### The brain is capable of free thought and volition—but that facility might not be so unique.

Consider the humble tennis ball. It sails gracefully over the net in the middle of the court, jumps off the racket on the other side, and starts moving swiftly back to where it came from.

Why do tennis balls choose to travel this way and not any other way?

If you read that question carefully, you'd probably raise objection to the word *choose*. Tennis balls don't *choose* to move a certain way or not; they move that way because that's how the laws of physics work. If you knew all the parameters—such as the weight of the ball, the angle at which the tennis players hit it and for how long, the current wind velocity, and any other forces acting on the ball—then you'd be able to predict precisely how it would travel: assuming your calculations were correct, seeing the ball travel in some other direction would be as impossible as 10 + 4 being equal to 2.

But tennis balls aren't the only objects forced to obey the laws of physics. Surely, the same laws must apply to the human brain as well?

The idea that one situation can be precisely predicted from an earlier one is called 'determinism'. Much has been said on the topic, but one way to understand it is through our tennis ball.

See where the ball is now: three inches off the racket, angled towards a point a bit above the net, with the force of gravity pulling it down and a light wind nudging it off course. That's the *state* of the system at the given time.

Now, you run your equations. If your ball weighs so many kilos and is travelling in such-and-such direction while being pulled and pushed on from there and here, then where will it end up? Plug in the numbers and an equation will tell you the answer: two inches clear of the net, and barrelling towards *there*. This is the new state of the system, and what you've just computed is the way the state changes or 'evolves' over time.

There is a catch: you need to know the situation with complete accuracy. Even a little bit of uncertainty can, over time, add up to a lot — as any bricklayer should be able to tell us.

Since the days of ancient Egypt, people have been using plumb lines. These are essentially ropes with some kind of weight stuck at the end. They are very useful for marking straight vertical lines: hold one end up, and let gravity do the rest.

Plumb lines are used to make sure walls are straight — which makes sense in ancient Egypt, but are they still needed in the modern world with its cement and neat rectangular bricks?

The answer, it turns out, is yes. Imagine laying bricks with only your eyes to guide you. Chances are you’d get them pretty straight: you just need to align those neat edges, right? But imagine that, for some reason, your alignment is off by half a centimetre to the left on the first row. And again on the second row. And on the third row. By this time, the third row of bricks is a centimetre and a half off from the first. In a 40-brick-high wall — which is reasonable for a house — you’d end up with the top part of the wall leaning 20 centimetres to one side compared to the bottom.

Not good.

That’s because, brick by brick, all the tiny and seemingly insignificant errors come up till they turn into something bigger. That’s why bricklayers always send their plumb line all the way down to the *bottom* of the wall. Even if they’re off by half a centimetre, it’s always the *same* half-centimetre and they aren’t adding up errors on the way.

Perhaps human behaviour is deterministic too, but there are so many factors and variable to keep track of that even the most powerful computer would be overwhelmed.

Or…it could be that human behaviour is *non*-deterministic, and predicting it with absolute certainty is simply impossible.

It can be argued that the latter is indeed the case: Nature sets no constraints on us, and we are free to do as we will. But if we accept this argument, it seems that systems much simpler than the brain *also* satisfy the same criteria that are needed for free thought and volition.

One can go into metaphysical intricacies, but below that lies the fundamental question: does Nature impose any restrictions on our thoughts?

Any activity can be thought of as time evolution of a system’s states, including that of the brain. If we are indeed free thinkers, then an earlier state of the brain shouldn’t deterministically control a later state of the brain. You shouldn’t be able to run equations and say that “if the brain is like *that* now, it will be exactly like *this* later” — because if that were the case, then it’d be the laws of nature doing the thinking, not you of your own volition.

Before we continue, let me make clear that we’re not concerned with external stimuli. When we talk about determining the tennis ball’s position, we ignore the cow that wandered onto the tennis court and bumped into the moving ball thereby disrupting the game. Similarly, when I talk about predicting the brain’s future state, I mean to predict it *assuming* nothing apart from what we have already measured and accounted for enters the system. We can always expand our deterministic formula to include a deterministic cow, but doing so would make the calculations needlessly complicated. If one is possible then so is the other: either way, my main argument remains the same.

So the problem of free will boils down to the question: is it possible, at least in principle, to predict how states of an isolated brain evolve?

If no, the brain’s later states develop without constraints from its past, and thoughts and volition emerge freely; if yes, there is no such thing as free will because we are executing only a preexisting blueprint.

Generally, to predict any system’s future state, two things are needed: one, full knowledge of the present state, and two, knowledge of the physical laws governing the system’s time evolution. The brain, of course, is the most complex system that we know of, but why could one not know with sufficient precision, at least in principle, its state at one point?

We know the physical laws relevant to the brain’s functioning. These laws come from electrodynamics and quantum theory, with possibly some added mechanics—all fields we know well. (Chemistry is encompassed within these fields of physics). Then, is free will in danger, not from the omnipotence of a supernatural being, but simply from the unyielding laws of Nature?

Not to worry, but before continuing with the brain, it is quite interesting to take a look first at a much simpler system.

Imagine a small Universe, empty of all but two planets revolving around each other through mutual gravitational attraction. Think Sun and Earth, or Earth and Moon, or any two celestial bodies you prefer. Such a system is very easy to follow: the law of its time evolution are simple mechanics, the oldest branch of physics. By knowing the initial state of the planets, you can calculate their future states forever.

But introduce a *third* planet into the mix, and things suddenly become vary chaotic. A planet could end up here — or there — or somewhere else, and before long you find your equations have taken you into extreme fuzziness, with no way to accurately say when a planet is where.

