Ceci N’est Pas Une Blog

Posted by jns on 20 February 2008

This is a blog posting about itself. According to the blog-software statistics, this is my one-thousandth posting [at my personal blog, the source for this essay] since the first one I posted on 18 October 2004.^* To be honest, I’m a bit surprised that I’m still writing here regularly three-plus years later. Evidently it works for me somehow.

I’ve noticed that one-thousand is an accepted milestone at which one is to reflect, look back, and perhaps look forward. Well, you can look back as easily as I can, and I don’t see much reason to try predicting the future since we’re going to go through it together anyway. Therefore I thought this article should be about itself.

Or, rather, the topic is things that are about themselves. So called self-referential (SR) things.

I believe that my introduction to SR things, at least as an idea, came when I read Douglas Hofstadter’s remarkable book “Gödel, Escher, Bach: An Eternal Golden Braid”. The book was published in 1979; my book of books tells me that I finished reading it on 17 August 1986, but I expect that that date is the second time that I read the book. I can remember conversations I had about the book taking place about 1980–and I didn’t start keeping my book of books until 1982.

Broadly speaking, GEB was about intelligence–possible consciousness–as an emergent property of complex systems or, in other words, about how the human brain can think about itself. Hofstadter described the book as a “metaphorical fugue” on the subject, and that’s a pretty fair description for so few words. Most of his points are made through analogy, metaphor, and allegory, and the weaving together of several themes. All in all, he took a very indirect approach to a topic that is hard to approach directly, and I thought it worked magnificently. In a rare fit of immodesty, I also thought that I was one of his few readers who would likely understand and appreciate the musical, mathematical, and artistic approaches he took to his thesis, not to mention how each was reflected in the structure of the book itself–a necessary nod to SR, I’d say, for a book that includes SR. There were parts of it that I thought didn’t work so successfully as other parts, but I find that acceptable in such an adventurous work. (The Wikipedia article on the book manages to give a sense of what went on between its covers, and mentions SR as well.)

The SR aspect comes about because Hofstadter feels that it may be central to the workings of consciousness, or at least central to one way of understanding it, which shouldn’t be too surprising since we think of consciousness as self-awareness. Bringing in SR for the sake of consciousness then explains why Kurt Gödel should get woven into the book: Gödel’s notorious “incompleteness theorems” is the great mathematical example of SR, not to mention possibly the pinnacle of modern mathematical thought.

Gödel published his results in 1931, not so long after Alfred Whitehead and Bertrand Russell published their Principia Mathematica (1910–1913). Their goal was to develop an axiomatic basis for all of number theory. They believed they had done it, but Gödel’s result proved that doing what they thought they’d done was impossible. How devastating! (More Wikipedia to the rescue: about Whitehead & Russel’s PM, and about Gödel’s Incompleteness Theorems.)

Gödel’s result says (in my words) that any sufficiently complex arithmetical system (i.e., the system of PM, which aimed to be complete) is necessarily incomplete, meaning that there are self-consistent statements of the system that can be made that are manifestly true but yet are unprovable within the system, which makes it incomplete, or that there are false statements that can be proven, which makes it inconsistent. Such statements are known as formally undecidable propositions.

This would seem to straying pretty far from SR and consciousness, but hold on. How did Gödel prove this remarkable result?^# The proof itself was the still-more remarkable result. Gödel showed how one could construct a proposition within the confines of the formal system, which is to say using the mathematical language of the arithmetical system, that said, in effect, “I am a true statement that cannot be proven”.

Pause to consider this SR proposition, and you’ll see that either 1) it is true that it cannot be proven, which makes it a true proposition of the formal system, therefore the formal system is incomplete; or 2) it can be proven, in which case the proposition is untrue and the formal system in inconsistent (contradictory). Do you feel that sinking, painted-into-the-corner feeling?

Of course, it’s the self-reference that causes the whole formal system to crumble. Suddenly the formal system is battling a paradox hemorrhage that feels rather like the Liar’s Paradox (“All Cretans are liars”) meets Russell’s own Barber Paradox (“the barber shaves all those in the town who do not shave themselves; who shaves the barber?”). When these things hit my brain it feels a little like stepping between parallel mirrors, or looking at a TV screen showing its own image taken with a TV camera: instant infinite regress and an intellectual feeling of free-fall.

Does Gödel’s Incompleteness Theorems and SR have anything to do with consciousness? Well, that’s hard to say, but that wasn’t Hofstadter’s point, really. Instead, he was using SR and the Incompleteness Theorems as metaphors for that nature of consciousness, to try to get a handle on how it is that consciousness could arise from a biologically deterministic brain, to take a reductionist viewpoint.

At about the same time I read GEB for the second time, I remember having a vivid experience of SR in action. I was reading a book, Loitering with Intent, by the extraordinary Muriel Spark (about whom more later someday). It is a novel, although at times one identifies the first-person narrative voice with the author. There came a moment about mid-way through the book when the narrator was describing having finished her book, which was in production with her publisher, how she submitted to having a publicity photograph taken for use on the back jacket of the book.

The description seemed eerier and eerier until I was forced to close the book for a moment and stare at the photograph on the back jacket. It mirrored exactly what had happened in the text, which was fiction, unless of course it wasn’t, etc. Reality and fiction vibrated against each other like blue printing on bright-orange paper. It was another creepy hall-of-mirrors moment, but also felt a moment of unrivaled verisimilitude. I think it marked the beginning of my devotion to Dame Muriel.

And that’s what this article is about. I suppose I could have used “untitled” as the title, but I’ve never figured out whether “untitled” is a title or a description. I think “undescribed” might be a still-bigger problem, though.

Now, on to the one-thousand-and-first.^%
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^* You’ll notice that neither the first nor the thousandth have serial numbers that correspond; the first is numbered “2″, the thousandth is numbered “1091″. Clearly I do not publish every article that I begin, evidently discarding, on average, about 9% of them. Some get started and never finished, and some seem less of a good idea when I’m finished with them than when I started.

^# And please note, this is mathematically proven, it is not a conjecture.

^% I’ve been reading stuff lately that described how arabic numerals were only adopted in the 15th century; can you imagine doing arithmetic with spelled-out numbers! Not only that, but before the invention of double-entry bookkeeping–also in the 15th century–and sometimes even after, business transactions were recorded in narrative form. Yikes!