Thinking too generally

Some recent results in physics force us to recognise more limits on what we can know

Photo by FlyD on Unsplash

When I write about needing to remain humble about what we know, I’m aware that this point can be understood in two very different ways. One is to assume that there are limits to our knowledge because we haven't yet figured everything out, but once we do then humility will become less important. The alternative is to hold that there are fundamental properties of us as humans and the universe that mean we will always face limits to what we can know. It is not even possible, in principle, for us to transcend our limits.

A classic formulation of the first view is expressed by what is known as Laplace's Demon. In 1814, Pierre-Simon Laplace argued that if there was an intellect that knew all the forces in the universe and the positions of everything in the universe, then it could perfectly predict the future. For Laplace, the universe was perfectly knowable and computable, even if humans don’t seem to be capable of knowing it.¹

However, developments in physical sciences, and other areas of our knowledge, have increasingly undermined the intuitive support for this view of the universe. This all reinforces my recent argument that all our knowledge is restricted as it is about something or some particular part of the world.

We keep discovering limits on what we know

I have covered some of these developments in previous articles. For a start, there are the limitations posed by chaos theory and genuine chaotic behaviour, where how a system changes is incredibly sensitive to tiny changes at the start. For example, even where we have only three bodies (such as planets) orbiting each other in space, there is no mathematical way to perfectly predict their future behaviour.

The conceptual weirdness of quantum mechanics poses similar challenges. The mathematics tells us that there are fundamental limits to what we can, even in theory, know - but making these assumptions then allows us to make incredibly precise and accurate predictions. Assuming that there are hard limits to what can be known about the position and velocity of particles (contra to Laplace's assumption) allows us to predict and know more than we could have otherwise.

There are also a wide range of physical systems where we are confident we understand the mathematics governing their behaviour, but we cannot solve the equations and so cannot make reliable predictions of how they will act in many circumstances. A typical example is fluid mechanics. If you ever notice, computer generated depictions of flowing water always look odd and unrealistic, because we cannot solve the equations we need to produce realistic modelling of fluids.

Problems from logic have spread into physics

A recent article titled ‘Next-Level’ Chaos Traces the True Limit of Predictability outlined a set of more recent scientific results with similar implications. These particularly interest me as they are derived from important results in the study of mathematics and logic.

In formal logic, there are a series of results (ultimately inspired by the Liar Paradox) that show there are hard limits on what we can prove or compute. Kurt Gödel showed in the early 1930s that, in any mathematical system with minimal mathematics, it is possible to create statements that have to be true but can never be proven. These ideas were extended by various people to questions of computability and the decidability of algorithms - in other words, whether an algorithm will give us a definitive answer.

The core result is known as the Halting Problem and was demonstrated by Alan Turing in 1936, even if he never used that term. He showed that it is not possible to create an algorithm that can always decide whether any other arbitrary algorithm will come to an answer, or halt. There is no universal method for deciding whether algorithms will work or not. It is, in essence, an application of Gödel’s results to computing.

The article referenced above, which is worth reading, explains how physicists in the last 10 years have discovered that the Halting Problem emerges in (idealised) physical situations, not just hypothetical algorithms. The key result emerged in 2015, when it was proven that a general solution to identifying a particular property of quantum particles - the spectrum gap - is equivalent to the halting problem, and so is undecidable. This proves that a particular problems in physics will never have a solution that works in all cases. Similar results in related fields have followed.

The lesson from these results is that the sort of universal knowledge and theories that Laplace’s Demon could obtain are not possible.² As one scientist quoted in the article said, "If you think too generally, you will fail." To rephrase this lesson, all our human knowledge is about something, rather than being a general result about everything - as I argued in my previous article.

The article notes that these results specifically apply to scientific theories, not the world itself. So it theoretically remains possible that the world itself is decidable or solvable and undecidability is just a limitation with our theories. In my view, this is a valid point, but is largely irrelevant to anything we might be interested in. As these results are derived from the Halting Problem, they emerge due to the logical structure of theories and computation, not specific features of particular theories. And as all our knowledge, especially scientific knowledge, is captured in theories, there is no conceivable way for humans to get around them.

In short, modern science keeps discovering ways there are structural limits to human knowledge that we cannot overcome. This is yet another reason to be humble about what we know.

1. While some consider that Laplace posed this type of universal knowledge as a goal for humans, he actually thought it was unattainable for humans. He also didn't refer to the intelligence as a 'demon', despite the common usage.

2. At least they are not possible for humans or any type of intelligence we can imagine. An omniscient God is still logically possible if, as we should expect, his existence is fundamentally different. For example, if God is outside time, as many theologies argue, then he wouldn't have to rely on calculating the future and so undecidability of theories is not a relevant issue.