"But if you can't picture it..."

The other day, I was talking to a couple of friends of mine - a literary theorist and a historian/political scientist - about the (apparently) recently proved Poincaré Conjecture. The Poincaré Conjecture involves a statement about the 3-sphere - that is to say, something like the usual sphere, but one dimension bigger. In order to visualise a 3-sphere, you would need to picture it in four dimensions, just as the usual sphere (2-sphere) is usually considered to be a two-dimensional surface, embedded in three-dimensional space. The idea of four dimensions can provoke puzzlement among humanities students of a sort to startle a blasé mathematician for whom such things are second nature. My friend the student of history and politics asked, with a slightly concerned look on his face,

"Can mathematicians actually picture four dimensional space?"

"Roger Penrose says he did it - briefly - once," I said, grinning [1].

"No, but are there people out there who can actually..."

"Not that I know of."

My historian friend was relieved. My literary theorist friend was confused. "If you can't picture it," he asked, "how could you have any intuition about it? I mean, you could just say whatever you wanted about it and no-one would be able to refute you."

The first way that students of mathematics start dealing with four (or more) dimensions is fairly simple. If you're in two dimensions, you can describe any point with, say, an x co-ordinate and a y co-ordinate.


To go to three dimensions, you need a third co-ordinate in, say, the z-direction (use the previous picture but imagine the third axis coming out of the screen towards you.). To go to four dimensions, add a fourth number, and there you have a way of describing four dimensional (flat/Euclidean) space.

It gets more complicated when you start dealing with curved surfaces, but the idea is similar. If, earlier, you wondered why we consider the surface of a sphere to be two-dimensional, think of it this way. The Earth is approximately a sphere. We can describe positions on the Earth's surface with two co-ordinates: latitude and longitude. Two co-ordinates, two dimensions. Exact definitions might worry about the poles, where there is no real longitude to speak of - it could be anything - but there are fairly simple ways of getting around that which need not concern us here.

Besides the question of four dimensions, there is a broader principle raised by my friend's comment. The extent to which intuition plays any role at all in the basic system of mathematical proof is arguable. This is a fascinating topic in its own right that deserves a post of its own, so for now I shall simply promise to address it later.

[1] He says it in The Emperor's New Mind.

What I Wish I'd Said About Women And Sex...

... because "what I wish I'd said" must surely turn up on blogs everywhere.

"I've never had sex," I admitted, one drunken night several months back.

"Oh, don't worry," he said. "You're not really missing anything."

I shall pause to give you time to choke at the idiocy of this.

Now I shall perhaps provoke confusion when I admit that my reply was not "Don't be silly" or "That's absurd" - but please remember that "That's sexist and absurd" is sometimes not an easy thing to say if you want to be taken seriously. So I replied with a heartfelt

"Oh, don't say that!"

Yeah? Makes sense to me, but it ended up meaning we were having a conversation about whether women can really get anything out of sex.

"Look, we'll ask A__"

A__ (not his real name, or even his real initial, actually; just the first letter of the alphabet) sobered up visibly and thought about the issue. "I think," he said, "... certainly within the context of a loving relationship, women can definitely get something from it, yeah."

Okay, be nice. The generous way to view this is to note that A__ has a long-standing girlfriend and is therefore (hopefully) speaking from experience when providing this counter-example to the idea that women don't get much from sex.

Agreement from the guy who had been speaking to me initially; A__ waffles a bit more and then asks me:

"I dunno - I mean, what do you think, what do you feel?"

A__ is, I suspect, consciously allowing me, as a woman, to define my own experience; he is a nice liberal man who generally tries and, yes, usually succeeds; this conversation does not really show him at his best. However, it is me who fails at this point.

"I don't know," I say, and it's true, of course; how could I know, of my own knowledge? "I mean, it's not like I'm waiting for any particularly special moment - I'm just waiting until I feel comfortable." Sufficiently comfortable, anyway; not, like, comfortable comfortable.

Actually, I'd give anything for a comfortable comfortable opportunity to have sex. I just don't know when I'll get one.

But look, let's go back.

