I’ve been doing some work on topological models of decision making recently (I do weird things for fun) and a result popped out of it that I was very surprised by, despite it being essentially just a restatement of some elementary definitions in topology.

The result is this: Under many common models of reality, there are no non-trivial deterministic experiments we can perform even if the underlying reality is deterministic.

The conclusion follows from two very simple assumptions (either of which may be wrong but both of which are very physically plausible).

- We are interested in some model of reality as a connected topological space \(X\) (e.g. \(\mathbb{R}^n\), some Hilbert space of operators, whatever).
- No experimental outcome can give us infinite precision about that model. i.e. any experimental outcome only tells us where we are up to membership of some open subset of \(X\).

Under this model, regardless of the underlying physical laws, any fully deterministic experiment tells us nothing about the underlying reality.

What does this mean?

Let \(X\) be our model of reality and let \(\mathcal{O}\) be some set of experimental outcomes. A deterministic experiment is some function \(f: X \to \mathcal{O}\).

By our finite precision assumption each of the sets \(U_o = \{x \in X: f(x) = o\}\) are open. But if \(f(x) = o\) and \(o \neq o’\) then \(f(x) \neq o’\) so \(x \not\in U_{o’}\). Therefore they’re disjoint.

But certainly \(x \in U_{f(x)}\), so they also cover \(X\).

But we assumed that \(X\) is connected. So we can’t cover it by disjoint non-empty open sets. Therefore at most one of these sets is non-empty, and thus \(X = U_o\) for some \(o\). i.e. \(f\) constantly takes the value \(o\) and as a result tells us nothing about where we are in \(X\).

Obviously this is a slightly toy model, and the conclusion is practically baked into the premise, so it might not map to reality that closely.

But how could it fail to do so?

One way it *can’t* fail to do so is that the underlying reality might “really” be disconnected. That doesn’t matter, because it’s not a result about the underlying reality, it’s a result about *models* of reality, and most of our models of reality are connected regardless of whether the underlying reality is. But certainly if our model is somehow disconnected (e.g. we live in some simulation by a cellular automaton) then this result doesn’t apply.

It could also fail because we have access to experiments that grant us infinite precision. That would be weird, and certainly doesn’t correspond to any sort of experiment I know about – mostly the thing we measure reality with is other reality, which tends to put a bound on how precise we can be.

It could also fail to be *interesting* in some cases. For example if our purpose is to measure a mathematical constant that we’re not sure how to calculate then we *want* the result of our experiment to be a constant function (but note that this is only for mathematical constants. Physical constants that vary depending on where in the space of our model we are don’t get this get out clause).

There are also classes of experiments that don’t fall into this construction: For example, it might be that \(O\) itself has some topology on it, our experiments are actually continuous functions into O, and that we can’t actually observe which point we get in \(O\), only its value up to some open set. Indeed, the experiments we’ve already considered are the special case where \(O\) is discrete. The problem with this is that then \(f(X)\) is a connected subset of \(O\), so we’ve just recursed to the problem of determining where we are in \(O\)!

You can also have experiments that are deterministic whenever they work but tell you nothing when they fail. So for example you could have an experiment that returns \(1\) or \(0\), and whenever it returns \(1\) you know you’re in some open set \(U\), but when it returns \(0\) you might or might not be in \(U\), you have no idea. This corresponds to the above case of \(O\) having a topology, where we let \(O\) be the Sierpinski space. This works by giving up on the idea that \(0\) and \(1\) are “distinguishable” elements of the output space – under this topology, the set \(\{0\}\) is not open, and so the set \(U_0\) need not be, and the connectivity argument falls apart.

And finally, and most interestingly, our experiment might just not be defined everywhere.

Consider a two parameter model of reality. e.g. our parameters are the mass of a neutron and the mass of a proton (I know these vary because binding energy or something, but lets pretend they don’t for simplicity of example). So our model space is \((0, \infty)^2\) – a model which is certainly connected, and it’s extremely plausible that we cannot determine each value to more than finite precision. Call these parameters \(u\) and \(v\).

We want an experiment to determine whether protons are more massive than neutrons.

This is “easy”. We perform the following sequence of experiments: We measure each of \(u\) and \(v\) to within a value of \(\frac{1}{n}\). If \(|u – v| > \frac{2}{n}\) then we know their masses precisely enough to answer the question and can stop and return the answer. If not, we increase \(n\) and try again.

Or, more abstractly, we know that the sets \(u > v\) and \(v < u\) are open subsets of our model, so we just return whichever one we’re in.

These work fine, except for the pesky case where \(u = v\) – protons and neutrons are equally massive. In that case our first series of experiments never terminates and our second one has no answer to return.

So we have deterministic experiments (assuming we can actually deterministically measure things to that precision, which is probably false but I’m prepared to pretend we can for the sake of the example) that give us the answer we want, but it only works in a subset of our model: The quarter plane with the diagonal removed, which is no longer a connected set!

Fundamentally, this is a result about *boundaries* in our models of reality – any attempt to create a deterministic experiment will run into a set like the above plane: Suppose we had a deterministic experiment which was defined only on some subset of \(X\). Then we could find some \(o\) with \(U_o\) a non-empty proper subset of \(X\). Then the set \(\overline{U} \cap U^c\) where the closure of \(U_o\) meets its complement (which is non-empty because \(X\) is connected) is a boundary like the diagonal above – on one side of it we know that \(f\) returns \(o\). On the other side we know that it *doesn’t* return \(o\), but in the middle at the boundary it is impossible for us to tell.

What are the implications?

Well, practically, not much. Nobody believed that any of the experiments we’re currently performing are fully deterministic anyway.

But philosophically this is interesting to me for a couple of reasons:

- I for one was very surprised that such a trivial topological result had such a non-trivial physical interpretation.
- The idea that non-determinism was some intrinsic property of measurement and not a consequence of underlying physical non-determinism is not one that had ever previously occurred to me.
- We need to be very careful about boundaries in our models of reality, because we often can’t really tell if we’re on them or not.
- It may in fact be desirable to assume that all interesting quantities are never equal unless we have a deep theoretical reason to believe them to be equal, which largely lets us avoid this problem except when our theory is wrong.

(As per usual, if you like this sort of thing, vote with your wallet and support my writing on Patreon! I mean, you’ll get weird maths posts either way, but you’ll get *more* weird maths posts, and access to upcoming drafts, if you do).