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For decades, Douglas Hofstadter has argued that consciousness can arise in physical systems such as the jumbles of neurons inside our heads. But if there is nothing magical about this process, do we possess free will? And what does it even mean to be free?

Posted on August 16, 2022.

If there is a bible among mathematicians, it may very well be Gödel, Escher, Bach. Douglas Hofstadter’s 1979 classic is still read today — although, I’ve been told, not necessarily to completion: the whole book is about 700 pages, and not all of it is light reading. That is a shame, because Hofstadter developed a truly original theory on how consciousness can arise in inanimate systems, with profound implications for the way we think about ourselves and other conscious entities.

So it’s time for a summary. If you want, you could skip Hofstadter’s theory by clicking here. That will send you straight to my thoughts on free will and freedom in a deterministic universe.

We have to start with formal systems. Such systems consist of two key ingredients: *axioms* and *rules*. *Axioms* are the fundamental, indisputable truths underlying the system.
*Rules* can use previously established truths (such as axioms) to generate new pieces of knowledge, known as theorems. Theorems are different from axioms because they are not *assumed* to be true, but they are — ultimately — derived from the axioms using the rules of inference.

As an example of a formal system, I will propose a slightly extended version of Hofstadter’s `pq`

-system. This system consists of just three symbols: `p`

, `q`

, and `-`

(that is, the hyphen). The system has just one axiom:

`pq`

-system`-p-q--`

We simply assume that the statement `-p-q--`

is true. From this axiom, we will derive some theorems. But for that we need rules, the first of which is given below.

`pq`

-system
If

`X`

, `Y`

, and `Z`

are strings of hyphens such that `X`p`Y`q`Z`

, then `X`-p`Y`q`Z`-

.
Consider our axiom `-p-q--`

, and suppose we want to apply Rule 1 to it. In this case,

and `X`

are both `Y``-`

, and

is `Z``--`

. Rule 1 then shows that the theorem `--p-q---`

holds. We have proven our first theorem!

We could apply Rule 1 to this theorem to find yet another theorem. The result would be the theorem `---p-q----`

. And of course, we might apply the rule to that theorem as well, and so on. But now, let us introduce the second and final rule of this small formal system.

`pq`

-system
If

`X`

, `Y`

, and `Z`

are strings of hyphens such that `X`p`Y`q`Z`

, then `Y`p`X`q`Z`

.
Remember how we derived the theorem `---p-q----`

above? Now suppose we apply Rule 2 to it. Since now

is `X``---`

,

is `Y``-`

, and

is `Z``----`

, Rule 2 gives us the new theorem `-p---q----`

.

We could apply Rule 2 again to `-p---q----`

, but this will just give us `---p-q----`

again. On the other hand, an application of Rule 1 to `-p---q----`

yields the theorem `--p---q-----`

.

Anyway, I could keep going for some time. The point is that with these rules, we can prove an infinite array of theorems. But I would not hold it against you if you considered all this meaningless symbol-shifting. You may be able to apply the rules like some downtrodden bureaucrat, but you may not understand why you would. What is the point of this system?

Here is one way to grant this system meaning: note that the number of hyphens before the `p`

plus the number of hyphens before the `q`

is always equal to the number of hyphens at the end of a theorem. In other words, there is a correspondence, a mapping, an isomorphism between the strings that the `pq`

-system churns out, and our familiar idea of the addition of numbers. The axiom of our system, `-p-q--`

, can through this isomorphism also be interpreted as $1+1=2$. The theorem `--p---q-----`

that we ultimately derived can also be read as $2+3=5$.

The rules of our system also make sense in the context of addition. Rule 1 is the system’s way of saying that if $x+y=z$, then $(x+1)+y = (z+1)$, which is clearly a valid rule about adding numbers. And Rule 2 simply says that if $x+y=z$, then $y+x=z$ as well. Because these rules align with our natural understanding of addition, the theorems that we can derive within the `pq`

-system always retain their correspondence with true statements about the sum of two numbers.

Of course, the simple formal system we discussed above only allows one to prove statements of the form

, where `X`p`Y`q`Z`

, `X`

, and `Y`

are strings of hyphens. We can use this system to prove statements such as `Z`$2+3=5$

and $23+7=30$

, but not $5$ is a prime number

or $13$ is the sum of two squares

.
If we wanted to prove those properties within a formal system, we would need a system that offers more flexibility, more symbols and rules. But in the end, the general structure will be the same: you start with some axioms, and using the rules of inference, you derive new theorems.