"Don't sat that!" Why? Because "you're not really missing anything" is, get this, exactly equivalent to the statement "sorry, Lynet, there is no satisfaction." It's a horrible thought - one of my worst irrational fears, actually. Apparently the man I was speaking to missed the point entirely. He probably honestly doesn't realise how it feels to be a woman in my position. I ought to have corrected his ignorance, but it's hard. It's very personal to talk about these things. Stating it here is an attempt to redress the balance; the best I can do. So let me tell you.

The worst times are when I ovulate. Seriously, at least when I'm getting my period I have a tummyache to distract me*. There have been times when I'm ovulating when I end up lying in bed, drenched in my own sweat and dissolving in my own sexual frustration, trying to masturbate though my clitoris aches 'cause I've been rubbing it too hard - I've been over once already and it's not enough - and when I shove myself up and over into orgasm I spontaneously burst into tears because even as my body shakes I feel as though it's really not doing anything, I still need sex just as badly (oh, gosh, I just burst into tears typing), and it just feels like nothing's going to help, nothing not ever.

*Ovulation and menstruation are the two times when a woman's libido is highest, in general. Most sources say the feeling is stronger on average during ovulation, but it's certainly perceptible at mestruation; sometimes I can tell when my period is arriving by the increased tendency to think about sex.

Now, I'm happy to say that this has only happened to me a couple of times - most months aren't that bad. I mean, there's no time of the month when I don't think about sex at all, and there is no time of the month when I don't want sex, but, while it's not unusual for ovulation to be a difficult time of the month, I don't always end up crying.

"You're not really missing anything"? NO! Please, please, no! Please, please, please, oh, please, no! Tell me it isn't true. Tell me it doesn't always feel this bad. Tell me one day I'll be okay in some sense of the word, I know I won't ever be free from sexual desire, but tell me that - I don't know - tell me something.

I'm crying again.

"In the context of a loving relationship"? Tell that to my cunt. The only thing I need a loving relationship for is to provide a situation in which I can have sex and still trust the person who I am having sex with to respect me. A situation which provides the latter but is not equal to the former would still be fine; I just don't like the idea of someone touching me if he thinks he is degrading me - or increasing his masculine power at my expense - in so doing. The idea makes me sick. Especially when I'm ovulating. And it would hurt me even worse than the sexual frustration that makes me cry.

But I didn't say that at the time.

It's hard to show people a part of yourself that's so very fragile.

Alphabet Meme

Alon hasn't tagged anyone - so I've just picked it up for myself.

Accent: New Zealand, overlaid by years of singing training.
Bible Book that I Like: Proverbs - except for all that stuff about believing in the Lord - and the stuff about how horrible loose women are... hmmm... maybe I should pick the Song of Solomon instead. It's cool how there's a part of the Bible that acknowledges the beauty of sex.
Chore I Don't Care For: No-one makes me do horrible chores at the moment. The worst I have to do is laundry - and that's usually a nice break from studying.
Dog or Cat: I left my cat Rosina behind when I left NZ to study in the UK. I miss her.
Essential Electronics: Laptop - what else?
Favourite Cologne: My least clued-in set of cousins gave me a set of two types of cologne once. I've had them for years, but on special occasions I'll sometimes wear one or the other. One smells of incense; the other one is more flowery.
Gold or Silver: Silver.
Handbag: Black leather, fairly small.
Insomnia: My sleeping pattern gets out of joint very, very quickly. Sometimes I'm definitely an insomniac; occasionally I manage to get back into a decent pattern.
Job Title: Student.
Kids: None, thank goodness - though I'd like to have some when I'm older, assuming I'm that lucky.
Living Arrangements: You'd call the place I live a 'dorm' in America, I suppose.
Most admirable trait: Honesty.
Naughtiest Childhood Behaviour: Throwing huge noisy tantrums. Especially when I did it in a public place.
Overnight Hospital Stays: Don't think I've ever had one.
Phobias: A slight, but manageable, fear of small spaces - but it has to be pretty small before it bothers me.
Quote: All who believe in daffodils while the snow falls about them are leading uneasily beating lives, their rhythm is the lost note that cannot or will not join the chord because although it will gain security and strength from being with the other notes it will at the same time forget the sound of itself, and therefore it stays alone in strange hollow places where there is no other music. The loneliness is the price and the reward. - Janet Frame, in her story Snowman, Snowman.
Religion: Atheist.
Siblings: Three younger sisters.
Time I Wake Up: Variable. See Insomnia.
Unusual Talent or Skill: Learning. Mwahahaha.
Vegetable I Refuse to Eat: Artichokes. Hate them.
Worst Trait: Well, I spend an awful lot of time fighting the urge to react to social insecurity by closing off from people. But I do manage to avoid that, most of the time.
X-Rays: One, when I broke my arm when I was four.
Yummy Stuff I Cook: I don't cook much, here, but I used to like cooking back home. I had a quiche recipe that always worked rather well...
Zoo Animal I Like Most: Meerkats.

