Now, we shift our attention to mathematics and logic and in doing so, we get to work on the other end of the big picture; hopefully we’ll be able to link back up with Inception somewhere in the middle.
Many of you will have, at some point, seen a demonstration of a classic mathematical fallacy. For example, a ‘proof’ that one is equal to two.
A small collection, some of which are not obvious, can be found here. The ‘proof’ above is incorrect because it relies on division by zero and the performance of this operation makes no sense.
When we see such ‘proofs’, we know immediately that they must be wrong because we all know that one does not equal two (although it is worth pointing out that knowing a claimed proof is wrong is not the same as actually identifying the fallacious logic). In summary, we know immediately that these proofs are wrong because we know that it is impossible for there to be something wrong with numbers in such an obvious and basic manner.
However, it becomes harder to be so sure when the mathematics is more advanced. It is conjectured, for example, that there are infinitely many prime numbers that are one greater than a square number, i.e. infinitely many primes of the form . Four is a square number because it is equal to two squared and therefore five, being one greater than four, is one such prime. Seventeen is another such prime. As is thirty-seven. Are there infinitely many such primes? The answer is thought to be ‘yes’, but is, in fact, unknown.
In mathematics we often start with foundational statements which we hold to be absolute, called axioms, and deduce other things (theorems) from the axioms. The proper name for what I’ve loosely described is called a formal deductive system – it is a set of rules, telling us the ways in which we are allowed to deduce one statement from another. The collection of all deducible statements is called a deductive theory. The familiar world of arithmetic is a deductive theory and it’s rules are based on familiar logic. The fact that one does not equal two is more or less part of the axioms that define arithmetic, but the problem of whether or not there exist infinitely many primes of a certain form is not the kind of thing that is described by the axioms. To decide whether or not it is true, we might try to prove it. This means that we’d start with the basic axioms and try to arrive at the statement we want to prove, using only the specific laws of deduction which are allowed in our formal deductive system.
Imagine that I prove the conjecture to be true. However, suppose also that you read up on the theory of prime numbers and arithmetic and are somehow able to disprove the conjecture. That is, you also start with some primitive facts and are somehow able to deduce that the conjecture is false. Apparently, we have a contradiction, since the conjecture cannot be both true and false!
If both proofs are indeed sound, then the issue does not lie with our deductions per se but with the theory of arithmetic itself. If, within the formal system of arithmetic, it is indeed possible to prove both a certain statement and its negation, we would say that arithmetic is inconsistent.
This is not an easy idea. What we are saying here is: What if arithmetic, the actual mathematical system of the natural numbers 1,2,3 etc. itself, simply isn’t logically consistent? What if it is possible to prove a specific statement and the negation of that same statement?
It would be a very strange state of affairs. Naturally, one would want arithmetic to be consistent…
Possessed of Some Radical Notions
If we suspect arithmetic is consistent, perhaps we should try to prove that it is. Perhaps we should try to prove that, in arithmetic, it is impossible to prove both a certain statement and the negation of that statement; i.e. to prove that you cannot prove a contradiction. Can one even prove such things as this? That is, prove statements about what you can and cannot prove? It is ambitious, to say the least.
Here’s when things start to get complicated. To answer this question we need to be careful. A proof is a sequence of deductions that take you from some axioms to a theorem. We want our theorem to be ‘arithmetic is consistent’, but what are our axioms? And what kind of deductions are we allowed to make? In short, what formal system are we working in?…
Kurt Gödel was a mathematician, logician and philosopher who became very famous for his so-called Incompleteness Theorems. It is difficult to overstate how much these theorems have influenced scientific and logical thinking. In particular, they have made us question the very foundations of mathematics. Here is an intimidating picture of Gödel.
An interesting discussion about his work can be heard on Melvyn Bragg’s BBC Radio 4 programme In Our Time.
I would like to mention Gödel’s Second Incompleteness Theorem. I won’t be going in to any details and so will merely present what I need in isolation, as an interesting fact.
Gödel’s Second Incompleteness Theorem implies the equivalence of two statements concerning any sufficiently sophisticated deductive theory (we won’t expand on or explain ‘sufficiently sophisticated’ and will assume all theories mentioned are thus). First we’ll look at the statements individually. The first statement is that the theory includes a proof of its own consistency. In our context, this means that you’d be able to prove that arithmetic were consistent using the formal system of arithmetic itself. I think almost anyone would agree that this idea alone is quite difficult to appreciate fully, not least because it throws up the rather technical issue of a theory expressing its own consistency, i.e. arithmetic somehow expressing the statement ‘arithmetic is consistent’. This can be done though and will (therefore) be glossed over. The second statement is that the formal system is inconsistent, in the sense that there is a theorem whose negation is also a theorem.
These two statements are equivalent. What? Yes, any theory which contains a proof of its own consistency is, in fact, inconsistent. Also, any inconsistent theory must necessarily contain a proof of its own consistency. So, a sufficiently sophisticated deductive theory includes a proof of its own consistency if and only if it is inconsistent. At this point, one can begin to guess just how influential these ideas are. In particular, it really drives home the fact that mathematical proofs don’t actually tell you what is true (because an inconsistent theory can prove its own consistency)! Proofs tell you what can be deduced from the axioms and nothing more. The relativity of ‘reality’, driven home by the final frames of the film, is the accordant descendant of this idea: that it’s not what’s true, it’s what you know, what you can be convinced of… what you can prove.
In late life, Gödel became increasingly mentally disturbed and suffered from persecutory delusions. Eventually, he wouldn’t eat unless his wife tasted his food for him. Consequently, when she was hospitalized for six months in 1977, he starved himself to death.
In the first half of the discussion (by which I mean the previous post), it was pointed out that the dreams of the film have inconsistencies themselves. Gödel’s work tells us that a theory being inconsistent is the same thing as a theory including a proof of its own consistency, analogous to the way in which the constructed dreams of the film must necessarily support the conviction that they are consistent.
But we won’t stop at that. Not only are the dreams within the film blurred reflections of deductive theories, the film itself is one as well…
Previously, in the ‘Fiction’ section of the previous post, I brought up the notion of unanswered questions/inconsistencies in the plot. I mention them now in order to say that none of them matter. One of the grandest things Inception pulls off is the fact that it covers its own tracks. The style of the film is quite gritty, but the plot is surprisingly frivolous, with lots of strange things miraculously falling into place throughout. Some of the things I hear mentioned most often in this regard being Saito’s unreasonably timely appearance during the Mombasa chase scene, the ease with which Ariadne accepts the notion of shared dreaming and Eames’ more or less unexplained ability to impersonate in Fischer’s dream (if this were so easy to do, wouldn’t it make extraction more or less trivial?). However, having opened the film in a dream within in a dream, in a world you’re never quite sure isn’t a dream, these contrived conveniences become part of the hypnagogic artistry of the film. None of the ‘issues’ in this category are plot holes or implausibilities, they’re surrealities of the kind which only really happen in dreams, or, to truly get to the bottom of it,… films.
What I’m saying is that this film proves its consistency to you while your watching it. Only later and on closer inspection, one sees that it is verging on inconsistent, but it’s all part of the game. If it didn’t have all its own little inconsistencies, all its imperfections, the stuff that fuels the theorists’ debates and the search for alternative interpretations, then it wouldn’t be convincing in the first place.