In the early twentieth century, Alfred North Whitehead and Bertrand Russell set out to create a formal system that would cover most mathematical topics, including number theory. This formal system, dubbed *Principia Mathematica*, offers enough flexibility to encode true statements such as $6$ is not a prime number

and $13$ is the sum of two squares

, but also false statements such as $6$ is a prime number

and $13$ is not the sum of two squares

. Of course, only true statements stood a chance to be proven within *Principia Mathematica*, but it was, in principle, possible to write down false statements as well. The symbols to do so were all there.

One of Russell and Whitehead’s main goals was to provide a foundation for mathematics that did not contain self-reference: for instance, sets containing themselves would not be allowed, as this might lead to paradoxes. So did they succeed? Well, a couple of decades after *Principia Mathematica* was published, Kurt Gödel showed that the system was so powerful that self-reference would *necessarily* appear. This is quite amazing: *Principia Mathematica* was constructed with the goal to keep its products from talking about themselves. Gödel showed that this was untenable, but more importantly, that it was not due to some mistake that Whitehead and Russell had made. Then, perhaps, the hole could have been plugged. Instead, Gödel showed that self-reference would pop up in *Principia Mathematica*, or any similar formal system, because of its *strength*.

Let us trace some of the major steps Gödel took. One of his insights was that the statements of *Principia Mathematica* could be transformed into (usually very large) numbers. We will call such a number the *Gödel number* of the statement. To illustrate this process, recall the `pq`

-system from before, which was of course much simpler than *Principia Mathematica*. It contained the three symbols `p`

, `q`

, and `-`

. Suppose we code each of these symbols a distinct number:

Symbol | Code |
---|---|

`p` |
$\color{blue}3$ |

`q` |
$\color{blue}2$ |

`-` |
$\color{blue}1$ |

We can now turn a string in the `pq`

-system, such as its axiom `-p-q--`

, into a number. One approach is to look at the first six prime numbers (because `-p-q--`

consists of six symbols), raise every prime number to the code associated with the respective symbols, and multiply these factors. So for `-p-q--`

, the Gödel number would be $$2^{\color{blue} 1} \times 3^{\color{blue}3} \times 5^{\color{blue}1} \times 7^{\color{blue}2} \times 11^{\color{blue}1} \times 13^{\color{blue}1} = 1,\!891,\!890.$$ (Pay attention to the blue exponents.) In this way, the statement `-p-q--`

is associated with a unique number. You can imagine that for longer statements, the Gödel numbers will be astronomical!

But of course, Gödel did not define these numbers for our silly `pq`

-system, but for *Principia Mathematica*. Remember that Whitehead and Russell’s system was able to prove statements such as $5$ is a prime number

and $13$ is the sum of two squares

, and many other properties of natural numbers. Gödel showed that one of those other properties that *Principia Mathematica* could prove was that a given number, such as $1,\!891,\!890$, was the Gödel number of a theorem of *Principia Mathematica* itself! Provability in *Principia Mathematica* thus became itself a matter of shifting ‘meaningless’ symbols around. From now on, the system would also be able to prove statements such as $1,\!891,\!890$ is the Gödel number of a theorem of

, or its negation, *Principia Mathematica*$1,\!891,\!890$ is not the Gödel number of a theorem of

.*Principia Mathematica*

At least, that is what you might hope. Mathematicians rely on proofs, because everything that can be proven is true. Conversely, mathematicians hope that whatever truth they are currently trying to establish can also be proven. Gödel’s ultimate aim was to show that this hope is sometimes in vain: there are true statements which cannot be proven.

How do we know this? Gödel managed to build a special statement in *Principia Mathematica*, which we will call $G$. This statement $G$ says

$g$ is not the Gödel number of a theorem of

*Principia Mathematica*,

where $g$ is the Gödel number of — wait for it — $G$ itself! So effectively, $G$ states

$G$ is not a theorem of

*Principia Mathematica*,

or, to put it even more succinctly,

This statement is not a theorem of

*Principia Mathematica*.

As mentioned above, not every string of *Principia Mathematica*-symbols corresponds to a true statement. So we should ask ourselves: is this statement $G$ even true? Suppose $G$ were false. Then $G$ would in fact be a theorem of *Principia Mathematica*. But that would mean *Principia Mathematica* can be used to prove false statements, such as $G$. Since the axioms and rules of inference of this system are all correct, this cannot be the case.

