The n-Category Café

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).

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 $\mathbb{R}[x]/(x^n + 1)$. Fallible Todd asks for your understanding…

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

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

Further mucking about reveals that $\mathbb{R}[x]/\langle x^n + 1 \rangle$ 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 $\mathbb{R}[x]/\langle x^n + 1 \rangle$ onto $\mathbb{R}$ for each real $n$th root of $-1$, and one onto $\mathbb{C}$ for each complex $n$th root of $-1$. So, for $n$ odd, we see there’s 1 copy of $\mathbb{R}$ and $(n-1)/2$ copies of $\mathbb{C}$ in $\mathbb{R}[x]/\langle x^n + 1 \rangle$.

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.

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.

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

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.

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.

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

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