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  • Design and chance

    Disclaimer

    See https://www.authorea.com/users/48154/articles/116937/_show_article for a newer version of this paper, at the moment when this was written it was a work in progress and it may still be so.

    The paper below started as a mathematical attempt to understand what it would mean to live in a world that is not designed, but, in the end, the mathematical part turned out to be rather small, containing only a few simple properties about set cardinalities and probabilities. I think that the non-mathematical ideas are fairly obvious consequences of the mathematical ones, so many people have already thought about them – I have also found quotes from various people that seem to hint at the idea below. However, I did not manage yet to find anyone drawing the same conclusions in the same way. The closest I could get is the idea that the order of the Universe implies or suggests that there is a God. The fine-tuning of the Universe is also close11https://en.wikipedia.org/wiki/Fine-tuned_Universe. However, I think that what I’m presenting in this paper is different from what I have read about both of these, maybe being complementary to the fine-tuning argument.

    For a description of the fine-tuning argument see http://plato.stanford.edu/entries/teleological-arguments/#CosFinTun22Ratzsch, Del and Koperski, Jeffrey, ”Teleological Arguments for God’s Existence”, The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.), URL = ¡http://plato.stanford.edu/archives/spr2016/entries/teleological-arguments/¿.. I think that the argument presented in this paper solves most, if not all of the fine-tuning objections in the quoted page, while improving the probability constraints, i.e. it shows that our Universe has a zero probability. However, this paper does not present an improvement of the fine-tuning argument, it describes a different way to compute the probability of our Universe, so it can have its own set of objections.

    Since it started as a mathematical paper I may use “we” instead of “I” more often than I should, but you should consider it an invitation to work together in discovering some ideas. And if some of those ideas are wrong or unclear33Given my lack of experience with philosophy this is more probable than I would like., I welcome counterarguments and feedback44Authorea allows everyone to comment on the paper. I may switch to a different commenting system if it turns out that something better is needed. You can also try e-mailing design dash and dash chance at poarta dot org..

    I recommend exporting this paper as, say, PDF because Authorea makes it hard to read the footnotes.

    Introduction

    This is an attempt to reason about why our world is the way it is and what we can reasonably believe about it.

    Many people believe that the world is designed and created and that it’s unreasonable to believe that any world can exist without being created, and I agree with them. However, these beliefs are not shared by everyone, so it’s worth thinking about what this means. If our world is created, then it’s likely to be the way it is because its Creator55Not everybody that believes that the world is created thinks that God created it. Still, I hope that they would agree that capitalizing the Creator of this world is reasonable. wanted it to have certain properties. In order to understand why our world works the way it does, one would need to understand the intent of its Creator. While that is interesting in itself, I will not try to pursue it here, except for a small related paragraph at the end.

    For the remainder of this paper, let us consider the other case and assume that our world was not designed and created.

    If that’s true then there may be other worlds66We don’t have any proof for the existence of other worlds, but one could expect them to exist for the same reason that ours exists. If ours has no reason at all for existing, which is likely if it is not created, then it’s likely that other worlds would also not need any reason for existing and would simply be. However, for the purpose of this paper it does not matter if there are other worlds or not and maybe we will never be able to tell if other worlds exist or not.. Even if there are no other worlds, one could easily imagine that ours worked in a different way, say that the speed of light is different or gravity works differently. We will denote by possible worlds these other worlds that either are or could have been.

    Argument summary

    The actual argument is more complex in order to avoid various pitfalls, but the basic idea is this:

    I am trying to compute the probability of our world given that it was not created. For any property \(p\) such that our world has this property, the probability of our world is at most the probability of \(p\), where the probability of \(p\) is defined as the probability of the group of worlds having the \(p\) property. Then, if there is such a property \(p\) whose probability is \(0\), our world’s probability is \(0\). I will also discuss why it is enough to look at just one property.

    The property \(p\) for which I will attempt to show that it has a \(0\) probability is “There is a mathematical theory that has a finite definition and is useful for making approximate predictions in a non-trivial part of our universe”.

    For clarity, I have split the argument in two parts.

