Math is Hard

(From my blog “Android Sisters”, May 1998)

When I start to think myself too much of a smartypants, I read about math. The problem is, that I get so dingdanged excited about it, it just makes me want to become a theoretical mathematician, or a theoretical physicist. Somedays, I think that would be ten times easier than trying to convince Janice Raymond that we should share the same bathroom.

My total favorite television show is “The Proof” which is this PBS special about the insanely complicated proof of Fermat’s Last Theorem. Though, there’s also this guy who says that he figured out the simple-dimple way of proving the theorem, and he also has 10 new articles about the Clinton Sex Scandal. But as a rule, I try not to automatically assume that someone is a wacko, since I get accused of being all sorts of things that I’m not. But that’s just the way it is for me.

There is also this interesting side story about Sophie Germain who was part of the long long line of mathemeticians who pounded their heads against the wall trying to come up with a proof (which Fermat failed to include with his theorem, saying that it wouldn’t fit in the margins of his notebook). It also makes me a little bit edgy, because it’s one of those stories that is about how she had to hide her identity by taking on the role of a man in order to learn mathematics and then to correspond with some of the guys who were working on the problem. It’s a pretty sad story, mostly because she is described as being so totally brilliant that she attracted too much attention and had to out herself to her instructor. To me, this story doesn’t have a huge amount of trans-ness in it, but more reflects the difficulties that women had and continue to have breaking into mathematics. Of course, the headlines play it up as some kind of Billie Tipton or Pope Joan kinda thing, but I guess that sells nowadays.

So, I was surfin’ around the other night and I ran into a paper by G. J. Chaitin called “The Decline & Fall of Reductionism in Pure Mathematics” which kept me up way past my bedtime, in a sort of wonderful haze that I don’t get very much since I stopped sniffing glue, that is really, really an interesting building upon the work of Alan Turing (The mathematician who theoretized the “Turing Machine” and who was convicted of homosexuality and subsequently committed suicide.)

Gosh, I do tend to go on weird tangents, don’t I? So, this whole presentation about Reductionism in Pure Mathematics is pretty easy to understand, and he talks a lot about Gödel’s Work and Turing Machines and Randomness and Incomputability. All of which have very little to do with Artificial Girls. Except that sometimes This Artificial Girl likes to think that maybe, just maybe, the ideas that are presented at the end of the piece have a link to her own life, and the understanding of it, that being that understanding isn’t necessarily important and that maybe something can just be assumed and we can move on from there, without having to prove it.

Of course, Turing Machines aren’t the same as theTuring Test. The test is basically created by Turing not so much as to create a test that will decide if computers can think, but to prove that it is possible for computers to think. By creating this test, Turing hopes to convince us what thought is. Thought is not something that is easy to describe, and there are a good number of reasons why “computers can’t think” that he brings up in his paper and then proceeds to argue against them. The concepts of Incomputability and Reductivism come up here, also.

At the beginning of the paper, Turing discusses the idea of a game in which a man and a woman who are hidden physically and only able to communicate using words, to attempt to convince a third person that each is a woman. It is up to the third person to determine which is the male. This game proposes that there is no essential difference between “real” thought and “imitated” thought. That the man may have fooled you into thinking he was the woman, but he’s still very much a man. But that doesn’t mean he didn’t “really” fool you. He did.

Same thing with thinking computers. If you thought the computer was thinking, then by gum, it was thinking. It may not be capable of “real” thought, but — and this is where Turing was really really revolutionary — there is no way of defining real thought. The converse of this would follow that there is no way of telling if you are really thinking. Yes, you’re reading this, but what is it in you that is thinking about it? Yes, you’re having independent thought in your head, but is it “real” thought?

Okay, that’s the land of the theoretical, time for an Artificial Girl Super-Whammy. A leap of deduction from the land of thought and “gender” to the land of body and “sex”. I might need a little bit of coffee before I can really make this work….

…time passes….

Okay, the coffee’s cookin’ and my mind is going crazy. There is kind of a rivalry between mathemeticians and physicists. Mathies live in the realm of the hard and fixed and physies live in the amorphous and chaotic. Only in the last century, really, have the two schools begun to overlap and switch places, with chaotic theory and the search for a unified field theorem. I’m going to try (cross your fingers, this is dangerous work, kids) to make some correlations between Sex and Gender and Math and Physics.

