Two wicked tautologies.

The midterm elections in the United States are two days away. Unfortunately, the importance of these elections is not limited to inhabitants of the US and to its expats (such as myself). The interconnectedness of national economies and political systems that defines the modern world means that insanity in the country I call home can spread and upset affairs all over the world. Moreover, issues such as environmental protection know no such thing as political borders, and have consequences all the world over. Increasingly, the vast majority of the world has been effectively disenfranchised by the disproportionate level of influence wielded by the US in world affairs.

It is thus that I feel justified in saying that what I write of today is of the utmost urgency and importance. As I write this, a group of unqualified and militaristic right-wing candidates is poised to take power in the legislative bodies of the US, which would render the already too-moderate Democratic party completely impotent in solving the myriad problems facing the US and the broader world. The impact of this potential swing in political power has already been addressed by others, perhaps most profoundly by Keith Olbermann in his most recent Special Comment.

There is, however, a point that has largely been missed in much of the discussion so far. I myself have said that the Tea Party has no coherent argument, philosophical basis, policy position or political stance. This is, unfortunately, not completely correct, though. The Tea Party does indeed have a coherent basis; not one of politics or policy, but of theology.

To explain what I mean by this necessarily involves a bit of a tangent. In his beautifully written and profoundly chilling book The Family (affiliate link), Jeff Sharlet documents the rise to power of a secretive group of theocrats, both in the United States and globally. The Family (also known as the Fellowship) is largely characterized by their adherence to a theology of power. According to Family doctrine, the powerful people in the world today are powerful by God's will, and thus are to be followed implicitly in recognition of God's choosing them as his agents. This creates, as Sharlet so eloquently puts it, a tautology of power, in which the powerful are powerful precisely because they are powerful. Their religious belief is thus directly connected to their support of many of the world's most despicable men and women; would the people that they support be powerful if not for some quality that God had identified and chosen them for?

In the Tea Party, we see a parallel theology-- a similarly wicked tautology-- driving their actions. If one takes the efficient market hypothesis extremely literally, then the wealthy are rich because the Free Market has chosen them for some quality that they must possess, even if it is invisible to mere mortals. This tautology of wealth states that the wealthy are wealthy precisely because they are wealthy. This doctrine of the Tea Party betrays the faux populism to their movement. Indeed, adherents of this tautology of wealthy are perhaps better referred to as the Tea and Crumpets Party (TCP), as their every policy action seems driven towards accelerating the expansion of the gap between the wealthy and everyone else. For all their cries against the redistribution of wealth, that is precisely what the TCP wants: that as much wealth be transfered to the already-wealthy as possible.

The primary difference between the TCP and the Family, as far as I can tell, is that the TCP is not in the slightest secretive. It is, after all, not as of yet disreputable to put literal religious faith in the efficient market hypothesis, nor to support the whims of billionaires. That we have as a society progressed to the point of understanding that giving aid to genocidal maniacs (such as the literal Nazis that what would eventually become the Family protected following WWII) necessitates a level of secrecy on the part of the Family that the TCP has no need of.

Of course, wealth and power are not uncorrelated. These two wicked tautologies thus interweave in horrible ways, of which I fear we have seen but the shyest echoes. I would rather not find out the terrifying ends produced through such an interweaving by handing the reins of power over to the TCP.

On Tuesday, however, there is a choice put to voters in the US: whether to endorse a wicked hypothesis that parallels that of the Family's power theology, or whether to stand up for policies formed on a rational basis. Those outside the US also have a part to play in the next two days, for in speaking up and reminding your friends and loved ones what is at stake, a further crisis may yet be averted. Please do not allow these wicked tautologies to win out over rationality.

On an eccentric use of volcabulary.

I want to write another blog post later today, but found myself wanting to use a word that I feared would be misunderstood. Hence, I am breaking my sickness and paperwork induced blog silence by talking not about something grandiose, but rather minute by comparison to my usual topics: two words, and why I prefer one to the other.

The first word, "evil," is one that I try to avoid using as much as possible. Not, mind you, because of some sense of moral relativism, but rather because of the connotations of the word. To many people, evil necessarily derives from some external evaluation of the world, be it by a god or authority figure. To me, however, being a utilitarian (at least to a rough approximation) means that any evaluation of what is good or bad must come from a rational argument and not the decree of another. Of course, this sense of the word "evil" is far from universally held, but it would be truly unfortunate for my writing to be misunderstood by virtue of such a colored meaning.

In preference, I choose to emulate Richard Dawkins in his use of the word "wicked." Specifically, I call wicked that which is directly, intentionally and needlessly harmful to other intelligent beings, their bodies and their rights. Moreover, I call wicked those value systems and philosophies that compel their adherents to wickedness towards others. On this latter point, I likely deviate from others, such as Dawkins, in applying the word to what is inherently a matter of thought rather than action. My motivation is not to ascribe to any individual a responsibility for the thoughts in their heads, but rather to examine what could compel an otherwise decent agent to act in a wicked manner towards their peers.

By using this word in preference to "evil," I hope to avoid my meaning being lost in the noise of cultural connotations. (Plus, the word sounds cooler, anyway.) It is important that my meaning makes it though, as something wicked truly this way comes.

Latest in the annals of false equivication.

If there's two things that the media loves, it's a good sex scandal, and a cautionary tale about how the Internet is a dangerous, dangerous thing. Bonus points if they can remind people that there's not really any difference between opposing ideological extremes. That's why I'm in no way surprised to find that NPR has hit all of these points yesterday.

Honestly, I'll make this a very short post. I don't have much to say about an article that compares gallivanting around as a Nazi with a completely harmless sexual expression. Rather, I must simply shake my head and wonder what kinds of mental gymnastics or twisted ethics are required to equate these two activities, to treat them as equally noteworthy and newsworthy stories. Frankly, I find Nazis to be much more intimidating and vile than dildos; it astounds me that this even needs to be said.

On the emergence of an accidental narrative.

Good writers of fiction will sometimes refer to being surprised at the actions their own characters take. After all, once a story takes a life of its own, in the way that good stories so often do, why should even the author be able to foresee everything that happens?

A similar effect is seen in tabletop gaming, where a good dungeon master (DM--- more correctly, game master) will present a carefully planned adventure to her players only to find that they fixate on details that the DM had thought to be inconsequential. Out of that interaction, a new narrative is drawn from the fibers laid down by the DM. Background material moves to the foreground as the story finds its own vibrancy at the hands of the players and the DM.

Up until recently, I had thought these kinds of emergent narratives to be the province of fiction. On Wednesday, however, I had the pleasure of attending a talk by Ben Schumacher on his experiment with teaching quantum mechanics to undergraduates from a quantum information perspective. In his talk, Dr. Schumacher described how the book that he and Dr. Michael Westmoreland wrote, Quantum Processes Systems, and Information (note: associate link), had a hidden narrative that emerged as they proceeded through the writing process. So as to not spoil the story, I'll refer interested readers to Schumacher's talk for details on the form that this narrative takes.

For my part, I have found that in writing this blog, I tend to write each post relatively independently, with little thought of how they fit together into some cohesive whole. Even the name, cgranade::streams, belies some of this approach. It seems plausible, then, that a narrative could emerge not from careful planning but through recognizing the common concerns which motivate me to write on disparate topics.

