Robert Recorde introduced the equal sign in 1557. He used parallel lines because "no two things can be more equal". And his equal sign was hilariously looooooong.
It's fun to fight your way through Recorde's text, with its old font and spellings. But if you give up, @mjd has transliterated it:
Howbeit, for easie alteration of equations. I will propounde a fewe exanples, bicause the extraction of their rootes, maie the more aptly bee wroughte. And to avoide the tediouse repetition of these woordes "is equalle to" I will sette as I doe often in woorke use, a pair of paralleles, or Gemowe lines of one lengthe, thus: =====, bicause noe 2 thynges, can be moare equalle.
The only real mystery here is "Gemowe", which means "identical" and comes from the same root as "Gemini": twins.
In the same book Robert Recorde introduced the mathematical term "zenzizenzizenzike", but I'm afraid for that you'll have to read @mjd's article!
Joe Gore did graduate work on musicology and is an expert on medieval harmony theory. But when he played guitar on the Tom Waits song "Goin' Out West", he got handed a cassette tape of Tom Waits singing while pounding drums in a steel shed - and Waits had him play over that on an electric guitar tuned a quarter tone flat.
A great interview! He describes how Tom Waits gets great music from players by pushing them way out of their comfort zone. And near the end he gets into some serious harmony theory! I know just enough to know he's not faking it. I like the idea that in Lydian, the sharp fourth functions as a kind of second leading tone - a half-tone below the fifth, just like the usual leading tone is a half-tone down below tonic.
An epidemiologist having a category-theoretic revelation. My colleague Nathaniel Osgood, discovering how the process of converting stock and flow diagrams into causal loop diagrams can be captured by a left pushforward functor between presheaf categories. These two kinds of diagrams are both important in the modeling tradition called 'system dynamics', which is used in epidemiology as well as economics and other disciplines.
I want to read this book: A Darwinian Survival Guide. Sounds like a realistic view of what we need to do now. You can read an interview with one author, the biologist Daniel Brooks. A quote:
...
Daniel Brooks: What can we begin doing now that will increase the chances that those elements of technologically-dependent humanity will survive a general collapse, if that happens as a result of our unwillingness to begin to do anything effective with respect to climate change and human existence?
Peter Watts: So to be clear, you’re not talking about forestalling the collapse —
Daniel Brooks: No.
Peter Watts: — you’re talking about passing through that bottleneck and coming out the other side with some semblance of what we value intact.
Daniel Brooks: Yeah, that’s right. It is conceivable that if all of humanity suddenly decided to change its behavior, right now, we would emerge after 2050 with most everything intact, and we would be “OK.” We don’t think that’s realistic. It is a possibility, but we don’t think that’s a realistic possibility. We think that, in fact, most of humanity is committed to business as usual, and that’s what we’re really talking about: What can we begin doing now to try to shorten the period of time after the collapse, before we “recover”? In other words — and this is in analogy with Asimov’s Foundation trilogy — if we do nothing, there’s going to be a collapse and it’ll take 30,000 years for the galaxy to recover. But if we start doing things now, then it maybe only takes 1,000 years to recover. So using that analogy, what can some human beings start to do now that would shorten the period of time necessary to recover?
"Stepping back a bit. Darwin told us in 1859 that what we had been doing for the last 10,000 or so years was not going to work. But people didn’t want to hear that message. So along came a sociologist who said, “It’s OK; I can fix Darwinism.” This guy’s name was Herbert Spencer, and he said, “I can fix Darwinism. We’ll just call it natural selection, but instead of survival of what’s-good-enough-to-survive-in-the-future, we’re going to call it survival of the fittest, and it’s whatever is best now.” Herbert Spencer was instrumental in convincing most biologists to change their perspective from “evolution is long-term survival” to “evolution is short-term adaptation.” And that was consistent with the notion of maximizing short term profits economically, maximizing your chances of being reelected, maximizing the collection plate every Sunday in the churches, and people were quite happy with this."
When global warming gets bad enough to create millions of refugees, just imagine how nasty the EU and US will become. They may just send people out to the desert to die.
