Yes, but who cared about it in 1830 (or earlier), and why did they imagine even for a moment that it might have some equivalence to the number line?
Looks like it was all about the quintics, somehow, but I don't know why they made the leap from that problem to geometry. I'm thinking maybe the equivalence to triangle-flipping is just like an amusing conceptual side-effect that happened by accident when working out stuff about permutations?
I don't think a Reuleaux roller functions very well if you flip it around a different axis, anyway, but I'll let you off because they're cute.
> Yes, but who cared about it in 1830 (or earlier)
Anybody with a Sphinx to move, struggling to make a purrfect circle.
> I don't think a Reuleaux roller functions very well if you flip it around a different axis
As an extruded 2D shape -> 3D solid it lacks a little in the mirror symmetry department, rotation is pretty much its limit .. admittedly a lacklustre submission, but cute indeed.
Galois was the first to realize the connection between polynomial roots and geometric symmetries through group theory.
Some simple examples can be found in subgroups of U(1). For instance in how Z_n is linked to regular polyhedra of order n and also n'th roots of complex numbers of the Unit Circle.
Kind of like how Z_12 is linked to an analog clock.
> why did they imagine even for a moment that it might have some equivalence to the number line?
A few of the groups which Galois introduced are what we now call Abelian (after Abel), which is to say that we can forget the order of elements within a product: AB == BA (if you get up to leave the beach and put on flip-flops then put on a shirt, you wind up in the same state as if you get up to leave the beach and put on a shirt then put on flip-flops)
Number theory studies products of primes, and here, always, although we generally must write down multiplicands in some order, it doesn't matter which: 2*3 == 3*2.
This connection would be enough for any modern undergraduate to consider applying general machinery built for quintics to the specific case of the number line, but in those days it took Euler (working before Galois) and Gauss[0] (working after? check this) to blaze the trails along this particular connection.
> maybe the equivalence to triangle-flipping is just like an amusing conceptual side-effect
Not just conceptual: spectroscopy is exactly why chemists are taught a little group theory, and triangle flipping is the simplest non-trivial[1] example.
Like programmers, who spend their days building up data structures and picking them apart[2], chemists are concerned with synthesis (building up molecules) and analysis (picking them apart; in principle this includes synthetic steps that make small molecules from bigger ones, but in practice this means checking your product at the end of synthesis to confirm that you made lots of what you were hoping to make[3], and little of what you didn't want to make[4]).
In particular, spectroscopy is a useful tool in chemical analysis, and very often[5] parts of a molecule will have a triangular symmetry, meaning that the peaks in a recorded spectrum[6] can be explained via a representation of the triangle-flipping group. If you set out to make, say, ammonia[7], but don't get any triangle-flipping parts in the spectrum when you run your tests ("characterise your product"), you know you failed[8].
[0] when Laplace was asked who the greatest german mathematician was, he replied "Pfaff". when asked why not Gauss, he explained "you asked for the greatest german mathematician; Gauss is the greatest european mathematician" (compare: the LUB of a set need not be a member of that set — EDIT: I guess the set of working mathematicians is always finite, so this comparison falls)
[1] in 0-D, a point has only the identity, so it's a degenerate group; in 1-D, a line segment does have a symmetry group (isomorphic to the booleans which are so important to CS) but unfortunately children do not learn about digons in elementary school, and must wait until they discover computer graphics to learn that edge AB is distinct from edge BA; indeed, they're explicitly taught to ignore that distinction in high school geometry, leaving the first non-trivial pedagogically-suitable example to be in 2-D: the triangle
[2] we've made some progress on also building up functions with the same aplomb as we handle data, but we're still not very comfortable when it comes to taking functions apart
[3] just as computer scientists often have a better-than-average knowledge of computer cracking, and physicists of bomb geometry, chemists tend to have a better-than-average knowledge of street syntheses. In particular, I have a second hand anecdote of undergraduates, who, having been in the process of characterising a synthetic product one evening, were interrupted by a grad student who, just by looking at the spectral lines, told them he hadn't seen anything that night but if they wished to continue exploring those particular [synthetic] pathways, they had better do so independently of university equipment.
[4] just as software engineers (who know what corners to cut to produce huge numbers of right answers and an acceptable number of wrong ones much more cheaply than only right answers) are generally paid better than researchers, ChemE's (who know what corners to cut to produce huge amounts of wanted product and an acceptable amount of unwanted) are generally paid better than their purer colleagues.