That said, I was still surprised to find that when responding to a comment by Sarah Kavassalis on one of my recent posts, a small narrative had started to emerge. In three of my last four posts, I have either alluded to or directly dealt with problems that children face in society, pointing out that they are told that their life experiences are less than real, that the authority figures that abuse them can be defended and even celebrated, and that their nascent sexuality is disrespected and disregarded. In all three of these cases, we see a common strand: children are not always seen as being fully human, and the effects of that are as real and destructive as for any group tarnished as being less than human. Their rights are trounced upon, just as with any marginalized group, illustrating the peril in this disregard.

In this way, a narrative about the disrespect of children by society at large serves as a poignant case study of why recognizing the humanity of those around us is so important. Whether the victims of our disregard by marked by sexual orientation, race, religion or nationality, the end effects share much in common.

I don't know if I'll carry this emergent narrative any farther, or if it has served its useful purpose here. Others write on the modern plights of children better than I do, so that my contribution is to entirely to tie it to other threads of thought. This mission, which I have accidentally worked at, is one of many worthwhile missions. If I have more to say on the topic, then, I will say it and will otherwise be content to hunt for other emergent narratives.

In which I argue against a named Test.

If you've been on the Internet more than a week, you've almost surely heard of the Bechdel Test. Named after artist that wrote Dykes to Watch Out For, the Bechdel Test is intended to filter out movies that fail to feature fully fledged female characters. A movie is said to pass the Test only if:

1. It has to have at least two women in it,
2. Who talk to each other,
3. About something besides a man.

There is some genuine insight here, as far too many movies do in fact fail to respectfully represent a whole half of humanity in their cast. The goal of encouraging more strong and interesting female characters is laudable, and likely a necessary step towards true equality. Thus, I without reservation say that the Test has contributed to the cultural conversation on equality.

On the other hand, I generally find invocations of the Test annoying, as it is far too easy to take such arguments too literally. I don't generally find much insight in discussing whether a particular movie passes or fails the Test, any more than I find that remarking upon a particularly warm day produces insight into climate change. Rather, I think the Test is best used as a means of making an argument, not as a metric.

To try and offer some support to this view, I want to list a few movies that I have seen and that definitely fail the Bechdel Test but which are nonetheless good movies which are hardly bastions of sexism. Such false negatives demonstrate to me that the Test cannot be taken overly literally without missing the rationale. Without further ado, then:

• 12 Angry Men: No female characters at all, as the movie is a jury drama set in a time when women on juries were very rare due to sexism.
• Dr. Strangelove: The movie is a comedy of errors about the leaders of major world powers in the 1960s--- a group that is not well known for its inclusion of women.
• Memento: The movie is narrated by a male character who speaks almost entirely to one person at a time.
• Moon: There's not enough characters in the cast to pass.
• Run Lola Run: This movie also has an incredibly tight cast. (And no, I don't count it when her neighbor asks Lola to pick up some shampoo.)
• Voices of a Distant Star: Again, only two characters with lines, each of opposite gender.
• Wall-E: The relative lack of human characters makes it hard for this movie to pass the Test, to say nothing of the movie being nearly a silent film.
• Hush (Buffy episode): Not a movie, I know, but I'm too sarcastic not to include it anyway.

Note that these movies fail not for their content, but for how the Test happens to be phrased. To hold it against Moon that Sam lives alone on his space station would be patently ridiculous, for instance, but it technically fails the Test for its isolated cast.

Even more ridiculous would be to hold it against Wall-E for failing the Test--- it doesn't even pass the Reverse Bechdel Test:

1. It has to have at least two men  in it,
2. Who talk to each other,
3. About something besides a woman.

No one should seriously think that the existence of movies which fail the reverse Test is evidence for a bias against men, so why is the mere existence of movies which fail the Test taken as evidence of such bias against women? Such a bias exists, to be sure, but it is not well demonstrated by applying the Test to this or that particular movie.

By continuing to fixate on the Test, I feel that we do ourselves a disservice. The problem of ensuring equality in media is not an easy problem, and isn't well suited to glib analysis. Arguments such as the Bechdel Test serve well to raise awareness, but at the end of the day, are a poor substitute for informed insight.

Sex negativity considered harmful, but to whom?

For all our capacity for compassion, we humans can be rather bloody rotten to each other. It shouldn't be surprising that this duality is apparent in children as well. Indeed, it's disturbing to read this story on Salon about how a video of a teenage girl being raped was shared via the Internet by other teenagers. The rape of any person is a tragedy and an outrage; the continuing exploitation of a victim in such a manner nearly defies description.

The reason I bring this article up, however, is because of something far more subtle and insidious [emphasis mine]:

This is the typical predicament law enforcement faces when it comes to online child pornography: Once it's out there, it's usually out there for good. The digital trail is just too difficult to trace. We've seen a similar thing with teen "sexting." A boyfriend gets angry when his girlfriend breaks up with him, so he texts a naked photo of her to all his buddies, they send it to all their buddies, and so on and so forth. In the end, it's hard to know just how many people have seen the image and where it's ended up.

If that paragraph doesn't strike you as deeply wrong, then I suggest giving it another read through. How else other than "deeply wrong" is one supposed to describe the comparison between a brutal rape, child pornography and the fully consensual exchange of suggestive pictures between children. This latter phenomenon can go sour indeed when the relationship between two children changes, and can lead to abusive situations, but that's not what the neologism "sexting" refers to.

By equating the consensual activity to which "sexting" refers with the form of abuse described, the author communicates a decidedly sex-negative position, objecting to the very presence of sexuality in the lives of teenage children. I suspect that this is unintended, or a product of miscommunication and misunderstanding, but it is a common enough position to take that it's worth discussing, to be sure.

Many others have described much more eloquently than I ever could how sex-negativity such as the anti-porn movement harms adults, but children undeniably suffer as well. Even beyond the direct consequences, sex-negativity can tie into other problems, such as sexism, leading to young girls being shamed for engaging in a very natural aspect of human experience.

Perhaps more disturbing is that sex-negative motivated approaches to education leave children ill-informed about their own sexuality, leading them to engage in riskier behavior. Indeed, children often live in a virtual sexual Prohibition, so should we be surprised to find them drinking of moonshine? That this approach of keeping children in the dark is also being applied to higher education by sex-negative advocates is something that we should find very disturbing.

There's another way, however, of dealing with the complexities and problems inherent in teenage sexual relations: treat children as competent, but in need of education. Don't hide them from the complexities of sex, and don't fall into the trap of well-meaning but sex-negative approaches to education. I am glad to see that even a state like Alaska can work towards implementing sex and relationship education programs that deal with such complexities. Rather than engaging in the kind of sex-negativity which so harms adults and children alike by teaching them to be ashamed of their own sexuality, harm reduction based education starts from the radical view that children are people, too.

A few immoderate things.

I like Jon Stewart. As far as understatements go, that's a pretty big one. Like with any other human being, though, there are some things that I don't agree with him on. One of these times occurred recently, when he announced a rally to restore sanity, where sanity is defined as political centrism. Glenn Greenwald rather well argues why this is problematic, and so I don't wish to flog that dead horse any further. Rather, I want to emphasize a somewhat tangential point: that the truth isn't always in the middle.

Sometimes the extreme position is the only correct option, or even the only morally defensible one. While I don't deny that the centrist assertion that truth always lies between the extremes is a decent heuristic, I flatly deny that is is true in full generality.

Take the pope's visit to the UK, for instance. What Richard Dawkins said about the pope is surely not moderate, but a moderate response would put one in the unconscionable position of defending someone who, by all available evidence, knowingly and deliberately protected those priests that raped children in their charge.

