I noticed this subtlety about group homomorphisms today; I have no idea whether it’s at all significant but I thought I’d record it anyway.

Given groups G, H and a map between their underlying sets f : G ⟶ H, we say f is a homomorphism if it preserves the multiplication on G and the identity of G, ie if

(1) f(ab) = f(a)f(b) for all a, b in G
(2) f(1) = 1

As it happens, any set map f satisfying (1) above must also satisfy (2). A set map just needs to preserve the group multiplication in order to qualify as a group homomorphism. (It follows from considering inclusion maps that any subset of G closed under the multiplication is a subgroup of G.)

To prove this we need a lemma that uses the invertibility of group elements:

lemma if a2 = a then a = 1
proof aa = a implies aaa-1 = aa-1 implies a1 = 1 implies a = 1

Now f(1)f(1) = f(1.1) = f(1), hence f(1) = 1 by taking a = f(1) and applying the lemma.

This result is not true, however, for monoids in general (ie “groups” where the elements need not be invertible). Here is a simple counterexample.

Let M be the two element monoid {1, a} where a ≠ 1 and a2 = a. Then {a} is a subset of M and a monoid under the inherited multiplication, but it is not a submonoid of M because a ≠ 1.

In particular, the inclusion map {a} ⟶ M satisfies (1) but not (2) above.

On reflection this shouldn’t surprise us. There is no reason to think that a set map which preserves one algebraic operation should preserve any others.

The correct definition of a homomorphism is a map between algebraic entities that preserves all their algebraic operations. The fact that in groups preservation of multiplication suffices is an artifact of the tremendous simplifying power of invertibility.

[on a similar note, see this earlier]

I'm amused that Darwin, at whom I've been taking another look, should say that he also applies the 'Malthusian' theory to plants and animals, as though in Mr Malthus's case the whole thing didn't lie in its not being applied to plants and animals, but only – with its geometric progression – to humans as against plants and animals. It is remarkable how Darwin rediscovers, among the beasts and plants, the society of England with its division of labour, competition, opening up of new markets, 'inventions' and Malthusian 'struggle for existence'. It is Hobbes’ bellum omnium contra omnes and is reminiscent of Hegel’s Phenomenology, in which civil society figures as an 'intellectual animal kingdom', whereas, in Darwin, the animal kingdom figures as civil society.

The whole Darwinian theory of the struggle for existence is simply the transference from society to animate nature of Hobbes' theory of the war of every man against every man and the bourgeois economic theory of competition, along with the Malthusian theory of population. This feat having been accomplished ... the same theories are next transferred back again from organic nature to history and their validity as eternal laws of human society declared to have been proved. The childishness of this procedure is obvious.

While the romantic numskulls of Nazi Germany are dreaming of restoring the old race of Europe's Dark Forest to its original purity, or rather its original filth, you Americans, after taking a firm grip on your economic machinery and your culture, will apply genuine scientific methods to the problem of eugenics. Within a century, out of your melting pot of races there will come a new breed of men – the first worthy of the name of Man.

Russia had a small but flourishing eugenics movement before the Bolshevik Revolution of 1917. With the advent of the Communist regime, some biologists hoped that the application of scientific principles to reproductive policies, as to agriculture, would receive official support. But many Soviet biologists, recognizing that complex human behaviors and social values cannot be ascribed to genes in any clear way, found the claims of Western eugenicists naïve and class-based. Moreover the "hard" hereditarian line promoted by most Western eugenicists was at odds with the Communist views of the malleability of human nature and thus appeared to provide no role for the environment in shaping human destiny. The Central Committee of the Communist Party outlawed work on eugenics in 1930, making the Soviet Union the only country where eugenics was officially denounced by governmental legislation.

Woke up this morning to the sad news of Whitney Houston's death at the age of 48. It's only too predictable that her life story is being cast into a "good girl gone bad" narrative, and on one level this fits like a glove.

