Are there examples in the literature of rigorous mathematical models of libertarian free will that take the laws of physics into account?

Question

What I'm looking for is a detailed description of the decision-making process of an agent that possesses libertarian free will, when this agent is on the verge of making a choice, at some time t.

For simplicity, we can assume that this agent is a human being, and therefore has (at the very least) a physical body and a brain, both made up of elementary particles. We can assume as well that these elementary particles obey the laws of physics. However, this does not mean that I am ruling out the presence of extra-physical elements at play (i.e. I'm happy if a model exists that doesn't assume physicalism to be true).

With all this in consideration, how exactly does this agent "decide" what to do at time t in a way that is not a function of the state of the universe up to t (so that the decision is "free", i.e., not causally determined)?

I previously prompted GPT-4 (a stronger version of "ChatGPT" with improved performance on multiple benchmarks, see technical report here) to provide a mathematical formalization of libertarian free will that considers the constraints imposed by the laws of physics. GPT-4 suggested key ideas and concepts such decision theory, quantum mechanics, the probability of a decision given the state of the universe, the indeterminacy of quantum mechanics, quantum superposition of all possible choices, collapse of the wave function, the "many-worlds" interpretation of quantum mechanics, "consciousness causes collapse", among others. However, judging by the comments, GPT-4's proposal didn't make much sense and failed to provide a precise description of how the decision is actually made, which was what I was ultimately interested in.

Has anyone in the literature tackled this challenging task and come up with a more successful model of the decision-making process of a libertarian free agent?

Answer 1

Two cents.

Libertarian free will is supposed to be a basic concept and process in libertarianism, not reduced to further mechanisms. This is simply the way it is.

An exact mechanism would make it determined and thus not libertarian free will. So this cannot happen. In a sense is like asking for a mathematical model for the next quantum transition. It is supposed to be irreducible.

If this satisfies you, maybe:

Libertarian free will: Among the realizable options O (given the current state of the universe U), choose freely one: o.

One cannot determine or reduce freely further and still remain free.

Libertarianism notes that the next state to happen is always under-determined (future is open) even given all possible past causes and determinations (agreeing with standard interpretations of current basic physics).

PS: One's reasons, preferences, whatnot can play a role in freely choosing, but this role is not fixed, it is also free (eg reasons can be re-arranged and re-prioritised freely).

PS2: What the free will theorem says, is that if one entity has free will, then, given the quantum nature of the universe, other entities must have it as well. So still free remains irreducible.

PS3: There are many authors with libertarian views, although they do not necessarily agree on everything. Some notable examples are a) Randolph Clarke, b) Robert Kane and c) Alfred Mele (Mele defends both libertarianism and compatibilism), check their works.

We are condemned to be free - Jean-Paul Sartre

Answer 2

Conway and Kochen[??] have offered a so-called "free will theorem" (note: the following quote is the Wikipedia summary of their conclusion):

The free will theorem of John H. Conway and Simon B. Kochen states that if we have a free will in the sense that our choices are not a function of the past, then, subject to certain assumptions, so must some elementary particles.

I can't find it right now, but Conifold noted in a comment on a previous question related to the above that there's some sort of "no-go" problem, here, however. At any rate, the FWT has been discussed in several other places on the PhilosophySE.

Whether the esoterica of temporal logic has much bearing on your inquiry, I am not confident to proclaim so much, but I will at least quote the following:

Much of Prior’s work on Tense Logic was initially motivated by the problems concerning the relationship between time and modality raised by the Master Argument of Diodorus Chronus and its fatalistic conclusion. Prior held a genuine interest in themes like indeterminism and the open future. One of his major concerns here was to allow for human freedom. In order to capture the notion of historical necessity underlying Diodorus’ argument, he opted for a branching representation of the interrelation of time and modality, which was first suggested to him in a letter by Kripke in 1958 (see Ploug and Øhrstrøm 2012). The Diodorean conception of modality as quantification over linear time was abandoned and replaced by the picture of a tree, whose branches depict alternative possibilities for the future. Prior (1967, Chapter 7) considered two different versions of branching time temporal logic, which he associated with the historical views of Peirce and Ockham, respectively.

So I would also like to mention some premises of the "stit [see-to-it-that] saga":

i. usually an agent is not able to select one possible future to become the unique actual future, but

ii. by his action he can make sure that certain futures, which before his action are possible, are no longer possible after his action.

