Logicomix

 

Logicomix

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Logicomix is a simply brilliant book written by Apostolos Doxiadis and Christos Papadimitriou and coloured/designed by Alecos Papadatos and Annie Di Donna. It is about Bertrand Russell‘s quest for the Foundations of Mathematics and the (tenuous and rather loosely interpreted) connections between the study of logic and madness. It also spends a lot of time on Russell’s life. It is not only self-referential, it has layered self-reference: we see the authors talking about the story, in their book, which features Russell giving a lecture where he recounts most of the major scenes in the book. The narrative structure even includes some rather edifying “time-outs” in between where the authors write pages of themselves arguing over the story and how they are presenting it, and also time-outs on the “middle level” where Russell pauses his story to explain some necessary background information or is interrupted by a question. (1) The authors do take a few liberties with the text, as they explain at the end, making up fictional encounters where the characters only corresponded or dramatizing certain elements of their personal lives.

The book is beautifully drawn -more should be said here, but I’m just not the right person to say it- and it’s very impressive how well they give a general idea of all the mathematical discussions going on at that time without actually using any maths, although for my part I would have gladly dealt with a few more symbols and numbers in exchange for some more detail on the substance of their debate. This is, however, a perfectly reasonable level of detail for a popular book and the fact that it deals with logic rather than another aspect of mathematics does mean that it manages to make the central arguments sufficiently clear.  When the package first arrived from flipkart I was pleasantly surprised at the heft of it, because I’m not used to graphic novels this size, but it certainly doesn’t go on any longer than it has to, and they manage to end a story that has such an inconclusive-at-best ending in a very pleasantly self-aware way, by picturing and commenting on the ancient Greek tragedy of Oresteia. (It makes sense in context.) Overall, definitely recommended.

PS: This was originally supposed to be part of a much longer post in which I also mixed in where Wittgenstein picked up from Russell, reviewed A.C. Grayling‘s “A Very Short Introduction to Wittgenstein” and explained very, very briefly all the little things I jotted down while reading that book, most of which Grayling also uses at the end when he criticises Wittgenstein’s theories. Then I realized that a) the dude wrote a lot of stuff and a lot of it is, quite frankly, meaningless, b) Logicomix is actually not very related to the sort of thing Wittgenstein was primarily concerned with, even though it’s about the Foundations of Mathematics and Wittgenstein ostensibly started off concerned about the Foundations of Mathematics(2) and c) at the end of the book even Grayling says that “the dude wrote a lot of stuff and most of it is, quite frankly, meaningless”.

OK, actually he points out a few fairly fundamental contradictions in Wittgenstein’s work, which for all his assertions and vagueness does contain some sort of positive theses, and says Wittgenstein is important in the sense that a small minority of dedicated “Wittgensteinians” wrote copious amounts of material where they praised Wittgenstein in very enthusiastic terms, but not important in the “perhaps more accurate” sense that his work meaningfully changed the future direction of philosophy, even though it’s too early to predict that at this point. Wittgenstein’s work (both early and late) is certainly quite exciting, in many ways, and it’s easy to see why his later work might be very appealing to some people, especially for rationally inclined people who feel the need to be at ease with their religious impulses. It certainly seemed appealing to me while I was reading it the first time around. I’m also very happy that I didn’t attempt to read Wittgenstein in the original, judging by the quotes in this book, although of course if Grayling had any insidious motives it could easily have been cherry-picking. And now that I’ve said all this, I don’t believe I’ll be doing a separate post either, if only because d) it would be real hubris to try to critique a philosopher’s ideas in just one blog post when the very notion of reading his unabridged work fills you with dread.

 

Footnotes: (Which wordpress should really provide a better way of using)

  1. Christos argues with Apostolos on 2 major points, namely the tenuousness of that connection and the extent to which the main quest was a failure and, therefore, the extent to which this book is a tragedy:
    Apostolos: “Russell himself called it a failure, and they never really got to the Foundations of Mathematics.”
    Christos: “But it provided a framework, and it led to the development of computers!”
    And it certainly seems likely that computer science would not have developed in the way that it did without all this as background, so yay! I guess?
  2. In fact, in the book Wittgenstein really only figures as yet another Giant of Logic who was “eccentric”/maybe the teeniest bit off his rocker, but as I already mentioned they do stretch this link between logic and madness quite a bit.