But something J at No Good Advice wrote last month suggests a different tack: that in Whitney the tragedy and neurosis were there from the start, and that her early carefree pop was anything but. Commenting on Whitney's 1987 smash I Wanna Dance With Somebody, J argues that it not a straightforwardly "happy song":

After the ecstatic release of the introduction, 'I Wanna Dance With Somebody' is pared down to a soft, melancholy twinkle. "Clock strikes upon the hour, and the sun begins to fade" – a lyric which could indicate anticipation and excitement is instead an indication of encroaching depression.

At the outset of the song, Houston is trying to "figure out how to chase my blues away". "I've done all right up to now, it's the light of the day that shows me how," she explains, "but when the night falls... my loneliness calls." Obviously, she's going out.

But the solution to her unhappiness isn't just partying, dancing and hedonism, it's real love – "a love that burns hot enough to last". And that pay-off at the end of the chorus  – she wants to dance with somebody, but not just anybody: "with somebody who loves me" – is as beautiful and perfect a pop moment as any I have ever heard [...]

On paper, then, 'I Wanna Dance With Somebody' is certainly not about feeling happy, or not in any direct way. At no point in the lyric does Houston say that she is feeling anything positive; she gives us no reason to think that she is dancing with somebody who loves her.

This hypothesis of a dark streak to Whitney's early hits is confirmed by an astonishingly haunting audio of her isolated vocal track from How Will I Know, posted up on Tumblr by Jake Fogelnet.

Stripped of its backing, Whitney's vocal is almost a case study in obsessional neurosis. Her insistent question addressed to the big Other – "How will I know?" – is all the starker when ringing out over a void. The curious line "I'm asking you cos you know about these things" only underlines the effect. Note also the counterpoint in the lyrics: "Don't trust your feelings", "Love can be deceiving", "Falling in love is so bittersweet".

how-will-I-know-vocal.mp3

(download)

Click here to download:

how-will-I-know-vocal.mp3 (4.78 MB)

A lot of this is of course drowned out by the 1985 song's jaunty synthpop production and the extraordinary day-glo colours of the video. But even here there are a series of odd details that jar the picture of wholesome happiness. The dancers all wear monochrome (Whitney herself is wearing a grey dress and still looks fabulous). One of the dancers is in a bizarre half-bride-half-groom costume. At one point we see Whitney hideously distorted in a fairground mirror. And occasionally the camera switches to an aerial view, revealing the dancers and Whitney to be scurrying about frantically, trapped in a maze like lab rats.

Note also that Whitney's constant questioning unease – how can she be sure that he really loves her? what guarantee can she have? – is typically associated with obsession rather than hysteria, with "masculine" rather than "feminine" neurosis. On this point I disagree with Mark Simpson's 2002 analysis of Whitney. I think Mark is right to point to a Nietzschean "will to power" in Whitney's voice, but this is interesting precisely insofar as it can't be reduced to "egotism" or diva clichés.

Whitney's obsessional quality comes out most strongly in what to me is her finest moment: 1999's comeback single It's Not Right But It's Okay. Again, the video tries to mask the darkness by playing up the I Will Survive style women's empowerment aspects of the song. Whitney leads a multiracial female chorus resolutely declaring its collective intention to soldier on. But again, this doesn't quite work – not least because the chorus consists entirely of young, beautiful and immaculately dressed "everywomen", which lends a certain implausibility to the proceedings.

And once more the details of the video undermine the positivity gloss. It's dominated by jet black, dark greys and metallic greens – perhaps a reference to the Nefertiti bust that Whitney's look brings to mind. We catch occasional glimpses of women clad in olive military costumes dancing in a strange formation.

The song came out in 1999 at the height of the UK garage scene, and was swiftly adopted as a club anthem. There were several 2step remixes, most of them preserving the upbeat elements mentioned earlier. But my favourite proceeds in a very different direction. It's a white label bootleg, adorned only with the letters "W.H" scrawled in marker pen. It sounds as though it's by X-Men, although I'm not certain of this.

The track is a good example of what I called at the time "vocal science" – extending to vocals the technological processes pioneered in the "breakbeat science" of the earlier jungle era. This in turn was part of general stylistic turn I called hypersoul, and Whitney gets a passing mention in my 2001 article on the concept.

