — but her paper and her lover Voltaire’s papers were published.
Châtelet was also surprisingly connected to the nascent mathematics and physics scene of the time. She had ongoing mathematical discussions with Pierre-Louis Maupertuis, of the principle of least action; Alexis Clairaut, who calculated the return of Halley’s Comet; Samuel König, author of a theorem relating systems of particles to their center of mass; and Bernard de Fontenelle, perpetual secretary of the Acadeémie des Sciences.
So for those interested in the history of mathematics and physics, and of women who are able to break through social restrictions to do good work, the podcast is worth a listen.
I spent much of the time waiting for a mention of Chatelier’s principle which never came. This because Chatelier’s principle’s — about the tendency of a system in equilibrium to resist changes — is named for Henry Louis Le Chatelier, a late 19th/early 20th century chemist with, so far as I know, no relation to Émile du Châtelet. I hope this spares you the confusion I felt.
I’m very slightly sorry to bump other things. But folks who like the history of mathematics, and how it links to other things, and who also like listening to stuff, might want to know. Peter Adamson, host of the History Of Philosophy Without Any Gaps podcast, this week talked for about twenty minutes about Girolamo Cardano.
Cardano is famous in mathematics circles for early work in probability. And, more, for pioneering the use of imaginary numbers. This along the way to a fantastic controversy about credit, and discovery, and secrets, and self-promotion.
Cardano was, as Adamson notes, a polymath; his day job was as a physician and he poked around in the philosophy of mind. That’s what makes him a fit subject for Adamson’s project. So if you’d like a different perspective on a person known, if vaguely, to many mathematics folks, and have a spot of time, you might enjoy.
The BBC’s general-discussion podcast In Our Time repeated another mathematics-themed session this week. The topic is P versus NP, a matter of computational complexity. P and NP here are shorthands to describe the amount of work needed to follow a procedure. And, particularly, how the amount of work changes as the size of the problem being worked on changes. We know there are problems of a complexity type P, and problems of a complexity type NP. What’s not known is whether those are, actually, the same, whether there’s a clever way of describing an NP problem so we can solve it with a P approach.
I do not remember whether I heard this program when it originally broadcast in 2015. And I haven’t had time to listen to this one yet. But these discussions are usually prett solid, and will get to discussing the ambiguities and limitations and qualifications of the field. So I feel comfortable recommending it even without a recent listen, which I will likely get to sometime during this week’s walks.
The panel this time is two philosophers and a mathematician, which is probably about the correct blend to get the topic down. The mathematician here is Marcus du Sautoy, with the University of Oxford, who’s a renowned mathematics popularizer in his own right. That said I think he falls into a trap that we STEM types often have in talking about Zeno, that of thinking the problem is merely “how can we talk about an infinity of something”. Or “how can we talk about an infinitesimal of something”. Mathematicians have got what seem to be a pretty good hold on how to do these calculations. But that we can provide a logically coherent way to talk about, say, how a line can be composed of points with no length does not tell us where the length of a line comes from. Still, du Sautoy knows rather a few things that I don’t. (The philosophers are Barbara Sattler, with the University of St Andrews, and James Warren, with the University of Cambridge. I know nothing further of either of them.)
The episode also discusses the Quantum Zeno Effect. This is physics, not mathematics, but it’s unsettling nonetheless. The time-evolution of certain systems can be stopped, or accelerated, by frequent measurements of the system. This is not something Zeno would have been pondering. But it is a challenge to our intuition about how change ought to work.
I’ve written some of my own thoughts about some of Zeno’s paradoxes, as well as on the Sorites paradox, which is discussed along the way in this episode. And the episode has prompted new thoughts in me, particularly about what it might mean to do infinitely many things. And what a “thing” might be. This is probably a topic Zeno was hoping listeners would ponder.
The program is three people, plus host Melvyn Bragg, talking about the life and work of Gauss. Gauss is one of those figures hard to exaggerate. He was extremely prolific and insightful. It is an exaggeration to say that he did foundational work in every field of mathematics, but only a slight exaggeration. (He compares to Leonhard Euler that way.) I’d imagine that anyone reading a pop mathematics blog knows something of Gauss. But you may learn something new, or a new perspective on something familiar.
This week the BBC podcast In Our Time, a not-quite-hourlong panel show discussing varied topics, came to Paul Dirac. It can be heard here, or from other podcast sources. I get it off iTunes myself. The discussion is partly about his career and about the magnitude of his work. It’s not going to make anyone suddenly understand how to do any of his groundbreaking work in quantum mechanics. But it is, after all, an hourlong podcast for the general audience about, in this case, a physicist. It couldn’t explain spinors.
