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December 30, 2007

Non-Mathematician Rediscovers Rn, Cn

Posted by John Baez

I did a lot of historical reading about Hamilton, quaternions and the like when writing my review article about the octonions. When Hamilton invented the quaternions in 1843, it was a big deal: people hadn’t realized you could just make up new rules for multiplication. A lot of people joined in the fun, inventing their own hypercomplex number systems. Most of these systems aren’t terribly interesting. Hamilton’s pal John T. Graves found one of the really special ones: the octonions. Clifford found a whole bunch. But in general, most mathematicians now prefer to study hypercomplex number systems en masse rather than individually. They’re now called “real algebras”, and there are lots of nice general theorems about them.

This does not prevent amateurs from continuing to invent hypercomplex number systems and become excited about them. An amusing example was brought to my attention by David Farrell:

Brace yourself: Rawlings links his discovery to the Objectivist philosophy of Ayn Rand!

A few quotes from the above ‘press release’ (apparently written by Rawylings himself) will convey the spirit of the thing:

Toronto ON, Canada, December 25, 2007 — A Toronto, Ontario, writer and editor has arrived at a system of creating hypercomplex numbers — numbers that extend the complex number system to more dimensions — using only high school algebra, as viewed through the lens of Ayn Rand’s philosophy of Objectivism. He contends that this has implications for mathematics and the philosophy of science.

Rodney Rawlings calls his multidimensional numbers “RADN numbers” — for “rotating any-dimensional numbers,” because they have a property of rotation exactly analogous to that of the complex numbers. They are also commutative and associative like them.

He says that he arrived at this result by asking himself what exactly numbers are, how they arise in the human mind, and what their relationship to reality is. But these questions were only so fruitful because he used a correct philosophy, he claims — Ayn Rand’s. Any other philosophy, such as the currently influential one of Karl Popper, he says, would not have led to such a result. “This has two implications: first, that Rand’s philosophy has a strong element of truth, at least in the area of epistemology; and second, that the type of numbers I discovered must have a special significance, seeing as how they are intimately related to the basic nature of numbers.”

[…]

Rawlings contends that the RADN program (which, he hastens to add, is not a new one but already known to mathematicians under a different name) must have a unique status among the hypercomplexes, because of the way he, a non-mathematician, arrived at them by means of extremely simple algebra absent any of the tools of modern analysis, but armed with a philosophy that takes a particular and unconventional view of the nature of concepts and of mathematics.

Accordingly, Rawlings decided to write up his thoughts and reasoning in an essay entitled “Understanding Imaginaries Through Hidden Numbers,” which he is currently offering at a low price…

Perhaps Objectivism encourages scientists to offer their findings only to people willing to pay for them. Luckily, we don’t need to buy Rawling’s book to discover which algebra he has reinvented, since elsewhere he has admitted that they’re the multicomplex numbers.

I hadn’t heard this term before! It turns out the multicomplex numbers 𝕄ℂ n\mathbb{MC}_n are the associative real algebra freely generated by an nnth root of 1-1. When nn is even, this algebra is just n/2\mathbb{C}^{n/2}: a direct sum of n/2n/2 copies of the complex numbers. When nn is odd, it’s n\mathbb{R}^n: a direct sum of nn copies of the real numbers.

To anyone familiar with mathematics, these algebras are the mathematical equivalent of white paint or the C major scale: very fundamental, very important, very thoroughly worked over, very familiar and bland, very hard to say anything new about. But, to anyone first discovering them, they must seem incredibly exciting!

I offer this item mainly as further evidence for the powerful hold mathematics has on our imagination. It draws us with irresistible force towards the simplest and most beautiful patterns. We can’t resist it. These patterns will be discovered over and over, across the universe, for as long as intelligent life ekes out an existence in any corner. Let’s enjoy it!

Puzzle: what’s the slickest, most elegant proof that the hypercomplex numbers are isomorphic to the algebras I described: n\mathbb{R}^n and n/2\mathbb{C}^{n/2}?

Posted at December 30, 2007 9:42 PM UTC

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30 Comments & 1 Trackback

Re: Non-Mathematician Rediscovers Rn, Cn

Hang on – are you sure that the multicomplexes are isomorphic to n\mathbb{R}^n in the odd case? I would have thought × (n1)/2\mathbb{R} \times \mathbb{C}^{(n-1)/2}.

I don’t know about simplest and most elegant, but off the cuff I’d factor x n+1x^n + 1 into irreducible polynomials over \mathbb{R} and use the Chinese remainder theorem. I’m feeling a little lazy about spelling it all out… and I hope someone will come up with something slicker.

