Free Will and Computational Intractability

Richard Dawkins, Sam Harris, Daniel Dennett and Christopher Hitchens are often referred to as the ‘Four Horsemen Of Atheism’. They haven’t had too many public disagreements but perhaps the most prominent ongoing dispute, relates to free will. This is a topic that is not unrelated to religion, of course. Most of the world’s faithful are taught that the difference between eternal paradise and eternal agony, lies in how we each exercise our free will.

Sam Harris is a strong determinist and he argues that free will is an illusion. That is, since the contents of the human skull are no more immune to the laws of physics than any other matter in the universe, our behaviour is always exactly commensurate with deterministic causality. Dan Dennett agrees that science comprehensively rejects the mind-body dualism on which religious ideas of free will are based. However, as a compatibilist he also argues that free will can be reconciled with the deterministic physical laws.

As it happens, I’m convinced by the determinist argument and I find compatibilist positions to be rather slippery. The conclusion that our decisions are determined by the matter in our brains obeying the laws of physics, rather than by some ethereal mind or soul, is one that is of great import. Since the dualism on which so much of religion depends is false, explaining the implications of that would seem to be a good use of any philosopher’s time. Instead, it seems that compatibilist philosophers quite often prefer to spend their time inventing new definitions of free will, which can accommodate determinism.


The Resemblance Between This Compatibilist Philosopher And Dan Dennett Is Surely A Coincidence?
The Resemblance Between This Compatibilist Philosopher And Dan Dennett Is Surely A Coincidence?


I think that a better analogue for our subjective experience of volition, is found not in new definitions of free will, but rather in computational intractability. When discussing this topic, philosophers often like to return to a famous Austin thought experiment involving a golfer. If a great golfer misses a simple putt, should we consider that she could have holed it, based on some degree of freedom that she had? Alternatively, if every atom in the golfer is returned exactly as it was before the missed putt and the shot is replayed, must she miss it again, as the invariant laws of physics consistently produce the same results? Many debates around free will and determinism have considered this question (including the often heated exchanges between Harris and Dennett).

Considering the issue from the perspective of computational intractability, may allow for a different starting point. For example, if a rain cloud moved over the golf course, could we calculate which blade of grass will be first to be struck by a raindrop? Certainly, the rain cloud does not have free will, so there will be one blade of grass that will be first to get wet and that will be the result of deterministic laws of physics acting on the relevant atoms and particles. If we were able to observe which blade of grass was struck by a raindrop first, and then replace all of the atoms in the atmosphere exactly as they were before the rain started, what would happen if we then replayed the events?

Of course, since the laws of physics don’t change, the same arrangement of atoms will give rise to the same results. Quantum indeterminacy can’t deliver free will either. Even if relevant at the scale of the particles and molecules that exist in human brains and rain clouds, quantum indeterminacy would offer some randomness but no free will. However, the vast number of particles and interactions involved in rain clouds, and the even greater complexity of human brains, means that any calculation to predict either outcome would be intractable. Even if we could measure the position and momentum of every relevant atom in advance, there will be no more practical way to see where the raindrops fall, or where the putt comes to rest, than to allow the laws of physics to play out and just watch.

Trying to prove mathematically whether the brain state of the golfer will result in her sinking her putt or not, is tangentially related to Hilbert’s Entscheidungsproblem (decision problem). In our case, the decision problem asks if we can analyse the complete configuration of the golfer’s brain along with the laws of physics, and prove in advance whether her putt will be holed or not. Alonso Church and Alan Turing showed that there is no solution to the Entscheidungsproblem (a previous blog on this site discussed Turing’s 1936 paper on the topic, which introduced the Turing Machine).


Cartoon Turing Machine
Cartoon Turing Machine


For example, if we view a rain cloud as a Universal Turing Machine that is configured to calculate the interactions between all of the relevant particles, then we cannot prove in advance which blade of grass will get wet first. The only practical approach is to ‘run the program’ on the machine (the cloud) and observe which blade of grass is first struck by a raindrop.

Similarly, if we consider the golfer to be a Universal Turing Machine, then Turing’s 1936 paper on computable numbers shows that even if we could know her complete configuration in advance, we still could not prove what the outcome of the putt would be. Additionally, if the golfer actually wanted to perform the full calculation to work out what kind of putt her brain state would produce, that would take exponentially more time than just taking the putt. That is, the problem is computationally intractable not just in practice but in principle. The only way to discover the outcome is to ‘run the program’ on the machine (the golfer) and watch where the ball goes. In Programming the Universe, Seth Lloyd describes the implications of Turing’s Entscheidungsproblem proof as follows …


“… once we set a train of thought in motion, we do not know whether it will lead anywhere at all. Even if it does lead somewhere, we don’t know where that somewhere is until we get there.”