More Stats from okcupid

These people are awesome…I don’t think I’ve linked to their previous few posts on the blog before, although I have shared their results before on Google Reader and/or Twitter. This is the latest, the real stuff white people like, on Gizmodo. Obviously even though this is for a bunch of races this is all (or almost all) Americans of those races. The writers have commented on the results for white, black, Latinos and Asians (which does not include Indians), so I’ll leave that alone, but here are some random points I thought were interesting:

  • Indian women, for some reason, refer to their passports and couches a lot more often than anyone else.
  • Asians, Middle Eastern people and Indians all seem to think “I’m a simple guy/girl” is a major selling point. This I will confess to being mystified by. The statement is basically signaling that they’re low maintenance, right? Or have low expectations? Are none of these groups “in demand” or at least “at par” on the dating market? Crap. Thankfully, it comes up a lot less for Indian men than for Asian/Middle Eastern men, and it’s not even on the list for Indian/Middle Eastern women.
  • Indian men like cricket and Indian women like bhangra. Also most of the men are software engineers or traders. I guess that’s not really surprising, but it’s always interesting to see stereotypes borne out.
  • Middle Eastern women have “different cultures” as their single most favourite thing. I feel there’s a joke to be made somewhere here, but… 🙂 . They are also far more likely to describe themselves as petite. They also like Darjeeling I guess they’re referring to the tea – which doesn’t come up for any other group, not even the Indians.
  • Pacific Islanders seem like pretty cool people 🙂 .

But of course, the real reason I’m linking is just so I have some excuse to share this:

Sidenote: reading level

Since we were parsing all this text anyway, we thought it would be cool to do some basic reading-level analysis on what people had written about themselves. We used the Coleman-Liau Index, and when we partitioned the essays by the race of the writers, we found this:

The Real 'Stuff White People Like’

Before anyone gets too charged-up about this, we also ran reading level by religion and found this:

The Real 'Stuff White People Like’

Is there a Comic Sans version of the Bible? There really should be. We subdivided this chart further, by how serious each person was about their beliefs:

The Real 'Stuff White People Like’

It’s interesting to note that for each of the faith-based belief systems I’ve listed, the people who are the least serious about them write at the highest level. On the other hand, the people who are most serious about not having faith (i.e. the “very serious” agnostics and atheists) score higher than any religious groups.

Interesting, right? Although it must be said that a higher grade level is not necessarily a good thing. It isn’t on most blogs, for instance. However, assuming you want to signal intelligence, a higher grade level for your profile is probably desirable. So if nothing else this at least signifies that more religious people don’t want to signal intelligence as much, which is also interesting

[Insert Scientology Joke Here]

I think this is a pretty ridiculous assertion overall, but it’s still interesting.

Why there is no Jewish Narnia:

To put it crudely, if Christianity is a fantasy religion, then Judaism is a science fiction religion. If the former is individualistic, magical, and salvationist, the latter is collective, technical, and this-worldly. Judaism’s divine drama is connected with a specific people in a specific place within a specific history. Its halakhic core is not, I think, convincingly represented in fantasy allegory. In its rabbinic elaboration, even the messianic idea is shorn of its mythic and apocalyptic potential. Whereas fantasy grows naturally out of Christian soil, Judaism’s more adamant separation from myth and magic render classic elements of the fantasy genre undeveloped or suspect in the Jewish imaginative tradition.

Via MR.

Since I Can’t Write My Professional Ethics exam anyway…

A friend of mine asked for my opinion on this article, and since I ended up writing so much, I felt I ought to share. Essentially, it is a Christian response to the “How can there be a God if there is so much evil in the world!?” question. My own opinion is that the question is rather irrelevant to whether there is a God or not-if there isn’t, it doesn’t matter, and if there is, I can find explanations that are not constrained by the specific attributes of the christian God.

The “problem of evil” has a lot of problems of its own. It’s a favorite of atheists and other people who argue against the God of Christianity, and it generally runs something like “if God is all good, and all powerful, then why is there suffering (or evil; or, why do bad things happen to good people)?” The implication, of course, is that God is either not all good, or not all powerful — either of which would destroy the Christian God’s exclusive claim to deity.