Similarly, though it isn't moderate to insist that Bush should be tried as a war criminal for his role in the torture of prisoners of the United States, a moderate position is one in which the laws and treaties that protect prisoners are fungible. Such a position ultimately allows for more people, both innocent and criminal, to be exposed to inhumane treatment in the future.

Should we sacrifice religious freedom on the altar of moderation by taking a position less extreme than that the Park51 facility (whatever kind of facility you wish to call it) be allowed? Should we put moderation above the well-being of future generations by taking the moderate "wait-and-see" approach to climate change? Should we deny the human rights of GLBT people by taking the moderate "civil unions, not marriage" route?

Don't get me wrong, though-- moderation in one's opinions is a fine and wonderful thing at times. What I cannot abide by, however, is when moderation is allowed to be the goal. One's opinions should, I submit, be aligned with reality, whether that reality is moderate or not.

Academia and the unreal.

A little while ago, someone offered me advice on how to get a job in the real world after grad school. This advice, though unsolicited, was undoubtedly well-intentioned, but hidden in the offer is the germ of an idea that I find quite poisonous. Implied is that the academic realm is somehow disjoint from the "real world." This phrase is often, in my experience, used in a condescending way to separate and denigrate various environments from some set of environments that are sufficiently "real" to merit recognition.

Consider one particularly harmful example of this. Since children are often told that things aren't like their school environment out there in the real world, reality as recognized by this phrase must surely exclude the first 20 or so years of our lives. Years in which we discover much about ourselves and in which our bodies change and betray us in myriad ways. Years in which we undergo challenges that we are, almost by definition, unprepared for. Years in which we experience emotions and pains which are all too real. To add to those burdens the condescending dismissal of unreality is a tragic perversion of the good intentions that must surely underlie the use of a phrase like "real world." What is seen by adults as a promise of a better tomorrow comes across as a failure to empathize with the problems of adolescence. This is why I call the ideas epitomized by the phrase "real world" poisonous: they pervert and distort our intentions and empathies.

How, then, does such a term come to be applied to academia? To many people not in academics, I suspect that the academic world is unfamiliar and arcane. Many people are not concerned with funding proposals, postdoc applications, tenure reviews, or any other of the myriad distractions from research. Even more fundamentally, the goals of an academic researcher are very different from the goals of most people employed in industry. It is all too easy, then, to fail to recognize these goals and concerns as being as real as those associated with other pursuits. Likewise, it is all too easy to compartmentalize the concerns of academics to some mythical ivory tower, locked away from daily life as surely as the princesses locked away in the towers of our more misogynistic fairy tales.

What could be more real than learning? In all walks of life, we must learn and grow to succeed, and it is this process that academia tries to incorporate and cultivate. When we lock this ideal, however imperfectly realized, out of our conception of the real world, we do ourselves a great disservice. Rather than responding to the foreignness of academia by drinking the poison of the real world, then, I encourage my friends and loved ones to ask questions of their academic friends. It can be difficult to bridge divides, to be sure, and those of us on the academia side of this divide aren't always the best at empathizing with the rest of society, but we can all do better than to dismiss so thoroughly the concerns of those around us.

The flip side of DBAD: we can call people dicks, too.

Who knew that Mass Treble's (Twitter) Golden Rule, don't be a dick, would be so controversial? Even stranger is that, due to how the word "dick" is interpreted so widely as to include many things which I find to be quite positive, I find myself often arguing against how DBAD is implemented. That said, I do in general think that not being a dick is a good and laudable goal, so long as one keeps a reasonable definition of the term.

In fact, I think it's a laudable enough goal that I'm quite willing to ask others to not act in asinine ways towards me and (more importantly) those people around me that I care about. To take one specific example, I'm quite content to demand that people not be misogynistic dicks. I consider it to be quite dickish to take prudish, sex-negative views about women and impose them on the world around you. Over at Daylight Atheism, Ebonmuse points out a few particularly appalling examples of this, including a picture that drives the point home:

Let me make this perfectly clear: if you are to escape being called a dick by people like me, you don't get to demand that women sit at the back of the bus any more than you get to demand that blacks do. I really don't give a flip if your religion says that you have the moral obligation to be dicks to those around you or not, so much as I care that this behavior causes real and physical harm to other human beings. If there is a more clear sign of being a dick than being willing to subjugate half of the human race to appease your own twisted morals, I don't know what it is.

At its most basic, much of why I write the words I write and spill the pixels that I spill over religion come down to the DBAD principle. Evidence such as these examples shows that religion is a rather efficient machine for either turning people into dicks, or at least amplifying dickish behavior by insulating it from analysis and criticism. After all, it takes religion to get one to say that maybe gay kids shouldn't be as protected from bullying as everyone else. It takes religion to lead otherwise decent people to oppose a lifesaving mode of defense against AIDS and other STDs.  And so on, ad nauseum.

In short, if we're to really take seriously, Mass' Rule, then that doesn't mean that atheists and other freethinkers should shut up, but rather, that we should be more vocal than ever about diskish and unjust acts, whether religiously motivated or not.

On ideas.

Yes, my blog posting has been slow as of late. Blame working on funding proposals, a lack of good sleep or the phase of the moon if you like. The real problem, though, has been one far more mundane and frustrating: a lack of good ideas.

Ideas are the currency of my career, really, along with hard work and technical skills. Some ideas can be turned into research directions, others into specific solutions to technical problems, and still others become blog posts. Research is, after all, a creative enterprise. What to do, then, when ideas run dry?

The short answer is that I don't think that ideas ever do run dry. We are surrounded by a whole ether of ideas, after all. The trick is finding those ideas which solve your problem, ideas which motivate you, ideas which others may find interesting. It is less, then, that I am short on ideas so much as I am short on ideas that are applicable to the tasks in front of me, including blogging.

Or is it that I am lacking the discipline and energy to refine those ideas into proper posts? After all, writing (like research) doesn't stop with ideas, as Gaiman so gracefully puts it in an essay on writing:

The Ideas aren't the hard bit. They're a small component of the whole. Creating believable people who do more or less what you tell them to is much harder. And hardest by far is the process of simply sitting down and putting one word after another to construct whatever it is you're trying to build: making it interesting, making it new.
[Source: "Where Do You Get Your Ideas," an e-book extra to the Kindle edition of Anansi Boys.]

If there's one thing that life on the Internet teaches, it's that any idea is interesting when placed in the right context and explored with passion. For instance, it seems that much of science fiction works by exploring perhaps mundane ideas in fresh contexts and thus making them new, as Gaiman puts it. Snow Crash doesn't thrive because (to pick a small example out of many from that book) Neal Stephenson invented the idea of a gated community, so much as because he placed it in the context of extreme corporatism and factionalism and thus makes the idea new again in a particularly terrifying way. Manifold: Time isn't a wonderful read because Stephen Baxter invented the idea of the end of the world, but because he puts it in the context of a physically manifest observable and explores the consequences.

Certainly, authors such as Sky McKinnon write based on wonderful ideas, but I think that they have something else to teach us as well: that successful writing is the synthesis of ideas and a dogged enthusiasm for exploring ideas. In that way, writing, be it for a blog or a book, isn't so different from research, even if the tools for exploring the consequences of an idea are very different.

All this leaves me, however, with nary an excuse for my protracted blog silence. Ideas aren't my problem, after all. Oh, well. Maybe I should write a post about how ideas aren't my problem?

Much needed closure.

Closure is an important concept in mathematics, and is deceptively simple. If you have a set of things and some operation acting on those things, then the closure of your set is the smallest set that contains your original set along with everything that operation gives you.