Oh wait, you don't have to imagine! It's happening already:
"A year-long joint investigation by The Washington Post, Lighthouse Reports and a consortium of international media outlets shows how the European Union and individual European nations are supporting and financing aggressive operations by governments in North Africa to detain tens of thousands of migrants each year and dump them in remote areas, often barren deserts.
European funds have been used to train personnel and buy equipment for units implicated in desert dumps and human rights abuses, records and interviews show. Migrants have been pushed back into the most inhospitable parts of North Africa, exposing them to abandonment with no food or water, kidnapping, extortion, sale as human chattel, torture, sexual violence and, in the worst instances, death."
"“There is Algeria, follow the light,” the Tunisian official barked at the Black migrants. “If you’re seen here, you’ll be shot.”
François, a 38-year-old Cameroonian, obeyed, jumping off the bed of a pickup truck near the desolate Algerian frontier. A day earlier, the rickety boat attempting to carry him and other hopeful sub-Saharans to Europe — including his wife and 6-year-old stepson — had been interdicted by the Tunisian coast guard in the cobalt blue waters off the coast. Still wet and cold, the group of 30 migrants, including two pregnant women, now walked toward their punishment: the desert.
Their ordeal — an odyssey of at least 345 miles from sea to sand, recounted by François and verified by matching GPS tracking on his phone with images and videos he captured during nine days of wandering — illustrates one example of the draconian practices being deployed in at least three North African nations to dissuade sub-Saharan migrants from risky crossings to Europe."
"Witness accounts and visuals reviewed by The Post place the Tunisian National Guard at the center of desert dump operations. Between 2015 and 2023, the German federal police deployed 449 staff members and spent more than 1 million euros to train nearly 4,000 Tunisian national guards. As the dumps were ongoing in November 2023, a 9 million euro border-management training center opened in Tunisia, funded by Austria, Denmark and the Netherlands."
@johncarlosbaez Terrible. I donate to Sea-Eye because it's ridiculous to let people die that could be easily saved. Do we now need rescue vessels on land too?!
A while back @pschwahn raised an interesting puzzle here:
You can define a well-behaved cross product of vectors only in 3 and 7 dimensions. The 7d cross product is weird because it's not preserved by all rotations of 7d space. But very smart people have told us there's a way to get the group of 𝟯𝗱 rotations to act on 7d space while preserving the 7d cross product. In fact, you can do it while also getting this group to act 'irreducibly', meaning the only subspaces of 7d space preserved by this action are {0} and the whole space!
The puzzle is: 𝗰𝗮𝗻 𝘆𝗼𝘂 𝘄𝗿𝗶𝘁𝗲 𝗱𝗼𝘄𝗻 𝘁𝗵𝗲 𝗳𝗼𝗿𝗺𝘂𝗹𝗮𝘀 𝗳𝗼𝗿 𝗵𝗼𝘄 𝘁𝗵𝗶𝘀 𝘄𝗼𝗿𝗸𝘀?
I got stuck on this so I asked some of my friends, and now Layra Idarani has outlined a nice way to do it:
Interestingly he doesn't actually give the formulas; he just tells you how to get them. So I will need to do some work to check his answer! If you want to help out, that would be great.
Layra said "The devil of the details is in the eating". I thought the proof was in the pudding. Now I'm hungry for devil's food cake.
@Dyoung@pschwahn - it's there: it's the group of rotations that rotates x, y, and z in the usual way. To understand this you need to start further back, e.g. with my failed attempt
I laid out the strategy nicely but failed to guess a good choice of 7 cubic polynomials - I called them 𝑒₁,...𝑒₇ . They lacked the properties I wanted to check, but worse they weren't even orthonormal. Layra's polynomials P, Q, R, S, T, U, V have all the right properties - or so he claims.
By the way, it's super-easy to construct the 7d cross product on the imaginary octonions: just define
𝑎 × 𝑏 = (𝑎𝑏−𝑏𝑎)/2
All the work here is constructing an SO(3) group that acts irreducibly on the imaginary octonions while preserving this cross product!
