I know nothing of math or physics but my unjustified intuitive prejudices have been hitting homeruns recently, so I'm going to guess that silver haired blue eyed alien lady who unlocks the universe by doing esoteric secret math on a yoga mat will turn out to be mostly bullshit. I could be wrong and have no way of knowing if I am right!

I had an octonion tied to my belt, which was the style at the time.

Quaternions are losing commutativity because of the Big Sort.

"The parentheses feel artificial."

That was a Seinfeld episode.

Anyway, I never thought of a plane of complex numbers before and I can't really develop a mental picture of it.

5: Does it have wings?

1: Neither will anyone else ever! I choose to believe in the silver haired blue eyed alien lady on the yoga mat!

Enough is enough! I've had it with these complex numbers on this complex plane!

Yes, but only ranch dress. No blue cheese.

8 is good but is this censored Samuel? No mf?

If you peel back the layers of the octonion, you'll find that complexity = mf.

9: Well, what's the problem then? Not so complex, and easy to picture.

Anyway, in my experience, you lose 90% of the people when you stick a modifier on numbers. "Real numbers" confuses thoughtful people because what is "real" about something like 49? "Complex numbers" sounds like something you could usefully ask for a definition of but "real numbers" sounds like somebody fucking with you over ephemera.

People who aren't particularly thoughtful probably just resent the idea that counting is something with a definition and various theories.

Much more bizarrely, the octonions are nonassociative, meaning (a × b) × c doesn't equal a × (b × c). "Nonassociative things are strongly disliked by mathematicians," said John Baez, a mathematical physicist at the University of California, Riverside, and a leading expert on the octonions.

I know nothing about any of this math, but that would definitely bug me. My approach and understanding of math has always been based on symbol manipulation (rather than, say, visual or spacial understanding) -- I'm good at algebraic manipulations, for example, and that reads to me as, "we will use the standard algebraic notation, but you can't perform the standard manipulations on it." I don't like that.

Failed to close my blockquote . . .

Also friend's dad is quoted in the article, which definitely means I'm special.

I debated for awhile whether to make a comment on the x-files time-warp quality of the lady-mathematician and ultimately decided it was gauche to comment on her appearance. But you all shmucks go right ahead.

My dad's friend was quoted in an article about binge drinking on a bus trip to Canada, but I can't find the link anymore.

Anyway, if people are going to create plausible but unverifiable theories and color them in academic garb, I wish they'd have the decency to call themselves Evolutionary Psychologists.

I like to think of numbers in terms of roots of polynomial equations. Suppose you start with polynomials with coefficients from the natural numbers (including zero). You discover the need for negative numbers, that is, the integers. Then consider polynomials with integer coefficients. You find out you need fractions (rational numbers). Polynomials with rational coefficients gets you real numbers. And polynomials with real coefficients gets you complex numbers. But polynomials with complex numbers doesn't get you any further.

This still leaves out most of the real numbers. But, I guess, one could think about limits of series of rational numbers or algebraic real numbers.

So I find quanterions mysterious, never mind octonions. I do not see how they are numbers. (I am aware that one can do some arithmetic with things that are not numbers: permutations, transformations, matrices.)

I believe in matrices because you need them to do regression.

Strangest episode of True Detective ever. Coal Fury, indeed.

Or is it Marvel comics?

"Get a Tilda Swinton clone to play a steely-eyed mathematician named Coal Fury."

-Stan Lee

That article makes the octonions sound way more mysterious than they are, but that's par for the course for pop math writing. They were invented in the 1840s, so they don't really push human understanding.

They do come up in a bunch of weird ways, though, for unclear reasons. There's a long list of theorems of the form "Here's a nice sensible list. Plus here's one or three weird examples because of the octonions."

I am now going to say a bunch of nonsense terms. The only alternative non-associative division algebras are the octonions. Lie groups over a field come in two infinite families, plus three exceptions that come from the octonions. Simple Jordan algebras are all boring, plus the octonions. Simple Moufang loops are either simple groups, or based on the octonions. They show up in topology too. Vector fields on the 7-sphere are parallelizable because of the octonions.

Through feats of navigation across vast distances using canoes made of real coefficients, the Polynomials discovered complex numbers.

Interestingly, the octonions are related to the E8 Lie group, which was the last "groundbreaking math that describes the universe" to get hyped. Coal Fury and Tony Lisi should get married and have mutant kids together.

Their first child must be named Fury Lisi.

24: thanks! I was hoping you'd come along and provide that kind of context.

Walt, do you understand (a good part) of the classification of finite simple groups? I only know this is something I will never understand.

26 beat me to the punch. I was just about to point out that since the surfer dude with the theory of everything based on the E8 group got his 15 minutes of fame, it's only fair that the yoga mat lady with the theory of everything based on octonions gets hers.