The Truth

I've finally discovered the truth.

Actually, I've always had something of a love for the truth of things. My mother tells me that when I was a toddler I initially didn't believe lying was possible, and then, when I was convinced, still had trouble understanding why anyone would want to lie.

Later, when I was four or so, I vowed to myself that I would never tell lies - apart from 'polite lies' ("I like my new top, thank you Gran"). I think it was my mother who suggested the exception.

I told my first, proper, protecting-myself-from-disapproval lie when I was five. I felt guilty about it for years afterwards. My sister and I had been sneaking out the side fence (I felt so daring for sneaking about a metre into the path that led off the street to the neighbours' gate). The lie was brief. It may even have been a reply like 'nothing' to a question about what we had been doing. At any rate, my sister, who was only two, and much more inclined to tell lies in general (she cares more for love than truth, always has), seems to have been convinced that since I was doing it, it must be okay, and blithely showed our mother the gap in the fence.

Mum wasn't mad; she would have hesitated before discouraging us from exploring, in general. And if she was conscious of my having lied initially, she didn't show it, which is why I think my lie may have been imperceptible to the less idealistic adult eye. I think I almost felt more terrible because no-one but me was aware of my crime (and there was no way I was going to tell anyone).

As I got older, my notion of honesty grew. Somebody once referred to the idea of giving some real part of yourself on stage as 'honesty', and the idea took root in my head and flourished beautifully into an idea of being true to some genuine feeling when you express yourself. When I see it in other people, I actually find it the sexiest thing on Earth.

On a slightly different note, which is more truthful: a factual statement from which people might get the wrong idea, or a statement which is not strictly correct, but which is more likely to give people the right general feel? That is, is the truth of a statement in ordinary conversation best measured by its content, or by the way it will be heard?

What about poetry? I used to have trouble with Keats:

A thing of beauty is a joy for ever...

Yep, that's right, folks, when the sun has turned into a red giant and long since engulfed the Earth, those crocuses on the way to college will still be a joy.

A thing of beauty is a joy for ever
Its loveliness increases, it will never
Pass into nothingness, but still will keep
A bower quiet for us, and a sleep
Full of sweet dreams, and health, and quiet breathing.

That's the beginning of Keats' Endymion, by the way. My quiet bower is up at the top of the rhododendron tree, with the sunlight streaming through the spring leaves of the ash tree, and me, twelve years old, relinquishing pain for a while and only seeing beauty.

A poem can be 'honest' in the same way that a freely given emotion can be honest; a rich way that goes beyond what can be factually contained in a given set of literally interpreted words. This increased breadth in what can be conveyed admittedly comes at the price of complete and definable accuracy. "For ever"? No, not really. But which of us knows forever?

We are getting close to what a friend of mine (who just happens to write poetry) calls "truth of the heart" as opposed to "truth of the mind". If it conveys some truth about the way you see the world, is it okay to paint your life with metaphors? With myths, like the ancients did, personifying the power of the storm and the fertility of the crop?

Can a religious ritual act as a physical metaphor? If so, does this mean there is "truth" in it, if it embodies some genuine feeling?

When people say that religion will die, that it's just too stupid, I don't know if they are right. I do know this. The non-scientific part of us that goes into poetry isn't going to die. And there will always be people who value that way of thinking above hard objectivity, who will discard the clear evidence of their own senses in favour of "melodies unheard" (Keats again, and that line in the second verse was the part of the poem that first caught my eye; it hits me like a sharper version of deja vu).

I think it is possible to care for both ways of thinking. I suspect that we require the more objective input of our senses before the imagination could even take flight. I know we need to trust our senses in order to survive. Still, the experience of beauty, the feeling of love - these things are part of what makes survival worth it in the first place. I love science, but I have never been short on imagination.