By the way, I’m having my coffee strong, black and bitter. Just the way I like it. (Yes, it’s 12:30 am, get off my back, I’m being!)

Okay, so Sex is Math and Gender is Physics. Sex was this majorly fixed thing, like Fermat’s Last Theorem (we can’t prove it, but we know it’s true!), yet these days is becoming more fuzzy: witness the Intersex movement validating the unfixed realities of people with body and chromosomal non-polarity and body-reconstruction techniques such as sex-reassignment surgery and hormone therapy. (Damn, this coffee is good!) While many would argue that Gender is the “fixed” system, I would point out that gender has always changed over time, and is different from culture to culture. Little more than one hundred years ago, in the west, men wore wigs and pantyhose.

It is difficult to completely remove the effects of Math on Physics or the effects of Physics on Math. Pure Mathematics is used to solve problems in Physics, much the way that Gender is ascribed to a person based on Sex.

It is here that I would like to note a possible divergence from my arguement, to follow along the lines of what I just said. If a Gender “problem” is solved by changing Sex, is that an effective way of dealing with that problem? (If a physics problem is solved with a new or updated formula in pure math…) Very astute of you to point out (and if you didn’t catch it, well there it is).

The “Anti-Transsexual” arguement is this: That the problem isn’t with the sex of the person, it is a problem with the gender. That the Math is correct, but that the Physics is flawed. Albert Einstein did this when he theorized the Cosmological Constant (which coincidently (?) is represented by the Greek Letter lambda). After Hubble noted that the universe is expanding, Einstein quickly withdrew the CC and called it his biggest blunder. Ah, if only Professor Raymond would be so sporting.

The very interesting thing about lambda is that it may exist after all, and could actually be _profoundly_ effecting everything that we consider real. My point is that, even naughty ole Professor Raymond makes a significant contribution to the dialog on sex and gender. However, to give too much weight to her views, or to only consider her views, or to completely disregard her views, would halt discourse and effectively end progress towards real understanding. And we have not reached anything anywhere near understanding, I think we can all agree on that.

I see the Transsexual “problem” as very much like Fermat’s Last Theorem. That sex changes came about as a scientific attempt to legitimately change the sex of a person, for legitimate reasons, but that in the ensuing firestorm of media attention, backlash and outcry, the Proof was lost.

There is a very interesting point that Turing makes in his paper on Computing Machinery and Intelligence. He identifies nine reasons why computers can’t think:

(1) The Theological Objection
(2) The “Heads in the Sand” Objection
(3) The Mathematical Objection
(4) The Argument from Consciousness
(5) Arguments from Various Disabilities
(6) Lady Lovelace’s Objection
(7) Argument from Continuity in the Nervous System
(8) The Argument from Informality of Behaviour 
(9) The Argument from Extrasensory Perception

I’m going to attempt (notice, she has no net underneath her, ladies and gentlemen….) to briefly sum up each of these points, and then show how each of them, in turn is also used to exclude an Artificial Girl from being fully, physically female.

(1) The Theological Objection

Basically, the notion of a “God-given” soul is not available to computers. Turing refutes this by noting that computer designers aren’t creating souls, but vessels for souls, which the Good Lord then infers with souls if He so wishes.

I’m about as reluctant as Turing to take this one one. Major religions are generally unsupportive of sex changes, either because God does not make mistakes, or because readings of sacred texts have pretty much nothing to say about sex-changes per se, but do tend to condemn castration, or because they come out pretty squarely against non-traditional gender representations and that’s pretty close to transsexual.

I’d like to remind you here, that my point is the same as Turing’s, not to actually make a rock solid arguement that shoots down this point, but to show that these points are also used to deny my physical reality, in the same way they are used to deny the intellectual reality of computers. The idea being, that if the two can be shown to be equivalent, the valid conclusion of Turing’s paper will also be my conclusion, and therefore, also valid.

On this, I would only state the transsexual motto, that we are born in the wrong bodies. Arguing for gender freedom does nothing to make this point, so I’ll not do it here. I won’t bother to argue the idea that when there is a physical problem with a person, the use of medical technology to repair that problem, is not only the correct moral choice, but the correct theological choice.