And even if you know a fair bit about Dirac and his work you might pick up something new. This might be slight: one of the panelists mentioned Dirac, in retirement, getting to know Sting. This is not something impossible, but it’s also not a meeting I would have ever imagined happening. So my week has been broadened a bit.
One of the podcasts I regularly listen to is the BBC’s In Our Time. This is a roughly 50-minute chat, each week, about some topic of general interest. It’s broad in its subjects; they can be historical, cultural, scientific, artistic, and even sometimes mathematical.
Recently they repeated an episode about Emmy Noether. I knew, before, that she was one of the great figures in our modern understanding of physics. Noether’s Theorem tells us how the geometry of a physics problem constrains the physics we have, and in useful ways. That, for example, what we understand as the conservation of angular momentum results from a physical problem being rotationally symmetric. (That if we rotated everything about the problem by the same angle around the same axis, we’d not see any different behaviors.) Similarly, that you could start a physics scenario at any time, sooner or later, without changing the results forces the physics scenario to have a conservation of energy. This is a powerful and stunning way to connect physics and geometry.
What I had not appreciated until listening to this episode was her work in group theory, and in organizing it in the way we still learn the subject. This startled and embarrassed me. It forced me to realize I knew little about the history of group theory. Group theory has over the past two centuries been a key piece of mathematics. It’s given us results as basic as showing there are polynomials that no quadratic formula-type expression will ever solve. It’s given results as esoteric as predicting what kinds of exotic subatomic particles we should expect to exist. And her work’s led into the modern understanding of the fundamentals of mathematics. So it’s exciting to learn some more about this.
Their plan was to make more exciting the discussion of some of Deep Space Nine‘s episodes by recording their reviews while drinking a lot. The plan was, for the fifteen episodes they had in the season, there would be a one-in-fifteen chance of doing any particular episode drunk. So how many drunk episodes would you expect to get, on this basis?
It’s a well-formed expectation value problem. There could be as few as zero or as many as fifteen, but some cases are more likely than others. Each episode could be recorded drunk or not-drunk. There’s an equal chance of each episode being recorded drunk. Whether one episode is drunk or not doesn’t depend on whether the one before was, and doesn’t affect whether the next one is. (I’ll come back to this.)
The most likely case was for there to be one drunk episode. The probability of exactly one drunk episode was a little over 38%. No drunk episodes was also a likely outcome. There was a better than 35% chance it would never have turned up. The chance of exactly two drunk episodes was about 19%. There drunk episodes had a slightly less than 6% chance of happening. Four drunk episodes a slightly more than 1% chance of happening. And after that you get into the deeply unlikely cases.
As the Deep Space Nine season turned out, this one-in-fifteen chance came up twice. It turned out they sort of did three drunk episodes, though. One of the drunk episodes turned out to be the first of two they planned to record that day. I’m not sure why they didn’t just swap what episode they recorded first, but I trust they had logistical reasons. As often happens with probability questions, the independence of events — whether a success for one affects the outcome of another — changes calculations.
There’s not going to be a second-season update to this. They’ve chosen to make a more elaborate recording game of things. They’ve set up a modified Snakes and Ladders type board with a handful of spots marked for stunts. Some sound like fun, such as recording without taking any notes about the episode. Some are, yes, drinking episodes. But this is all a very different and more complicated thing to project. If I were going to tackle that it’d probably be by running a bunch of simulations and taking averages from that.
Also I trust they’ve been warned about the episode where Quark has a sex change so he can meet a top Ferengi soda magnate after accidentally giving his mother a heart attack because gads but that was a thing that happened somehow.
Among my entertainments is listening to the Greatest Generation podcast, hosted by Benjamin Ahr Harrison and Adam Pranica. They recently finished reviewing all the Star Trek: The Next Generation episodes, and have started Deep Space Nine. To add some fun and risk to episode podcasts the hosts proposed to record some episodes while drinking heavily. I am not a fun of recreational over-drinking, but I understand their feelings. There’s an episode where Quark has a sex-change operation because he gave his mother a heart attack right before a politically charged meeting with a leading Ferengi soda executive. Nobody should face that mess sober.
At the end of the episode reviewing “Babel”, Harrison proposed: there’s 15 episodes left in the season. Use a random number generator to pick a number from 1 to 15; if it’s one, they do the next episode (“Captive Pursuit”) drunk. And it was; what are the odds? One in fifteen. I just said.