Is it really a favourite of atheists? I’ve heard more apologetics trying to do their thing with this question than I’ve heard atheists seriously offering it up as a proof of their beliefs. But moving on…

There are quite a few interesting things about this idea. First, of course, is the obvious assumption on the part of our atheist, that suffering is evil and therefore wrong. Usually it’s a good idea to start by taking this apart. What is wrong with suffering? What is evil as opposed to good? This separates the men from the boys right away, because the atheist has to realize that he is mounting an internal critique of Christianity.

Er. Not really. Good and evil maybe predominantly christian concepts, but are neither uniquely nor originally (I think; not sure, though) christian concepts, and can in any case be phrased more appropriately in terms of a more comprehensive moral framework. Why not consider a simple “do no harm” utilitarianism? Events caused by unconscious or non-volitional(is there a better word?) systems (like natural disasters) are arguably outside the realm of good and evil. I don’t believe even most christians would argue otherwise.

An external critique on this basis is hard to justify, because in a naturalistic world of survival of the fittest and the pretensions of meat machines to higher function, “good” and “evil” are either meaningless, or to be defined in terms of what is best for (a) me, or (b) my offspring.

Or c) everyone. Which brings us back to utilitarianism*.

And nobody can deny that suffering and “evil” are powerful applications of nature to cull the weak and encourage the ongoing life and multiplication of the strong. Let’s not fight straw men, though; some atheists believe that they have a workable system of ethics without religion. Whether they actually do isn’t part of the scope of this post.

See above.

So, our atheist is trying to mount an internal critique of Christianity.

Why, if he admits that some atheists (believe they) have a workable non-religious system of ethics, is he going back to this “internal critique”? Why would one confine oneself to a single, not necessarily coherent, unproven set of postulates when the answers may quite possibly lie beyond them?
What happened to “let’s not fight straw men”???

Perhaps we should rephrase the problem for him: if God is good by His own (revealed) standards, and all powerful, then why is there evil in the world? If God really loved His creation, then would He allow crack-addicted babies to be born, or six million Jews to be brutally murdered in a space of a few short years, or people to go to eternal conscious torment? The obvious answer to the atheist is usually that He cannot be good and allow those things, and therefore Christianity is either a religion of horrors serving a God Who delights in misery, or an inconsistent system of beliefs.

Actually, I don’t think I know any atheists who say the former. Seems like something fundamentalists from another religion could try, though.

Bahnsen points out that the syllogism need not end like this. He recommends that we actually try to grasp this internally, like so: God is good. God is all powerful. There is evil in the world. Therefore God has a morally sufficient reason for the existence of evil. This is consistent with the Scriptures, and logical.

This might be rephrased without too much exaggeration: There’s a reason, but we’re not going to tell you until you’re old enough to know. If then. Hey, maybe never! (Do christian scriptures, in general, place the same emphasis on “enlightenment” as Buddhist or Hindu scriptures? Some emphasis is given, I think, so “maybe never” is probably unfair. This comment could be appreciated more if the reader understands that I’m trying to neutralize the sermon with an equal amount of snark.)

However, the atheist has a problem with that, because he does not see how there can be a morally sufficient reason for crack babies, holocausts, or hell. And generally speaking, the Bible does not provide direct answers for such things, beyond the glory of God (doubly repugnant to somebody who is already disinclined to glorify Him). So, the atheist judges God’s standard of goodness, and finds it wanting…

And at this point, the atheist’s attempt at an internal critique fails, because he had to step outside the critique to come to his conclusion (that God is not good).

Again. What is with this insistence on an “internal critique”? If a flat-earther came and demanded an “internal critique” of his arguments, how many would oblige?

How can I say this?

Consider: the internal critique begins with the premise that God is good. Everything that God does is good; one might say that good is defined by the character and action of God. Under the circumstances, the God of the Bible is in fact the only One capable of judging the goodness of a thing, but nobody is good except for Him. And anything He does, and anything He chooses, is good. When the atheist decides that something God has ordained or allowed is not good, he is effectively taking God’s place as the judge of what is good, and setting himself above God. The atheist is on the bench, and God is in the dock once again. That is usually where the atheist chooses to be, but we must point out that there can be no internal critique anymore, because internally, the system of Christian theology presupposes that God is good, and the atheist must declare that, no, he will be the judge of what is good, effectively switching places with God, and breaking any hope of consistency.