(I strongly recommend reading over the linked section of that SEP article; it would be rather time-consuming to try to copy it into the SE word processor.) Insofar as one might hold these logical analyses to be "sensitive to" the laws of physics, one might appeal to a many-worlds interpretation of quantum physics (though note that Everett himself does not seem to have clearly favored the "many-worlds" interpretation of his thesis).

For what it's worth, Stephen Wolfram thinks:

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I should add that there is an entire SEP article on the difference between chance and randomness, which modulo logical pluralism opens the question as to whether libertarian free will could be modeled as an alternative to those (when those are otherwise taken for a dichotomy). Then the chanciness of the laws of quantum physics (so to speak), and the randomness of various real numbers' decimal expansions (and their implicit role in physical quantification over the continuum of spacetime), might lend themselves to a contrastive mathematics of free will. The functions such as this would encode (or be encoded by) could not be expected to work as mechanically as composition of functions as we know them on standard numbers. But it is sometimes said that the value and justification of love's true power is grounded in irreplaceability:

To be fungible is to be replaceable by another relevantly similar object without any loss of value. Thus, money is fungible: I can give you two $5 bills in exchange for a $10 bill, and neither of us has lost anything. Is the object of love fungible? That is, can I simply switch from loving one person to loving another relevantly similar person without any loss? The worry about fungibility is commonly put this way: if we accept that love can be justified by appealing to properties of the beloved, then it may seem that in loving someone for certain reasons, I love him not simply as the individual he is, but as instantiating those properties. And this may imply that any other person instantiating those same properties would do just as well: my beloved would be fungible. Indeed, it may be that another person exhibits the properties that ground my love to a greater degree than my current beloved does, and so it may seem that in such a case I have reason to “trade up”—to switch my love to the new, better person. However, it seems clear that the objects of our loves are not fungible: love seems to involve a deeply personal commitment to a particular person, a commitment that is antithetical to the idea that our beloveds are fungible or to the idea that we ought to be willing to trade up when possible.

This perspective can mean understanding irreplaceability in terms of haecceities:

I explain what I understand by individuation or numerical unity or singularity: Certainly not the indeterminate unity by which anything in a species is said to be one in number. Rather, I mean designated unity as a this, so that just as it was said above that an individual is incompossible with being divided into subjective parts and the reason for that incompossibility is asked there, so too I say here that an individual is incompossible with not being a designated this by this singularity and the cause is asked not of singularity in general but of this designated singularity in particular – that is, as it is determinately this.

The intersection of the concept of haecceities with modern understanding of quantification, sets, and numbers then should appear as the concept of a haecceifer for ur-elements, which are not distinct from empty sets (or zero, then) or even each other by the discursive first-order properties but by their haecceities. In other words, the numerical, and hence mathematical, order for love's value proceeds not from zero/empty sets but has some other nature. If love's value is a kind of ethical value (or at least overlaps this to a good enough extent), then given the relationship between free will and ethics, we might be able to construct a mathematical representation of the pure will, albeit not one that is determinative in the usual way (i.e. not one where we get special formulas for computing outcomes in various cases).

Answer 3

Any intelligent agent, free or not free, will attempt to contemplate the available information and prior knowledge, establish a model of relevant elements of the world, consider multiple different alternatives and their likely outcomes and rank the alternatives based on utility based on some factors like (short and long-term) potential gains, risks, morality, laws, then take the best ranked decision at a given time.

This is done by all intelligent agents, no matter if they have libertarian free will or not, libertarianism cannot deny AI those abilities.

A nonfree agent however will be programmed in that way, meaning the eventual decision is a mere function of physical inputs and physical programming, like a computer program.

A free agent however is not restricted to eventual decision determined by physical inputs and physical programming, it has a mysterious independent contributing factor in addition to all that a non-free agent has, which is not reducable to either determinism or random.

In practical observation of agents that makes no difference, as this additional factor does not improve the free agents decision compared to their non-free competitors in any competition (regardless if we compete for being most economical, most beautiful or most compassionate). No matter which dimension of quality we would like to optimize, we could do so on a nonfree agent by improving the programming, no optimization would require adding libertarian free will.

Libertarian free will is becoming increasingly irrelevant philosophically as artificial AI becomes more powerful and psychology and neurology explain the brain and mind more, revealing severe flaws and limitations.