I've had another thought: there's nothing that really limits this sort of considerations to the 𝑐𝑜𝑚𝑝𝑎𝑐𝑡 real form of G₂. Say we replace octonions by split-octonions (not a division algebra, but a so-called composition algebra), then the automorphism group G₂* is not compact any more - it does not preserve an Euclidean inner product on ℝ⁷, but instead one of signature (3,4).
Dynkin's classification of maximal subalgebras is undisturbed by this, since G₂ and G₂* have the same complexification. But it is not clear to me what the corresponding subgroups are.
It is known that the maximal compact subgroup of G₂* is SO(4) (which is also maximal as just a subgroup). This suggests that the analogues of SU(3) and SO(3)ᵢᵣᵣ are 𝑛𝑜𝑛𝑐𝑜𝑚𝑝𝑎𝑐𝑡 - but what are they exactly, and how do they act on the imaginary split-octonions?
Since May 1st, a small team of category theorists, computer scientists and epidemiologists have been meeting daily in James Clerk Maxwell’s childhood home in Edinburgh. We’re hard at work on our project called New Mathematics and Software for Agent-Based Models.
We're creating a general software framework for agent-based models in epidemiology. By now we've really entered the zone where all these ideas come together. We're equally likely to talk about details of opioid abuse models or using coproducts of representables to make our software more efficient. It's exciting!
First we came up with a general framework for 'stochastic C-set rewriting systems'. These are models where graphs or more complicated combinatorial structures change in a random way through local 'rewrite rules'. Each rewrite rule says that when a bit of your structure matches some pattern, you can 'rewrite' it to some specific new pattern. See the pictures below for a couple examples.
'Stochastic' is a fancy word for 'random'. In our models you specify the randomness in a carefully crafted way by associating to each rewrite rule a 'timer'. The timer says the probability with which the rule is applied - as a function of time. A timer starts whenever a new match to the rule appears.
Kris Brown has already created a program that lets you run these stochastic C-set rewriting systems in AlgebraicJulia. This is a Julia package for scientific computing with categories. But we're just getting started!
@bks - Alas, Rosen's work has very little substance to it and has not influenced applied category theory very much. Very roughly speaking, successful applications of category theory to a wide variety of practical areas started with computer science, then spread to quantum physics, and continued growing, meriting its own conference series in 2018. You can hear my story of this here:
When Maxwell realized in 1862 that light consists of waves in the electromagnetic field, why didn't anyone try to use electricity to make such waves right away? Why did Hertz succeed only 24 years later?
According to 𝘛𝘩𝘦 𝘔𝘢𝘹𝘸𝘦𝘭𝘭𝘪𝘢𝘯𝘴:
"Since he regarded the production of light as an essentially molecular and mechanical process, prior, in a sense, to electromagnetic laws, Maxwell could elaborate an electromagnetic account of the propagation of light without ever supposing that ether waves were produced purely electromagnetically."
In 1879, a physicist named Lodge realized that in theory one could make "electromagnetic light". But he didn't think of creating waves of lower frequency:
"Send through the helix an intermittent current (best alternately reversed) but the alternations must be very rapid, several billion per sec."
He mentioned this idea to Fitzgerald, who believed he could prove it was impossible. Unfortunately Fitzgerald managed to convince Lodge. But later he realized his mistake:
"It was FitzGerald himself who found the flaws in his "proofs." He then proceeded to put the subject on a sound theoretical basis, so that by 1883 he understood quite clearly how electromagnetic waves could be produced and what their characteristics would be. But the waves remained inaccessible; FitzGerald, along with everyone else, was stymied by the lack of any way to detect them."
In 1883, Fitzgerald gave a talk called "On a Method of Producing Electromagnetic Disturbances of Comparatively Short Wavelengths". But he couldn't figure out how to 𝘥𝘦𝘵𝘦𝘤𝘵 these waves. Hertz figured that out in 1886.
@johncarlosbaez@BashStKid "The antenna was excited by pulses of high voltage of about 30 kilovolts applied between the two sides from a Ruhmkorff coil. He received the waves with a resonant single-loop antenna with a micrometer spark gap between the ends."
This time it corresponds to the diagram but it's an experiment set up after the involuntary observation described in the message above.