While some of us are still troubled and hoping for some sort of consistency (with christian doctrine, that is; consistency with the observable world is considered an absolute criterion), I think it should be clear from even a cursory examination of any New Atheist book that most atheists want nothing to do with it, so offering it as some sort of reward isn’t really going to accomplish much. The attempt to frighten the blasphemers with that last bit of rhetorical flourish-“effectively switching places with God”- is subject to the same argument. (Although more likely to succeed; fear is usually more effective.)

To put it simply, the atheist is not comparing himself to the Christian God if he thinks that he can judge God. The Christian God, and the revelation that He has delivered, exclude any possibility of that fact.

I would imagine that the atheist, by definition, is (if (s)he considers at all) considering the existence of some supernatural being with as-yet-unidentified attributes, not some “precisely imprecisely” drawn out picture of a “Christian God”.

Once the atheist is outside the paradigm of Christian ethics again, he has the same problem we outlined above; he must defend his concept of good and evil without God. Inside the paradigm, the atheist finds himself begging the question of God’s transcendental goodness.

A dilemma indeed, until one bothers to consider seriously everything philosophers have written for centuries about a non-theistic system of ethics. “Defending the concept of good and evil without God” is really not that hard! Such a moral framework might consider “good and evil” in a more clinical fashion than most believers, but it should still prove comprehensive. I think.

Truly, though, we have a point of contact with the atheist at this stage. He can’t get away from the idea that there is such a thing as good and evil. Now it is only a matter of proclaiming to him the truth of the God Whose ethical system he is borrowing in ignorance.

I wonder if the author would concede that it could possibly work the other way around?

Do check out the comments on the page, specifically this link, which has arguments that most of us will already have seen about altruism and morality in a non-theistic sense.

* I should probably explain that I prefer my utilitarianism garnished with some virtue ethics, and maybe a little deontology on the side, too. I haven’t worked out if this is merely because of an instinctive prejudice that pure utilitarianism/consequentialism is “meaninglessly” hedonistic, or what implications that has. There are refinements galore to all these theories which make them more acceptable to me than in their standard forms, but most are outside the scope of this post.

Quote of the Week: the Inevitability of Original Sin

[God] puts an apple tree in the middle of [the Garden of Eden] and says, do what you like guys, oh, but don’t eat the apple. Surprise surprise, they eat it and he leaps out from behind a bush shouting “Gotcha.” It wouldn’t have made any difference if they hadn’t eaten it…Because if you’re dealing with somebody who has the sort of mentality which likes leaving hats on the pavement with bricks under them you know perfectly well they won’t give up. They’ll get you in the end.

Objective Reality, Rationalism and the Nature of Belief

I was in the middle of an utterly normal conversation with a friend when it suddenly veered off into a discussion of whether beliefs are a matter of choice. I said, obviously not, and she said, “of course they are!”. And then we quickly got into a tangential discussion which ended with her comparing me to Hitler*. So for the elucidation of anyone who cares to read it, here is my take.

The first question we need to settle is whether there is such a thing as objective reality. This is apparently not a very obvious question, but most of (modern, as in post-Kant) Western philosophy agrees that there is. Quite a lot of Eastern Philosophy doesn’t, I think, but we are sidestepping this question for a moment until more authoritative sources can contribute. Kant himself was cited as an objection, but his point was simply that the “true” reality was unknowable, not that it doesn’t exist in the first place. We will assume that there is such a thing, which I think is a comfortable assumption to make, because this is the assumption that all of modern science depends on. (No references to quantum physics or Fritjof Capra or anyone else will be accepted without substantiation; and I do think that is fair.) There can be no logic without an objective reality, or mathematics(again, any references to Riemann geometry or anything similar must elaborate). For a somewhat quirky “proof”, see here.

Belief is the psychological state in which an individual holds a proposition or premise to be true, according to wikipedia. When you believe something, you are essentially holding a concept in your head that you “believe” corresponds to The Way Things Are (or Should Be, or Will Be, or Might Be, or Were-we are not particularly concerned about that.) This means that to a rational mind something you believe must be either true or false. “Truth” might be unknowable, as Kant says, or it might just be unknown at present, in which case one can hold no beliefs about what one doesn’t know. This doesn’t mean that one can just believe anything. (It is possible to designate a “current best candidate” to a certain “degree of belief”, as one does in Bayesianism. One then performs “Bayesian updating’ taking into account the strength of new evidence and the prior degree of belief. This is essentially what the true rationalist/Bayesian always does, although it is invariably astoundingly difficult to explain to any “true believers” 🙂 )I can believe something to be true that isn’t, “actually”, but if I realize that it isn’t then I have an unavoidable obligation to change my belief.

