Possible Worlds: The Fifth Dimension
By bblfish on Sep 03, 2006
One of the great logical discoveries of the last 30 years is that of the 5th Dimension. This is still very little known outside logical/philosophical circles, though it is starting to have an influence on physical interpretations of quantum mechanics, the arts and history, and soon the internet.
Einstein made us all aware of the 4 dimensions of space time (3 dimensions for space and 1 for time). This lead to the further discoveries of quantum mechanics and its discovery of quantum indeterminacy, whereby it is impossible to know both the position and the speed of a quantum particle. This found its realisation in the creation of the Atomic bomb, with which we have all been living for the past 60 years, and which changed the political landscape for ever.
Einstein is well known to have said that "God does not play dice", a statement that is fundamentally opposed to the discoveries of quantum physics. And yet it seems both the theory of relativity and quantum physics are both needed today to describe the world. Neither quantum physics nor relativity give us a complete description of the world.
The 5th, and higher, dimensions, though they did not find their origins in an attempt to reconcile these two views, but from an analysis of language and of logic, do as I understand it, resolve this dilemma. (note: Before continuing to read what I have to say here, it would certainly be best to view the tenth dimension video which will make what I am saying here a lot easier to understand. I only discovered this video after posting this, but it completely confirms the points made here.)
So let us trace back quickly the evolution of this idea. At the end of the 19th century Frege, a German logician, set the foundations for mathematical logic. This logic comes in three parts:
- a syntax: how signs can be combined to form sentences. Starting from signs such as "Henry" "is in" and "Berlin" we can create sentences such as "Henry is in Berlin".
- a semantics: how these signs related to the world. "Henry" refers to me. "is in" is a relation that relates individual objects to spacial regions. "Berlin" is a specific spacial region. The sentence together refers, according to Frege to the True, ie it is true that "Henry is in Berlin" since the objects refered to by the elements in the sentence do in fact stand in the relation stated above. Ie: I am in Berlin (no quotes).
- The above led to the formal discovery of meaning, as something more than the relation between a word and its referent. This is demonstrated by a the following simple thought process. If the word's meaning were wholy contained in its referent, the thinking goes, then the sentence "Berlin is the capital of Germany" would have the same meaning as "Berlin is Berlin" since "Berlin" refers to a thing that is the same thing as "the capital of Germany". Yet it is clear that the information content of the first sentence is not the same as that of the second sentence. Frege deduced therefore that not only did words possess a referent, they also possessed a "sense".
I will skip quickly over 70 years of philosophical/logical thinking and jump straight to David Lewis, who in is book "Counterfactuals", set out and succeeded in giving a logical analysis of sentences such as "If kangaroos had no tails they would topple over". This type of sentence constitutes a very big problem for the simple Fregean logic, since the antecedent of the condition "Kangaroos have no tails" is false. In Fregean mathematical logic this leads inevitably to the conditional being evaluated to be true, an initially somewhat counterintuitive analysis of an "if ... then ..." conditional, but one that works in fact very well for mathematical statements and many others. Material conditionals as the Fregean conditionals are called are designed to transfer truth. We have the same in programming when we write
if (2==1) then System.out.println("ouch"). Since the antecedent is false (two is never 1) the "ouch" is never printed out. No pain.
Ok so back to our kangaroos. Since kangaroos do in fact have tails, the sentence "If it were the case that kangaroos have no tails then it would be the case that kangaroos topple over" would be immediately true, were we to interpret the logical connective "If it were the case that ... then it would be the case that ..." as a material conditional. And yet I have had some very fun and instructive debates about what would or would not happen if kangaroos had no tails, the question not being settled in any way by recourse to any notion of material conditionals.
Having introduced this problem David Lewis then goes on to give an brilliant analysis of such sentences. To do this he introduces the notion of a possible world.
A possible world he says, is a world just like this actual one, the one I am writing this in and the one you are reading this in. Except that it is different in some factual aspect. So there is a possible world where I am writing this from another café in Berlin. Of course you're not going to have two worlds that differ from another world by just one fact. Usually there is a difference in the past events that led up to this. So in the world where I am writing this from a another café, is a world where I walked some different path, asked a different bar maid for a beer, am sitting at a different table, and perhaps would have, as a result written this text somewhat differently, and perhaps my future will be very different too in this other world as a result of making a fateful encounter in that café that I don't make in this one. Anyway, these worlds are like this one in that if we go with Einsteinian physics and accept that this world is 4 dimensional space-time continuum, then the other world too can be a 4 dimensional space time continuum. Yet these two worlds are at no physical distance from each other. You can't travel from one to the other. Light does not travel between the two. Each world is a factually complete 4 dimensional world (there could be worlds with more dimensions, but let us leave this for now).