@johncarlosbaez@BashStKid "Hertz did produce an analysis of Maxwell's equations during his time at Kiel, showing they did have more validity than the then prevalent "action at a distance" theories."
It was what I thought tonight: was it a "action at a distance" or a "transport"?
@NickPizzoOceans - great, thanks: this looks good! I've been reading Helmholtz's work "On the Sensations of Tone as a Physiological Basis for the Theory of Music" since I'm studying harmony, and it will fun to learn more about what he did in electromagnetism. I know he pushed Hertz to create and detect electromagnetic waves.
The precise location of the boundary between the knowable and the unknowable is itself unknowable. But we 𝑑𝑜 know some details about 𝑤ℎ𝑦 this is true, at least within mathematics. It's being studied rigorously in a branch of theoretical computer science called 'meta-complexity theory'.
For some reason it's hard to show that math problems are hard. In meta-complexity theory, people try to understand why.
For example, most of us believe P ≠ NP: merely being able to 𝑐ℎ𝑒𝑐𝑘 the answer to a problem efficiently doesn't imply you can 𝑠𝑜𝑙𝑣𝑒 it efficiently. It seems obvious. But despite a vast amount of work, nobody has been able to prove it!
And in one of the founding results of meta-complexity theory, Razborov and Rudich showed that if a certain attractive class of strategies for proving P ≠ NP worked, then it would be possible to efficiently crack all codes! None of us think 𝑡ℎ𝑎𝑡'𝑠 possible. So their result shows there's a barrier to knowing P ≠ NP.
I'm simplifying a lot of stuff here. But this is the basic idea: they proved that it's probably hard to prove that a bunch of seemingly hard problems are really hard.
But note the 'probably' here! Nobody has 𝑝𝑟𝑜𝑣𝑒𝑑 we can't efficiently crack all codes. And this too, seems very hard to prove.
So the boundary between the knowable and unknowable is itself shrouded in unknowability. But amazingly, we can prove theorems about it!
@leemph - regarding 1, you're exactly right: in math most of us have accepted that "mathematical knowledge" is relative to a set of axioms and deduction rules. This started with the discovery of non-Euclidean geometry, where "truths" of Euclidean geometry turned out to be consequences of axioms that no longer hold if you switch to a different sort of geometry. Later Goedel showed that any sufficiently powerful consistent finitely axiomatizable theory could never prove or disprove all statements formulated in its own language: there's always some statement P such that we can add either P or not(P) to the axioms and get a new such set of axioms that's still consistent.
In constructivist logic to prove something exists means that you can, at least in principle, exhibit an example. In classical logic there are cases where you can prove something exists but not exhibit an example. So your attitude to this question is closely allied to whether you prefer classical or constructivist logic. Most really smart mathematicians realize that this, too, is another case of the relativity in part 1. I.e., neither constructivism or classical logic is "really true": they are just alternative sets of axioms, and it's worth exploring both.
"If we prove T then do we have a proof that it is provable?" In the logics I know, exhibiting an example of something counts as a proof that it exists. So yes, in both classical and constructivist logic, we can get from a proof of P to a proof that P is provable. Part 2 was about the more problematic converse.
"In which deductive system should that meta-proof be carried out?" There are many choices. This is another instance of the relativity in 1.
@leemph wrote: "<<For example, we know that if Goldbach's conjecture is true but unprovable, it's also impossible to prove that it's unprovable. So there are cases where unknowability shrouds itself in unknowability.>>"
Sorry, this was a stupid sentence, and you were right to be confused. Here's what I was trying to say:
"For example, if Goldbach's conjecture is true but unprovable, it's also impossible to prove that it's unprovable. So there are cases where unknowability shrouds itself in unknowability."
And normally I avoid using the word "true" in this context, since it doesn't really mean much to say a mathematical statement is "true" except as a shorthand for it being provable. If I were trying to be precise, I would have said this:
"For example, if neither Goldbach's conjecture nor its negation is provable, it's also impossible to prove that either of those is unprovable. So there are cases where unknowability shrouds itself in unknowability."