[5] why? (hint: it's the same mechanism —related to Natural primes— that makes binary taxonomies so popular)
[6] indeed, "spectrum" has been reborrowed back into maths to refer to something in algebraic geometry which is currently beyond my ken. If you poke around these areas long enough, you'll also find that von Neumann (who had physical, computational, and mathematical reasons to be interested) has had the "von Neumann regular rings" named after him, and rings are nothing but a pair of groups which interact in a certain manner. (the "regular" here being related to the "regular" in regular expressions, btw) Exercise: do regular expressions contain any rings?
[7] as the last century taught us, being able to make ammonia is very powerful, having applications both desirable and undesirable.
[8] Exercise: if you do see signs of triangle-flipping, is that enough to be sure you just made ammonia?
Thank you for the extensive reply. There's a hint in there about commutativity inspiring the connection, but my mental model is now simply "Euler did it", which somewhat like creationism relieves me from having to ask further questions.
Something I used to imagine: what if we were radically different creatures, like ant colonies, and by habit we communicated non-linearly (with thousands of limbs and organs swarming in parallel all over our mathematical work, perhaps written in 3D)? That could make equations mostly trivial, since our symbols wouldn't be constrained to any particular arrangement in the first place: and that seems kind of advantageous. But then grasping the concept of "non-commutative" would be a real strain for these poor ant-hills. They'd have to deliberately reintroduce linear ordering, maybe with special symbols to mark precedence.
(a) if you haven't read it, Chiang, Story of Your Life (1998) might have interesting aliens.
(b) the special symbols is a good point. Reading older maths papers is cool because you get to see all sorts of things people tried before we settled on what we use now. Two works that come immediately to mind: Principia Mathematica (1910) uses various numbers of dots instead of parentheses to mark precedence, while Peano, Arithmetices principia: nova methodo exposita (1889) uses very modern-looking notation, including parens as we would use them, but its expository text is all in latin!
(c) I don't think your aliens would have any more trouble with non-commutative than we have with commutative. Have you heard of the Boom Hierarchy? It starts with trees; when we add an associative law, so (AB)C == A(BC), then we only have flat lists; when we add a commutative law, so AB == BA, then we only have unordered bags; and finally when we add an idempotent law, so AA == A, then we have unduplicated sets. It turns out (exercise!) that if we have information encoded in any of these representations, we always have at least one way to represent the same information in all the other representations, such that we can "round trip" between any two levels of this hierarchy without losing any information.
So for programming, where we care about time and space, picking ordered or unordered representations can be very important, but for maths, where all that matters is the existence of invertible functions between all these representations, that decision is unimportant. Does that make sense?
> That could make equations mostly trivial, since our symbols wouldn't be constrained to any particular arrangement in the first place
You might find this little tidbit[1] from C.S. Peirce by way of John Sowa interesting then. Existential Graphs (EG) are an unordered diagramatic representation of mathematical logic. And Peirce is the real deal. His more conventional notation was adopted by Peano (who substituted the familiar symbols for capital sigma and pi (which created confusion when used in the context of broader proofs, even though sigma and pi pretty much directly correspond to what existence and universality mean)) and he is credited as an independent co-discoverer of both quantifiers with Frege.
For EGs, only one axiom is necessary: a blank sheet of assertion, from which all the axioms and rules of inference by Frege, Whitehead, and Russell can be proved by Peirce’s rules. As an example, Frege’s first axiom, a⊃(b⊃a), can be proved in five steps by Peirce’s rules
Peirce gives us the entire predicate calculus with three rules and one axiom. And of course it's all built on NAND.
And another teaser:
In the Principia Mathematica, Whitehead and Russell proved the following theorem, which Leibniz called the Praeclarum Theorema (Splendid Theorem). It is one of the last and most complex theorems in propositional logic in the Principia, and the proof required a total of 43 steps ... With Peirce’s rules, this theorem can be proved in just seven steps starting with a blank sheet of paper.
John Sowa himself is also no slouch, having been one of the leading lights of the earlier AI push. I expect advances in modern AI will come when we stop trying to do everything with ngrams and start building on richer models of knowledge representation.
I dont want to get too far into a joint and doritos speculation here, but I wonder if the fact that humans have a very small capacity for holding multiple objects in their minds, a small number of physical digits, and excellent visual acuity is why we do a lot of math the way we do. Group theory comes out of symmetry for example. Algorithms come out of linear stepwise problem solving. It takes us considerable mental effort to think about problems in ways that are not like this.
An example that stayed with me for years is when Adelman of RSA fame considered whether DNA could be used as a computer. (Spoiler: yes). It basically does all the computations at once, and then discards all the non optimal solutions.
