We’re used to thinking of space as continuous.
A stone can be anywhere in space. It can be here. Or it can be an inch to the left. Or it can be half an inch further to the left. Or it can be an infinitesimal fraction of an inch even further to the left. Space is infinitely divisible.
The graphs of Wolfram Physics, however, are discrete.
If, as Stephen Wolfram proposes, the universe is a graph, then you can’t be just anywhere in space. It makes sense to think about a node of the graph as a position in space. It makes no sense to think about anywhere in between the nodes as positions in space. This space is not infinitely divisible.
It’s as if a stone could be here in space, or here in space, but nowhere in between.
So which is it?
Has every physicist from Leucippus to Einstein been right to insist that space is continuous?
Or is Wolfram right to up-end millennia of settled science and insist that space is discrete?
Space seems smooth
Let’s agree that space is more or less continuous on a large scale.
How do we know this?
Well, I really can move a stone a tiny fraction of an inch, so that’s some evidence, at least.
And, well, there’s General Relativity. Einstein’s equations consider space – or, to be more precise, space-time – to be continuous. And Einstein’s equations work. They give an astonishingly accurate account of how everything from photons to planets move through space.
How about smaller scales?
Well, General Relativity works right down to scales of around 10-35 metres.
That’s a pretty small scale. A hair’s breadth – the diameter of a human hair – is maybe 1/10th of a millimetre. That’s 10-4 metres. So to imagine how small 10-35 metres is, think of a hair’s breadth, then think 1000 times smaller than that, then think 1000 times smaller than that, then think 1000 times smaller than that, then think 1000 times smaller than that, then think 1000 times smaller than that, and you’re not even close. Think 1000 times smaller than that, then think 1000 times smaller than that, then think 1000 times smaller than that, then think 1000 times smaller than that, then think 1000 times smaller than that... actually, who am I kidding? 10-35 metres is an unimaginably small scale.
This tiny, tiny, tiny scale of 10-35 metres is called the Planck length.
Nothing to do with the length of a plank. Planks of wood, in my experience, tend to be longer than 10-35 metres. It’s called the Planck length because it was proposed by the physicist Max Planck.
Because General Relativity works right down to scales of around 10-35 metres, we can be surmise that space is more or less continuous at scales larger than this Planck length.
Nothing to see here
How about scales smaller than the Planck length?
Well, physicists are a bit cagey about what space is like at such small scales. They concede that Einstein’s equations break down at these scales, but they’re somewhat reluctant to conclude that maybe space is no longer continuous at these scales.
Instead, they refer to the Planck length as a “minimum length”. They suggest that “you can’t go smaller”.
This is the academic equivalent of putting your fingers in your ears and saying “la-la-la-la-la-la-la-la I don’t want to think about it I don’t want to think about it”.
I’ve noticed that scientists have a habit, when they reach the limits of their knowledge, of pretending that what they don’t know is unknowable. Humans, it seems, scientists included, hate to admit that there’s anything they could know, but don’t.
Obviously, you can go smaller than the Planck length. It would be a strange universe indeed if it were governed by laws of physics down to scales of 10-35 metres, but no further.
I concede that it’s hard to see beyond the Planck length. I concede that, given the way our tools for seeing break down at such small scales, there’s a danger of slipping into speculation when we theorize about physics at smaller scales.
But to give up and say there’s nothing to see here is defeatist.
The fact is, theories of physics that assume that space is continuous just don’t work at these scales.
Maybe, just maybe, that’s because space is not continuous at these scales.
How big are we?
To understand why we find it so hard to contemplate that space might not be continuous, we’re going to have to confront an unpalatable truth about ourselves as humans.
We’re biased.
As large creatures, we humans are deeply prejudiced against small scales.
When I say “large”, I don’t mean that we’re large compared to iguanas or starfish. I don’t even mean that we’re large compared to bacteria or viruses. I mean that we’re large compared to the Planck length.
If we look at it less egocentrically, it’s not so much that the Planck length is really, really small, it’s that we humans are really, really large.
And since space is more or less continuous on a large scale – on our scale – we can’t help but think that the answer to the question “Is space continuous or discrete?” is bleeding obvious.
On a human scale, space seems continuous. Therefore space is continuous. OK, so the continuity of space breaks down at a tiny, tiny, tiny scale? La-la-la-la-la-la-la-la I don’t want to think about it I don’t want to think about it.
Go with the flow
We’ve been here before.
Think about water. It seems continuous, right? If I pour water from one glass to another, it flows in a continuous stream.
Turns out that, at a small scale, water isn’t continuous.
It’s made up of discrete molecules. Each of the molecules is made up of two hydrogen atoms and an oxygen atom. Each of those atoms is made up of protons, neutrons and electrons. Each of the protons and neutrons is made up of quarks.
It’s only at a large scale that water seems continuous. At smaller scales – below about 10-9 metres – it’s discrete all the way down.
We could choose to deny this discreteness, and treat water as continuous, since it’s more or less continuous at a human scale, which, obviously, is the only scale that matters.
Or we could concede that water is discrete, and choose to explore the implications of a universe in which molecules, atoms, protons, neutrons, electrons and quarks are the underlying reality.
Be discrete
If, as Stephen Wolfram proposes, the universe is a graph, then space is discrete.
We should stop pretending that continuous space is the underlying reality, and concede that it’s merely an approximation that applies only on a large scale.
We should stop trying to wrangle continuous theories to fit the awkward discrepancies we run into at scales around the Planck length and below, and start trying to formulate discrete theories that approximate to those continuous theories at scales around the Planck length and above.
Wolfram Physics might be just the discrete theory we’re looking for.
His team are making progress in proving that his physics approximate to Einstein’s equations of General Relativity at large scales.
Is space continuous or discrete?
If I were a betting man, I’d be putting my money on discrete.