(2) The “Heads in the Sand” Objection

The consequences of “men becoming women, or women becoming men” would be too dreadful. Let us hope and believe that they cannot do so.

Turing is being darn witty here and I urge you to read his response, if you haven’t already. The sentence above was written by Turing, and I replaced “machines thinking” with the part in quotes. I, too, will just let this stand as being unworthy of refutation, but leave the note that I believe it is one of the most commonly thought reasons why transsexuals are considered unreal.

(3) The Mathematical Objection

That there must be some uncomputable portion of thought that separates machine thought from human thought, thereby making machine thought inferior to human thought.

This is a really good one for people who like to say that I can’t be fully female, because there are portions of femaleness that I am unable to experience, such as menstruation. Turing says, yes, there will be questions that computers are unable to answer that humans will be able to answer. Then he goes on to say, “Whenever one of these machines is asked the appropriate critical question, and gives a definite answer, we know that this answer must be wrong, and this gives us a certain feeling of superiority.” This superiority is simply over one machine at one instant, as is the superiority of a woman who menstruates monthly over me. However, what of the woman born who never menstruates? And what does that superiority confer that specifically _discludes_ a transsexual woman from the reality of womanhood?

Turing ends this discussion by noting that those who hold to this arguement would “mostly be willing to accept the imitation game as a basis for discussion.” To which I would also say to anyone who specifically holds to this arguement, be willing to accept some sort of Turing Test for transsexual people.

(4) The Argument from Consciousness

“This argument appears to be a denial of the validity of our test. According to the most extreme form of this view the only way by which one could be sure that machine thinks is to be the machine and to feel oneself thinking.”

Gosh, this is a good one. And believe me, I’d love to give some transsexual-hater a weekend in my body.

“One could then describe these feelings to the world, but of course no one would be justified in taking any notice. Likewise according to this view the only way to know that a man thinks is to be that particular man. It is in fact the solipsist point of view.”

Turing seems to be saying that this arguement is a non-arguement. That for one to take the extreme solipsist point of view, one is trying to avoid the computer taking the “easy contrivance” of merely recording what it is to think and then outputting that as an example of thought.

As for Artificial Men (cause Turing is talking about men, so we’ll stay in that vein for this, even though I’d rather talk about women, cause I have a thing for women, don’tchaknow), well, I think that this is the strongest arguement for the separation of “contrivance” of gender from the raw physicality of sex. Are trans men simply taking on the gender of maleness in order to fool the viewer into believing they are men, without actually being men? To take the solipsistic arguement, one would barely ever know, but to step back a little bit from that and watch the full expression of transexual men’s lives…. well, that might be a little bit more compelling.

Turing says that there are mysteries about consciousness, and I’d jump to the conclusion that there are equivalent mysteries of physicality. He states that he doesn’t think we need to define or solve these mysteries in order to answer the questions of whether machines can think. Again, I’ll take this same tack (though, I might have less leeway here), and suggest that we don’t need to lock down and define sex before we can accept the possibility that transsexual men are physically and valid men.

(5) Arguments from Various Disabilities

This seems to be an extension of the Mathematical Objection, or the Arguement from Consciousness. Basically, it says that if the observer can think of a thing that a computer can’t do, and if it can’t do it _right_now_, then the Turing Test has failed. Turing states that the test is not something that will necessarily be testable at the current point in time, but will be at some future point in time.

To this, I say, that someday, people will be able to change sexes with the press of a button. It may be a hundred years from now, or a thousand. But the physicality of sex is not so unattainable as to make it an imposibility. To losely reconstruct Turing’s arguement; because it is possible in the future, then it is a valid concept now.

But seriously folks. These are the early early days of sex changes. You have the option to get over it now, or get over it later. Why not save everyone a lot of pain and frustration and get over it now?

(6) Lady Lovelace’s Objection

Lady Lovelace helped Babbage define the Analytical Engine. She said of the Analytical Engine, that “It can do whatever we know how to order it to perform.” And no more than that. That computers are incapable or original, creative thought, and therefore not capable of imitating real thought.

I don’t know how to directly relate this to trans people’s issues. Perhaps, the idea that trans women don’t contribute in a meaningful way to women’s political discourse, or that trans men are unable to attain the same patriarchal strata of societies that non-trans men are. Turing has some interesting thoughts on creativity and where creativity comes from. He presumes that the end-result of fetishizing the creative process as something “mystical” leaves one assuming that “there is no virtue in the mere working out of consequences from data and general principles.”