The remix strips out most familiar elements of the song, foregrounding the vocal in a manner similar to the How Will I Know rip above. The beats are strikingly minimal, skittish and abstract, pointing forward to the grime and funky scenes that would later eclipse garage. 

All of this accentuates the deranged, cocaine-paranoid aspects of the song. Whitney no longer sounds confident that "it's okay". She is trapped and unsure about everything. The surveillance technology that surrounds her – a new phenomenon back then – only serves to make her more uneasy. She checks her errant man's credit card receipts, the caller ID on his phone. At one point she asks: Why do you turn and look at me? It's a beautiful treatment of a beatiful song, and one that stands up as an elegy to an extraordinary lost talent.

[Here's the unedited version of my Socialist Worker film review. The published version contains some useful background on the Nazi predelictions of Wallis and Edward culled from Paul Foot's fabulous 1998 London Review of Books article.]

Predictably, almost everything. Madonna's W.E. has been rightly greeted with almost unanimously derisory reviews.

Not least for her baffling decision to intercut the historical narrative of Edward VIII and Wallis Simpson with a pointless tale of a posh New Yorker - also called Wallis - and her tedious search for love (advised along the way by Ghost Simpson).

The film jumps back and forth confusingly between Historical Wallis and Pointless Wallis. The dialogue is ridiculous. The acting is dreadful. The politics are atrocious.

The Nazi sympathies of the Duke and Duchess of Windsor are at best glossed over and at worst defended. Early on we are told that they are Nazis - but by Modern Wallis's cartoonishly villainous husband.

So some people think they were Nazis, but those people are bad people who shag around and beat their wives. Got that, kids?

Andrea Riseborough phones in her performance as Wallis Simpson. She plays her as a dull and bitter narcissist. Given that this is meant to be a sympathetic portrayal, one dreads to think what the Duchess was like In Real Life.

What the reviews don't prepare you for is how boring the film is. Much of it consists of endless scenes of rich idiots mixing Martinis or fawning over jewellery.

One of the very few socially useful functions performed by the very rich is to act as an outlet for our fantasies of glamour and power. This lot don't even manage that.

And it's all curiously sexless. This is meant to be the romance of the century - the King renouncing his kingdom for love. But it's all utterly prim and passionless.

But every now and then the tedium is punctuated by a scene so cringeworthy and badly judged it leaves you gasping.

At one point Edward visits a poverty stricken Welsh mining village. The proles doff their caps, tug their forelocks and politely bemoan their lot. Edward loftily declares that they are "decent" people and that "something must be done".

This, apparently, was "socialism" on his part and the reason he was drummed out of office. Wallis chips in that Germany has got unemployment all sorted. No doubt she also praises the punctuality of the trains.

There are other such moments, such as when Modern Wallis meets Mohammed al-Fayed for a chinwag about the beastly treatment of foreigners who have the temerity to shag royals.

But the prize has to go to the party scene where the Duke and Duchess and all their chums get off their faces on speed. And dance with their funky black supermodel friends. To the Sex Pistols.

You think I'm making this up, don't you? I wish I were. The moment the opening chords of Pretty Vacant drift in through the loudspeakers your mind cries NO NO NO MADGE DON'T DO IT. But she does it. And you're left speechless.

Alas these flashes of what-were-you-thinking inadvertent surrealism hardly make up for the depressing awfulness of the rest of the film. 

There has been a bit of a pop cultural turn towards poshos of late. Downton Abbey. The King's Speech. Even Doctor Who has replaced Russell T Davies' multiracial council estate with a picture postcard English village and a Tory time lord.

But Madonna's W.E. is so bland it makes the rest of the sorry pack look like Brideshead Revisited. The most exciting moment in the film is at a Sotheby's auction where Modern Wallis spends $10,000 on Nazi Wallis's gloves. Which she later gives back to the Ghost Wallis. No, don't ask.

Curiously this film left me feeling a bit nostalgic for the culture wars of the 1980s. Roxy Music's Avalon was reactionary tripe but at least it served up something that could genuinely be called aspirational - an impossible aristocratic grandeur, a summery ennui.