I tend to think that the line between an imagined story and the truth or an attempt thereof should never be blurred. [EDIT: Even when I'm taking some belief as given, I almost always make a little note in the back of my mind that that's what I'm doing. Somehow that doesn't get in the way of the purpose of believing it in the first place]. Some people disagree, and always will. That doesn't make them right, but many of them are worthy people, nevertheless, who contribute to the world in other ways.

Back to Ode on a Grecian Urn:

When old age shall this generation waste,
Thou shalt remain, in midst of other woe
Than ours, a friend to man, to whom thou say’st,
“Beauty is truth, truth beauty,”—that is all
Ye know on earth, and all ye need to know.

No, no, Keats, no! Rightfully lauded poet though you are, I will never forgive you for that one!

Edited slightly for The Humanist Symposium - but all I did was cut a basically irrelevant paragraph, alter some punctuation here and there, and add the sentence above that is declared as an edit.

Why do men seem so much more eager for sex?

Being new to blogging (even though I've been hanging around for a while) means that I keep seeing fairly old posts. This one is about 1/9 of the way down the huge pile of comments on this thread:

In casual social situations of single people, if a woman suggests sexual activity to most any man there, he's willing, probably eager, and more than likely thrilled. If most any man there suggests the same to a woman, she'll say no, probably be offended, and very likely will decide that he's a shallow jerk, regardless of whatever interests and attitudes he might have.

I quote it not to refute that statement in particular, but because I have my own thoughts on this phenomenon, and I was finding it difficult to describe without sounding silly. I kept saying things like "People think women can have sex whenever they want, whereas men have to keep asking and asking", which is ridiculous, everyone knows there are plenty of sexually frustrated women out there (don't they?), and the reason men have to 'ask' is because that's the convention and women don't usually break it (my discussion below should suggest some reasons why they don't break it to anyone with sufficient imagination). Described as above, though, it at least seems to be getting at something that looks like it might have a grain of truth in it. I do think there is a(some?) genuine social phenomenon(s) here, buried under the inevitable generalisation.

Before I get to my main point, though, I wouldn't be me if I didn't nitpick:

In casual social situations of single people, if a woman suggests sexual activity to most any man there, he's willing, probably eager, and more than likely thrilled.

Ok, can I just ask something? How many men have actually been in that situation? I mean, when men say they'd be thrilled, can they point to any examples? And if so, was the man in question in fact thrilled? I'm not a man, let alone an "average man", so I don't know if the average man would be thrilled if the average woman suggested sex. I have had at least one male friend mention being propositioned with the remark "not flattering, trust me", so I know it's not always true. The question needs to be asked if it is true in general. I'm wondering if part of the reason men can say that is because they've never seen the reality, but I don't actually know if that's a reasonable suggestion.

Now that I've got the minor question out of the way, I'll get to my main thought on this matter.

As far as I can tell, in a "casual social situation between single people" it is just about impossible for a man to lose status by having sex.

There are some notable exceptions to this, the cruellest and most obvious being "unless the girl is fat", but, in the main, I think it holds.

Similarly, it is just about impossible for a woman to gain status by having sex.

Exceptions to this rule tend to be both equivocal and reliant on there being some commitment between the two people. For example, a woman might gain status by having a successful husband (personally I think such 'status' would probably annoy the heck out of me, but some might disagree). However, this is a far cry from a casual sex situation. In general, a girl having casual sex is going to have to fight to keep the status she has, let alone gain any. Sometimes I scare myself by wondering if the word 'slut' has more synonyms than the word 'good'. If it's true, I don't want to know. The idea behind the word is pervasive, and it means more than 'promiscuous', it means more than 'woman who has sex'. Mostly, the meaning which bothers me is that of woman who is used - used sexually by men. And if being perceived as being used doesn't lower your status, I don't know what does.

A woman is less likely to be perceived as being used if she makes a show of being choosy. That's a point in itself and I'll leave it there.

Another point, though, is that if a woman appears eager for sex, she might end up with less control over the situation. Unless she trusts the man, she can't be sure he won't just do whatever makes him happy with no care for what she wants. Ideally, she would have a say in these things no matter what, but there are men who would be oblivious to their own sefishness (especially if she was begging for sex in the first place), and there are women who would find it hard to protest.

Add that to the fact that the man is essentially asking her to put the way people see her on the line in a way which simply isn't true for him, and the fact that it still isn't conventional for the woman to do the asking; add the stereotype that women are less interested in sex...