Well, I'm glad that someone is paying more attention to octonions, isn't the whole deal that with octonions you have more slots to stick your fields into (or is it your particles?) and therefore you can have a better match between the properties you need to define and the "numbers" that describe them?

At the same time, physicists are wondering why no one can find any dark matter, and the idea of tweaking the gravitational equations is raising its head again. I don't think anyone has actually seen any evidence of sterile neutrinos, but they're hoping!

My friend the iOS developer uses quaternions a lot to model those fancy 3D transition effects on the phones. And there's a bit in Finnegans Wake about Hamilton carving the quaternion multiplication formula on Broom Bridge. But I still don't understand how they work. Quaternions and me, a flash fiction.

I have a sterile neutrino in my office. Does anybody want it? I'm moving and getting ready to throw it away.

From wikipedia:

From the mid-1880s, quaternions began to be displaced by vector analysis, which had been developed by Josiah Willard Gibbs, Oliver Heaviside, and Hermann von Helmholtz. Vector analysis described the same phenomena as quaternions, so it borrowed some ideas and terminology liberally from the literature of quaternions. However, vector analysis was conceptually simpler and notationally cleaner, and eventually quaternions were relegated to a minor role in mathematics and physics. A side-effect of this transition is that Hamilton's work is difficult to comprehend for many modern readers.

Fun fact, Thomas Pynchon used the rivalry between quaternion and vector partisans as a minor subplot in Against the Day.

That's right, I knew I'd read about quaternions somewhere else.

I was going to ask why you can't combine octonions to get 16-ions, but I looked and sedenions are a thing. Only they're another degree of weird up from the octonions, which means... too weird by definition to explain the universe from a yoga mat? Or you need an octagonal yoga mat.

KAYAK I AM SO FUCKING FAR AHEAD OF YOU I MADE LITERAL MILLIONS OFF THAT IDEA.

36: In some ways you can get anything. You can define something of any dimension where you add, subtract, and multiply. If you want to divide, though, you can only do it in dimensions 1, 2, 4, and 8. (You can give other examples besides the complexes, quaternions, and octonions, though. Those are just the ones that ever come up without you looking for them.)

Classification of finite simple groups request seconded.

29: I think only 5 people who really understand it, and they are starting to die of old age. They are trying to write a "second generation" proof, and it's really a race against death.

I think the actual groups on the list are understandable. Most of them are readily realized as groups of matrices that have analogues over the reals. What's hard to understand is why are there extra ones, and how do we know there's not more of them?

I can say more about the finite simple groups, if you tell me what you're looking for. In a random coincidence, I read a survey about the finite simple groups last week (and one about the octonions a few weeks ago). I've been curious about this "Here's a nice list, plus a few extra" phenomenon.

I've been curious about this "Here's a nice list, plus a few extra" phenomenon.

The Lagniappe Conjecture?

Who is Lagniappe and why would he conjecture about Walt's mind?

Because I'm the philosophical zombie that philosophers have worried about. I lack conscious experience, qualia, or sentience.

I lack conscious experience, qualia, or sentience.

would hit that

What Is it Like to Be a Walt?

So I find quanterions mysterious, never mind octonions. I do not see how they are numbers.
Possibly there is a sense in which the quaternions are a completion of the complex numbers, that being the Caley-Dickson construction. Viewing the complex numbers as pairs of reals manipulated through CD, the quaternions arise when you iterate CD on pairs of complex numbers (and so on through the Octonions). The quaternions can then be viewed as roots of multinomials perhaps?

Also put me on the classification of simple groups list. I'd really like to know why the Monster Group, and all the other weird ones, fall out.

Vectors are much nicer to work with than quaternions, and generalize better. I've never used them for graphics work, but I've never felt the need to.

As for what counts as a number, enh, anything that's vaguely ring-ish feels numbery to me. Associativity is important but not a deal breaker.

Is it safe to say that there's probably some group corresponding to a meaningful organizing symmetry underlying the structure particle physics?

I think only 5 people who really understand it, and they are starting to die of old age. They are trying to write a "second generation" proof, and it's really a race against death.
This is fascinating. Why the die-out? Is the written work alone insufficient to preserve the knowledge? Is the knowledge worth preserving?

So this is my impression of how the proof goes. The proof is just a bunch of cases where you prove that a group fitting a certain description is already on your list. Most simple groups are naturally given by n-by-n matrices over a finite field. These are called "of Lie type". Each field has a prime number, p, associated with it. So step one is to identify a candidate p for your group. The way you do it is you identify something that looks like "upper triangular matrices" of characteristic p. Then you identify a candidate n, for the n-by-n part. This is called the rank.

The classification theorem proves that if the rank is high enough, then you really do have a group of Lie type. Below that rank somehow you have less room to maneuver, and this is where the sporadics come from.