This, of course, is only in an ideal situation. People do find it quite easy, in real life, to hold contradictory ideas in their minds. People find it easy to forcibly believe various pieces of dogma that they are “required” to believe, according to their church, their parents, their advisors or the state. I just finished reading 1984 yesterday, and was treated to some fascinating descriptions and explanations of doublethink. I have read, if only in a very amateur capacity, about cognitive dissonance.I know that people can be made to think that 2+2=5. So I fully understand that people are capable of voluntarily believing in things, although to see whether they believe it at all levels of consciousness is a question for psychologists-despite the horrifying, frightening ending of the book, I would like to think that there is some subconscious set of core beliefs that one simply cannot alter. My point is that it is not rational to do so.

(“Why rationality?”, you might ask. I would reply, “What else?” For more eloquent arguments-although I can’t see why, because someone who doesn’t believe in rationality will, by definition, not be swayed by any arguments other than appeals to brute instincts of pleasure and pain-you can check out any number of websites. Overcoming Bias and Less Wrong are comprehensive if tedious places to start, although the first is no longer updated, at least about this. Or just check any number of competent atheist websites.)

Not surprisingly, one of the areas where this question must be dealt with is atheism, or the voluntary “conversion” to atheism(I specifically mention voluntary as a means of excluding those cases where you are simply substituting another piece of dogma, such as communism). This post explains the concept of involutarism and voluntarism in the question of believing in God. It is written by an atheist, and at the end the author confesses that he tends very strongly towards involuntarism.

According to Terence Penelhum, there are two general schools of thought when it comes to how beliefs originate: voluntarist and involuntarist. The voluntarists take the position that belief is a matter of will: we have control over what we believe much in the way we have control over our actions.

Theists often seem to be voluntarists and Christians in particular commonly argue the voluntarist position. Some of history’s most prolific theologians like Thomas Aquinas and Soren Kierkegaard have written that believing — or at least believing religious dogma — is a free act of will. This isn’t surprising, because only if we bear some responsibility for our beliefs can disbelief be considered a sin punishable by sending nonbelievers to hell. Nonbelievers encounter this perspective when evangelists exhort others to “just believe” and to “choose Jesus,” reminding us that our atheism is a sin and a path to damnation.

Involuntarists, on the other hand, argue that we cannot really choose to just believe anything. According to them, a belief is not an action and thus cannot be attained by command. For example, everyone realizes that even after a person has concluded beyond any doubt what they must do, that doesn’t mean that they will automatically do it. Beyond their conclusion is the fact that extra steps must be taken to actually make the action happen.

This post, on the other hand, is written by a Christian, at a site that seeks to answer rationalist skeptics of Christianity (and religion in general, I assume). It , naturally enough, chooses the side of voluntarism. Aside from saying that “The Bible Teaches We Can Choose Our Beliefs”, it attacks the idea that a belief cannot be changed by exertion of will:

As an example, for many years, people held to the belief that those of African descent were somehow less human than other people on the planet and could be bought and sold as property. It was only after a dedicated minority of people such as William Wilberforce began to insist through discourse and argument that beliefs began to change. Even today, I know of many people who had some type of belief or conviction that they were taught but now no longer hold since they thought through the arguments. So, I would see this as proof that a specifically held belief can be changed by the exchange of ideas and some hard thinking on the topic.

Of course, this argument completely misses the point. “Thinking through the arguments”, as he puts it, implies that one is either acquiring additional evidence or better interpreting existing evidence. It is completely rational to change one’s beliefs when one no longer has evidence for them, or has evidence against them. When we say that belief is involuntary, what we mean is that we cannot “choose” to believe something against the evidence. If they hadn’t changed their views, that would be evidence of voluntarism.

In conclusion: I can see no way in which one can rationally argue that belief is a matter of choice. This isn’t as overarching a concept as one might think, however. As I was surprised at having to explain, it does not in any way affect my libertarianism. I do not suggest that some one or some group of people go around correcting all wrong beliefs. All beliefs might have value, or they might not-in an area where we do not have conclusive proof, anything can go. However, this does not mean that one abandons rationality, it just means that one adopts a more sophisticated but equally obvious method of reasoning, such as Bayesianism.