The reason why we can’t predict the actions of a human being with full confidence, is not because people have free will or an ethereal soul. It is because either under a classical deterministic universe or under a quantum probabilistic universe, the problem is computationally intractable. Free will is an illusion and our actions are fully determined by the laws of physics. The path of a golf ball rolling on an undulating green is also fully determined by the laws of physics and easily predictable. However, Turing distinguishes between some problems that are computable and others that are not. The path of a golf ball on an undulating green is an easily computable differential equation. The proof of a decision problem for the brain state of Austin’s golfer either sinking or missing a putt, is impossible in principle.

Austin’s golfer will miss the putt every time. Even if before each replay, a computer is given her complete configuration in advance, Turing shows that it will still always be impossible to prove the outcome beforehand. Each time we replace every atom in the golfer exactly as it was, we will still need to replay the putt by ‘running the program’ on the machine (Austin’s golfer) and watching her miss.

Religion says that god gave man free will, but Turing shows that it only seems like we have free will, because we’re computationally intractable and so we cannot know what we’re going to do next.


Alan Mathison Turing
Alan Mathison Turing

John Hamill.

One thought on “Free Will and Computational Intractability

  1. Do you Believe in Free Will?
    Most people believe that they have a choice over the decisions they make, but there are good arguments from logic that seem to prove them wrong. Here, when discussing free will, the argument is aimed at showing that the biblical concept of free will, meaning that we are free agents who make our own choices on a case by case basis cannot be correct. Even if there were a god involved in the creation of our universe, the case for free will has fundamental flaws arising from the way the natural world works..
    The exact mechanism of the beginning of our universe and time and space in our universe is still a mystery, although we already know that since particles can pop in and out of existence in the vacuum of space, it is clear that something can come from nothing without any known outside force that we know of, making it happen.
    So back to free will. When our universe popped into existence, possibly with billions of other universes, our universe happened be a type of universe with physical laws that would allow stars, planets, and galaxies to form. Other universes might be different, in which case they could have different physical laws such as having different values for the strong and weak nuclear forces. It is unlikely that life could form on that type of universe.
    But our universe happened to be just right for life, and given that fact, and the fact that the elements necessary for carbon-based life such as water, carbon and oxygen are scattered throughout our universe, life is likely to, or at least may have, spread throughout our universe in any solar system that has a suitable planet.
    Religious apologists are fond of going back to the first cause as an explanation for God. In the scientific view, the first cause was our universe popping into existence, and therefore the starting conditions of our universe must have been such that in time life would develop. Since we are here to observe the universe, that seems to be a point that cannot have a good argument against it. Given those starting conditions, it is not an option for life not to develop, because each thing flows from what happened before. So the universe expanded and is still doing so, and what follows, follows logically from that.
    After the initial expansion, whether or not there was what people have come to call the Big Bang, the universe cooled enough to allow subatomic particles to form, and later simple atoms like hydrogen and helium. These atoms were initially in the form of gas, permeating as far as the universe reached at that time. But we know that gravity causes atoms to stick together, and so vast clouds of dust began to form, and the gravity of those clouds pulled the atoms together to form the first stars, which were short lived supergiants. We know that we owe our existence to those initial supergiant stars, because – simplifying the process – under their intense gravity, nuclear fusion took place, and protons were fused together, sometimes keeping electrons, to form most of the other elements that make up the periodic table of elements.
    Stars are still mainly hydrogen and helium, but when they explode, the elements created inside them are released, and another process takes place called supernova nucleosynthesis, in which other elements are formed, and the guts of the star and the elements produced during the supernova nucleosynthesis and which are necessary for life to exist, are blasted into space to eventually coalesce into smaller stars like our sun, and the planets around it, usually as part of a galaxy like our own massive Milky Way or other galactic types.
    What has this to do with free will, you may ask? The answer is relatively simple. So far what has been described occurring in the birth of our universe was inevitable, given the starting conditions and corresponding laws of physics that exist in our universe, and for all we know in every universe in a multiplex of universes, or perhaps just in some universes.
    Whatever the case, planets came into existence around Suns, and on our planet, and probably billions more, primitive life formed roughly a billion years after our planet coalesced into existence and over time it evolved into different species, surviving a number of major extinction events.
    Again, that process was inevitable given the starting conditions of our universe. One thing follows another inevitably. Life had no option but to come into being given the right conditions, which were supplied by the exploding supermassive stars, which were the result of the subatomic particles that resulted from our universe popping into existence. As more and more complex animals evolved, and speciation occurred because of the laws of natural selection, humans evolved from primitive ancestors, just as every other life form on earth evolved from less well adapted ancestors. Again, that was an inevitable process.
    Some people might argue that earth could have ended up like Venus, with a runaway greenhouse effect that made our planet uninhabitable, but earth had no option but to follow the laws of nature and form where it did, and be as it is, with the necessary elements for what we know as life.
