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I'm of the firm opinion that free will (the idea of choice we exercise in an event) at least in the daily usage of the word is incompatible with the "laws of the universe" as we currently know them.

Not even probability from quantum mechanics provides room for free will. As we do not say the electron in a double slit experiment has free will. Nor a quantum computer performing Shor's algorithm for that matter.

My question is: If I allow a mathematician to create toy model universes can he create a notion of free will which is not an illusion?

I use the word mathematician of their use of rigour and reducing things to axioms.

What do the philosophers have to say about rigorously defining free will given the freedom of choice in the laws of universe?

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I think this question would be clarified as "Is free will an even logical possibility with respect to some set of laws of nature we could in principle design, or is it impossible in every possible world". What you're getting at is a very deep, very interesting question, but the "mathematician" somewhat obscures it the core idea here.

That out the way, it's a clever, but unsuccessful, attempt to side step some of the conceptual issues at play in the free will debate. We still need to agree on a definition of "free will" to see whether we can design a possible world with features that allows for it. And so you'll be dragged, inevitably, kicking and screaming perhaps, into the compatibilism debate. Does it REALLY follow that, if an arbitrary action of mine, X, is determined, but part of that determiniation is things we consider part of our "will", that I haven't willed it? And so on. Standford encyclopedia will be helpful there.

But your strategy is a fruitful one if we do the following. Suppose we list every definition of freewill within the free will debate, and then try to be "mathematicians", as you put it, and try to design a possible world that satisfies it. I imagine that's been done by someone, but it's a useful thing to do for yourself: see which definitions of free will are logically possible. But the two crucial questions still remain. In order of depth (ascending), they are:

  1. Is our world one which satisfies the criteria of being a world W, where W is the set of worlds that allows some definition D of free will to exist?

  2. Even if this definition, D, of free will is possible on our world (that is, our world is a member of the set W with respect to D), is this definition the "right one"? What is even the criteria for the "right" definition?

As you can see, your line of reasoning is absolutely worthwhile, but can only get us so far.

I hope this is helpful.

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  • So for 1. I'd leave it to the physicists. Fortunately we don't have to necessarily wait to arrive on a theory of everything due to the idea of approximation! If free will can exist it might exist at a subatomic scale we haven't been able to probe yet and then you can have measurable deviations from that which may lead to deviation from classical outcomes given enough time. However, it is a bit hard to believe but who knows maybe "quantum biology" will become "quantum field theory biology" and maybe even "quantum gravity biology."
    – More Anonymous
    Sep 22, 2019 at 17:59
  • I don't even consider 2. to be a problem because we don't need a definition the "right one". As an analogy consider the idea of conditional probability in quantum mechanics. There isn't really a unique formula that doesn't reduce to the Bayes theorem. So a better strategy is to use the call the all different names like Luder's rule, etc and use them as modelling tools whose effectiveness depends on the context of the situation. This would be my pragmatic view for the whole "the criteria for the "right" definition?"
    – More Anonymous
    Sep 22, 2019 at 18:06
  • @MoreAnonymous To your comment of question 1, I'd slightly disagree. I certainly think physics knowledge is relevant, but the task would be to see whether a definition is satisfiable within a system of rules/laws. But consider; we'd be postulating physical laws that don't necessarily exist: we're not bounded by findings in physics in terms of what can be proposed. It's a conceptual question about whether X is possible within a certain, abstract logical space (does that phrase make sense?), and that's a metaphycian's speciality. But I have no doubt having a physist on board would be useful
    – Daniel Prendergast
    Sep 22, 2019 at 18:22
  • I've actually done a physics (not philosophy) degree. Usually physicist don't find it useful to create an axiomatic system (counter example: axioms of quantum mechanics but not "everyone" is happy with them). Because there are way too many of reformulations and when moving from an approximate to a more fundamental law it's not as easy to know which one to drop and which to keep. It's not that they aren't capable it's just not as useful. Also I'm not sure I agree with "X is possible within a certain, abstract logical space" isn't this "abstract" case the initial conditions?
    – More Anonymous
    Sep 22, 2019 at 18:32
  • @MoreAnonymous On 2, I don't think that's the right metaphor. When principles reduce to one principle, it's often appropriate to choose between them on pragmatic grounds. In logic, it's often easier just to use an inference rule like modus ponens even though we could prove it from the axioms: modus ponens is just an intuitive presentation of how the axioms work in some contexts. But consider: "could have done otherwise at t given the exact set of facts" vs "if what was done was in accordance to an agent's desire". These cant both be applied: they're inconsistent, so we have to choose.
    – Daniel Prendergast
    Sep 22, 2019 at 18:35
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One approach to free will from a mathematical perspective is the Conway and Kochen Free Will Theorem. Here is Wikipedia's description of it:

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. Conway and Kochen's paper was published in Foundations of Physics in 2006. In 2009 they published a stronger version of the theorem in the Notices of the AMS.

This appeared in two papers: The Free Will Theorem and The Strong Free Will Theorem. Conway is an English mathematician and Kochen is a Canadian mathematician. They were not, however, creating a "toy universe", but describing a notion of free will which is not an illusion in the actual universe.

For a philosophical perspective on free will see Mele, A. R. (2014). Free: why science hasn't disproved free will. Oxford University Press. I discussed this book in an answer to a different question.

Wikipedia contributors. (2019, August 20). Free will theorem. In Wikipedia, The Free Encyclopedia. Retrieved 03:58, September 22, 2019, from https://en.wikipedia.org/w/index.php?title=Free_will_theorem&oldid=911648414

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  • I'm not sure if this is a semantics game being played here or something more profound. If we use the notion of free will as: "the idea of choice we exercise in an event". In quantum mechanics the notion of probability comes from the Born rule. A different debate is if the brain harnesses quantum mechanics (or is the classical limit enough to describe the computations it does). But even if I say it does. It seems to me that one has modified the definition of free will to say "so must some elementary particles (have free will)."
    – More Anonymous
    Sep 22, 2019 at 4:56
  • The paper seems to be along the lines of no to super-determinism implies particles have freewill? Also do you have any idea of follow up papers in light of (measurement not being mysterious) : arxiv.org/abs/quant-ph/0405161 For a wonderful discussion of the measurement see: math.columbia.edu/~woit/wordpress/?p=10533&cpage=1
    – More Anonymous
    Sep 22, 2019 at 4:56
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    Is "not a function of the past" potentially equivalent to pure randomness? If so I think that's not really what the OP was asking about, since they specifically said that quantum randomness, even if genuine, wouldn't qualify as free will to them. Likewise the book by Mele seems to be just arguing against certain arguments for choices being pre-determined prior to our awareness of making a choice, judging by your summary on that other answer, not arguing for a concept of free will which is distinct from both determinism and randomness, which is what I took the OP's question to be about.
    – Hypnosifl
    Sep 22, 2019 at 16:44
  • Another example of your consistently good answers, and some interesting references I had no idea about. Thanks.
    – Daniel Prendergast
    Sep 22, 2019 at 17:47
  • @Hypnosifl Free will would be distinct from both determinism and randomness. I don't think free will can be reduced to determinism and randomness, but those two might be reduced to free will at different levels within a system. The problem with free will is that it presumes consciousness to make a choice at whatever level it occurs. What Conway and Kochen are showing is that one could consider quantum systems as having free which implies panpsychism or some other explanation. I am not saying that they have this, but the OP is asking for mathematicians who have actually considered free will.
    – Frank Hubeny
    Sep 22, 2019 at 18:16
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If you want a notion of free will which is distinct from both compatibilism and quantum randomness, perhaps one way would be to create an axiomatic system describing a simulated universe in which there would be axioms for the initial conditions and natural laws governing unconscious matter, but where each choice made by a conscious being within the simulation is determined by a new axiom which is not derivable (or even predictable statistically) from prior states + the laws for unconscious matter. Of course, the very act of choosing what those axioms should be by a mathematician in the real universe would involve some combination of determinism and randomness in their brain, but one could imagine some set of axioms picked out in some metaphysical way from the platonic space of all possible axioms, although one could argue (and personally I would argue myself) this just displaces the problem to another level.