Perhaps the transsexual experience currently hampers the trans person from actively taking part in much of the similarly-sexed discourse. The communities that might openly embrace us tend to be where we don’t spend our time, and if we do, we are not allowed to fully take part in the similarly-sexed activities that we’re being denied our realities for not taking part in.

I think the kernel of wisdom here, though, is that Lady Lovelace and Babbage failed to see that, while their Analytical Engine was incapable of creative thought, the idea that a sufficiently powerful Analytical Engine could be capable of the “creative mental act”. I think that this very same argument holds true for Artificial People. That we are not allowed to freely intermix with similarly-sexed people, does not mean that we aren’t capable of it.

(7) Argument from Continuity in the Nervous System

That the Central Nervous System (the body) is also part of human thought, and that the CNS is not a discrete-state machine. That inherent in real thought is a randomness that cannot be imitated by a computer.

Turing’s argument is that that close enough is close enough. Even though the CNS may be impossible for a computer to recreate, an effectively realistic model can be used to simulate the randomness and chaos inherent in a non-discrete system.

For the Artificial Person also, close enough is close enough. If they come up with a procedure that, for instance, might allow me to menstruate only every other month, would that be enough for me to qualify as female?

The Turing Test is the simplest way of proving the concept that imitated thought is equivalent to non-imitated thought. If we’re able to develop a Turing Test for the _reality_ of Artificial People, then perhaps we can give this same amount of leeway. This is based in the idea that there is no way of coming up with an essentialism that will fit all currently believed “real” people while specifically leaving out imitated people.

(8) The Argument from Informality of Behaviour

Turing doesn’t like this one, and doesn’t seem to do a very good job of defining it. Basically, that humans can’t be machines because there is no set of rules defining human behavior. Turing, I believe, is of the opinion that there _are_ rules for human behavior, but that those rules are so complicated they are unknown at this time.

There is a bit about what a human would do if confronted with a situation that is unguided by laws of human conduct, that this is unknown, and therefore humans are unpredictable and therefore ungoverned by rules. Turing brings up the point that one would have a difficult, if not impossible, task to predict — without knowing the programming of the computer — how a computer will act when presented with a situation that is not governed by its rules.

I think there are a number of correlations here. First, that there could very well be rules for determination of sex, but we are unaware of their scope. Not knowing their scope and therefore implying that the simple signs point a certain direction is not good enough. If the rules are unknown, that is not the same as there being no rules.

I might also offer this up as some sort of hope (or further confusion) for transsexual people who are unsure of their “true” sex, or define as undefined for reasons of not having a factual basis upon which to base their sex. Not knowing your sex is not the same as not having a sex, and it is most certainly not the same as having one imposed on you.

On a completely unrelated note, say that a computer is developed that can think, and is then posed with the question, “What sex are you?” I predict the making of a the first “transsexual” computer.

(9) The Argument from Extrasensory Perception

Huh? What? Oh, yea, right. Well, duh, if the person determining if the computer can really think has ESP, then that person would of course know that the computer was a computer and it would therefore fail the Turing Test. This wouldn’t be a fair test. So, “To put the competitors into a ‘telepathy-proof room’ would satisfy all requirements.”

If you had ESP, then you would know a lot more about me than I’d really ever want you to know. For me, being an Artificial Girl who still has a lot of the characteristics of the spare parts that were used to build me, this is just like you having ESP. You can instantly tell from looking at me that I was build using “bits of boy”™. But that doesn’t make it fair, does it?

My version of a “telepathy-proof room” would be some sort of acceptance of different-bodied women, from where I could then be seen not as “male-bodied”, but as alternatively-female-bodied. The initial pigeonholing me as “male” would then be circumvented, and then your job of deciding if I was doing a good job of imitating your reality, would be much much more difficult.

Hey, tough luck, math is hard.

Finally, Turing makes some points in favor of his test, and of the results that his test would prove. He brings up a wonderful notion of, “recitations tending to produce belief,” as opposed to convincing arguements. I think that we need to stand up for this. Transsexual people at this point in time have an important need for these produced beliefs. Because, like Turing in his day, he could not actually produce a machine that was capable of passing his test.

