Design and chance


See for a newer version of this paper, at the moment when this was written it was a work in progress and it may still be so.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 close11 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, Del and Koperski, Jeffrey, ”Teleological Arguments for God’s Existence”, The Stanford Encyclopedia of Philosophy (Spring 2016 Edition), Edward N. Zalta (ed.), URL = ¡¿.. 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.


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 u