> ...partly because there isn't a whole lot of benefit to his approach if you're just doing things in 2D and 3D— in those cases, the alternative approaches are in a sense simpler and more straightforward
This is the crux: Grassmann had discovered and explored several mountain passes while others had not yet settled the valleys which they connected? Even if he hadn't been an outsider, he seems to have been (from a career standpoint) "too early".
I'd change the metaphor a bit. Math at the start of 19th century was like a country many of whose high mountains (say theorems) had been climbed but whose basic geography (foundations) hadn't been charted. Grassman was perhaps the first to begin the process of true foundational studies - creating structures that formally tied the mountains together (different from intuitively tying the mountains together, notably).
Grassman's approach (incorporated as "exterior algebra") was klunky and difficult most likely specifically because he was doing something new, something never done before. That is, he was working when no formal linear algebra and indeed neither formal abstract algebra nor formal foundations of real numbers nor formalization of calculus existed (though many things were talked about informally).
And Grassman was resisted because of the (probably necessary) klunkiness, but also because many mathematicians didn't see the need for this and no small number actively resisted the need for formal. IE, saw the qualities of numbers and collections of thing as best handled intuitively rather than formally and especially disliked the way formal formulations allowed and required counter-intuitive structures (transfinite sets, the axiom of choice, etc). This resistance didn't just touch Grassman but affect Cantor and Hilbert, etc.
Today, math still has a tension between the provers of hard theorems and the structure builders, that would be the mountain climbers and the geographers, the Erdős' and the Grothendiecks. Moreover, the history of that tension isn't as clear as it could be since fields of math are generally taught as fully formulated systems with few nods to the state of the field before a given formulation (I just learned today that van der Waerden's Moderne Alegebra, published in 1930, was the first text to put abstract algebra in anything like it's present form).
More on the history of the formalization of mathematics can found in Jan van Plato's The Great Formal Machinery Works (Grassman is the first writer covered).
Van Plato's book seems interesting! I see that it has a chapter on natural deduction and sequent calculus. I found the focus on introduction and elimination rules in natural deduction always somewhat mystifying, and wondered about what exactly natural deduction is. As I discovered just in the last few weeks, even if, like me, you don't know anything about the proof theoretic foundations of natural deduction and sequent calculus, they pop up naturally as proof systems for abstraction logic (AL): natural deduction is the proof system if your truth values form a complete lattice, and sequent calculus is the proof system if your truth values form even a complete bi-Heyting algebra. Note that in AL, there are no a-priori constants (such as ∧ or ⇒), so there are also no a-priori rules for elimination and introduction, but just the essence of natural deduction and sequent calculus.
Grassmann's approach is not "klunky and difficult". It's clear and straightforward and makes proving some difficult results much easier than the previous methods, it just takes a bit of a leap of faith to practice enough with it to build fluency. (The subject itself is also challenging, irrespective of approach. When you strip away some of the incidental complexity of a mismatched representation, the inherent complexity of the subject still must be contended with.)
I'd recommend people try to directly read both of Grassmann's two books; there are translations of both into English. (Disclaimer: I haven't made it through either one beyond a skim and reading the first few chapters. Then I got distracted by some other project. Like any math books, reading them takes a lot of slow careful effort.)
It's kind of like taking a bunch of people used to riding horses and handing them a bicycle. The bicycle is not "klunky and difficult" compared to the horse, it's just unfamiliar.
This reminds me of the Grothendieck quote about his approach to solving problems (in contrast with Erdős, who I suppose in this analogy would look for a big hammer with which to smash the nut open quickly!):
“I can illustrate the ... approach with the ... image of a nut to be opened. The first analogy that came to my mind is of immersing the nut in some softening liquid, and why not simply water? From time to time you rub so the liquid penetrates better, and otherwise, you let time pass. The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough, the shell opens like a perfectly ripened avocado! A different image came to me a few weeks ago. The unknown thing to be known appeared to me as some stretch of earth or hard marble, resisting penetration ... the sea advances insensibly in silence, nothing seems to happen, nothing moves, the water is so far off you hardly hear it ... yet finally it surrounds the resistant substance.”
With the caveat that Deligne (?) complained that his work was too moral, ie, a lot of crucial details had to be put in by others to prove anything nontrivial.
Yes, agreed. But he could have done a better job outlining and explaining the valleys to people first so they would be in a position to appreciate the mountain passes! Luckily, credit in academia tends to reward such people better than the commercial market, even if it ends up happening long after death. Look at Galois for example.
True. Even were it to have been making a virtue of necessity, he did note "some day these ideas, even if in an altered form, will reappear and with the passage of time will participate in a lively intellectual exchange. For truth ... remains even if the garments in which feeble men clothe it fall into dust."
It's from the forward to Grassmann's 1962 Ausdehnungslehre[1]:
einst dieſe Ideen, wenn auch in veränderter Form, neu erstehen und mit der Zeitentwickelung in lebendige Wechſelwirkung treten werden. Denn die Wahrheit … bleibt bestehen, wenn auch das Gewand, in welches schwache Menschen ſie kleiden, in Staub zerfälit.