Posted by: Todd Trimble on December 30, 2007 11:07 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

You must be wrong, Todd, since my source was the always infallible Wikipedia, which cites an article on the carefully refereed arXiv (see page 26 of the PDF file, which is page 21 of the article).

Posted by: John Baez on December 30, 2007 11:37 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

O mighty Wiki! I am not worthy enough to gather the crumbs under thine table… I humbly beseech thee, have mercy upon my soul.

Seriously: did I misunderstand something? I read the problem as describing the structure of [x]/(x n+1)\mathbb{R}[x]/(x^n + 1). Fallible Todd asks for your understanding…

Posted by: Todd Trimble on December 30, 2007 11:55 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

That’s the problem all right: describing [x]/x n+1\mathbb{R}[x]/\langle x^n + 1 \rangle.

I think the mighty Wiki must be wrong, along with the omnipotent arXiv. You may have crushed a couple of small theories of physics, too.

When n>1n \gt 1, there’s a homomorphism from [x]/x n+1\mathbb{R}[x]/\langle x^n + 1 \rangle to \mathbb{C} sending xx to your favorite non-real nnth root of 1-1. And since this root ain’t real, the homomorphism has to be onto. But there’s no homomorphism from n\mathbb{R}^n onto \mathbb{C}, since the kernel of such a thing must be a maximal ideal, and the maximal ideals of n\mathbb{R}^n all look like n1\mathbb{R}^{n-1}. So, the algebras [x]/x n+1\mathbb{R}[x]/\langle x^n + 1 \rangle and n\mathbb{R}^n are not isomorphic, contrary to what I said.

Further mucking about reveals that [x]/x n+1\mathbb{R}[x]/\langle x^n + 1 \rangle must be semisimple, since it’s a quotient of the group algebra of a finite group. Since it’s commutative, it must be a sum of copies of \mathbb{R} and \mathbb{C}. To figure out how many of each, we can just count its homomorphisms onto \mathbb{R} and \mathbb{C} — and divide by 2 in the second case, since complex conjugation gives two homorphisms with the same kernel. There’s one homomorphism from [x]/x n+1\mathbb{R}[x]/\langle x^n + 1 \rangle onto \mathbb{R} for each real nnth root of 1-1, and one onto \mathbb{C} for each complex nnth root of 1-1. So, for nn odd, we see there’s 1 copy of \mathbb{R} and (n1)/2(n-1)/2 copies of \mathbb{C} in [x]/x n+1\mathbb{R}[x]/\langle x^n + 1 \rangle.

So, I think you’re right! And thanks to you, perhaps Rawling’s press release will actually have a beneficial effect. After some time for contemplation, I will remove the last error blemishing the otherwise flawless Wikipedia.

Posted by: John Baez on December 31, 2007 12:42 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

Neither bird, nor plane, nor even frog, it’s just little ol’ me…

Posted by: Todd Trimble on December 31, 2007 12:48 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

By the way, my argument was not supposed to be the slickest, most elegant way of setting up an isomorphism between multicomplex numbers and some more familiar algebras. There must be a nice 2-line way to do it. I think I can even guess what it is…

Posted by: John Baez on January 2, 2008 12:37 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

Recall that any proof is one line if you start sufficiently far to the left. Your proof looks sufficiently elegant except for the reference to the group algebra of a finite group. In fact, it seems essentially the same as what Todd Trimble outlined at the outset:

[x]/(x n+1)[x]/(f i(x))\mathbb{R}[x]/(x^n+1) \simeq \oplus \mathbb{R}[x]/(f_i(x))

where the f i(x)f_i(x) are the \mathbb{R}-irreducible factors of x n+1x^n+1, and hence, degree two or one. Since x n+1x^n+1 has one or zero real roots, for nn odd or even respectively, the result follows.

Hmm, two lines.