The question: how many episodes would they be doing drunk? As they discussed in the next episode, this would imply they’d always get smashed for the last episode of the season. This is a straightforward expectation-value problem. The expectation value of a thing is the sum of all the possible outcomes times the chance of each outcome. Here, the possible outcome is adding 1 to the number of drunk episodes. The chance of any particular episode being a drunk episode is 1 divided by ‘N’, if ‘N’ is the number of episodes remaining. So the next-to-the-last episode has 1 chance in 2 of being drunk. The second-from-the-last has 1 chance in 3 of being drunk. And so on.
This expectation value isn’t hard to calculate. If we start counting from the last episode of the season, then it’s easy. Add up , ending when we get up to one divided by the number of episodes in the season. 25 or 26, for most seasons of Deep Space Nine. 15, from when they counted here. This is the start of the harmonic series.
The harmonic series gets taught in sequences and series in calculus because it does some neat stuff if you let it go on forever. For example, every term in this sequence gets smaller and smaller. (The “sequence” is the terms that go into the sum: , and so on. The “series” is the sum of a sequence, a single number. I agree it seems weird to call a “series” that sum, but it’s the word we’re stuck with. If it helps, consider: when we talk about “a TV series” we usually mean the whole body of work, not individual episodes.) You can pick any number, however tiny you like. I can then respond with the last term in the sequence bigger than your number. Infinitely many terms in the sequence will be smaller than your pick. And yet: you can pick any number you like, however big. And I can take a finite number of terms in this sequence to make a sum bigger than whatever number you liked. The sum will eventually be bigger than 10, bigger than 100, bigger than a googolplex. These two facts are easy to prove, but they seem like they ought to be contradictory. You can see why infinite series are fun and produce much screaming on the part of students.
No Star Trek show has a season has infinitely many episodes, though, however long the second season of Enterprise seemed to drag out. So we don’t have to worry about infinitely many drunk episodes.
Since there were 15 episodes up for drunkenness in the first season of Deep Space Nine the calculation’s easy. I still did it on the computer. For the first season we could expect drunk episodes. This is a number a little bigger than 3.318. So, more likely three drunk episodes, four being likely. For the 25-episode seasons (seasons four and seven, if I’m reading this right), we could expect or just over 3.816 drunk episodes. Likely four, maybe three. For the 26-episode seasons (seasons two, five, and six), we could expect drunk episodes. That’s just over 3.854.
The number of drunk episodes to expect keeps growing. The harmonic series grows without bounds. But it keeps growing slower, compared to the number of terms you add together. You need a 31-episode season to be able to expect at four drunk episodes. To expect five drunk episodes you’d need an 83-episode season. If the guys at Worst Episode Ever, reviewing The Simpsons, did all 625-so-far episodes by this rule we could only expect seven drunk episodes.
Still, three, maybe four, drunk episodes of the 15 remaining first season is a fair number. They shouldn’t likely be evenly spaced. The chance of a drunk episode rises the closer they get to the end of the season. Expected length between drunk episodes is interesting but I don’t want to deal with that. I’ll just say that it probably isn’t the five episodes the quickest, easiest suggested by taking 15 divided by 3.
And it’s moot anyway. The hosts discussed it just before starting “Captive Pursuit”. Pranica pointed out, for example, the smashed-last-episode problem. What they decided they meant was there would be a 1-in-15 chance of recording each episode this season drunk. For the 25- or 26-episode seasons, each episode would get its 1-in-25 or 1-in-26 chance.
That changes the calculations. Not in spirit: that’s still the same. Count the number of possible outcomes and the chance of each one being a drunk episode and add that all up. But the work gets simpler. Each episode has a 1-in-15 chance of adding 1 to the total of drunk episodes. So the expected number of drunk episodes is the number of episodes (15) times the chance each is a drunk episode (1 divided by 15). We should expect 1 drunk episode. The same reasoning holds for all the other seasons; we should expect 1 drunk episode per season.
Still, since each episode gets an independent draw, there might be two drunk episodes. Could be three. There’s no reason that all 15 couldn’t be drunk. (Except that at the end of reviewing “Captive Pursuit” they drew for the next episode and it’s not to be a drunk one.) What are the chances there’s no drunk episodes? What are the chances there’s two, or three, or eight drunk episodes?