This is commonly known as the three-body system or problem. The reason it becomes unpredictable is because, when you try to do the math, you never get a closed-form expression. Think of it like calculating the digits of pi, but for equations: you can compute the outcome to a certain level of accuracy, but there’s still more left that you’ll never reach.

And then, like the bricklayer without a plumb line, the errors will start to add up.

In the case of the three-body problem, the equations amplify small uncertainties in a system’s state: in other words, they’re very sensitive to changes. Saying ‘2.000’ will give you a very different result from ‘2.001’ which means that figuring out the futures of these systems is impossible even in theory. While the equations themselves are straightforward — discovered centuries ago by Newton — their solution, in a three-body problem, leads to a time evolution that is characterised as being ‘chaotic’.

The defining attribute of a chaotic system is that any uncertainty regarding the system increases exponentially with time. So if the first calculation is fuzzy with a degree of 6, the fuzziness may increase to 18 next time, and 162 after that, and keep rising faster and faster from there. Pretty soon, the answer is so fuzzy it’s as good at saying “the answer is there somewhere in this Universe, but I don’t know exactly where”.

This is, of course, the very opposite of being deterministic. No longer can one predict future states by the current one, or say the current state was caused by the past, because any given state can end up as…literally anything.

The popular name of such behaviour is the “butterfly effect” — based on the idea that a butterfly flapping its wings on one side of the world could trigger a storm on the other, as the tiny perturbations add up. Yes — the weather is an extremely complex and chaotic system too. So is the tennis ball, which we’ve been thinking of as deterministic till now. Indeed, it is now a generally accepted scientific truth that the whole world is chaotic.

It turns out that, the simplicity of the three-body system notwithstanding, its state for an indefinite future can’t be known even in principle. And the same applies to pretty much every other real-world system too. It’s still convenient to *pretend* the world is deterministic, and use it to say — for example — how the ball will fly till the next return, or whether a thunderstorm is likely in the next few minutes.

But try to go too much beyond that, and you’ll witness the creation of chaos out of order.

One can think of a chaotic process as being surrounded by fog. The further one looks, either backward or forward, the less visible the path becomes. As time goes on, eventually, the system has no information whatsoever about its past and looking far enough ahead about toward what state it is heading for. The growing uncertainties are washing out everything.

Thus, the only way a chaotic system’s future could be deterministically known is if at one time-instance its state could be determined with zero uncertainty. But, such infinite precision is not possible even in principle.

The inability to predict the future of a three-body system is not due to a lack of our capabilities; no, the laws governing its evolution are such that information about its future state simply *does not exist* in its present state. Thus, with time, the states of a chaotic system somehow emerge as if by themselves.

Now that we illustrated chaos with the three-body system let’s return to the brain. Obviously, given its complexity and the laws governing it, the brain must be an *extreme* case of a chaotic system.

We can’t show this simply with calculations as we can for the three-body system: the brain as a whole is way too complex for that. But we know that some of its functioning, like the firing of neurons, involves extreme non-linearities. Non-linearity and chaotic outcome go hand in hand.

Since, for us, the brain’s evolution in time manifests itself as thoughts, this means that the common natural phenomenon of chaos is the one that affords the freedom of our thinking and volition!

Considering the three-body system again, is there a fundamental difference between its behaviour and that of the brain? The brain is more complicated, but they are both chaotic; thus, the future emerges on its own accord for both. Could one say that the three-body system also has free will? That it can do things of its own volition?

In fact, given almost all phenomena in Nature are chaotic, *does it mean free will is also universal?*

The big difference between the brain and simpler systems lies in how long it takes for this freedom under the aegis of chaos to manifest itself.

In the brain, a rough guess could link such freedom to the rate neurons communicate with each other. With each neuron typically connecting to more than 1000 others and firing on average every few milliseconds, the brains’ time to freedom might be sub-microsecond. From the point of view of what we are sensing: instantaneous. While in the three-body system, the same time scale might be that of a thousand slow, ponderous revolutions.

But who chose our timescale as opposed to a slower one? After all, the brains of a fly run significantly faster than ours, while trees live and grow and react on a much slower timescale.

What if we normalise times with a measure of the system’s complexity — something like multiplying the time to freedom with the number of states that the system at any moment can enter into? Maybe such normalised times to freedom are more or less the same independently from any given system’s peculiarities. This would mean that all chaotic systems are free to the same degree after having travelled through roughly the same amount of phase space.

Accordingly, the difference between the brain and a three-body system is in their complexity and not inherently in their capacity to act free.

The brain has no exclusivity; the differences are in quantity, not in quality.

Of course, one presumes that although a three-body system seems to have the ability to behave according to its own choice, it is not self-conscious. The interesting question is, then, how complex a system has to be for sentience to emerge? Will Artificial Intelligence, or in general anything we humans are capable of creating, ever reach a complexity to become sentient? One could not be faulted for wagering against such an outcome.

An aside, pure quantum systems, even complicated ones, evolve predictably. However, in Nature, pure quantum systems fall prey to what is called decoherence. The time for a macroscopic system like the brain, or parts of the brain, to stay in a pure quantum state is exceedingly short. Thus the predictability of a quantum state does not stand in the way of classical physics’ chaotic behaviour. The only exception may be the whole universe’s quantum state, which by definition has no outside disturbance to cause its decoherence.

Do we seem to be then in the paradox that while the universe itself is strictly deterministic, minor parts of it, like our brain, or the Milky Way, have free will? Maybe all these little freedoms are somehow correlated to add up to determinism on the largest of scales?

Or *did* the tennis-ball decide to travel one way and not another after all?

**Credits:** An earlier version of this article was published at http://www.infinitetime.org.