The only option is therefore that $G$ is true. But then $G$ is indeed not a theorem of *Principia Mathematica*. In other words, there is some true statement that we could never show within the formal system *Principia Mathematica*. The system is, as the kids say, incomplete.

What does any of this have to do with consciousness, free will, and freedom? This is at the core of Hofstadter’s book Gödel, Escher, Bach. In it, he argues that the logical structure that makes $G$ unprovable in *Principia Mathematica* is the same one that allows us to feel like conscious decision-makers.

By encoding statements of *Principia Mathematica* as numbers, Gödel creates a new level of meaning: there is the old level of meaning about number theory, but there is also a new meta-level about statements in *Principia Mathematica*. On the low level of meaning, $G$ looks like nothing special. It just seems to assert some vague number-theoretical property of some huge number. If this statement were true (which we know it is), you would expect it to be provable. But when we move to the second level of meaning, when we view $G$ as this statement is not provable

, then we see that statement is both true and unprovable. It is this new level of meaning Gödel devised that becomes the reason that *Principia Mathematica* is incomplete. Only on this second level does it become clear that $G$ cannot be proven.

Here we see an example of, as Hofstadter calls it, upside-down causality. The second level of meaning was constructed using the tools of the lower level: the meaningless symbols shifted around by following meaningless rules. A reductionist might expect everything that happens on that second level to be a consequence of what happens below. A car drives or a clock ticks because all of their parts perform the tasks they are supposed to do. But Gödel’s proof reveals something strange. Here it is the second level of meaning of $G$ that drives the nail in the coffin of completeness. In Hofstadter’s words:

It is as if the sentence’s hidden Gödelian meaning had some kind of power over the vacuous symbol-shunting, meaning-impervious rules of the system, preventing them from ever putting together a demonstration of $G$, no matter what they do.

The thesis of Gödel, Escher, Bach is that this upside-down causality also lies at the root of consciousness. The human brain consists of neurons — at the end of the day, physical stuff. From this perspective, our thoughts are nothing more than specific sets of these neurons firing in certain patterns. Hofstadter calls these patterns symbols. We have a symbol for doll

, for singing

, and perhaps now you have a symbol for incompleteness

. As we grow up, we learn about the consequences of our own actions: if I drop the doll, I will no longer have the doll; if I sing, people will be cross with me, and so on. All this feedback on our actions is collected and determines what we know about ourselves, shaping the I

-symbol.

Symbols form a level of meaning on top of that of the neural level. An individual neuron firing has little meaning, but at least we know that symbols are made up of firing neurons. One might try to explain the symbols that are actually being thought in someone’s brain by looking at the neurons’ level. But Hofstadter argues that it may be more informative to think of causality the other way round: as symbols pushing around the neurons. Specifically, the I

-symbol being the force pushing around neurons (and ultimately our bodies), similar to how $G$’s second level of meaning drives the unprovability of $G$.

If we follow the line of reasoning above, human consciousness is nothing special. It is an epiphenomenon of a sufficiently complex network of neurons, but the neurons are not particularly important. We could also create consciousness using transistors, or even well-arranged dominos. The important thing is that we can form sufficiently complicated symbols mirroring the outside world. Eventually, a second level of meaning will arise, and it will step into the driver’s seat.

Humans are thus nothing but stacks of carbon, hydrogen, and a bunch of other atoms. Those atoms are arranged in a clever way, sure, but at the base level, it’s all physical. Asking Why did you become angry when I kicked you?

is the same as asking a boulder why it rolled down a hill when I kicked it. The answer is: because it was the only possibility that the laws of physics allowed. I would therefore argue that the decisions we make are not really free: if we made an exact copy of the universe as it exists right now, and pressed play on this copy, you would have to behave the same way there as in our original world.

Hofstadter takes a different position, arguing that something has free will if it can be said to *make choices*. But where do we draw the line, exactly? He would say that a robot navigating a maze by a simple decision rule (alternating between going left and right, for instance) is not really making choices. A robot with a sense of self, however, whose I

-symbol causes the decisions the robot takes, does make choices in Hofstadter’s view, and is therefore endowed with what we call free will.

To me, this seems to define a choice to be one of the outputs of a necessarily conscious system. But I would argue that a choice is just the answer to a multiple-choice question: do you want to go left or right?