    The first one uses ”There is a mathematical theory that has a finite definition and fully models the universe” as the above property and shows that the probability of a non-created universe to have this property is 0. However, this result is not really useful for a number of reasons, including that we may need to have an infinite definition only if we want infinite precision in our predictions, but for most or even all practical purposes we could not tell the difference between predictions with extremely good precision and predictions with infinite precision.

    In the second part I will also consider theories which do not fully model the universe and I will show that in a non-created universe we can’t have a non-zero probability for a finite theory that works in a non-trivial part of a universe.

    Then the first part argument has the following steps:

    1. 1.

      If our world is not created then either there are other worlds, or our world could have been different.

    2. 2.

      We will consider only worlds which are ”well behaved”, e.g. they can be modelled mathematically (for a reasonable definition of modelling that focuses on predictions), they can have intelligent life, there is a concept of time and so on.

    3. 3.

      We will consider all the possible theories that could model such worlds. Their set has the same cardinal as the real numbers.

    4. 4.

      For any reasonable statistical distribution, the set of finite theories has zero probability.

    5. 5.

      Therefore the probability of \(p\) is \(0\).

    The second part of the argument reuses the first steps above, rephrased to allow partial modelling, but also has a few extra ones:

    1. 5.

      In order to have intelligent beings one needs finite theories that are useful.

    2. 6.

      In order to have a finite theory with a non-zero probability the only option is to have a theory that only works in a small part of the universe, so small that it covers a \(0\) fraction of the universe.

    3. 7.

      Therefore \(p\)’s probability is \(0\).

    Modelling possible worlds

    First, let us note that there can’t be any causal interaction between two different possible worlds. If two worlds are interacting, it’s more reasonable to say that they are actually a single possible world with two parts.

    How would a possible world look like? It could have exactly the same fundamental laws as ours, but with matter organized differently. It could have similar laws, but with different universal constants. It could have different fundamental particles (or whatever the basic building blocks of our universe are, assuming that there are any). Or it could be completely different, i.e. different in all possible ways.

    It could be that our logic and reasoning are universal instruments, but it could also be that some of these possible worlds could be beyond what our reasoning can grasp and others could have properties for which our logic is flawed. Aknowledging that, let us see if we can say anything about the possible worlds that we could understand and could model in some way. In the following, the possible worlds term will denote only the possible worlds which we could model. But the “we could model” term here should not mean that we estimate our maximum capacity as humans in this world to build a model, since that is an arbitrary limit, we should also allow models that still follow our rules for reasoning, but for which we would need an infinite amount of resources to build and use.

    This notion of model is not precise enough. Let us restrict the possible worlds term even more, to the possible worlds that we could model mathematically, even if that may leave out some worlds. We will also do the reverse and say that all the models of any set of mathematical axioms which is at most countable77We could also go beyond countable axiom sets, but that would complicate things without any benefit. and has at least a model are possible worlds88Even if some of those models seem outlandish, there could be something that follows those rules and that is completely separated from anything else, not interacting with any other universe in any way. I would say that this something would be an universe. If you don’t like it, in the following I am going to restrict what I call a possible universe..

    If nobody designed our world then we have no way of preferring a possible world over another, except that there is one of them in which we live. In other words, if our world is the only one that exists, then any other possible world is as likely to have existed as well. If multiple worlds exist, by picking a random world, ours has a the same chance of being picked as any other.

    It seems that reasoning about all the possible worlds could be very hard, but maybe we could do something easier, maybe we could reason about the mathematical axioms that model the worlds.

    Let us consider the set of axioms that define, say, a monoid. All groups are models for this set of axioms, but intuitively a group is something more interesting than a monoid and we should include extra axioms for defining it. On the other hand, if we relax the countability requirement, we could include all possible axioms for each model (e.g. for each monoid), uniquely identifying it, but again, intuitively this is not a useful way of modeling. We will call these the too-general problem and the too-specific problem.

    For a given world, a good set of axioms would be one that would allow us to make all possible correct predictions for that world. The term prediction is not a clear one. To make it more clear, let us restrict again the possible worlds term. One option would be to make it to denote all possible worlds which have a concept of time and a concept of the state of the world at a given time and for which describing the state of the world at all possible times is equivalent to describing the world.