The words get in the way, though, so let's consider an example. If you have the numbers zero and one, then their closure under addition would be all positive integers. Why? Because you can get to any positive integer by adding one to itself over and over. For instance, 2 is in the closure, since addition produces 2 from our set: 1 + 1 = 2. By the same argument, 3 is in the closure since 1 + 2 = 3, and since 2 must be in our closure.

We say that this set is the closure of our original set since it is the smallest set which is, well, closed. If, in our previous example, we omitted the number 2, our set wouldn't be closed any more, since addition could take us outside of the set.

As of late, however, the way that mathematicians use the word closure has started to be seen well outside of mathematics. Witness the rise of "epistemic closure" (closely related to deductive closure) as a useful term in political science. The word finds much use even outside of mathematics, as it gets to the heart of a very powerful technique in rational thinking: asking what, given some tool, one can produce. In epistemic closure, the tool is reasoning itself, while in our more pedestrian example, our tool was basic addition. In both cases, however, what remains is the use of closure as a mechanism for understanding and characterizing an operation.

In the spirit, then, of exploring closure, I'd like to bring some much needed closure forward. Specifically, I'd like to consider a kind of causal closure. If we consider some set of events which may or may not be causally related, we can for any specific event ask what events may be caused and what other events may cause it. Both of these are a kind of operation; extrapolating both directions in time to understand the causal structure of your set of events. The causal closure, then, of a set of events is the full set of events which caused the original set, along with the full set of effects caused by these events.

Luckily, we already have a term for this kind of causal closure. What we mean when we say that two events are causally related is that they lie within the same universe, so that the universe can be thought of as the set of all events which are causally related to an event representing our powers of observation. Under this realization, if A causes B, then A cannot be in a different universe than B.

All this is well and good, but why do I bring it up now? In a recent post, I asserted that religious claims were of a material nature, and thus amenable to the methods of science. That this is the case can be easily seen by invoking a principle of causal closure; if religious claims include any causal relation to the material universe, then they must, by closure, be entirely about the material universe. Of course, that alone does not mean that such claims are subject to scientific understanding, but that is an argument I have made before and don't wish to repeat here. Rather, my intent was simply to bring some much needed closure to bear on an argument that has gone on too long.

What is a gate? (Part 1)

It's been a bit since my last "what is" post, but I'd like to return to talking about science by taking a pause from my build-up to quantum states and quantum computation to instead discuss something more classical: the notion of a logic gate.

One way of modeling classical computation is as a sequence of operations performed on some data. We can then consider each operation independently. Just as we can build up complicated equations from simple arithmetic operations, these computational operations, typically called gates, can be used to build up arbitrarily complicated computations.

Take a specific example, the NOT gate, also written ¬. This gate takes a bit and produced a bit with the opposite value. Since each bit can only have one of two possible values (either 0 or 1), we can completely specify the behavior of the NOT gate by listing what it does to each of these inputs. That is, if I tell you that ¬ 0 = 1 and that ¬ 1 = 0, then in principle, I have told you everything that there is to know about the NOT gate. If this reminds you of a basis, then your intuition serves you well— we will explore that connection more in due time.

For now, though, I would like to discuss a few more examples of gates: the AND and OR gates, often written as ∧ and ∨, respectively (if these symbols seem arcane, it may think of them in terms of set unions and intersections). Each of these gates takes two bits as inputs and produces one output. AND produces 1 if and only if both its inputs are 1 (1 ∧ 1 = 1, 0 ∧ 0 = 0 ∧ 1 = 1 ∧ 0 = 0), while OR produces 1 if and only if at least one input is 1 (0 ∨ 0 = 0, 0 ∨ 1 = 1 ∨ 0 = 1 ∨ 1 = 1). Finally, the XOR gate (short for exclusive or and written ⊕) returns 1 if and only if exactly one input is 1 (0 ⊕ 0 = 1 ⊕ 1 = 0, 0 ⊕ 1 = 1 ⊕ 0 = 1).

With these four gates, we can build up any arbitrarily complicated Boolean function; that is, a function from strings of bits to a single bit. Functions returning multiple bits can in turn be built up by representing each output bit as a Boolean function. We could actually do with less kinds of gates, but that's besides the point. Rather, the point is that together, NOT, AND, OR and XOR are universal for classical computation.

It takes some effort to prove this, but an example helps to make things concrete. The full adder circuit in particular can be used to add two one-bit numbers, and is built up entirely from two XOR gates, two AND and one OR gate, as shown in this circuit diagram from Wikipedia. These full adders in turn can be combined to add arbitrarily long integers. From addition, one can get to subtraction and multiplication, demonstrating the usefulness of the gate model in capturing arithmetic.

Even more compellingly, we can efficiently simulate Turing machines with these few gates, meaning that NOT, AND, OR and XOR are at least as expressive as Turing machines. Thinking about gates, then, is a powerful way of thinking about classical computation. As we shall see, this power carries very nicely to the quantum case as well.

On loyalty of a peculiar kind.

The Scientific American podcast highlighted today research showing that members of "Generation X" (why is that Godawful name still around?) are on average more loyal to religion than are members of their parents' generation. Setting aside the question of how reliable this report is, since there are very few details given, let us instead treat the article as a launching point for discussion.

Indeed, it is uncontroversial that loyalty to religion exists in some sense. What does such a loyalty mean, however? What courses of action are demanded by such a loyalty? This is at best a problematic question to answer, as it is belief in a set of material claims that may be taken as comprising a religion. Under this view, religions are not adopted as matters of principle, but rather as a matter of that belief. One may as well ask what a loyalty to The Lord of the Rings implies.

Not to put too fine a point on it, but it is a bizarre notion that one can be loyal or disloyal to a set of claims about material reality. Such claims are ideally decided by consulting that self-same material reality, rather than loyalty to one or another set of claims. One that is loyal to a set of religious claims is then someone that is compelled by this loyalty to assert the primacy of their claims over evidence. Such loyalty is distinct from loyalty to a person, ideal or value in that it cannot be a matter of principle without falling into the trap of argument by appeal to consequences.

This sort of religious loyalty is, on the other hand, one onto which principles can be grafted. If some of the material claims to a religion are that certain modes of conduct are inherently morally superior to others by divine edict, then adoption of principles reinforcing those modes of conduct is a consequence of loyalty to those claims. Since correspondence to reality is not demanded from these religious claims about reality, such claims may be manipulated so as to imply any of a wide range of mutually contradictory principles. That is, religious loyalty is not a matter of principle so much as a vessel into which principle can be poured.

Witness, for instance, the latest absurdity from Glenn Beck, his I Have A Scheme speech. As others have noted, the attendees were indeed fiercely loyal, but to no particular principle. Rather, the Tea Party, in buying into and expressing loyalty towards material claims that are demonstrably false, has made themselves into so many empty vessels into which the hateful principles of the GOP may be poured.

Just as with religious loyalty, this political loyalty neither demands nor exhibits correspondence with reality, neither demands nor exhibits principle. As such, it is difficult if not impossible to apply reason to these views. This sort of blind loyalty, made without deference to what actually is, must be seen as a problem if we are to progress in our moral thinking as a society.

All that is sacred.