It does not really matter if our robot is very simple or highly intelligent. If we treat the robot as a black box decision-maker which we give all the information it desires, and ask it to choose left or right, I would call any outcome a choice. I think it would be more appropriate to say a system has free will if, given the state of the world (including itself), it could respond in multiple ways. But then I will immediately concede that no robot, human, or well-arranged system of dominos could ever clear that bar.

Hofstadter is onto something here, though: the silly, basic robot certainly does not *feel* that it has free will, while the sophisticated robot might. Certainly humans feel they have free will, that they could make different choices in certain situations. His criterion seems to distinguish between those who *believe* they have free will, and those who do not. But to my mind, free will itself is an illusion. The laws of physics control us all, like puppets on a string.

With the reader perhaps somewhat depressed, let me end by trying to define freedom in a world without free will. The word freedom

can mean different things to different people, so I will focus on Isaiah Berlin’s Two Concepts of Liberty. In this essay, he distinguishes positive liberty from negative liberty, arguing that the two are not only different, but can also contradict each other. For our purposes, though, this distinction is useful insofar as it allows us to understand what the different definitions of freedom are. In fact, many authors’ conception of freedom can be traced back to either positive or negative liberty. It will suffice to interpret these two concepts in a context without presuming free will.

Positive liberty is the degree to which an individual is his or her own master. As Berlin puts it, I wish my life and decisions to depend on myself, not on external forces of whatever kind.

Humans should be in control of their own existence:

It is true that to offer political rights, or safeguards against intervention by the state, to men who are half naked, illiterate, underfed, and diseased is to mock their condition; they need medical help or education before they can understand, or make use of an increase in their freedom.

This notion of freedom therefore suggests that a person should have the power to affect reality in meaningful ways. The state of the world should depend, to a significant degree, on what you decide.

As we saw, given a state of the world, the choices you make follow from a deterministic process. Your decisions are not free, but they should have some impact before we can speak of positive liberty. If, through divine intervention, you made a different decision, that would have to affect the world in a way we could consider significant. So the test I propose for positive liberty is this: at some point in time, through a deterministic process, your mind makes a certain choice — to go left at a junction, to kick a boulder, whatever. Imagine we could create an alternative universe, the only difference with the original being that here, we send jolts of electricity to some of your brain’s neurons and inhibit others, with the sole effect that this particular choice would be different: in this alternative universe, you choose to go right, or leave the boulder alone. Would life play out significantly differently? If so, you possess at least some positive liberty. This alternative universe can never become reality, but it makes a major difference for our happiness to imagine that it could.

Negative liberty is the degree to which someone’s actions are not interfered with by other people. To again quote Berlin:

If I am prevented by others from doing what I could otherwise do, I am to that degree unfree; and if this area is contracted by other men beyond a certain minimum, I can be described as being coerced, or, it may be, enslaved. Coercion is not, however, a term that covers every form of inability. If I say that I am unable to jump more than ten feet in the air, or cannot read because I am blind, or cannot understand the darker pages of Hegel, it would be eccentric to say that I am to that degree enslaved or coerced. Coercion implies the deliberate interference of other human beings within the area in which I could otherwise act. You lack political liberty or freedom only if you are prevented from attaining a goal by human beings. Mere incapacity to attain a goal is not lack of political freedom.

This definition strikes me as a bit anthropocentric. Of course, political freedom should only depend on the actions of other humans: that is what it means to be political. But could other inabilities also not be seen as limiting one’s freedom? If I am blind, I do not have the option of reading a book — let alone Hegel, who my limited brain could never *choose* to understand.

But let’s stick with Berlin’s definition; it will not matter much. So, negative freedom means that other humans should not limit the set of options you feel you have, even if many of those options will never be chosen — and owing to a lack of free will, could never be chosen. That is the one qualification that I want to make here. If someone declares that it is forbidden to feed the ducks in a pond, that limits my negative freedom, even if I was never going to feed the ducks in the first place. Negative freedom is impacted if certain actions are forbidden, regardless of whether I was ever going to take them or not.

There is a tension here between imagination and reality. Freedom asks us to imagine that multiple worlds are within reach. Otherwise it would not matter if they got cut off from us by outside forces (limiting negative freedom), or that we are unable to make them come true through our actions (limiting positive freedom). But I would argue that this is an illusion. A very pleasant, important illusion, but an illusion all the same. We can imagine other worlds, and that imagination may even be said to drive our actions — but those will have been our only possible actions.

At best, we are free to dream.