    This ignores some important issues like the fact that it’s reasonable to have a concept of time without having a well defined concept of the state of the world at a given time, so we could rephrase the definition above to include many other reasonable notions of space and time, e.g. we can include worlds where point is a concept and we can know which (point, time) pairs are before a given (point, time) pair.

    Then, in the following, we will say that we can predict something (\(S\)) whenever we have a set of axioms for which \(S\) is uniquely determined by the state of the world at a subset of the previous points in time99Will be extended to statistical predictions in the next paragraph.. If we are interested in predicting the state of the world at a given point \(P\) and time \(t\), a good choice for this subset could be a full section through \(P\)’s past (e.g. a plane which intersects it’s past cone), i.e. a subset that separates \(P\)’s past in two parts, one which is before the subset and one which is after the subset1010This means that all lines which fully lie in \(P\)’s past and connect a point which is before the subset with a point which is after the subset must go through the subset.. One could think of similar definitions for predicting the entire state of the world. If needed, this definition could be changed to work for more concepts of space and time.

    In a deterministic universe, if we know the laws of the universe and its full state at a given time, we could, in theory, fully predict any future state. But an universe does not have to be deterministic and, even if it is, the only reasonable model available at a certain time may be a statistical one. Then we will allow using a set of axioms which only gives a statistical distribution for the state of the universe given its past (I’ll call this a statistical axiom set). For the purpose of this paper we don’t need to make a difference between a non-deterministic universe and a deterministic one but for which we only have a statistical model.

    Let us restrict the possible worlds term to the worlds where we can make predictions and let us use only sets of axioms that allow predictions. As defined above, for a given world, a good set of axioms is one which allows us to make all possible correct predictions for that world (statistical or not). Using only good sets of axioms solves the too-general problem since such a set would describe its possible worlds in a way that does not leave out important things. Still, there is nothing that prevents such a set from going into too much detail.

    Let us chose any formalism for specifying axioms that uses a finite alphabet and let us choose an arbitrary order relation on that alphabet. Also, let us consider only axiom sets which, when written on an infinite paper, use at most a countable number of symbols. Then, for each possible world, we could say that the best set of axioms is the smallest good one, smallest being defined as having the smallest length when written on the infinite paper. This is not a fully defined order relation, so let us also say that for sets of axioms having finite length we will break any ties by using the lexicographic order. For worlds having only sets of axioms of infinite length we can’t use the same trick to break ties, but they all have the same length (countably infinite) so we will choose any such set of axioms and say that it’s the best one1111Note that we could try a few things to find a canonical set for the infinite case, e.g. whenever a set of axioms is implied by a smaller finite set we could replace it, but it’s not clear that any such method will be enough.. With an axiom set chosen in this way we would also solve the too-specific problem in the finite case. In the infinite case we have made an arbitrary decision, but then the cardinality of the axiom set matches the complexity of the world anyway, even if there is some redundancy.

    If \(U\) is an universe and \(A\) is the axiom set chosen as above, then we would say than \(A\) is the optimal set of axioms for \(U\). If \(A\) is a set of axioms which is optimal for some universe \(U\) then we say that \(A\) is an optimal set of axioms.

    Now let us see if we actually need infinite length sets. We can have infinite sets1212Note that for a given finite alphabet, i.e. in the current context, the infinite-length-set notion is identical to infinite-set. of axioms, and there is no good reason to reject such sets and to ignore their possible worlds, so we will take them into account. It is less clear that we can’t replace these infinite sets with finite ones. Indeed, let us use any encoding allowing us to represent these sets as strings over a finite alphabet, e.g. as binary functions over the set of natural numbers (it’s not important to use binary strings, the construction below would work with many other mathematical objects). For the given example, the encoding of a set of axioms \(A\), \(encoding(A)\), would be a function from the natural numbers to a binary set, giving the value of the bit for each position in the encoding, \(encoding(A):\mathbb{N}\longrightarrow\left\{0,1\right\}\). Then the following scenario becomes possible: for any universe \(U\) with an infinite set of axioms \(A\), we can consider \(U+encoding(A)\) to be an universe in itself which has \(encoding(A)\) as part of its state at any moment in time. Then it’s likely that we can find a finite set of axioms which allows predictions for such an universe. While, strictly speaking, this would be a different universe than the one we had at the beginning, it may seem similar enough to it so one may be tempted to use only finite sets of axioms.