(adj) sacred (worthy of respect or dedication) "saw motherhood as woman's sacred calling" [WordNet]

Leaving aside the implicit misogyny of the example given, the citation from WordNet for the word "sacred" demonstrates something very important: we can divorce what it means to be sacred from any sort of religious sentiment. Indeed, if we are to leave irrationality behind us, I assert that we must do so. Thus, I'd like to talk a bit about what is sacred to me. That is, what I find to be inherently worthy of respect or dedication.

In that spirit, then, knowledge is sacred to me in ways that nothing else is. Were I to be asked to identify the most quintessentially defining aspect of all that is good about humanity, I would likely respond that our ability to accumulate and record knowledge is what allows us to transcend not only the ignorance into which we are all born, but also the limits of our physical brains. All other human achievements are enabled by our accrual of knowledge in ways that outlast any individual human. At once, the acquisition of knowledge is a highly individual and highly collective pursuit, epitomizing what it means to achieve something of permanence. Towers crumble, words remain.

It is difficult for me to adequately justify my valuation of knowledge as uniquely sacred, as it is fundamental to the person that I've become. As someone that values rationality, however, I must work to increasingly do just that. If this valuation cannot be supported on its own merits, then it is no better than faith and other such anti-virtues. That said, I am in the awkward position that every rational person eventually finds themselves in of not knowing all the answers. Like everything else in my life, this valuation must be amenable to rational analysis, and yet I must have some notions which guide my actions in the interim. Put differently, I must employ some set of heuristics that I use to evaluate both my own choices and those of the society around me. These heuristics must then be refined by learning additional facts and must be discarded to the extent that they contradict reality.

On a more tangential note, I tend to suspect that it is these heuristics that often get confused with religious-minded beliefs, driving the "science is faith" fallacy that I find so detestable. The key difference is that the heuristics adopted by someone that values rationality are recognized as being mere approximations, and thus are malleable to the extent that the underlying reality is not known. Thus, while such heuristics superficially resemble beliefs, they are quite different in practice.

Of course, it's very easy to simply say that something is sacred; a more pressing question for someone dedicated to rationality and materialism is what this assertion implies. A heuristic which does not either directly imply action or imply other values and heuristics which in turn imply action is by hypothesis a vacuous and useless heuristic. Exploring this notion, then, consider what a heuristic of sacred knowledge leads me to aspire to.

Though it is somewhat circular, the first and perhaps most important consequence of this heuristic is the valuation of science, the formalization of the pursuit of knowledge. Disentangling this apparent bit of circular reasoning would take me still further afield, so I will be content to leave it for now with the claim that the sacred-knowledge heuristic and the scientific method are synergistic rather than truly circularly dependent.

Another important consequence of this heuristic is the additional heuristic that knowledge should be shared-- after all, knowledge locked away is knowledge that cannot help in the further pursuit of other knowledge. This is a large part of why the open access and open source movements excite me so, and why I oppose the locking away of human knowledge behind paywalls, military secrecy or other such artificial barriers. Additionally, knowledge kept secret is knowledge that is much more difficult to preserve.

With new approaches to information storage, computation and communication, we are blessed (if you'll forgive the pun and not read too much into it) with new opportunities to safeguard our knowledge against the relentless march of time. To exercise these opportunities, however, archivists must be recognized as a critical part of our societal infrastructure and knowledge must be accessible for preservation.

While I could continue in this vein, I think that this short exploration of the consequences of my sacred-knowledge heuristic is sufficient to demonstrate an essential point: rationality requires rather than precludes the adoption of strong principles to be applied to the world around us, insofar as these principles are derived from sources amenable to rational analysis. We cannot afford for religion to maintain a cultural monopoly on the respect and dedication that underlie the word "sacred," but rather must build our own sacredness in a rational way. All that is sacred, in short, must still lie within the realm of that which can be reasoned about if we are to maintain the primacy of rationality.

Poisoned ethics and used video games.

Thanks to @saverqueen (blog) for inspiring this discussion.

Recently, I bought some used video games. I love buying used, as it saves me money and keeps perfectly fine games out of landfills, not to mention preventing more from having to be printed in the first place. These days, at least half of the video games I buy are used. Similar goes for me and movies.

With this particular purchase, however, some mixed feelings were brought forth. You see, these games were purchased as a gift. It seems almost instinctual that one doesn't give used games, movies, books, etc. as gifts. To do so is almost as bad a sin as playing with a toy before gifting it, or wearing clothes intended for someone else. At least, that's what the societal norm seems to be. Love is buying new things, goes the chorus.

I think we must, however, take a step back and ask if that is really the kind of ethical norm that we wish to adopt as our own. Why should our love for one another be expressed by continuing a destructive consumeristic cycle, where newness is its own reward? It is not even consumerism itself that I find so objectionable as the pointlessness of making consumerism the goal rather than the means. The philosophy of buying new for its own sake seems dangerously close to the vapid philosophy once espoused by a classmate of mine: "the meaning of life is to have kids!"

This is why my family has made a decision: used gifts are just fine with us! That decision affords me new opportunities to find unexpected gifts, such as classic video games for my brother that he wouldn't have found on his own, or out-of-print novels for my parents. Occasionally, yes, I do buy new things as gifts, even within the family, but when I do, I'd like to think it's because it's my decision to and not because I have let my sense of ethics become poisoned by the obsession with a growth-based economy. I give gifts to loved ones to bring them happiness; isn't that enough?

Accommodationism: A vexing asymmetry.

In my last argumentative post, I slipped in a bit of a sarcastic point at the end that I feel is worth treating more seriously. In that post, I said that:

It is truly unfortunate, however, that [his] approach to arguing for this controversial claim is to build such silly and distorted strawmen of atheists who might otherwise be more inclined to ally themselves with him in fighting the woo that he so rightfully expresses a passion to fight.

When I originally put those words to pixels, I intended only a cheap laugh at the thesis that atheists should keep quiet so as to not scare off the religious from the goal of (for instance) science education. This thesis, broadly called accommodationism by its detractors, including myself, has been been quite pervasive as of late (making it all the way to the AAAS, for instance), and has been the center of much discussion.

What bothers me most about accommodationism, however, is something that is too seldom remarked upon: its strange and vexing asymmetry. While it is often claimed that anti-religious sentiment scares off the religious from worthwhile causes, irregardless of how well or poorly it is supported by rational argument, I have never heard it argued that people need to be more accepting of atheists for fear of scaring us away from these same worthwhile causes. Does it not cause accommodationists consternation that referring to "fundamentalist atheists" may be the precise kind of incivility that that they fear poisons communities? Anecdotally, at least, I can confidently state that I have a harder time partaking in communities where my atheism is rejected out of hand and treated with derision rather than argued against.

Don't get me wrong, however, as I wouldn't dream of asking for special privilege and exemption from criticism. Criticism, when delivered in an honest and clear manner, is the lifeblood of an intellectual community. Rather, I find disturbing the comparative lack of concern at the derision pointed at atheists that one would expect from an intellectually consistent position. Is it the case, then, that atheists are seen as less desirable by such accommodationists than are the religious? Is it that atheists are seen as a more direct threat to the goals of promoting science in society than are the true fundamentalists?

There is another possibility that seems much more palatable to me. Atheists are seen as mature enough being able to take such derision in stride along with the criticism. For obvious and self-centered reasons, I should like to think that this is the case. Why, then, is the assumption that people of faith are less able to deal with both legitimate criticism and the sort of derision that comes with any emotional issue? Such an assumption seems to me to be more insulting than any of the derision thrown about by the atheists.

All this is a long-winded way of saying that I think we should not let the valid and laudable pursuit of civility and mutual respect lead us into the sort of asymmetric mire that is accommodationism.