Posted by: Minhyong Kim on January 2, 2008 3:48 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

It’s also true that any proof takes one line if you know enough lemmas. I came to algebra late in life so there are probably some obvious things that aren’t obvious to me. So, I could easily see that for any real polynomial ff we’ve got homomorphisms

[x]/f[x]/f i\mathbb{R}[x]/\langle f \rangle \to \mathbb{R}[x]/\langle f_i \rangle

where f if_i are the \mathbb{R}-irreducible factors of ff. But, I felt a need to know that [x]/f\mathbb{R}[x]/\langle f \rangle is semisimple to conclude

[x]/f i[x]/f i\mathbb{R}[x]/\langle f \rangle \cong \oplus_i \mathbb{R}[x]/\langle f_i \rangle

For example, this ain’t true when ff has repeated roots. It probably is true whenever ff doesn’t have repeated roots — that’s probably something smart kids learn in school. But, I couldn’t instantly see why, so I decided to grab ahold of some other proof that [x]/f\mathbb{R}[x]/\langle f \rangle is semisimple, which just happens to be available when f(x)=x n+1f(x) = x^n + 1.

Posted by: John Baez on January 2, 2008 4:06 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

You can still conclude that isomorphism if all the f if_i are mutually prime (even if not necessarily prime themselves), e.g., when the f if_i are the prime powers in the prime factorization of a polynomial like x n+1x^n + 1. That’s what I had in mind when I said “Chinese remainder”. Then of course argue as Minhyong Kim just did.

Posted by: Todd Trimble on January 2, 2008 4:28 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

Both of you are right of course. I was actually thinking of the repeated roots criterion myself: Since (x^n+1)’=nx^{n-1}, it can’t share any roots with x^n+1 whose roots all have size 1. I guess I should have included that to increase the number of lines to three.


Posted by: Minhyong Kim on January 2, 2008 9:34 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

Sorta reminds me of the time we were sitting around in high-school physics and I got to thinking, “Could mass be a vector?” I nearly reinvented tensors… nearly.

And I did reinvent the Euclidean distance metric for Cartesian coordinates a few years before that, because I needed some way for the microbes in my artificial life simulation to know how far they were from one another. Ah, the days of MS-DOS and BASIC!

Posted by: Blake Stacey on December 31, 2007 3:15 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

Is anyone prepared to pay £1.38 to find out in 38 pages why Rand is so much more helpful than Popper?

Back at my old blog, I asked mathematicians to tell me which philosophy had been most useful for them.

Posted by: David Corfield on December 31, 2007 11:08 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

You could just email the guy and ask him, or post a question on that forum. I’ve never met an Objectivist who was reticent about explaining the advantages of Ayn Rand’s philosophy.

Posted by: John Baez on December 31, 2007 10:49 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

I do know of someone quite knowledgeable in category theory, who teaches how to become an Objectivist in Ten Easy Steps, for those people who don’t feel up to plowing through the Fountainhead and Atlas Shrugged. I’ve no idea if he reads this blog however, and I don’t know him well enough to ask him outright if he could assist us.

Posted by: Todd Trimble on January 1, 2008 12:34 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

That site is hilarious, especially the reader responses page. And I wonder if I would have stayed awake in Philosophy 100 if instead of Plato we’d had more passages to study like this:

She took off her nightgown, stepped to the window, sighed deeply and said: “The mind is the origin of all meaning. I mean what I say. Words in themselves are empty.” He lit a cigarette and admired her naked body which was illuminated by silver moonlight.

Posted by: Greg Egan on January 1, 2008 4:04 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

Happy New Year, Greg! You’re right, that page is pretty funny.

Posted by: John Baez on January 1, 2008 10:54 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

There used to be some other choice stuff on that guy’s website I mentioned: at one point you could even join a fan club (mostly populated by his female students, and with pictures of him smiling and dimple-cheeked, surrounded by hearts)!

Sadly, the Wayback Machine doesn’t go back quite that far, but you can get an idea of it here.

Posted by: Todd Trimble on January 1, 2008 8:40 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

I guess I’m a luddite.
No philosophy other than that there’s stuff out there to be discovered - and invented?

Posted by: jim stasheff on January 1, 2008 3:12 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

A few year ago, I went to a lecture of Daniel Dennett. He spent a while dwelling on ‘scientists as philosophers.’ The list included Einstein and Darwin, the latter being especially important for Dennett’s own program. However, he made it clear that he was interested primarily in the impact that such people had on philosophy, which generally had little to do with what they thought of as ‘their philosophy.’ In fact, Dennett was mildly derisive about the parts of a scientist’s writings where he ‘waxes philosophical.’ (‘That’s not the good stuff.’)