Chemistry is like physics where the particles have personalities - and chemists love talking about the really nasty ones. It makes for fun reading, like Derek Lowe's column "Things I Won't Work With". For example, bromine compounds:
"Most any working chemist will immediately recognize bromine because we don't commonly encounter too many opaque red liquids with a fog of corrosive orange fumes above them in the container. Which is good."
And that's just plain bromine. Then we get compounds like bromine fluorine dioxide.
"You have now prepared the colorless solid bromine fluorine dioxide. What to do with it? Well, what you don't do is let it warm up too far past +10C, because it's almost certainly going to explode. Keep that phrase in mind, it's going to come in handy in this sort of work. Prof. Seppelt, as the first person with a reliable supply of the pure stuff, set forth to react it with a whole list of things and has produced a whole string of weird compounds with brow-furrowing crystal structures. I don't even know what to call these beasts."
Fun article by John Psmith featuring some ferociously competitive mathematicians and physicists. A quote:
.....
In the 1696 edition of Acta Eruditorum, Johann Bernoulli threw down the gauntlet:
"I, Johann Bernoulli, address the most brilliant mathematicians in the world. Nothing is more attractive to intelligent people than an honest, challenging problem, whose possible solution will bestow fame and remain as a lasting monument. Following the example set by Pascal, Fermat, etc., I hope to gain the gratitude of the whole scientific community by placing before the finest mathematicians of our time a problem which will test their methods and the strength of their intellect. If someone communicates to me the solution of the proposed problem, I shall publicly declare him worthy of praise.
Given two points A and B in a vertical plane,
what is the curve traced out by a point acted on only by gravity,
which starts at A and reaches B in the shortest time."
This became known as the brachistochrone problem, and it occupied the best minds of Europe for, well, for less time than Johann Bernoulli hoped. The legend goes that he issued that pompous challenge I quoted above, and shortly afterward discovered that his own solution to the problem was incorrect. Worse, in short order he received five copies of the actually correct solution to the problem, supposedly all on the same day. The responses came from Newton, Leibniz, l’Hôpital, Tschirnhaus, and worst of all, his own brother Jakob Bernoulli, who had upstaged him yet again.
(1/2) (The fun part about Newton comes in part 2.)
There's a dot product and cross product of vectors in 3 dimensions. But there's also a dot product and cross product in 7 dimensions obeying a lot of the same identities! There's nothing really like this in other dimensions.
We can get the dot and cross product in 3 dimensions by taking the imaginary quaternions and defining
v⋅w= -½(vw + wv), v×w = ½(vw - wv)
We can get the dot and cross product in 7 dimensions using the same formulas, but starting with the imaginary octonions.
The following stuff is pretty well-known: the group of linear transformations of ℝ³ preserving the dot and cross product is called the 3d rotation group, SO(3). We say SO(3) has an 'irreducible representation' on ℝ³ because there's no linear subspace of ℝ³ that's mapped to itself by every transformation in SO(3).
Much to my surprise, it seems that SO(3) also has an irreducible representation on ℝ⁷ where every transformation preserves the dot product and cross product in 7 dimensions!
It's not news that SO(3) has an irreducible representation on ℝ⁷. In physics we call ℝ³ the spin-1 representation of SO(3), or at least a real form thereof, while ℝ⁷ is called the spin-3 representation. It's also not news that the spin-3 representation of SO(3) on ℝ⁷ preserves the dot product. But I didn't know it also preserves the cross product on ℝ⁷, which is a much more exotic thing!
In fact I still don't know it for sure. But @pschwahn asked me a question that led me to guess it's true:
@johncarlosbaez It seems this maximal SO(3) is more mysterious than anticipated!
Looking at Dynkin's paper (Table 16), there are four conjugacy classes of 3-dimensional subalgebras of 𝔤₂. Two of those are the 𝔰𝔲(2) factors of the 𝔰𝔬(4) subalgebra, one belongs to the SO(3)⊂SO(4) acting reducibly on Im 𝕆, and one belongs to the maximal subgroup SO(3)ᵢᵣᵣ.
Each 𝔰𝔬(3) subalgebra contains what Dynkin calls a "defining vector", that is, an element in a fixed maximal torus of 𝔤₂. Not just any element of the torus can be a defining vector: Dynkin gives the possible coordinates in the basis of simple roots.