What is a matrix? (Part 2)

Now we have a new kind of mathematical toy to play with, the matrix. As I said in the previous post, the easiest way to get a sense of what a matrices do is to use them for a while. In this post, then, I just want to go over a couple useful examples.
Suppose you wish to make all vectors in ℝ² longer or shorter by some factor s ≠ 0. You can represent this by a function f(v) = sv. With a moment's work, we can verify that this is a linear function because of the distributive law. Thus, we can represent f by a matrix. To do so, remember that we calculate f for each element of a basis. For simplicity, we will use the elementary basis {x, y}. Then, f(x) = sx and f(y) = sy. By using coordinates, we can write this as f([1; 0]) = [s; 0] and f([0; 1]) = [0; s]. The matrix representation of f then becomes:

Note that if s = 1, the function f doesn't do anything. Representing f(v) = v as a matrix, we get the very special matrix called the identity matrix, written as I, 𝟙 or 𝕀:

The identity matrix has the property that for any matrix M, M𝟙 = 𝟙M = M, much like the number 1 acts.

Of course, there's no requirement that we stretch x and y by the same amount. The matrix [a 0; 0 b], for instance, stretches x by a and y by b. If one or both of a and b is negative, then we flip the direction of x or y, respectively, since -v is the vector of the same length as v but pointing in the opposite direction.

A more complicated example shows how matrices can "mix up" the different parts of a vector by rotating one into the other. Consider, for instance, a rotation of the 2D plane by some angle θ (counterclockwise, of course). This is more difficult to write down as a function, and so a picture may be useful:

By referencing this picture, we see that f(x) = cos θ x + sin θ y, while f(y) = - sin θ x + cos θ y. Thus, we can obtain the famous rotation matrix:

As a sanity check, note that if θ = 0, then R_θ = 𝟙, as we would expect for a matrix that "does nothing."
One very important note that needs to be made about matrices is that multiplication of matrices is not always (or even often) commutative. To see this we let the matrix S swap the roles of x and y; that is, S = [0 1; 1 0]. Then, consider A = SR_θ and B = S. Since applying S twice does nothing (that is, S² = 𝟙), we have that BA = R_θ. On the other hand, if we calculate AB = SR_θS, we find that AB = R_-θ:

(Sorry for the formatting problems with that equation.)We conclude that AB ≠ BA unless sin θ = 0, neatly demonstrating that not all the typical rules of multiplication carry over to matrices.

I'll leave it here for now, but hopefully seeing a few useful matrices makes them seem less mysterious. Until next time!

What is a matrix? (Part 1)

Functions are an important tool in mathematics, and are used to represent many different kinds of processes in nature. Like so many mathematical objects, however, functions can be difficult to use without making some simplifying assumptions. One particularly nice assumption that we will often make is that a function is linear in its arguments:

One can think of a linear function as one that leaves addition and scalar multiplication alone. To see where the name comes from, let's look at a few properties of a linear function f:

This implies that f(0) = 0 for any linear function. Next, suppose that f(x) = 1 for some x. Then:

This means that if f represents a line passing through 0 having slope m = 1 / x.

So what does all this have to do with matrices? Suppose we have a linear function which takes vectors as inputs. (To avoid formatting problems, I'll write vectors as lowercase letters that are italicized and underlined when they appear in text, such as v.) In particular, let's consider a vector v in ℝ². If we use the {x, y} basis discussed last time, then we can write v = ax + by. Now, suppose we have a linear function f : ℝ² → ℝ² (that means that takes ℝ² vectors as inputs and produces ℝ² vectors as output). We can use the linear property to specify how f acts on any arbitrary vector by just specifying a few values:

This makes it plain that f(x) and f(y) contain all of the necessary information to describe f. Since each of these may itself be written in the {x, y} basis, we may as well just keep the coefficients of f(x) and f(y) in that basis:

We call the object F made up of the coefficients of f(x) and f(y) a matrix, and say that it has four elements. The element in the ith row and jth column is often written F_ij. Application of the function f to a vector v can now be written as the matrix F multiplied by the column vector representation of v:

We can take this as defining how a matrix gets multiplied by a vector, in fact. This approach gives us a lot of power. For instance, if we have a second linear function g : ℝ² → ℝ², then we can write out the composition (g ∘ f)(v) = g(f(v)) in the same way:

That means that we can find a matrix for g ∘ f from the matrices for g and f. The process for doing so is what we call matrix multiplication. Concretely, if we want to find (AB)_ij, the element in the ith row and jth column of the product AB, we take the dot product of the ith row of A and the jth column of B, where the dot product of two lists of numbers is the sum of their products:

To find the dot product of any two vectors, we write them each out in the same basis and use this formula. It can be shown that which basis you use doesn't change the answer.

If this all seems arcane, then try reading through it a few times, but rest assured, it makes a lot of sense with some more practice. Next time, we'll look at some particular matrices that have some very useful applications.

Rebuttal: The Difference Between Religion and Woo

I have tried to resist writing about science and religion for awhile; at least, dial back the frequency a bit. My ideas are not hidden, but they're also not terribly unique. Much of the time, I suspect my voice only marginally adds to the conversation, if at all.

All this aside, there are times when I find it extremely difficult to resist. It is particularly hard for me to let something lay when someone else makes an issue of it. This is precisely the case of Rob Knop's latest post, in which he attempts to insert a wedge between religion and woo while still maintaining the validity and importance of science. His post is such a quintessential example of that protected status for religion that I find so harmful to our society that I find myself drawn into yet another Web-delivered argument. I don't write this post with the hopes that my argument with Knop will go any better than last time, but rather because it is important to me that I try.

Without further ado, then, let us look at what Knop has to say. It's a long post, so by necessity I will pick out the bits I feel most deserving of response-- go read it for the full context of his remarks.

Why do I mention this? Because I see a lot of those who call themselves skeptics making exactly the same mistake— judging another field of intellectual inquiry on what they believe to be the one true way of reason. They dismiss things as trivial or childish based on criteria that fail to be relevant to the field of human intellectual activity they’re trivializing. Specifically, there are a lot of people out there who will imply, or state, that the only form of knowledge that really can be called knowledge is scientific knowledge; that if it is not knowledge gained through the scientific method, it’s ultimately all crap.

At this point, Knop has made it clear that he intends on revisiting his false equivocation between religious fundamentalists and "fundamentalist atheists" (full disclosure: Knop apparently considers me to be a member of this group). By using phrases like "one true way of reason," Knop conveniently ignores that skepticism, atheism and rationality have no central dogma beyond a sort of pragmatic honesty: if you are going to claim that your methodology (or way of reason, in Knop's vernacular) works, then it had damn well better work. As a part of that, yes, you must be able to verify that your "way of knowing" produces useful results, or else you cannot legitimately say that your methodology is a valid one.

The scientific method, then, which Knop elevates to the level of dogmatism in order to build his straw man, is not a dogma at all but a formalization of those ways of learning that have been shown to work. Far from being immutable or the "one true" way, science is adaptive and self-correcting. Already, then, Knop's equivocation fails on the basis that he's not describing skepticism as is espoused by the atheists he is so reviled by, but rather his own funhouse mirror version. We've got a lot of post left to cover, though, so let's press on:

What makes Robert Frost so much more important to human culture than the stories I wrote when I was 7? It’s not a scientific question, but it is a question that is trivially obvious to those who study literature, culture, and history. And, yet, using my 7-year-old story to dismiss all of literature as crap makes as much sense as using the notion of believing in a teapot between Earth and Mars as a means of dismissing all of religion.

