Another post on Mr B is coming. Soon I hope, though I’m being carefully

vague even with myself on exactly when. Before Christmas. I have two other

essays to write before the end of term, realistically one had better be

written by the middle of next week, so Mr B is low-priority.

In the meantime I’ve been having another look at *The Difference
Engine*. I originally posted a review back in July of last year, which

concluded:

[D]on’t bother reading this book. It’s not bad, it’s

just rather thin on the ground. Once you’ve had the basic concept explained

to you, you’ve pretty much gotten what there is to be had.

Which

is a rather comprehensive dismissal of a book by two towering figures on the

SF scene of 20 years ago, when the book was written.

Something which struck me about that initial review was that I was terribly

interested in *character* and *story*, but less interested in

political commentary or philosophy, and as for world-building, well that’s a

low priority even when reading SF and Fantasy proper. But I propose that a

key part of reading any text is in figuring out what its own parameters and

ground rules are. Whether that works for you personally or not, I guess it’s

incumbent upon you to explore that. I have been aided in

this revisitation by a classmate of mine with a more recent mathematical

education than mine.

Let’s start off this time by looking at the book not as a novel at all – not

as something designed primarily to entertain. I’d like to think about it as

an allegory instead. An allegory is basically an explanation rendered into

symbolic terms. You know what I’m talking about: Aslan, Rasselas, etc.

The allegory in this instance is about systems of control. This is a central

preoccupation of the Cyberpunk generation. Cyberpunk is explicitly about

operators outside of a system trying both to use and abolish that system. In

*Neuromancer*, Case is both the prototype and the exemplar: a

marginalized hacker, he has effectively dropped out of the corporate system,

but the skills he retains see him hired to attack that system. His ultimate

goal however, is basically to rejoin the system which he is substantially

undermining. *The Matrix* really seems like the logical end-point of

this preoccupation, where humanity only exists at all in the system’s

cracks, and despite their opposition to it, even Neo and friends are

basically dependant on it because without it they are utterly without

meaning. Perhaps the great failing of the sequels was in not understanding

the necessity of this dichotomy, or how it worked.

In *The Difference Engine* the system of control is a pseudo-police

state, where your “number” is stored and makes accessible all pertinent

details of your life. Compared with the height of modern invasive

monitoring, it seems pretty tame, but it is conceptually totalitarian.

Moreover, it is overtly judgemental – the modern system of control we live

in doesn’t really care what you do, as long as you don’t attempt to hurt the

system. The Victorian hypocrisy drives a less benign society and use for the

machine. The object of the system is eventually to be all encompassing: to

say everything about everyone. Through total information, total power.

Enter the macguffin which is the central plot point that joins the four

novellas – the Modus. The modus is a proof of Godel’s Incompleteness

Theorem. The incompleteness theorem basically says that for any system of

mathematics, you can’t prove the validity of the axioms. In simple formal

logic terms, you can use a formula like: All A are B, All C are B, therefore

all C are A; if I’ve got that right. What you can’t prove is the validity of

those initial premises. The incompleteness theorem is basically the reason

that Kirk or the Doctor are able to cause computers to self-destruct: the

unprovability of the axioms implies the possibility of inconsistency (though

I should state that the incompleteness theorem does not necessitate any such

inconsistency) which they inevitably find.

To a society based around complete information, complete control, the modus

represents the glaring imperfection that is inevitable. Thematically it

means that however much they know, however invasive they become, some sliver

of blindness will remain. That undermines the absolutist philosophy of the

powers that be inside the fiction. In practical terms it is insignificant,

but it calls into question the basis of the society and so is tremendously

dangerous.

In *The Matrix*, Neo is the equivalent of the modus, at least if you

believe the Architect, which you are not necessarily bound to do. He

represents the inability of the system to be complete, and that’s why he

does not obey its rules, and why he represents a danger to the matrix

itself. Again, while the sequels toy with this idea, I don’t think they

really push home its natural result and conclusion.