This is just one example, but over the years, I’ve had frequent contact with philosophers who felt that their actual practice was much misunderstood. I may be mistaken about this, but the basic thrust of the claimed misunderstanding was almost opposite to the misunderstanding of mathematics that David Corfield appears to be interested in. That is, in a different post David points out the neglect of mathematics’ higher aspirations. Meanwhile, several philosophers I’ve met seemed eager to point out that philosophy is not only about higher aspirations. Viewed from a slightly different angle, it occurred to me that for many scientists who read philosophy, it can easily be the portion that’s most important for the philosopher, that is, the part that represents some real progress in the tradition of the field, that fails to be ‘the good stuff.’ I believe this phenomenon is already visible in some of the comments that appear above.

There are a number of philosophers that I suspect, rightly or wrongly, were an important influence on me. And then, there must be quite a few whose ideas I reflect unwittingly. However, the supposed misunderstandings in both directions seem to be just a small hint of the complex structure of the ongoing discourse.

Posted by: Minhyong Kim on January 1, 2008 1:19 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

I am certainly in favour of more complex discourses between philosophers and scientists, and wary of certain philosophical projects which have left the possibility of any form of external influence way behind.

Dennett, as a philosopher of mind, has expressed similar sentiments, that fellow philosophers should engage more closely with psychology and brain science.

To give a sense of possible forms of interaction, my complaint of the neglect of mathematicians’ higher aspirations is part of a larger complaint of the failure to recognise what is at stake regarding the rationality of a tradition of enquiry. My conception of this rationality informs my participation in this blog. At the same time I get to learn wonderful new ideas about space, symmetry and quantity.

Posted by: David Corfield on January 1, 2008 2:25 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

What I find most fascinating is the view (mentioned a couple of times in the forum thread) that the RADN/multicomplexes should be judged particularly significant because of the philosophical rectitude of the reasoning that led to the definitions rather than upon how nice/compact/useful the implications of those definitions are. (Indeed there’s a brief criticism of other mathematical objects’ definitions for being less clearly justified.)

It’s very much a “generative” rather than an “extractive” view of philosophy/mathematics. At least in the area I’m familiar with, concepts and their definitions get proposed from vaguely observed phenomena and then iteratively molded by assessing whether changes in the definitions “improve” the “nice/compact/useful”-ness consequences.

A sort of example is the “is 1 prime?” question. My understanding of prevailing thought is that this isn’t decided by arguing whether or not the definition “it is divisible by exactly only itself and 1” applies or not, but that there are so many theorems (eg, about finite fields) that need “prime” replacing by the more complicated “prime (not equal to 1)” in their statements, so defining 1 not to be prime is the better course.

Posted by: bane on January 4, 2008 11:51 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

This is an interesting point, and one which aligns you with the philosopher Imre Lakatos. The point of ‘Proofs and Refutations’ is that concepts are properly formed and modified in mathematical dialogue.

There perhaps is still space for what you call ‘philosophical rectitude’, but it must take place within the context of a mathematical discussion, such as embodied in this blog.

Perhaps with the complexity of modern mathematics, the days of reflecting on the form of our spatial intuition as a means to making progress in geometry are over. We can no longer be Hermann von Helmholtz. But there certainly is still the need to stand back and reflect, in a mathematically informed way, on the fundamental concepts of the field.

Posted by: David Corfield on January 5, 2008 11:29 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

NB: I haven’t shelled out to read the detailed document.

I wasn’t making a point about whether looking at things from a philosophical viewpoint was bad (which I wouldn’t claim is true, and probably “rectitude” is a more negative word than I wanted) but something more banal. Rather that the fact that a definition was arrived at by strong logical/philosophical reasoning should be a stronger criterion for it being important than, once you’ve somehow hit upon a good definition that gives nice consequences figuring out a good logical/philosophical status for it afterwards. In either case, one could advance things by discussion and debate.

To put it very crudely, it’s the idea that you can figure out the totally optimal definitions/concepts before you even look at their consequences. The only philosophy book I ever struggled through was Proofs and Refutations (a decade ago) and I seem to recall debate about getting nice properties for platonic solids causing revisions to initial definitions, so I think I agree with that.

Posted by: bane on January 6, 2008 2:16 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

Not just Platonic solids. It was about defining ‘polyhedron’ properly to make sense of the observation that VE+F=2V - E + F = 2 for many familiar solids. It ends up at the analysis situs of Poincaré (having missed some rather important Riemannian contributions).

I can see that prospective philosophical thinking would be more impressive than post hoc rationalisation. My suspicion, though, is that any such prospective work will today require a considerable mathematical education.

Posted by: David Corfield on January 7, 2008 9:46 AM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

By the way, I like how Rawling’s acronym for his “rotating any-dimensional numbers” is an anagram of RAND. Did y’all notice that?