I don’t know if I pass the test of whether I am fully, functionally, “really” a woman, but I think the point is that I have the possibility of becoming that at any moment. Should the great scientific breakthrough in gene therapy take place tomorrow, you can bet I’m going to be up early and be the first in line.

I get a lot of shit for not having “grown up female”. First of all, I really resent this, because I didn’t have a lot of choice in the matter. But there are women and men who never went through the puberty of their incorrect sex. These are incredibly lucky people. (Though, it’s also a tremendously painful process, most of the women I know who did this did it after being kicked out of the house, by pimping themselves for a place to live and money to buy illegal hormones. Most were terribly abused. Some luck.) But often times, they are given more of a “right” to be seen as their correct sex, because they don’t carry as much male privilege (if they’re female) or have too much male privilege (if they’re male). I think this is the corolary to the Theological Objection, but the religion in this case is Academic Feminism.

I know a lot of people don’t want to face up to the idea that Feminism contains elements of bigotry and perpetuates oppression. It is not my intent to tear down the house of Feminism, but instead to build it up by pointing out the rafters that have been eaten away by termites. Unless we are willing to evaluate our processes and critically expose weaknesses in our theories, instead of clinging to them tighter, they will cause the whole house to fall down around us.

This would be a sad thing, a sad thing indeed.

I don’t want to go on about this too much, I am a Feminist, I find find Feminism to be a powerful tool in eliminating bigotry. I just wanted to point out that there are pots burning on the stove and that we might want to do something about that. Just mentioning it. I’d get up and turn off the stove, but I can’t seem to get past the really mean woman standing at the kitchen door looking at me like I’m going to end up with a severe bruise if I attempt to gain entry to the kitchen.

I would also like to note that there are far more vile political theories and mindsets that I personally wouldn’t mind seeing burn to the ground. Heck, I’d be happy to be the one caught holding the match.

Okay, so Turing devised this test to see if a computer can imitate a human. And again, this was not so much a test that had to be taken _right_then_and_there_ by whatever computer happened to be plugged in nearby, but more the idea of a test for the idea of a computer that challenged the notion of what thought is. I think I’ve made a pretty good point that thought can be equated with gender — actually, that Turing made that point — but I pointed out his point.

If a man can represent himself as a woman, and fool you, then he has fulfilled the test that Turing made to test if a computer can imitate thought.

Now, a test to see if an Artificial Girl can imitate your body.

How to present this?

Are there imaginable transsexual women which would do well at imitating women? if they were taken out of the secret room and allowed full access to a person with medical diagnostic equipment, and allowed to ask any question of the artificial girl to determine if she was a “real” woman or not?

This proceeds from Turing’s restatement of the question, “Can computer’s think?” as “Are there imaginable digital computers which would do well in the imitation game?” (This is a different imitation game than the gender imitation game, it is if the computer can imitate a human, one follows from the other.) If the imitation game is based on hidden physical reality, thus allowing a man to pretend he is a woman in order to pass as a woman for the purposes of the test, and we remove that hidden physical reality, then we have moved from the realm of just thought, to physical reality. To the realm of sex.

If we take the computer out of the closet and give it a real human form, does it then actually _become_ a satisfying immitation of humanness, so satisfying that the differences are unimportant to the business of being real?

Turing proposed nine opinions that were opposed to his own. I went over those nine points and made notes about how they could be conferred to the physical realities of transsexual woman, who I am positing are the least likely to be conferred with the status of “real” woman.

As a brief note, the points that Turing makes in favor of his point, that computers _can_ think are so easily applied to Artificial People, I’ll not spend time bashing you over the head with them.

Does it not follow that there is a Turing Test for the female reality of transsexual women? One that would allow complete access to the trans woman, physically, emotionally, mentally?

If so, then because of Turing’s work, and even though no one has come up with a computer that can pass the Turing Test, the fact that it has not been done yet is not a proof that it is not possible. Because Reductivism is growing out of style in Pure Mathematics.

Maybe it should be growing out of style in real life, too?

I’ll close with Turing’s own words at the end of his paper on “Computing Machinery and Intelligence”,

We can only see a short distance ahead, but we can see plenty there that needs to be done.

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J. Wilder

My friends call me Jonnie. You can too.

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