But who on earth would want to aspire to this? Why would anyone want to be like this? This film is how bourgeois culture ends - not with a bang but a whimper. Time to burn it down.

I tweeted a link yesterday to this story which suggests that Brighton & Hove Greens are looking to cut back on trade union facility time, with the Tories cheering them on. On closer inspection it turns out to be a false alarm. Here's testimony from a Brighton & Hove union rep (thanks to Jonny Jones for bringing this to my attention):

A couple of points about this Sun story and similar articles linking Breivik to fascists in Britain.

• Breivik writes in his manifesto that he spent months on Facebook building up 5,000 contacts with fascists and racists across Europe. Having "harvested" some 1,000 email addresses, he deleted his profile. He then sent out his manifesto to these addresses shortly before going on his murder spree. Some 250 of them are believed to be UK based. So it would be surprising if leading BNP and EDL figures had not received a copy.

• Regarding Breivik's tales of "Knights Templar" and being mentored by an Englishman called "Richard", this stuff is pretty tenuous and has a strong whiff of fantasy about it. Even if "Richard" is real, it's highly unlikely that he is the idiotic Paul Ray. Mind you, Ray himself has fingered Alan Lake as a possible candidate. This is at least plausible, though it is still pure speculation.

Having said all that, I don't think the EDL will ever shake off their association with Breivik – and rightly so. They sup from the same political and ideological well, regardless of what formal links there were between them.

Breivik certainly was in contact and communication with EDL members via Facebook and the EDL forum (he was forum member 3614 under pseudonym Sigurd Jorsalfare). There's testimony from EDL members that he came over for at least one of their demos last year (the Geert Wilders one outside parliament and the Newcastle one have been mentioned). Some recall meeting him, though this could be overactive imaginations on their part.

But the crucial point is the ideological affinity – in terms of obsessions, symbolism, rhetoric, strategy etc you couldn't put a cigarette paper between them. The only differences Breivik has with the EDL are over tactics. He explicitly sides with their "crusader nationalism" over the various other strands of contemporary fascist ideology that he discusses in detail in his dossier. In this respect, the EDL – and the cabal of Islamophobic columnists who spread their worldview – are guilty as hell.

Meditation 12 of Alain Badiou's Being & Event introduces the concept of a natural multiple, or ordinal. It ends with a demonstration that "Nature does not exist", ie the natural multiples cannot themselves be composed into a multiple: the ordinals do not form a set.

The demonstration given by Badiou hinges on the fact that any hypothetical set of ordinals would itself be an ordinal and thus be a member of itself. But "auto-belonging is forbidden" by the axiom of foundation, hence this set cannot possibly exist.

There is, however, a problem with this line of argument: it suggests that the prohibition of Nature is a consequence of foundation, which in turn suggests that Nature could perhaps be admitted into being in the event of the foundation axiom's suspension.

But the interdiction of Nature runs much deeper than this, and holds even if we set the axiom of foundation aside. In fact the paradox of Nature's inexistence was discovered by Cesare Burali-Forti in 1897, making it one of the earliest of set theoretical paradoxes to be uncovered. It certainly predates the axiom of foundation, which was only introduced by John von Neumann in 1925.

The crucial point is not so much that a set of all ordinals would belong to itself, but rather that such a set would itself be well-ordered, and thus be isomorphic to a particular ordinal β. But β is in turn equal to the set of all ordinals below β. Hence the set of all ordinals would be isomorphic to a proper initial segment of itself – which is a contradiction whether or not we accept foundation.

One final point to note: this proof relies on the fact that any ordinal enumerates the ordinals strictly below it. This fact is trivial in Zermelo-Fraenkel set theory, where an ordinal is quite simply defined to be the set of ordinals preceding it.

But in other forms of set theory, notably Quine's New Foundations, this fact does not hold. And as it happens, the set of all ordinals does indeed exist in Quine's version of set theory; as does a host of other sets typically considered paradoxical, such as the set of all sets. See TE Forster, Set Theory with a Universal Set (OUP 1992), for more details.