ONE of the grand aims of science is to explain every aspect of nature in terms of simple, fundamental laws – but is this possible? A team of physicists claims to have found a hint that some things simply cannot be computed, and that nature could be more than the sum of its parts.
The idea of reductionism, a key tool in science for centuries, holds that everything in nature can ultimately be understood by gaining knowledge of its constituent parts. The laws of fluid flows, for example, can be derived from the deeper laws of atomic and molecular motion, which in turn follow from quantum physics.
In 1972, physicist Philip Anderson pointed out that there could be a problem with this approach. Anderson suggested that some systems may be more than the sum of their parts. He championed “emergence” – the notion that important kinds of organisation might emerge in systems of many interacting parts, but not follow in any way from the properties of those parts. If so, then even perfect knowledge of the physics at one level would be inadequate for understanding organisation at higher levels. This conjecture has been debated ever since.
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“Even perfect knowledge of the physics at one level may be inadequate for understanding organisation at higher levels”
Now Mile Gu at the University of Queensland in Brisbane, Australia, and colleagues, claim that it may be possible to prove Anderson’s idea. They studied a basic mathematical model called the , which is often used to study how magnetism arises in iron and other materials from the collective organisation of their atoms.
To picture the Ising model, imagine a 3D lattice of atoms. Each atom acts like a tiny magnet to those around it, and adopts a particular orientation depending on the forces between atoms (see diagram). This mirrors what happens in real-world materials, where the atoms adopt different patterns of orientation depending on the atomic forces. In iron, for instance, the atoms will sometimes point in a similar direction – making the material magnetic overall – whereas in alloys the pattern is more complex.
Using the model, the team focused on whether the pattern that the atoms adopt under various scenarios, such as a state of lowest energy, could be calculated from knowledge of those forces. They found that in some scenarios, the pattern of atoms could not be calculated from knowledge of the forces – even given unlimited computing power. In mathematical terms, the system is considered “formally undecidable”.
“We were able to find a number of properties that were simply decoupled from the fundamental interactions,” says Gu. Even some really simple properties of the model, such as the fraction of atoms oriented in one direction, cannot be computed.
This result, says Gu, shows that some of the models scientists use to simulate physical systems may actually have properties that cannot be linked to the behaviour of their parts (). This, in turn, may help explain why our description of nature operates at many levels, rather than working from just one. “A ‘theory of everything’ might not explain all natural phenomena,” says Gu. “Real understanding may require further experiments and intuition at every level.”
Some physicists think the work offers a promising scientific boost for the delicate issue of emergence, which tends to get swamped with philosophical arguments. John Barrow at the University of Cambridge calls the results “really interesting”, but thinks one element of the proof needs further study. He points out that Gu and colleagues derived their result by studying an infinite system, rather than one of large but finite size, like most natural systems. “So it’s not entirely clear what their results mean for actual finite systems,” says Barrow.
Gu agrees, but points out that this was not the team’s goal. He also argues that the idealised mathematical laws that scientists routinely use to describe the world often refer to infinite systems. “Our results suggest that some of these laws probably cannot be derived from first principles,” he says.