... and biological determinism really has only so much explaining left to do.

EDIT: It occurs to me that I didn't really need to reference biological determinism to make my point. The stuff I've shown gives reasons why women might appear less interested in sex that do not rely on any actual difference in sexual interest between the sexes.

There may well be a difference between male/female sex drive, on average, but I think people probably overestimate it - there are just so many reasons why they would.

Things are becoming clear.

More on a similar topic, because I'm still thinking about this 'objectification' thing, and my thoughts keep being refined.

Maia at Alas! A Blog gives an insightful commentary in which she states

There are many, many different ways women are taught that for us being sexual is being desired, rather than desiring.

Desired/desiring. Object/subject, grammatically speaking, and this is part of what 'objectification' means. Of course, there is also a link between 'objectification' and treating women as objects. The extent to which women are dehumanised - treated as objects - and the extent to which the idea of a woman being desired ends up eclipsing the idea of a woman desiring are both important issues, and they are to some extent related. However, I know very well that in the past I have confused myself by mistakenly conflating the two issues. Thanks, Maia. Things are becoming clear.

"Female Chauvinist Pigs" review at The F-Word

The F-Word has a review up of Ariel Levy's Female Chauvinist Pigs. The book is strident in its views, so I was glad to see that they had viewpoints from several women, rather than simply posting one woman's view on the subject.

I particularly liked this remark from Holly Combe

“objectification” (as I understand it) is only dehumanising when it excludes all other possibilities

She goes on to say that she can see why the incredibly large amount of objectified images of women

leads some of us to say a big final “no more” to anything that could be perceived as “raunch”.

I think her original remark is spot-on, though. The biggest problem is not that there are images out there that objectify women sexually. The biggest problem is that there are not enough images out there that portray women sexually without objectifying them.

I don't think we can get rid of the former without first creating the latter.

I think trying to get rid of the former sometimes prevents us from creating the latter because we are too afraid we will mis-fire and end up back at the dreaded, demonised objectification of women. A big final "no more" can get in the way, sometimes.

But I have hope.

Carnival of Mathematics

Alon Levy at Abstract Nonsense has put together the inaugural Carnival of Mathematics. I suppose I really ought to try to get something together to help the gender balance along for next time...

Hey - I hadn't heard about this!

A success story! Forget what I said on the 2nd about "too hard to make a change" - somebody forced Girls Gone Wild out of town! Read I Blame the Patriarchy, or go to the Myspace page of the woman who pulled it together.

Matters of Identity

It was after a fairly amicable break-up. But I have to confess it, when he said I was "kind of masculine" because I'm "analytic about things", I did not react by thinking "Why, yes. As a result of socialisation which may or may not be aided by natural differences between the sexes, women do appear to be less analytical than men, on average, and I am certainly an exception to this rule." No, what I thought was basically "Well, f*ck you, too!" Why is this?

On some level, I want to be feminine. That is to say, I want to identify as a woman, want to count as a genuine member of my sex. It's a matter of identity. I think most people feel this way about their gender. As a result, statements of the form "Men are usually..." or "It is feminine to..." are almost never able to be nothing more than statements about the way men or women are. Inevitably, they end up containing some idea that this is the way men or women ought to be.

This means that when people make entirely scientific statements about women, on average, scoring less than men on maths tests (or having a smaller sexual appetite, or whatever) I find it hard to believe that someone, somewhere, is not taking that as a normative statement; a statement about what they should do to fit in with their treasured identity. I'm not saying that people shouldn't do these studies, or report what they find, but I think it's important that people consider what they are playing with when they make gender statements. They are not to be made lightly or without basis.

Of course, it's often possible to flout the 'rules' that apply to masculinity or femininity. I do it all the time. I think it ought to be encouraged. That will make reporting on these kinds of scientific tests less dangerous; it will also decrease the effect of the far more frequent casual statements about gender. Above all, it gives people more freedom to be themselves.

Still, I have to admit, when I look around my room and see the dusky pink silk scarf hanging up on one wall and the sloppy picture of Mount Cook daisies on the other, I can't help feeling glad that, quite without conscious effort on my part, you can tell my room's a girl's room, physics texts and all.

It's important to be yourself. It's very comforting, though, when the person you are can naturally fit in with the labels you want to hold.