    But again, back to free will. We all have a common ancestor in the first life form that came into being, however it came into being. We know that because we share DNA with all the life forms we have found on earth so far, from cabbages to seaweed, to dolphins, birds, and bacteria.
    Our evolution into humans is an inevitable consequence of all the life forms that came before us, the things that happened to our planet, and the laws of nature. We had no choice in the matter. Nature is entirely indifferent to what happens to individuals or species, but it is the inevitable driving force of change.
    We have thus inherited characteristics from our parents, who inherited characteristics from their parents and so on to the beginning of life on our planet. And each of our ancestors was a winner in evolutionary competition. They all survived long enough to reproduce, or we would not be here. We come from millions of generations of winners.
    Take those inevitable inherited characteristics, talents, intelligence and circumstances, and add environmental pressures, upbringing, fears and challenges, sadnesses and triumphs, loves and hates, and the rest of the mass of variables that make us who we are, and we become individuals different to anyone else on the planet. Nevertheless all those factors give us particular tendencies. We cannot think other than the way we do think. We cannot change our mind about something without all those complex factors coming into play and making us change our mind. New information coming to our attention does not change who we are and how that new information will affect us. If we do change our mind about something, everything from our past experiences, our genetic makeup from our ancestors, and our own well being are factors that have caused us to change our mind. No free will is involved in the matter, because we cannot help but act as we do from everything that has gone before.
    You can toss a coin to make a decision, but the result of that toss is inevitable. A million random factors come into play, from the weight bias of the coin, to how hard we flip it, to the history of the world that caused that coin to come into our possession, and the factors in our makeup that decided to make a decision based on the toss of the coin. The fact remains that the starting conditions of our universe moving us through one event after another combined with billions of other intertwined past and present factors will cause the coin to land the way it does. So we had no free will about how the coin landed.
    Thus the illusion of free will dissipates into oblivion. Everything that happens, from the words we speak, think or write, to whom we fall in love with and what type of person we become is the inevitable consequence of all that has gone before. And all that has gone before and all that will ever happen, the wars and famines, the natural disasters and our responses to them, the time of our birth and death, the natures of our children and grandchildren, the extinction events and survival rates, the creation of artificial intelligence, the monsters of our society and the rise and fall of civilisations, the stars in the sky and the flowers in our garden are the inevitable consequences of the starting conditions of our universe and the laws of nature and physics that came into existence with our universe.
    The question must be asked. Is there any escape from our destiny? Is not our effort to change things an example of our free will? The answer can only be no. Those efforts are an inevitable result of all that has gone before, all the influences that affect us, from the past to the present to what we hope for in the future.
    However, despite the absence of free will, we can do our best to live the good life, to do as little harm as possible, and to try to advance our understanding, learn as much as we can and try to make life better for the humans and other species we share the world with, in the short blink of a lifetime that the universe has lined up for us.
    If that is the course we choose, that too was inevitable, but it is not a bad way to live a life.
    There is a further point here. People who worship the Judeo-Christian-Islamic God use the concept of free will to justify the inaction of their god in the presence of suffering, injustice, war, and evil. On the one hand they believe they can influence the mind of their god through prayer, praise, and certain rituals that differ according to the concept of god they hold to be the correct one. On the other hand, they claim that their god gave us free will, and it is up to people to change things.
    A god who does nothing to influence matters in the world cannot be discerned, and therefore we can have no knowledge of such a god. Looked at this way, that type of god is more of an invisible observer than a participant in this universe.  But a god who does something to help somone who gives him/her sufficient worship, whilst allowing others to be tortured to death is a deeply flawed god, with the morals of a selfish child.
    What are we to become with such an example – are we to we favour those who praise us and ignore the suffering of those who don’t know us, or fail to praise us enough? Are we to punish those who snub us with unending torture? Are we to condemn forever those who do not believe in us, from the example more than once quoted in the Bible “… he that believeth not shall be damned”. (Mark 16:16, John 3:18)
    So much for turning the other cheek…  Christians often claim the bible as a guide to moral behaviour. But if we were to take our code of morality from the god of the old testament, the slavery condoning warlord, the one who, according to his own book, sent Moses to kill babies, (Numbers 31:1-35) or even the less warlike god of the new testament, we would be harsh and cruel.
    Believing in a god who answers prayers of believers and worshippers means believing in a deeply flawed god. Taking your moral stance from such a god would make a deeply flawed person.
    But we are not, as a rule, like that. Atheists, Christians, Muslims, and Jews are not like that. We are all, save for a few twisted individuals, better than that. With this in mind, the faithful have to know, if they think about their beliefs at all, that they know better than the god they worship, what is right, and what is wrong, and that they have higher ideals than the god they worship – they are not so cruel, not so unthinking, not so inhumane, and they have a sense of justice that would not punish someone forever for any crime, no matter what the crime. And if they had the power to do so, unlike the god they worship, they would stop torture, end cancer, get rid of the parasitic worms that blind hundreds of thousands in Africa every year, end AIDS, stop priests, mullahs, and rabbis from abusing children, and they would make this world a better place.

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