Simple groups have lots of involutions, which you can think of as being like reflections. The reflections are like a characteristic 2 part, and you analyze the interplay between the characteristic 2 part and the characteristic p part. But p can equal 2, and then this breaks down.

So the proof splits into four cases: low rank p = 2, high rank p =2, low rank p odd, high rank p odd. Each case requires separate techniques. The low rank p = 2 case was the hardest, and took 1500 pages.

Both the low versus high rank split and small or big p splits happen in other settings. Oddly, axiomatic geometry is an example of the rank case, where dimension 3 and up is much easier. (This is a bit related, since you can picture matrices as transformations that preserve lines in space.) The octonions are kind of another example.

49: The proof was mainly worked out in the 70s and the 80s. The people who did it are just getting old. In theory, the published work is sufficient, but people are human and they make mistakes. They have discovered gaps in the proof, but there were experts on hand to fix the gaps. People have published papers filling in gaps as recently as 2008.

The whole proof is something like 12,000 pages, and my impression is that a lot of it is pretty boring -- you enumerate possibilities, and then you eliminate them one by one. Given that it seems a settled question, what incentive does a younger researcher have to carefully read it? If you could prove that the proof was wrong and there were more simple groups, then that would be epic, but if the proof is fine then you won't get tenure and you'll have to break out the yoga mat.

51: So not do further work, okay. But don't people learn parts of it to use for other stuff?

You're not sure that the 12,000 page proof is boring? Maybe a couple thousand more and you'd know without reading it.

And were all 12,000 proved by hand?

That was me.

15: If you want an example of communtative but non-associative arithmetic that you can see with an ordinary pocket calculator, there's floating point numbers (due to intermediate rounding). For example, on my Windows 10 Calculator app in Scientific mode, (-1e+50 + 1e+50) + 1e-50 = 1e-50, but -1e+50 + (1e+50 + 1e-50) = 0.

(You can also get some pretty hilarious results due to rounding with hidden guard digits - that same app insists that -1e+75 + (1e+75 + 1e-75) = 1.5878899595757734378621471801169e+37, which is off by over 100 orders of magnitude.)

FYI - if you type in 8008 into your scientific calculator and turn it upside down it says boob

How does 8008 look different upside down?

it's a complex boob-creating number

42 is good, but I came here to say that, although you would be forgiven for thinking otherwise, I did not write 45.

I like to think of numbers in terms of roots of polynomial equations. … So I find quanterions mysterious, never mind octonions. I do not see how they are numbers

I'm no mathematician but I feel like this is a "doctor, it hurts when I do this" kind of situation.

I'd have said 9009, but I'm no mathematician.

I am quite impressed with your post, hope you will have more posts.
duck life

The spambots have been been denied full-time employment and forced into the gig economy! The poor dears.

"These are called "of Lie type""

The modern term is FAKE NEWS.

It's relieving to know that no matter how smooth the fake news is, it will still be differentiable.

Silly, ajay, Lie comes from the name of their inventor, Sophus Lie... Sophus sounds like it comes from "sophist", a person who is good with clever but false arguments... I'm beginning to see what you mean...

57 is surprisingly on topic because one of the key figures in finite simple groups is Jacques Tits.

Thinking about it a bit more, the quaternions are roots of integer-coefficient polynomials, in the same way that complex numbers are; j^2-1=0. But they're apparently unnecessary roots, so now you can't factor a polynomial of degree n into n linear factors uniquely.

I'm surprised no one has told the story of 55378008 yet. Was the talk of fifth grade, or whenever we started using calculators.

52: There's a new book out called "Applying the Classification
of Finite Simple Groups: A User's Guide". It's less than 300 pages. A big part of the proof is: "Here's something that in theory can happen... (600 pages go by). It looks like that can't happen after all." The statement "X can't happen" is really all you need in practice.

The theorem that simple groups have involutions (the Feit-Thompson theorem) was the first really long proof in group theory, and the only thing you need to know to use it is that you can always assume there's an involution.

53: We're talking "boring to mathematicians". There's a website called the Stacks Project that has over 2000 pages, and people are so excited by it I'm pretty sure they masturbate to it.

54: Parts used computers, in that they had statements like "Maybe there's a group of size n wih this structure" and then they used a computer to check. Most of the computer parts have been replaced, but I don't know if all of them are. For big sporadic groups like the Monster, it's pretty routine to use computer calculations.

Thanks Walt.

My name links to a post in which I try to explain to myself what a finite simple group is. This doesn't get you very far.

In the comments to my post, somebody suggested Robert A. Wilson's The Finite Simple Groups. I bought this. I've read the start of a couple of chapters, but don't ever expect to understand.

Sorry. If anybody cares, try this link from my name.