If there is one sure way of pissing me off, it's to tell me that something "isn't a scientific question." Given that science is the methodology of pragmatism, such claims are no more than a way of giving up reasoned analysis. As someone who has made a career out of cultivating and exploring his own curiosity, few things are more offensive to me than someone putting such ultimate limits along my path. I don't expect that Knop refrain from doing things that offend me, however, as that would make the world a much more boring place--- rather, I would hope that as a fellow scientist, Knop would feel the same curiosity and lust for knowledge that renders such a claim so offensive to me.

The burden, however, of demonstrating that analyzing Frost versus the 7-year-old writings of Knop lies within the realm of science is one that I shall have to take on to truly make my point. In that vein, then, note that in addition to the "hard" sciences such as physics and chemistry, we have a full array of social sciences that are dedicated to applying scientific (that is, useful) methods to social questions. Such questions inevitably deal with the behaviors of entities each composed of many more than 10²³ particles, so that the "hard" sciences are completely overwhelmed by the sheer scale of the questions. Thus, we have found it useful to develop alternate methodologies that sacrifice some degree of exactness and objectivity in exchange for an enhanced ability to cope with such overwhelming questions as that proposed by Knop.

Ultimately, though, we must expect that the methods of analyzing Frost must lie within science for one simple reason: Frost existed within this reality, was a physical being and produced tangible objects that are amenable to study. Frost was, just like you or I, a citizen of the physical universe. Even if one posits the existence of a soul to try and escape this fact, the soul then influences the physical world by some mechanism that is not completely random, and thus can be examined. Knop's question is as scientific a question as any that could be asked, in that it is a question that concerns physical objects and that can be answered using useful and robust methodologies.

Of course, this is all a distraction from Knop's apparent point to mentioning Frost and his younger self. Rather, Knop accuses those who make reference to Russell's teapot of being akin to those strawmen that would discard Frost as useless due to the apparent uselessness of stories written by seven-year-old children. Indeed, Knop makes this accusation quite clear:

If you cannot see the difference between Russell’s teapot and the great world religions, then you’re no more qualified to talk about religion than the fellow who thinks that cultural bias is the only reason any of us believe in the Big Bang is qualified to talk about cosmology.

Pray tell, then, what is the difference between Russell's teapot and, just to make the discussion concrete, Christianity? Besides, of course, that the teapot is a gendanken intended to provide an easy example of the kinds of arguments that can and should be made against religion. All of Knop's strawmen aside, I have never heard of anyone claiming that Russell's teapot invalidates all of the world's religions, but rather that the gendanken explains why we should insist upon claims being testable. Religion is, in actuality, a complex and multi-faceted thing which many atheists and skeptics take a great deal of effort to understand. That along the way we find such examples as Russell's teapot useful is far from using the teapot as "a means of dismissing all of religion."

If Knop is interested in dragging atheists through the mud for overly reductionist arguments, then perhaps he should start by not reducing us to such a caricature of our actual arguments. That would include, for instance, not saying things like this:

There are quite a number of skeptics who openly say that they cannot see the difference between religion and belief in UFOs, Homeopathy, or any of the rest of the laundry list of woo that exists in modern culture.

There is of course a difference between religion and homeopathy: there's a hell of a lot more religious people in the world. Mind you, that's not the only difference, but the most immediately important one. As a consequence, religion alone has earned itself a special status in our society as immune to rational analysis and criticism. The point that I and others that agree with me tend to make isn't that religion and woo are the same, but rather that they draw from there is an important commonality to be found in their mutual rejection of rationality. This hypothetical reductionist that is blind to anything but that commonality, important as it is, is no more representative of actual atheists than any other strawman presented thus far.

On the other hand, Knop pretty much nails it with his next claim:

The assertion is that being religious is a sign of a deep intellectual flaw, that these people are not thinking rationally, not applying reason.

Yes, that is precisely what I have said here and in many other venues, though presented in much more judgmental terms than I find are appropriate to the assertion being made. Rather, I would put it differently by asserting that religion is not philosophically compatible or logically consistent with rationality.

Of course, the part of this assertion that  people repeat far less often is that religion is not unique in that regard. There are many other intellectual "flaws," a great many of which I will admit that I am afflicted by. Why I focus on religion, then, is that it is relatively unique in being celebrated and enshrined despite that it is defunct as a means of learning-- of accumulating accurate knowledge.

I could go into much more detail on this point, but for now let me leave it for now, as I would like to get onto Knop's next point:

It’s fine to believe [that religion is a sign of a deep intellectual flaw], just as it’s fine to believe that the Big Bang theory is a self-delusional social construction of a Judeo-Christian culture. But it’s also wrong.

Read that again, please. Knop is saying that it is fine to believe something that is wrong, and is it is with that assertion that I most passionately disagree with him. In my life, I strive to ensure that I believe only things which are true, and so I will admit that I have very little basis for understanding Knop's assertion here. Even moreso, when Knop continues thusly:

Yes, there is absolutely no scientific reason to believe in a God or in anything spiritual beyond the real world that we can see and measure with science.

This is a statement which is not new to me, but which I have made no recent progress towards understanding. I doubt that Knop intends to say that his god is impotent in that it is incapable of affecting the material world, and so I presume that Knop is asserting the existence of an untestable and yet still physical phenomenon. As I said before, however, this is where I must take earnest and profound offense: learning does not stop where it is convenient for the religious, and so we should not impose a priori limits on understanding the world just because of someone's god. Either Knop's god is impotent or it is material in the sense that it affects the material world; if we insist upon the latter, than the methods of science (sometimes called "methodological naturalism" in this context) must be able to study the patterns by which his god affects the world.

It is in the context of this assertion that I find Knop's closing comments so difficult to agree with:

But that does not mean that those who do believe in some of those things can’t be every bit as much a skeptic who wants people to understand solid scientific reasoning as a card-carrying atheist.

Knop has admitted in his post that there are a priori and impregnable limits to the limits of rationality, something which I do not admit or agree with. In doing so, there is at least one "bit" with which I am more willing to be a skeptic than is Knop. While overall, Knop may be more or less skeptical than I am (I really don't know which is the case), I cannot agree with the claim that his endorsing of religion is compatible with the skepticism he practices elsewhere in his life.

It is truly unfortunate, however, that Knop's approach to arguing for this controversial claim is to build such silly and distorted strawmen of atheists who might otherwise be more inclined to ally themselves with him in fighting the woo that he so rightfully expresses a passion to fight.

Note: Rob Knop said a great many things in his post I did not address, in the interests of brevity (believe it or not). Please don't take this posting as being a fair summary of the entirety of his argument, as it is intended only as a response to those points I found most objectionable.

Role for initiative.

Since I can't resist doing more cultural and philosophical blogging, I figure I should chime in on one more of the recent topics to set the Internet abuzz. In particular, Greta Christina wrote a pair of excellent articles on ten unfair and sexist things expected of men (part 1, part 2). As is her custom, Greta Christina nails beautifully the points that she attempted to hit in her columns. For my part, then, I'd like to expound just a bit, and to be presumptuous enough to add a stupid thing of my own to the list of stupid things society expects of men.

Preemptive Apology: The rest of this post will be unintentionally heteronormative, as that is the set of experience from which I can best draw. As such, I apologize to those outside the narrow range of sexual identities discussed here for neglecting their experiences.