Posted by: John Baez on December 31, 2007 10:51 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

I noticed it, and it made me wonder how smart he can be to not come up with “Rotations in Any Number of Dimensions”

Posted by: John Armstrong on December 31, 2007 11:03 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

Well, you know how subtle Objectivists like to be.

Posted by: Blake Stacey on January 1, 2008 4:58 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

I have naturally been watching for reactions to my press release, and came across this discussion. I thought it might be to my benefit to make a few clarifications here about the essay to which my release refers. (And yes, RADN does refer to Rand as well as rotations; the fact that it also resembles my name clinched the matter for me!)

The aspect of my piece that I thought was newsworthy was not mathematical, but philosophical and specifically epistemological. As I mentioned in the release and in the essay, the RADN numbers, as I have called them, have already been identified by mathematicians as a certain type of hypercomplex number with real coefficients. I admit that when I first hit upon them, I thought it possible they were a new idea, but deep down I suspected that they were not. Extensive Internet investigation confirmed my suspicion.

So the interesting and significant thing is not WHAT the RADN numbers are, but HOW I arrived at them–by purely epistemological thinking about the nature of numbers as such, and the significance of the complex numbers in their relation to reality. The fact is, I did not even know about hypercomplex numbers, or even that the complex numbers had been extended by mathematicians. The idea that they must be able to be extended was something that arrived at me after I had thought out the reality-connection of complex numbers.

Another important thing, I thought, was that all this happened in the course of my effort to extend the thinking of Ayn Rand into the area of mathematics, a project she herself had undertaken in the last years of her life. (She had always been of the opinion that mathematics had a deep connection to concept-formation, a topic with which she had dealt in some detail.)

Finally, the fact that I arrived at one, and only one, type of hypercomplex number out of all the established or possible types was also interesting. If my thinking on the nature of numbers was correct, then this meant that the RADN numbers have some sort of special status. The first post above lends some confirmation to this, but I think there may be more to be said, and if I get the time I intend to look into the matter, using what math ability I have. (If my thinking is fruitful, I will create a revised version of the essay.)

In other words, while some mathematicians may not think this class of numbers is “interesting” by a certain set of standards, from the standpoint of epistemology and perhaps other aspects, they may be very interesting indeed.

I think it is obvious that my motivation is not to make a big profit. I just thought that since my essay does (in my opinion) have a certain value, it would be nice to make a bit of pin money out of it. (There are other factors in my decision also.) It is certainly no implication of Ayn Rand’s philosophy that thinkers should not offer what they regard as innovative ideas for free in the journals of their discipline. (In my case, of course, the relevant journal would not be a mathematical one, but one devoted to philosophy or epistemology.) I quite understand the reluctance of professionals in any field to pay money in exchange for the thoughts of someone totally unknown to them. However, I have made several sales, possibly to persons specifically interested in Ayn Rand’s philosophy. Perhaps discussion generated by them will make it possible for others, such as yourselves, to learn more about what I have to say.

Posted by: R. Rawlings on January 1, 2008 5:00 PM | Permalink | Reply to this

Re: Non-Mathematician Rediscovers Rn, Cn

The above will be my only post here. I just thought it especially important to dispel some possible misunderstanding.

Posted by: R. Rawlings on January 1, 2008 5:06 PM | Permalink | Reply to this
Read the post Two Cultures in the Philosophy of Mathematics?
Weblog: The n-Category Café
Excerpt: A friend of mine, Brendan Larvor, and I are wondering whether it would be a good idea to stage a conference which would bring together philosophers of mathematics from different camps. Brendan is the author of Lakatos: An Introduction, and...
Tracked: January 2, 2008 7:21 PM

Re: Non-Mathematician Rediscovers Rn, Cn

After revisiting Andrej Bauer’s excellent send-up of Objectivist theory, I had a question (for Andrej or anyone else who happens to be knowledgeable about the lore):

The page says Ayn Rand submitted an article to Acta Mathematica on a result in Euclidean geometry that she discovered in 1936. I assume that’s actually the case, but wasn’t able to find anything on this through Google except references to Andrej’s page. Where is this fact recorded? And what was the content of the journal’s reply to Ayn Rand, besides the fact her article wasn’t accepted?

I did research the result enough to discover that it’s part of the (rather extraordinary) phenomenon of the “nine-point circle”, and that it’s essentially been known since around the early years of the 19th century (so Rand would have been more than 100 years late).

Posted by: Todd Trimble on January 10, 2013 12:42 AM | Permalink | Reply to this

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