In particular libertarianism increasingly fails to show any benefit such an agent would have over nonfree agents. Libertarian free will has been a fill-gap belief to elevate humans above other animals and computers, as an explanation why humans are "so much better". But because AI gradually becomes as good as or better than humans in all dimensions, and human decisional freedom and quality is revealed to be not that great at all, no more explanation is needed why humans are "so much better", because they aren't. As such libertarianism becomes just a straw that spiritual people will cling to as proof of their "immortal soul".

Answer 4

There are no examples of such models. Decision-making simply cannot be modeled mathematically. Libertarian free will has nothing to do with physics. Mental processes like decision-making have no measurable physical properties. Laws of physics do not apply.

Psychology is the study of the mind. There you may find some non-mathematical models based on concepts like need, preference, value, emotion, imagination, instinct, estimation, anticipation and future planning. All mental stuff, but nonetheless no less real than physical stuff.

Answer 5

(This doesn't directly answer the Title question, but perhaps it gives you an idea of how it could be modeled.)

Physical

If everything is strictly physical, then free-will, consciousness, self-awareness, and other aspects of the mind are illusions, nothing but artefacts of electrochemical brain activity. Everything we do, all our decisions, are determined by physical actions (electrical, chemical, and even mechanical). Our decisions are causal and certain, though being a result of our immediate physical state, they are not necessarily predictable.

Consider an animal, such as a mouse. Its brain continually receives various inputs:

• low blood sugar: hunger — look for food!
• smell: food — its nearby, find it.
• sight: round object with bells on it — familiar.
• memory: a similar object was soon followed by the appearance of a cat — run away!
• etc.

Some part of the brain receives all of these signals, each with its own level of urgency, and it reduces them to a single course of action:

• If the hunger is low, and the smell is weak, the cat memory will win and the mouse will go somewhere else.
• If the hunger is high and the smell is strong, the danger signal will be overruled, and the mouse will look for the food.

A decision has been made, but no thinking was required; the inputs could have been reduced to mathematical values and a formula applied to determine the best action to take. (Consider the simplicity of Conway's Game of Life, or how little it takes to simulate the behaviour of schools of fish or flocks of birds.)

The same mechanism controls the actions of all animals, though at a much simpler level in some and at a much more complex level in others.

This includes humans, who have the most complex circuits, including an additional one that we call consciousness. It's like a feed-back loop that observes the automatic decision making process, adds running commentary, and remembers it for future use. It doesn't make decisions, but it does rationalize them.

Between stimulus and response there is a space. In that space is our power to choose our response. In our response lies our growth and our freedom.

But again, even this is still a result of purely physical reactions that could be mathematically modeled.

Even if one modeled some version of free-will choices, ultimately it would reduce to the final decision being made randomly (whether by the flip of a coin or the parity of the current time in microseconds).

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Beyond Physical:

Some people though, believe that there is also a super-natural component to the human mind, one that cannot be explained in physical terms.

Free-will not only allows our minds to observe and rationalize, it also allows it to intervene. All the sensory inputs, including memory, may strongly indicate one course of action, but the mind is able to override this gut-instinct natural choice, and it can decide to do something different (perhaps simply by providing a much stronger input to the decision making mechanism). It may later rationalize or explain its choice, and it may store this decision and its result in memory, but the fact is that this added non-physical component caused the person to do one thing when all natural instincts and experience were saying to do otherwise.

This of course still doesn't explain how the free-will decision was made.

This super-natural aspect is central to effectively all religions. Some call it a "soul", some call it a "spirit", some an "inner self", some a "thetan", and some an "ego" (as opposed to the instinctual "id").

Various beliefs exist about it:

• Ancient Israelites believed in only one life, followed by non-existence.
• Catholics believe in only one life, but that after death the immortal soul is either eternally rewarded or eternally punished for one's behaviour.
• Some Christians believe that one's spirit must be improved by consciously building a character like God's.
• Hindus believe that after physical death, it is reincarnated in a better or worse host, depending upon karma accumulated in the current life.
• Buddhists believe in reincarnation, and that experiencing physical suffering and losing personal desires eventually leads to nirvana.
• Scientologists also believe in reincarnation, and that one must work to clear bad memories acquired from previous lives.
• etc.

The one common factor seems to be a goal of doing the best that one can do to improve oneself as a person, to build one's character so that it always makes the morally best choice.