To put it bluntly, modern society still seems (anecdotally, anyway) to maintain the antiquated expectation that men take the initiative in forming relationships. It is common that we put the pressure on men to initiate relationships ("have you asked her out yet?"), and that we encourage women to wait. While this standard is manifestly unfair to women, as it strips them of yet mode of personal decision making, I posit that it is just as manifestly unfair to men. Taking the first step is bloody hard, after all. You must be willing to put your feelings on the line, to be honest in the face of intimidating awkwardness, and perhaps most frighteningly, to be wrong about your feelings.

All these is leaving aside, of course, the mire of ambiguities and potential misinterpretations built up from cultural expectations of a male privileged society. In any action, one must take into account the cultural context of that action in order to respect the humanity of those around them. It is no different in the case of romantic initiative, save for that the context is that much more overwhelming.

Thankfully, we see the signs of this unfair standard starting to break down, as both men and women alike are encouraged to seek the pleasure of another person's company. In time, then, and with the introspection granted by such discussions as that started by Greta Christina, perhaps we can decouple the role from the initiative.

Looking back at what is.

Over the past few weeks, I've tried to tunnel through the potential barriers to actual science blogging, with mixed success. One of my bigger oversights thus far has been to omit any sort of a "big picture" from my posts, leaving them as little islands in a vast sea of scientific ideas. Today, I'd like to correct that.

My most immediate goal in science blogging has been to explain how physical states function in the beautiful formalism of quantum mechanics. The language of quantum mechanics, however, is one of probabilities, of complex numbers and of linear algebra. Arguably the most fundamental part of the language of quantum mechanics, linear algebra may be roughly thought of as the study of vectors, and how they transform. As I shall discuss in a future post, by using the idea of a basis, we can represent a special kind of vector transformation by an object called a matrix (more generally, an operator). Thus, what we have discussed thus far is not a set of disparate islands so much as a set of stepping stones. If you prefer a more concrete metaphor, we have poured a foundation for future discussions, including a discussion of the quantum state itself.

Once we have the idea of a quantum state, the horizon opens wide for exploration. The quantum state gives us a language in which we can understand seemingly arcane consequences of a world described by quantum mechanics, such as entanglement or superposition. With the formal tools of mathematics at our disposal, we can overcome the limitations of our intuition, so that we can understand even such tricky concepts as these.

One downside to my stepping-stone approach, however, is to seemingly put the concept of a quantum state on a pedestal, inaccessible without a high degree of mathematical maturity. Little could be further from the case. Indeed, the mathematics with which we understand quantum states are not so difficult as they are esoteric. It is my own opinion that these areas of math need not be esoteric, save for that it has been arbitrarily decided upon (at least in my home, the United States) that Math Is Hard, and that concepts such as those discussed here Should Be Left To the Professionals. Bollocks. We live in a probabilistic world, and one in which statistics guide nearly every aspect of society, so why should understanding probability be so inaccessible? While complex numbers are not so manifestly real, even to the point that i is called the imaginary unit, it takes but a small amount of study to see that the complex numbers form an integral part of how we describe reality. Similarly, the concept of a vector may seem too far removed from reality for the layman to pursue, but in many ways, vectors formalize and encode much of our intuition about geometry, and are just as accessible as the sort of geometry that is taught in many grade schools.

No, quantum states are there for those who want them. My goal is to bring the concepts just a little bit closer, and to let the mathematical beauty underlying them shine through just a little bit brighter. In doing so, I won't always go from point to point in most straightforward way, but I ask your patience, for I am going somewhere. With a bit of looking back at what is, I hope you'll agree that we're going somewhere interesting.

What is a basis?

Consider a vector. Just to make things concrete, consider a vector on the 2-D plane. In fact, let's consider this one (call it v⃑):

It's a vector, to be sure, but it's hardly clear how one is supposed to work with it. It doesn't make sense to pull out a ruler and pencil every time we want to add our vector to something; mathematics is supposed to be a model of the world, and thus we should be able to understand things about that model without recourse to physical measurements. To solve this problem for vectors on the plane, we can introduce two new vectors, x̂ and ŷ, then use vector addition to write v⃑ as a sum:

Now we can write v⃑ = ax̂ + bŷ̂, which doesn't at first seem to buy us much. Note, however, that we can write any vector on the 2D plane as a sum of these two new vectors in various linear combinations. Mathematically, we write this as ℝ² = span {x̂, ŷ}. Whenever a space can be written this way for some set of vectors B, we say that B is a basis for the space.

Once we have a basis picked out, we can work with the coefficients (a and b in our example) instead of the vector itself, as they completely characterize the vector. For example, adding vectors becomes a matter of adding their respective coefficients.

In spaces other than the 2-D plane, we can also apply the same idea to find bases for representing vectors. Consider, for instance, the space of column vectors such as [a; b] (pretend they're stacked in a column, OK?). Then, a perfectly fine basis would be the set:

It's easy to see that we can write any other 2-dimensional column vector as a sum of the form a[1; 0] + b[0; 1] = [a; 0] + [b; 0] = [a; b].

A point that can get lost in this kind of discussion, however, is that there's absolutely nothing special about the bases I've given here as examples. We could just as well used [1; 1] and [1; -1] as a basis for column vectors, or just as well used a different pair of vectors in the plane:

Put differently, a basis is a largely arbitrary choice that you make when working with vectors. The relevant operations work regardless of what basis you use, since each of the vectors in a basis can itself be expanded. For example, [1; 0] = ½([1;1] + [1; -1]) and [0; 1] = ½([1; 1] - [1; -1]), so that we have a way of converting from a representation in the {[1; 0], [0; 1]} basis to the {[1; 1], [1; -1]} basis.

While there is much, much more to be said on the topic of bases for vectorspaces, I'm happy to say a few words about bases. As we shall see when we get into discussing linear operations, the existence of bases for vectorspaces is a large part of what gives us so much power in linear algebra. We shall need this power in the quantum realm, as linear algebra may well be said to be the language of quantum mechanics. Hopefully I'll get a few more words in on the subject before my vacation!

What are vectors?

As I've said before, science is social-- oops. Wrong mantra. What I meant to say is that vectors are an abstract way of describing a pattern. Specifically, the vectorspace axioms formally describe a kind of mathematical object, the vector, that encapsulates the geometric and algebraic properties of a large class of seemingly disparate objects. By using the vectorspace axioms, we will be able see that lists of numbers such as are vectors, as are arrows on the 2D plane.

Rather than describe how to do so myself, though, I will try something different. Vectors are important in much of physics, and so lots of people have already written much about them. Thus, for the bulk of the work in describing vectors, I will defer to these other writings. A very physics-oriented approach can be found over at Dot Physics, starting with a trig-based introduction to vectors, followed by a discussion of how to represent vectors. An alternate physics-motivated discussion of vectors can be found at HyperPhysics.

For the more mathematically motivated amongst us, Wikipedia has a good page describing a very special family of vector spaces called ℝⁿ that is used to describe points in Euclidean space. MathWorld has a few good articles on vectors, including a technical definition and listing of properties and a more concise listing of the vectorspace axioms. Finally, the Unapologetic Mathematician derives vectorspaces from a more general construction called a module (warning: not for the feint of math).

To understand why we care about vectors in quantum information and computation, however, takes one more observation. A quantum state can be written as a linear combination of some set of basis states. For example, an arbitrary qubit state can be written as . This important property means that quantum states are a kind of vector in what we call a Hilbert space. This has some profound implications for how we think of and manipulate quantum states, as we shall explore in forthcoming posts.