*The Difference Engine* keeps the modus as its central preoccupation,

but through the four main novellas, its significance is kept unclear. Its

enormous thematic importance is held in reserve, and so to all intents and

purposes while actually reading the novel, it is a mysterious object which

could indeed be anything. After the big reveal, I think you are supposed to

re-examine the whole text in light of it, and what it signifies. To that

extent it qualifies as a twist – not a plot twist, but a narrative one. Like

realizing Bruce Willis is dead, it casts what you have experienced in a

different light.

Structurally, the implication is that each successive novella represents an

expansion of the world. In the first part, there is lots hinted at – you

read between the lines to fill in what you don’t know, but you’re guessing.

The second part shows you a much larger chunk of the world, and explains a

lot more of the mechanisms of that world to you: in light of the second

part, many elements in the first make a lot more sense. And so on through

the remaining two parts. At the end, the reveal comes, but not in a lot of

ways a resolution: how can the world be finally resolved? It can’t. That

becomes the touchstone of the work.

This kind of perspective renders the work much more successful than my first

reading, trying to take it on what are effectively my own terms. It

transforms this work from being a failure, as I originally claimed, to being

a very clever and successful work: more like an intricate clockwork engine

than a story really. But unfortunately, I think my end conclusion remains

the same: it’s a lot of work to put in for what you get, and the lack of a

real pay-off on the typical requirements for literature greatly reduces the

value of reading this book. While it is tremendously clever in this second

reading, I can’t help but feel that it shouldn’t sacrifice the other

potentials I have previously outlined in order to deliver this sterile

commentary.

The incompleteness theorem basically says that for any system of mathematics, you can’t prove the validity of the axioms. In simple formal logic terms, you can use a formula like: All A are B, All C are B, therefore all C are A; if I’ve got that right. What you can’t prove is the validity of those initial premises.

Actually, no, that’s basically wrong. In no system can you prove axioms; axioms are (by definition) things that you assume to be true. For instance, Euclid set out five axioms of geometry, but he thought the fifth might not be a true axiom: it might be provable from the other four. It was not until 1868 that mathematicians finally proved that the fifth axiom was truly independent of the first four.

Gödel’s incompleteness theorem says, roughly, that in any sufficiently complicated system of mathematics, there exist true statements that cannot be proven.

Basically, if you’re a logician, you might sit down one day and invent a new logic, or a new formal system. You’d define what truth means in your logic, you’d define how prove statements, and so forth. Once you’ve done this, you’ll try to prove that your logic is sound and complete.

“Sound” means: if you can prove a statement, then it is true. No one is interested in logics that let you prove false statements, so you will never come across an unsound logic.

“Complete” is the converse: if a statement is true, then you can prove it. Simple logics may be complete; for example, propositional logic is complete. This is the logic you may have met in first year university: the logic of AND, OR, NOT, and IMPLIES, where you can prove things using modus ponens, and check the truth of statements using truth tables.

First order logic (essentially, propositional logic plus the quantifiers “FOR ALL” and “THERE EXISTS”), on the other hand, is not complete.

So, back to Gödel: any logic, or system of mathematics, that is sufficiently complex to allow basic arithmetic, is not complete.

He proves it basically by using numbers to represent statements, and then arithmetic on those numbers becomes statements about mathematics. Then he constructs a statement that is equivalent to: “This statement is not provable in the system”. This statement must be either true or false. Suppose it is false. Then it is provable in the system. But we know that mathematics is sound, so false statements cannot be provable. Thus, it must be true. And hence, it is not provable, and mathematics is not complete.

The theorem goes a bit further: even if you add that statement as an axiom, you can still always build another one like it. So it is not possible to “repair” maths by adding new axioms, and Hilbert’s great dream came crashing down.

First order logic (essentially, propositional logic plus the quantifiers “FOR ALL” and “THERE EXISTS”), on the other hand, is not complete.

Actually, having checked wikipedia, this is wrong too. First order logic is complete. Second order logic (which lets you quantify over sets) is not complete.

And if any real logicians were reading this, they would probably quibble with that statement too 🙂

Thanks for clarifying that. I knew I was on slightly shaky ground, but I trusted my recollection. Next time it’s off to wikipedia. 🙂

I am not sure how that correction changes the argument, because the key idea is simply that no system can be complete and correct/consistent, which still nails the worldview of the Victorians in the Difference Engine and the Matrix.

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