Counting on your Fingers

In my post on bases of number systems I asked, rhetorically, "Why ten?" Why do we use base ten rather than another base? It's worth noting that this is not just a cultural thing - most cultures use 5, 10, or 20 as a base for their way of counting. The obvious answer is that we have five fingers on each hand; ten fingers total; twenty fingers and toes. This is supported by the fact that, in some cultures, the word for 'five' is related to the word for 'hand'. Indeed, in English, the word 'digit' can also mean 'finger'. People have been counting on their fingers for a long time.

The only exception that I know of, off the top of my head, is the Babylonians. Their system was essentially in base 60. There is an advantage to this.

Why is the five times table easy? 5, 10, 15, 20, 25... it's a nice pattern. The reason the pattern exists is because 5 x 2 = 10. As a direct result of this, every element in the sequence of multiples of five has the same final digit as the element two steps back. Similarly, every element in the sequence of multiples of two has the same final digit as the element five steps back (check this if it doesn't seem immediately obvious). Times tables are easier for numbers which are factors of the base you are using. Indeed, multiplication in general is easier for numbers which are factors of the base you are using. For instance, a quick way of multiplying by five is to multiply by ten and then divide by 2. And fractions? 1/5 = 0.2 and 1/2 = 0.5, but if you want to divide by three or seven or nine - by anything which is not a multiple of 2 or 5 - you're going to end up with an infinite (recurring) decimal expansion. In fact, six is a multiple of 2 but 1/6 still has an infinite decimal expansion because the fact that it is also a multiple of 3 mucks things up.

I hope you can see that using a base that had a large number of factors would give you considerable advantages. Ten is pretty useless in this regard actually. Short of using a prime number (which has no factors apart from the obvious factorisation into one times the number itself) we really couldn't have done much worse. Sixty, on the other hand... sixty is rather large, but let me show you why the Babylonians liked it:

60
= 2 x 30
= 3 x 10
= 4 x 15
= 5 x 12
= 6 x 10

2, 3, 4, 5 and 6 all go nicely into 60, and bring some other numbers along for the ride. That's why the Babylonians used it.

Closer to our usual ten, twelve would have been pretty good:

12
= 2 x 6
= 3 x 4

2, 3, 4, and 6. For a small number like 12, that's about as good as you could hope for. Why couldn't we have had six fingers?

Actually...

There's a lot to be said for counting on your fingers in base six! Look at it this way. In base 10 we have 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. On our hands, we have only five fingers, but we can also make a zero by putting up no fingers. So we have 'digits' 0, 1, 2, 3, 4, 5, representable by holding up that number of fingers on a single hand. We count like this:

ONE: hold up one finger (on the right hand, say)
TWO: hold up two fingers on the right hand
THREE: three fingers on the right hand
...
FIVE: five fingers on the right hand
SIX: zero fingers on the right hand, one finger on the left

Can you see how this works? In order to recover the original number, we multiply the number of fingers on the left hand by six, and add that to the number of fingers on the right hand (multiplied by one). This is similar to the way 23 is equal to the digit on the left multiplied by ten, added to the digit on the right (multiplied by one). We are in base six rather than base ten. We continue:

SEVEN: one finger on the right hand, one finger on the left
EIGHT: two fingers on the right hand, one finger on the left
...
TWELVE: zero fingers on the right hand, two fingers on the left

The highest number possible has five fingers on both hands. That's five times six, plus five = 35. 6 x 6 = 36 plays a similar role in this system to the one played by 10 x 10 = 100 in the usual one. We would need a third hand ("digit") to represent it.

Base 6 would be better than base 10 in some ways. 6 is smaller than 10, but it has the same number of factors. The only number less than 6 that isn't a multiple of a factor of 6 is 5. Maybe we were created by a being that knew more about maths than about how to create people that would be good enough at maths to notice from the first that 6 is a good base to use for counting on your fingers. I am obliged to confess that I had been counting on my fingers in base five for years before a friend of mine pointed out that six would be more sensible!

Manet's Olympia

It's interesting that Manet's Olympia is simultaneously an example of 'realist' painting and also a work that sparked incredible controversy not least because of the lack of submissiveness or shyness from the nude portrayed.

I think, or at least hope, that this goes some way to showing that real women are strong and sexual at the same time.