Now it had already been shown by another very famous logician Kripke, that the notion of possible worlds was enough to explain the concepts of metaphysical necessity and possibility - two notions that had seriously gotten into disrepute in the early 20th century - and not only that, he showed that these metaphysical notions were very useful in solving some difficult philosophical problems. A sentence he argued is necessarily true, if it is true in all possible worlds. So for example mathematical truths are necessarily true, since they are true in all possible worlds. A sentence in possibly true, if there is at least one possible world in which it is true. For example it is possible that you win the lottery is to say that there is a possible world in which you do. Depending on how you define the identity of a person across possible worlds, you will then come to different conclusions as to whether it is possible that you have a different mother and father. Kripke took personal identity to be given by the union of a particular sperm and egg. As such it follows that that it was impossible that you have different parents.
David Lewis though adds something very important to Kripke's notion of possible worlds. He gives them an ordering of similarity. Some worlds are more similar to ours than others. This is not a linear ordering, meaning that two worlds could be equally similar to ours. The world where you did not read this article is a lot more similar to this one that the world in which pigs fly (through their own volition of course). With this ordering David Lewis is then able to analyse our sentence about kangaroos as: In the closest possible worlds in which kangaroos have no tails they topple over. He then analyses sentence such as "If Kangaroos have no tails they could topple over" as having the weaker meaning of: in the closest possible worlds in which kangaroos have no tails, there is one of them at least in which kangaroos do in fact topple over. All of this is given formal mathematical backing, and is very much worth reading.
This analysis of counterfacuals proved to be immensely helpful in further helping us analyse causality, knowledge, meaning, necessity, actuality, probability, time, mind and many other notions that had previously eluded philosophers. I will come back to these other notions in future posts. The point being that if a theory is to be judged by the quality of the explanations it leads to, David Lewis' philosophy has been extremely successful.
For a technically minded person though, resolving philosophical puzzles may not seem to be concrete enough, which is why I mention a little longer the relation to quantum physics. In one of his papers David Lewis also gives an analysis of quantum indeterminacy by allowing the actual world to be not just one world but a set of indeterminately similar worlds. I can't remember the precise details off hand, and my books are in some box some where in France, but this is probably equivalent though viewed from the outside, to possible worlds interpretations of quantum physics. I say it is viewed from the outside, because as I understand, possible worlds interpretation of quantum physics imagine that new possible worlds are created each time a quantum choice is made, whereas David Lewis would say they all exist, it is just that the choice that is made forces a particular set of worlds to be the actual ones. In a way, quantum indeterminacy is what we would call choosing which world we live in. If we follow Roger Penrose's thinking in The emperor's new mind, that the brain is using quantum effects, then we could see more precicely how, when we decide to do one thing or another, we are choosing which possible world we are going to live in. We are in a sense travelling through possible worlds by the choices we make.
To summarize, the 5th dimension springs out of an analysis of language, the key topic of the 20th century research. Every great discovery has its physical realisation, to follow the thinking Jean Gebser. If the discovery of space and our situation in space led to the discovery of the Americas, if relativity theory and quantum theory led to the atomic bomb, so logic led to computing, and the theory of language is being instantiated by the internet, the web, and the just enfolding Semantic Web. Possible worlds theory is being concretized in games, which allow a player to save a game, try different possibilities, come back to a past state and try again. Even films have started illustrating this topic. Run Laura, Run a German film that came out five years ago, goes through the same story of an attempted bank robbery by showing how small changes in someone's decision can make huge changes to the ending of the film. Recently the French comedy Jean Philippe which imagines a fan of the famous French singer Johnny Hallyday waking up after an accident in a slightly altered parallel world where his star never quite made it, and puts all his energy into convincing him to try again.
To the seven classical arts
- Film/comic strips
- Computer Games
David Lewis also considers, and endorses making the similarity relation between possible worlds continuous, so that between any two possible worlds there is always a third possible world that is more similar to the first than the second. This is in part what the indeterminacy of the actual world is about, since as in mathematics any number n + delta d (an infintely small numer) is indistinguisheable from n. What we have then is a space-time-possibilia continuum.
Ok so having said this, and having started my explanation by pointing out to Frege's discovery of sense, I'll fill in a sketch of David Lewis's theory of language. For David Lewis the meaning of a sentence is the set of possible worlds in which it is true. So the reason "Berlin is the capital of Germany" is different in meaning from "Berlin is Berlin", is that we can very well imagine a possible world in which the capital of Germany is not Berlin (in fact it only recently became Berlin again). Meaning is thus given a extensional semantics, which has an intentional feel (since we can only imagine these possiblia).
Yesterday, Sunday 3rd September 2006, must have been dimension day. Because at the same time I wrote this Nova Spivack wrote about I'm Addicted to StumbleUpon where he points to the amazing Guided visualisation of 10 dimensional space which makes exactly the point I describe here, namely that the 5th dimension is that of possible worlds. I highly recommend watching that video.
In fact looking at the video above, it is clear that possible worlds in David Lewis's sense englobe the 5th to the 10th dimension. As I understand now, thinking of the 5th dimension only give you a subset of all possible worlds. You need higher dimensions to get the possible worlds that have different starting points from our own (and so presumably different laws of physics, etc). From that video it is now clear, that the possible worlds interpretation of quantum physics is a lot more serious than I had hitherto conceived. It is the core of it (as far as string theory is concerned, if the video is a good introduction to it).