The translation is from Kannenberg, Extension Theory, AMS, 2000[2].
Double thanks! (Always glad to see the German taking "Elizabethan" s-f glyph pair to its ambiguous conclusion.) My very earliest exposure to the Ausdehnungslehre was in Sternberg+Bamberg. Until coming across Penrose diagrams, this book staved off the disgust I felt for tensor calculus (as used by physicists).
Mainly shocked by the compression of Maxwell's equations to a single line.
Otherwise.. that was.. (regrettably cryptic) a reference to the eerily resonant Kolmogorov option, roughly motivated by Hilbert's slogan "truth does not need martyrs".
More precisely, I am sympathetic to the idea that the path that Grassmann took to release his discoveries into the world was actually close to the optimal one.
Other paths to consider are thus Grothendieck's,Kaczynski's, Galois' (generalizing to purely social bits, even Nash's)
Two-sided intellectual exchange with a finished work across spacetime, that's also something.
Since you are possibly not terrible at unearthing obscure but Critical rocksnacks, exactly why, Archangelsk Governorate? I mean, if besides that there it being harder to tell noon from midnight, e.g. neighboring Finnics having obnoxiously dissimilar mythology )
Since the remainder of the thread has been infested by "noonday demons", there's a nice line by a singer-songwriter about a SF novel in the Noon Universe[0]: "of course, freedom and brotherhood[1] will come / But it will happen without us."[2]
[1] aka « liberté et fraternité ». A cynic would say the Arkanarians already have « égalité », in that they're all stuck on a planet cycling endlessly between mere anti-intellectualism and outright theocracy.
I was more interested in pointing to you the measly followups to Aaronson's question, which consists solely of forays into nonlinear quantum mechanics & superluminal neutrinos.
Hmm... what are the finnic equivalents/dysquivalents of the Południca/полуднице[0]?
My poisk-fu will probably be insufficient, but my wild guess at the moment is that due to circumstances mentioned in https://en.wikipedia.org/wiki/Arkhangelsk#Literature , coincidentally the reference was collected there. (Midnight to me is just a local play on being easier to temporally localise in summer than Noon[1], being just south of the arctic circle[2]. There's another russian folk tale I can't remember specifically in which young folk, on a particular saint's day, go looking for a specific species of flower[3] and if they find a —phenotypically non-existent— colour, it grants invisibility: the rural peasantry's equivalent of ex falso quodlibet?)
[1] I'll note that in my local laws, 12.00-13.00 are still quiet hours; one doesn't rely on any Mittagsgespenst but instead could just threaten to call the cops on errant workers.
[3] at least as long as they're not deflowering each other instead. Another just-so story for a Halfday Witch is related: older members of the village can easily be kept in check by threatened retaliation, but when the younger kids (acting as temporary fieldhands for the harvest) might be in danger of intruding upon Masha and Sasha's privacy[4] over midday, they are threatened with the Midday Witch, and if the small kids ask why Masha and Sasha are not afraid of her, they get told M&S are taking their chances but know enough about agronomy to answer her questions — or would Masha's mother be told she didn't make it home for lunch because she had been caught by the Midday Witch and had to recite flax culture 101?
Unfortunately also a mutually suspicious reference to Don Rumata's universe (what would be the curvature of the M-K-K curve at noon, or more fascinatingly, in each of the first book, points in the engagements of von Seydlitz, considering the battle orders, elders, implicit social rules &c ).
Being fully precognizant of the scene at the Arctic circle (haha) I was going more from vague preintuitions writ Finnic riddles. If you met a Finnic noondemon, you might be more likely to dismiss them than the Sphinx, or (aunt) Dasha, or the catcher in the rye. Slavic demigods/semidemons are more clearly natural phenomena, Finnic demigods seem to be better described as more evolved humans (1 tale of Väinämöinen vs a Christian Alia, 2 compare with the neoEgyptian scifi of the 90s-naughts 3 temporality distortion fields)
As to [1] Switzerland has only 2/3 in SMEs(?!) so I'm more inclined to guess Liechtenstein (or it's brethren) or somewhere close to an Alpine lake. Too small though..(clearly not Luxembourg).. you beat me to the comment than
the Eitgenoze is the most underrated example of an anarchic contract. Consider your (collective) privacy protected by my laziness.
The only aunt Dasha I can think of is Annie's (belle-)aunt Darya Alexandrovna? Or am I barking up the wrong (unhappy) family tree?