Bases of Number Systems

Take a number. Take 462, for example. That's 4 x 10^2 + 6 x 10^1 + 2 x 10^0. Yes?

Or take 43.5. That's 4 x 10^1 + 3 x 10^0 + 5 x 10^(-1).

If your knowledge of raising things to a power is a little rusty and you think that having something to the power of zero or negative one looks strange, think about it this way. Ten cubed is 1000, right? Divide by ten and you get ten squared. Divide that by ten and you get ten to the power of one. Divide that by ten and you get... ten to the power of zero (which equals one). Divide by ten again to see that 1/10 = 10^(-1).

The point is, this is the way we choose to write our numbers: the first digit to the left of the decimal point takes its value multiplied by one, the next one along to the left adds its value multiplied by 10, the next digit adds its value numtiplied by ten squared, and so on. If we proceed to the right of the decimal point the digits start adding their values multiplied by 1/10, 1/100 and so on. A number system which uses powers of ten in this way is said to be in base 10.

Why do we use ten? Theoretically, it doesn't have to be ten. It can be two. This is called base 2, or binary. There are only two possible digits, zero and one. If you want to write the number two, you have to remember that the digit which is second to the left of the decimal point now takes its value multiplied by two. So we write:

10 = 1 x (two)^(one) + 0 x (two)^(zero) = two

I have chosen not to write two as '2' in the above because this symbol is part of the base 10 system, and it could cause confusion in the above if the left of the equation is written in base 2 and the right of the equation is written in base 10.

What if you want to write three?

11 = 1 x (two)^(one) + 1 x (two)^(zero) = three

Four?

100 = 1 x (two)^(two) + 0 x (two)^(one) + 0 x (two)^(zero) = four

And so on.

You can use any natural number greater than one as a base. If you want to use the number thirteen, the fact that it is greater than ten will mean you will need to come up with new digits to represent 10, 11 and 12.

Writing a number in a different base doesn't change any of its arithmetic properties (unless those properties involve something like adding up the digits, of course!) Two plus two is four whether you write that as 2 + 2 = 4 or 10 + 10 = 100. Two times three equals six whether you write that as 2 x 3 = 6 or 10 x 11 = 110 (work out that last one: 1 x (two)^(two) + 1 x (two)^ (one) ).

It's just a question of how you choose to write it.

Feminism and the 'Personal'

Feminism is personal as well as political. Yes, I know sometimes the personal is political. Thing is, sometimes we push the 'political' side so far that we forget the 'personal' side.

It's too easy to decry the portrayal of women in popular culture. I hate music videos the most; you can pick your own pet hate. It's too easy to denounce these things and too hard to make a change.

Sometimes, it's so hard to make a change, all we can think of to do is to keep up the criticism. Keep it up! Shout louder! Shout more! Find more things to shout about, if necessary. Taken too far, this can lead to debates over whether lipstick belongs in the category of things to be denounced. Because, obviously, if we can just decide that, getting rid of those images of women gyrating in front of the cameras will be so much easier.

No. No, it won't. But here's another problem - a personal problem. I'm nearly twenty-two, and I still don't really feel like I have a fully developed sexual identity. I reject ideas like 'sex kitten' and I've never been that fond of 'dominatrix'; both have their place, but I'm just not happy with them. Partly I may just be slow to learn; I've certainly been slow to get into actually having sex. But I have found ideas here and there that have helped me along. Sheila Kitzinger's Woman's Experience of Sex was a wonderful introduction to the realm of sex beyond the basic mechanics; Girl With A One-Track Mind gave me a completely different and somewhat more detailed view several years later. I don't agree with everything in those two sources. The point is that they had enough stuff in there that I could agree with to broaden my scope.

They say it's easier to destroy than to create, but when it comes to ideas, that simply isn't true. It really does seem to me that destroying the idea of women as sexual objects would be much, much harder than helping to create ideas about women as sexual subjects. When young women say they have no problem with objectification because they like to be sexy, convincing them otherwise is going to be pretty hard, but if we can create more notions of women as sexual subjects, we will at least give them - give us, give me - another option. Bad ideas die more easily when the needs they service can be addressed by a different idea.

Sometimes feminism needs to be personal. It needs to give true examples of ways to be a woman, instead of just being political about the personal by closing options off.