From this point of view all 𝔰𝔬(3) subalgebras are on equal footing, so I'm not sure whether one can speak of "generic" objects here. At a glance it looks like they all have the same degrees of freedom, namely a choice of maximal torus in 𝔤₂ plus a choice of simple roots.
@pschwahn - There is a space of all 𝔰𝔬(3) subalgebras of 𝔤₂, and I was guessing that those subalgebras that act irreducibly on Im(𝕆) are 𝑑𝑒𝑛𝑠𝑒 in this space. That's what "generic" means in this context.
Another thing I guess is that the conjugacy class of 𝔰𝔬(3) subalgebras acting irreducibly on Im(𝕆) has higher dimension than the other 3 conjugacy classes. This should be a lot easier to check.
We were listening to bouzouki music, and the conversation naturally turned to bazookas. It turns out that there's also bazooka music!
The comedian Bob Burns invented a horn-like instrument in the 1910s - he's shown with it below. It actually caught on in jazz in the 1930s. Someone jokingly called it the "bazooka" after the word "bazoo", which was slang for "mouth".
Later, in World War II, "bazooka" became the name for a new American anti-tank weapon, because it looked like this instrument.
Ironically, the slang word "bazoo", for "mouth", probably came from the word "buisine", which was the name of a medieval trumpet! And that comes from "buccina", a brass horn used by the Roman army.
In case you're wondering, he word "bouzouki" is unrelated. It comes from the Turkish word "bozuk", meaning "broken" or "modified", which refers to a particular way of tuning a string instrument where the notes are not arranged from low to high.
@TruthSandwich - it seems strange to name some bubble gum after a portable anti-tank missile. What's next? RPG chewing gum? HIMARS candies?
The article doesn't explain this name choice except that it happened shortly after WW2 and the ad campaign featured one "Bazooka Joe". But Bazooka Joe wasn't a soldier: he was a kid with an eyepatch.
I'm left hoping that the original use of "bazoo" to mean "mouth" was still relevant....
Suppose raindrops are falling on your head, randomly and independently, at an average rate of one per minute. What's the average of the 𝑐𝑢𝑏𝑒 of the number of raindrops that fall on your head in one minute?
The probability that (k) raindrops fall on your head in a minute is given by the Poisson distribution of mean 1, so it's
[ \frac{1}{ek!} ]
I could explain this but let's move on. The puzzle asks us to compute the expected value of (k^3) for this probability distribution, which is
[ \sum_{k=0}^\infty \frac{k^3}{ek!} ]
The heart of the puzzle is to figure out this sum. It turns out that
[ \sum_{k = 0}^\infty \frac{k^n}{k!} = B_n e ]
where (B_n) is the (n)th 'Bell number': the number of partitions of an (n)-element set into nonempty subsets. This is called 'Dobiński's formula'. I'll prove it in my next post. Now let's just use it!
We're interested in the case (n = 3). There are 5 partitions of a 3-element set
[ {{1,2,3}}, ]
[ {{1,2}, {3}}, ; {{2,3}, {1}}, ; {{3,1}, {2}}, ]
[ {{1}, {2}, {3}} ]
so (B_3 = 5).
So, the average of the cube of the number of raindrops that fall on your head in one minute is 𝟓.
Wild, huh? From probability theory to combinatorics.
@dougmerritt - I'm glad you know 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑛𝑔𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑜𝑙𝑜𝑔𝑦 - it's a blast. Flajolet's book is also lots of fun, full of concrete examples of how to use generating functions. I think it gently brings in the fact that what you're computing an (exponential) generating function 𝑜𝑓 is actually a species: a functor from the groupoid of finite sets to Set, sending each finite set to some set of structures you can put on it. The book by Bergeron, Labelle, and Leroux digs a bit deeper into the functorial approach to species. But none of these gets anywhere near the full glory of that viewpoint.... you'll probably be relieved to know. I got into that more deeply in this course:
@uxor - nice, I don't know the Wyman Moser asymptotic formula, but it sounds like a fun example of approximating something by a Gaussian, and I bet there is a lot of deep combinatorial significance lurking beneath it.