There’s a rule for this. This kind of problem is a mathematically-famous one. We get our results from the “binomial distribution”. This applies whenever there’s a bunch of attempts at something. And each attempt can either clearly succeed or clearly fail. And the chance of success (or failure) each attempt is always the same. That’s what applies here. If there’s ‘N’ episodes, and the chance is ‘p’ that any one will be drunk, then we get the chance ‘y’ of turning up exactly ‘k’ drunk episodes by the formula:
That looks a bit ugly, yeah. (I don’t like using ‘y’ as the name for a probability. I ran out of good letters and didn’t want to do subscripts.) It’s just tedious to calculate is all. Factorials and everything. Better to let the computer work it out. There is a formula that’s easy enough to work with, though. That’s because the chance of a drunk episode is the same each episode. I don’t know a formula to get the chance of exactly zero or one or four drunk episodes with the first, one-in-N chance. Probably the only thing to do is run a lot of simulations and trust that’s approximately right.
But for this rule it’s easy enough. There’s this formula, like I said. I figured out the chance of all the possible drunk episode combinations for the seasons. I mean I had the computer work it out. All I figured out was how to make it give me the results in a format I liked. Here’s what I got.
The chance of these many drunk episodes
In a 15-episode season is
Sorry it’s so dull, but the chance of a one-in-fifteen event happening 15 times in a row? You’d expect that to be pretty small. It’s got a probability of something like 0.000 000 000 000 000 002 28 of happening. Not technically impossible, but yeah, impossible.
How about for the 25- and 26-episode seasons? Here’s the chance of all the outcomes:
The chance of these many drunk episodes
In a 25-episode season is
8 or more
And things are a tiny bit different for a 26-episode season.
The chance of these many drunk episodes
In a 26-episode season is
8 or more
Yes, there’s a greater chance of no drunk episodes. The difference is really slight. It only looks so big because of rounding. A no-drunk 25 episode season has a chance of about 0.3604, while a no-drunk 26 episodes season has a chance of about 0.3607. The difference comes from the chance of lots of drunk episodes all being even worse somehow.
And there’s some neat implications through this. There’s a slightly better than one in three chance that each of the second through seventh seasons won’t have any drunk episodes. We could expect two dry seasons, hopefully not the one with Quark’s sex-change episode. We can reasonably expect at least one season with two drunk episodes. There’s a slightly more than 40 percent chance that some season will have three drunk episodes. There’s just under a 10 percent chance some season will have four drunk episodes.
There’s no guarantees, though. Probability has a curious blend. There’s no predicting when any drunk episode will come. But we can make meaningful predictions about groups of episodes. These properties seem like they should be contradictions. And they’re not, and that’s wonderful.
I concede January was a month around here that could be characterized as “lazy”. Not that I particularly skimped on the Reading the Comics posts. But they’re relatively easy to do: the comics tell me what to write about, and I could do a couple paragraphs on most anything, apparently.
While I get a couple things planned out for the coming month, though, here’s some reading for other people.
The above links to a paper in the Proceedings of the National Academy of Sciences. It’s about something I’ve mentioned when talking about knot before. And it’s about something everyone with computer cables or, like the tweet suggests, holiday lights finds. The things coil up. Spontaneous knotting of an agitated string by Dorian M Raymer and Douglas E Smith examines when these knots are likely to form, and how likely they are. It’s not a paper for the lay audience, but there are a bunch of fine pictures. The paper doesn’t talk about Christmas lights, no matter what the tweet does, but the mathematics carries over to this.
MathsByAGirl, meanwhile, had a post midmonth listing a couple of mathematics podcasts. I’m familiar with one of them, BBC Radio 4’s A Brief History of Mathematics, which was a set of ten-to-twenty-minute sketches of historically important mathematics figures. I’ll trust MathsByAGirl’s taste on other podcasts. I’d spent most of this month finishing off a couple of audio books (David Hackett Fischer’s Washington’s Crossing which I started listening to while I was in Trenton for a week, because that’s the sort of thing I think is funny, and Robert Louis Stevenson’s Doctor Jekyll and Mister Hyde And Other Stories) and so fell behind on podcasts. But now there’s some more stuff to listen forward to.
And then I’ll wrap up with this from KeplerLounge. It looks to be the start of some essays about something outside the scope of my Why Stuff Can Orbit series. (Which I figure to resume soon.) We start off talking about orbits as if planets were “point masses”. Which is what the name suggests: a mass that fills up a single point, with no volume, no shape, no features. This makes the mathematics easier. The mathematics is just as easy if the planets are perfect spheres, whether hollow or solid. But real planets are not perfect spheres. They’re a tiny bit blobby. And they’re a little lumpy as well. We can ignore that if we’re doing rough estimates of how orbits work. But if we want to get them right we can’t ignore that anymore. And this essay describes some of how we go about dealing with that.