Robert Anton Wilson wrote about Finite Simple Groups? I bet that's a trippy read.

If I remember correctly, in The Cosmic Trigger, he wrote about his college experience (EE and Math), and was pretty enthusiastic about some of it (mostly Markov chains, cf. also the character Markoff Chaney in the Schrödinger's Cat Trilogy). So you actually can find some Wilson trippy reads about math stuff, but I don't know how much of it is anywhere near accurate - I read those books before I learned the smattering of actual math I currently know.

Standpipe says that of course, the one writing about Finite Simple Groups is a different R.A.W.

Quick explanation of what finite simple groups are, for those who are not mathy but still curious:

A group is a mathematical structure that consists of a set of things along with a binary operation defined on it that behaves a lot like addition. That means, using additive notation:

1) It's associative. (a+b) + c = a + (b+c)
2) It has an identity, 0, such that for every element a + 0 = 0 + a = a.
3) Each element has an inverse element, -a, such that a + (-a) = (-a) + a = 0.

Groups are all over math; most things of algebraic interest are more specialized groups. Most kinds of multiplication behave like groups, if you set aside zero and use 1 as the identity. Basically, groups are just a nice, intuitive, yet abstract place to be working in.

A finite group is just a group that has a finite number of things. If you recall that discussion a week or two ago about rotating flags, the number of ways you can rotate a flag is finite and forms a group. Modular addition is also a finite group.

It's a bit harder to say what a simple group is. First, groups can have things called subgroups: that's just a smaller space inside the group such that if you apply the group operation to any two things in the smaller space, you stay in that space; since the sum of any two integers is an integer, the integers are a subgroup of the rationals.

So the cool thing is that for sufficiently nice subgroups, called normal subgroups, you can "divide out" that subgroup and reveal some structure of the original group. The even numbers are a subgroup of the integers, so if you divide the integers by the even numbers you get the group of numbers modulo two. Which is a really tiny group, just 0 and 1.

A simple group is a group that doesn't have any normal subgroups, so you can't divide anything out. If you take a finite group and keep on dividing subgroups out until you're left with a simple group, it turns out that it doesn't matter how you do this, you'll get the exact same set of simple groups in the end (up to isomorphism, which is the mathematical way of saying "enh, forget about it!").

And that's really cool because that's exactly how prime number factorization works. The simple groups are the primes of the group world.

Not only are there infinitely many simple groups, but they come in a bunch of different flavors. Some of them make sense, but some of them are friggin' weird, and so that's why all those old mathematicians made it their life goal to figure out what's going on.

Most kinds of multiplication behave like groups, if you set aside zero

Why do you have to set aside zero? (Oh because zero doesn't have an inverse element. Nevermind! But you could add in an inverse-of-zero element, right, and stipulate that (inv-zero * b) *c = inv-zero * (b * c) = inv-zero = inv-zero * 1 = 1 * inv-zero?)

I would just like to say that I'm pretty sure I've had more classroom mathematics than Walt, but he is eons beyond me in terms of intrinsic interest and autodidactedness, and also ability to retain what he knows.

78: Don't feel bad, heebie. Lacking conscious experience, qualia, or sentience makes Walt a natural at math.

77: You totally could do that if you want to, but usually when we're dealing with multiplication we're also dealing with addition (with their relationship described by a distributive law), and the result is not going to behave well. (Assuming that your second = was supposed to be an "and")

78: We probably had the same amount. I dropped out in the middle of writing my dissertation.

To be explicit, Suppose 0*Z = 1, and we also have addition with all the usual rules.

0+0 = 0
(0+0)*Z = 0*Z
0*Z + 0*Z = 0*Z
1 + 1 = 1
1 = 0

And that's a rather unpleasant situation, from which you can derive that everything is 0. That gives you the trivial ring. In some ways this looks like a field but it's sufficiently weird to not be considered one.

You also lose associativity. (Z * 0) * 5 = 5, while Z * (0 * 5) = 1.

Actually, I don't think I understand what you have in mind.

This thread is becoming an exception to my "I read all the comments on Unfogged" policy.

If 1 = 0, 5 = 0, so 5 = 1, so you keep associativity.

Don't try to help

from which you can derive that everything is 0

We are in agreement.

Actually, I don't think I understand what you have in mind.

No, on further thought you're right.

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The inhabitants of the hot and temperate zones were seen by the bogotanos as 'an ugly, colourless race which works little and grubs around amidst lush vegetation'.

Tough, but fair.

The section is headed "Social Darwinism".

Asocial Darwinism is when everybody playing videogames and masturbating doesn't have children until nobody with the gene for fixing the computer exists.

40: The revenge of Schleswig-Holstein.

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As noted at the other place, Sarah Coefield is back. https://www.missoulacounty.us/government/health/health-department/home-environment/air-quality/current-air-quality

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