    On the other hand, using only finite sets of axioms in this way seems to be some sort of cheating. In order to get a more honest system of axioms, we should define the state of the universe as what could be changed from inside. As an alternative, we could work with very specific sets of axioms, e.g. we could only talk about worlds which have \(\mathbb{R}^{4}\) as their space and time, whose objects are things similar to the wave functions used by quantum mechanics and so on.

    Modelling from inside

    Many of these possible worlds would not be able to have intelligent life inside of them, which makes them less relevant. Let us focus only on the ones which could contain intelligent beings and let us think about how the intelligent beings would model their world. Even more, let us include only worlds where the intelligent beings would be similar enough to us in that they can use logic and mathematics, but they wouldn’t be able to process an infinite amount of that in a finite time. Let us note that in worlds with an infinite optimal set of axioms these intelligent beings will never have a complete description of how their world works, but they may be able to build better and better models.

    Let us assume that those intelligent beings are continuously trying to find better models for their world and that they are reasonably efficient at this.

    As a parenthesis, note that until now we restricted the possible world concept several times. The argument below also works with larger possible world concepts as long as those worlds have a few basic properties (e.g. one can make predictions and it can contain intelligent beings) and, at the same time, it is plausible that our world is such a possible world.

    First, let us note that having intelligent beings in an universe likely means that their intelligence is needed to allow them to live in that universe, which likely means that they can have a partial model of the universe. That model does not have to be precise, e.g. it could be made of simple rules like “If I pick fruits then I can eat them. If I eat them then I live”, and it can cover only a small part of their world, but it should predict1313This is the only place where predict means that the beings can actually say something about the future instead of a theoretical way of making predictions. Everywhere else we’re using the previous definition of prediction which only requires that prediction is possible for a being which can take full snapshots of the universe and can go through all the possible models of an axiom set. something. Of course, these predictions do not have to be deterministic. Also, they might not be able to perceive the entire universe.

    A mathematical model for a universe needs a set of measuring units. For each of the universes containing intelligent beings, let us take a fixed set of measuring units covering everything that those beings would measure. As an example, when measuring distance in our space we could use meters, light seconds or various other measuring units. The measuring unit is not important as long as we pick something.

    We can define the fraction of the world that is modelled by an axiom set in at least three ways:

    1. 1.

      As the fraction of the observable space for which the axiom set predicts something with a reasonable error margin.

    2. 2.

      As the fraction of the optimal set of axioms that is implied by the current axiom set.

    3. 3.

      As something between the first two cases, where we use a weighted fraction of the optimal axiom set, each axiom having a weight proportional to the fraction of the world where it applies. As an example, let us say that we have an infinite set of axioms, and that for each point in space we can predict everything that happens using only three axioms (of course, two different points may need different axiom triplets). Let us also assume that there is a finite set of axioms \(S\) such as each point in space has an axiom in \(S\) among its three axioms. Then \(S\) would model at least a third of the entire world.

    In all of these cases, predictions made only from the artificial constraints imposed by this paper, e.g. that the world can be modelled mathematically or that it contains intelligent beings, should not count towards the fraction of the world that is modelled by an axiom set. In other words, this fraction of the world is actually the fraction of the world that is modelled above what is absolutely needed because of the constraints imposed here.

    We can use any of these definitions (and many other reasonable ones) for the remainder of this paper. Then we would have three possible cases1414All of these assume that the intelligent beings use a single axiom set for predicting. It could happen that they use multiple axiom sets which can’t be merged into one. One could rewrite this paper to also handle this case, but it’s easy to see that the finite/infinite distinction below would be the similar..

    First, those intelligent beings could, at some point in time, find an axiom set which gives the best predictions that they could have for their world, i.e. which predicts everything that they can observe and they wouldn’t be able to find anything which is not modelled by their axiom set. We could also include here axiom sets that are good enough for all practical purposes. As an example, for an universe based on real numbers, knowing the axioms precisely with the exception of some constants and measuring all constants with a billion digits precision might (or might not) be good enough. Only caring about things which occur frequently enough, e.g. more than once in a million years, could also be good enough.

    Second, those intelligent beings could r