*My theory was wrong, at least according to another site that I visited just now. It was a thought experiment more than anything else, anyway. I was talking out of my ass, as less polite people would call it.

PS: This is certainly one of the best Christian sites that I have seen, I must say, because it accepts a rationalist worldview and concedes the obvious, and then makes Christianity consistent by claiming the facts of the Resurrection:

This same concept holds true in all our beliefs about God and the world. Beliefs make truth claims that can be examined to see if they are internally consistent and externally consistent. An internally inconsistent belief is one where its own claims contradict each other. Logical positivism is a good example of this. Logical Positivists held a principle called the verification principle which said that for a non analytical statement to be meaningful, it must be empirically verifiable.2 Of course, the statement “for a non-analytical statement to be meaningful, it must be empirically verifiable” is neither analytic nor able to be proven empirically3, so it becomes internally inconsistent.

A belief is externally inconsistent if any of its claims contradict what we know to be true. Baha’i beliefs hold that “God has revealed Himself to humanity through a series of divine Messengers, each of Whom has founded a great religion.”4 However, if the fundamental tenants of each of these “great religions” contradicts each other, then their claim that cannot be true.

Jesus held to the concept of a single God, while other faiths like Hinduism hold to multiple gods and yet others like Buddhism don’t believe in a personal God at all. All these views of God are in competition with each other – it is impossible for all of them to be true. Further, Jesus taught that God had a son, whom He called “His only begotten son”, while Muhammad taught that God does not beget nor is He begotten”. Again, these teachings stand in direct opposition to each other, therefore, Baha’ism is externally inconsistent. It makes claims that don’t match the reality of the world.

The concept of consistency is a good way to begin to judge the truthfulness of any belief system, including the one in which a person is raised. If a belief is internally inconsistent, you don’t need to be exposed to any other beliefs to recognize the problems with the inconsistent belief. As an example, look at the way our understanding of medicine and the human body have changed. We no longer believe in blood-letting and such things – and even if we don’t know what the diagnosis is, we wouldn’t resort to treating a patient using those concepts.

Christianity is unique in its challenge to its adherents to test its claims and see if they are true. Paul says that the entire faith if the Christian rests on the historical fact of the resurrection. If that’s not true, then we are liars and we should be rejected. To this end, God has given us our reasoning faculties so that we don’t simply follow whatever we’re taught. To do so doesn’t show maturity and wisdom as a person.

It so happens that they conveniently ignore some of the other parts of christianity, as laid out in the Bible and the guidelines of the church, which call for blind faith. But that is an entirely different issue, and one which I have no interest in going into right now.

Dangerous Dungeons and Dragons

So, I was reading these, which are truly, amazingly fantastic, and I got really excited about Dungeons and Dragons, and then I went and found I had a complete collection of the rulebooks for Advanced Dungeons and Dragons 2, and then I found that it’s really, really tedious. Which made me give it up.

And then I saw these, and I’m thinking, cool 😀 :

dd2

Hey, I sympathize. The D&D manual is a pain in the ass

Ok, I know Chick tracts are just “easy game” and everyone’s probably seen it already and I shouldn’t even link to one any more. But DUUUDE! Can you believe it? This guy was serious!

One of many such panels

One of many such panels

PS: OK, no more Iranian solidarity. I never intended it to be permanent anyway. This is better, right? Anyone out there wants to suggest ideas for a new blog header?

Quote of the Week: The Anti-Testicle-Grabbing Clause

I mean, come on, how could I possibly resist this?

“If two men, a man and his countryman, are struggling together, and the wife of one comes near to deliver her husband from the hand of the one who is striking him, and puts out her hand and seizes his genitals, then you shall cut off her hand; you shall not show pity.”

Deuteronomy 25:11-12.

From this amazing article. Read.

Quote of the Week: Oh, for Pete’s Sake…

“So many of them are living in fear of spirits, of malign and threatening powers. In their bewilderment they even end up condemning street children and the elderly as alleged sorcerers”

That guy with the direct line to the Big Guy.

Like Mathew said, this is seriously wearing out my irony meter.