Mamas don't let your babies grow up to be Vronsky
'Cause he'll always flee home and he's only alone
Even with someone he loves
(not that I would have done any better than Alexei Kirillovich: while I always claim in writing that Yuri would've done much better to just go to Paris with Tonya, and I bear no resemblance to Mr Sharif myself, somehow my muses have tended to resemble Ms Christie far more than they have Ms Chaplin)
Lagniappe: https://www.youtube.com/watch?v=gpV-Ii0TJm8&t=370s (Amy Kane deals death at High Noon, but she has more in common with Han Solo than a noonday demon, as she doesn't ask any questions first)
Freeman Dyson did a nice job of summing up the situation with Grassmann:
"In the year 1844 two remarkable events occurred, the publication by Hamilton of his discovery of quaternions, and the publication by Grassmann of his “Ausdehnungslehre.” With the advantage of hindsight we can see that Grassmann’s was the greater contribution to mathematics, containing the germ of many of the concepts of modern algebra, and including vector analysis as a special case. However, Grassmann was an obscure high-school teacher in Stettin, while Hamilton was the world-famous mathematician whose official titles occupy six lines of print after his name at the beginning of his 1844 paper. So it is regrettable, but not surprising, that quaternions were hailed as a great discovery, while Grassmann had to wait 23 years before his work received any recognition at all from professional mathematicians. When Grassmann’s work finally became known, mathematicians were divided into quaternionists and antiquaternionists, and were spending more energy in polemical arguments for and against quaternions than in trying to understand how Grassmann and Hamilton might be fitted together into a larger scheme of things." (Source: "Missed opportunities")
In The Karenin Stratification Jason Bourne must cope with an occult enemy (but instead of Fhtagn! they say Myob! and F'iw!) whose vessels silently drift on cross-flowing benthic rip currents to arrive at and depart the ancient hydrothermal ancap settlement of R'lyeh without detection by the maritime patrol aircraft of more traditional jurisdictions.
Being too early is a privilege. Not everyone CAN invent things that are valuable but ahead of their time.
If you can then on the one hand don't expect tremendous accolade but also do continue working on what you're doing if it feels obvious to you that this is important. Today with the pace at which technology is advancing people tend to receive credit in their lifetime. In Grassman's day that was just NOT the case.
If more people took risks building out ideas they "felt" were the right thing to do the world would be a substantially better place.
Very nice write-up. Can you expand a bit on one of the phrases you used: tensors as essentially matrices of matrices? Also, the connection between the wedge product and tensors is still pretty fuzzy for me.
"Tensors as matrices of matrices" isn't particularly deep; basically you can think of a 3D tensor of being a "cube" of entries, where each cross section in the direction going into the page is just a regular 2D matrix. For example, if you wanted to store the density of discrete voxels in a region of space, you could use such a tensor. This is in fact how MRI scan data is stored, as a bunch of cross sections that are glued together to make a 3D structure. You can then extend this to 4 dimensions, by taking an array of those 3D tensors, and so on.
Understanding the link between wedge products and tensors, I think the key thing is the anti-symmetry of the wedge product, where if you reverse the order, the sign changes-- this is related to how the determinant is an "oriented" measure of area, and can take on negative values. The results of combining elements with the wedge product are called differential forms, and these can be thought of as "antisymmetric tensors." I agree that it's all a bit confusing. You can read more here:
aha, Plücker is how one gets Grassmann back down to dot/cross memory traffic?
K is developer Whitney's, or
Wit is threefold Halving, dedicatedly?
I am disappointed there is no Spice Girls schlager/cover band named Die Würze in der Schürze.
(although if I'm browsing YT for 90s covers, I guess that strongly implies my stage name should be "Old Spice")
March: when the elite need to seed their wheat to beat the heat. (no more chill? get the drill!)
EDIT: Sheaves, stalks, epis, germs, and fields — it just occurred to me that your noon demon in the rye might be metaphorical? (what is temperature in mathematics? the ratio of entropy delta, which makes sense, to energy delta, which requires a suitable def'n...)
I have a rudimentary understanding of differential geometry, but aren't tensors a bit more special than nd-arrays? I recall a tensor being an object that changes under coordinate transforms according to the jacobian of the transformation.
Yes, you're right, I was talking more about how to visualize a tensor. They also do have special transformation rules under coordinate changes where you have to compute the Jacobian to account for how the tensor "warps" regular Euclidean space locally.
>"...these ideas were totally revolutionary, and he ended up collecting them all several years later in his groundbreaking work, Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (https://dn790002.ca.archive.org/0/items/dielinealeausde00gra...), published in 1844 when he was 35. Grassmann essentially came up with a totally original and new conception of linear algebra. Instead of the more traditional development of systems of linear equations, vectors, matrices, determinants, etc., Grassmann had a more general and abstract conception of the subject, which focused on what is now known as an exterior algebra, also known as the "wedge product."
Unlike in the traditional development, where we use simpler operations like the dot product or cross product, in Grassmann's presentation of the subject, everything is couched in terms of the wedge product, which is a bit more abstract and harder to explain. In a nutshell, the wedge product of two vectors can be thought of as the region spanned by the vectors in space using the traditional parallelogram rule; it's basically what you probably already know as the determinant, but
instead of representing the quantity of the area spanned by this space, it's the space itself.
Basically,
the wedge product generalizes the determinant to higher dimensions
This is the crux: Grassmann had discovered and explored several mountain passes while others had not yet settled the valleys which they connected? Even if he hadn't been an outsider, he seems to have been (from a career standpoint) "too early".
(compare https://news.ycombinator.com/item?id=40312021 )