I bring it up because it resembles the Christian position known as "Molinism" which argues that God's omniscience includes middle knowledge of "counterfactuals of freedom", meaning the choices all souls (including possible souls God does not actually create) would make in all conceivable external circumstances they might find themselves in. The idea is that this is distinct from God's knowledge of all naturally possible woulds, so although if I found myself in circumstances X it might be naturally possible for me to choose Y or Z, God's middle knowledge would tell him that in these circumstances I would in fact choose Y, and this truth would not be reducible to any natural truth. These truths would in a sense be timeless and unchangeable even by God, much like the way most theistic philosophers don't believe God can change the truths of mathematics, God's omnipotence would consist of the ability to pick any of the possible universes permitted by both natural laws and counterfactuals of freedom and make it real (this allows Molinists to avoid the worry that some people condemned to hell could have been saved if only some external circumstances beyond their control had been different, see the discussion here and here by a Molinist on the notion 'trans-world damnation', that God only allows people to go to hell if they would have been damned in all possible worlds allowed by the counterfactuals of freedom).

Also, this notion of choices as just new axiomatic inputs to the collection of facts about the universe and its history is similar to the argument in this post which has a good discussion of what would be required to have a form of "free will" that was neither deterministic nor just a random element chosen from some probability distribution determined by the laws of nature (see also the paper here that More Anonymous posted in a comment). To me it seems like a kind of reductio ad absurdum since these choices do not in any way depend on your prior preferences or thoughts or feelings, but it seems like the best attempt to create a steel man of this notion of free will.

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    I feel like your making the argument that if free will does not exist then the human brain does not have computational integrity. I think this is false. Just as a classical calculator will deterministically produce 24 when 6 is multiplied by 4. Just because it is deterministic does not reduce it's computational integrity. And we get this computational integrity due to evolution.
    – More Anonymous
    Sep 22, 2019 at 17:07
  • @MoreAnonymous - I don't understand what you mean by "computational integrity", can you elaborate? And what part of my answer made you think I was making such an argument? I'm perfectly happy to believe in compatibilism, I wasn't intending to suggest there is anything negative about the fact that we don't have "free will" in the sense of something non-deterministic or non-random, as I said I find the whole idea leads to absurd conclusions and is probably conceptually incoherent even if it doesn't lead to logical self-contradiction.
    – Hypnosifl
    Sep 22, 2019 at 17:15
  • By "computational integrity" I meant the output of a computational device produces results of integrity. These results should be affected by logic. A calculator would be a good example of a device having computational integrity.Also the lines that gave me that impression is: "mathematician in the real universe would involve some combination of determinism and randomness in their brain" and "leads to absurd conclusions and is probably conceptually incoherent "
    – More Anonymous
    Sep 22, 2019 at 17:26
  • In that 1st quote I was talking about a mathematician intentionally trying to pick an arbitrary set of axioms which would tell you the choices of beings in a simulated universe (I've edited my first paragraph to clarify a bit), I wasn't suggesting that randomness would make mathematician's proofs of theorems from some set of axioms (including standard ones like the Peano axioms) suspect. As for the 2nd quote with "leads to absurd conclusions", in that sentence I was talking about the notion of free will which is distinct from determinism or randomness, not talking about real human brains.
    – Hypnosifl
    Sep 22, 2019 at 17:32
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I take free will to be a moral concept. Rather than something of metaphysical interest in itself; whatever it is I doubt we'll find its meaning in naturalistic terms (perhaps that's your issue with it).

As a theory-neutral point of departure, then, free will can be defined as the unique ability of persons to exercise control over their conduct in the manner necessary for moral responsibility.

Here's an article from the Atlantic newspaper

There is, however, one important variety of free will that I do reject: the one that has it as an unearthly power; some kind of mysterious force standing outside of science as we know it, and allowing us to make choices that are not caused by our brains. ... [Maybe] the jury is out on whether we have free will, because it depends on the forthcoming findings of physics as to whether there is randomness in the decision-making processes in our brains...

Maybe if we can be morally responsible, it’s for some other reason entirely. I [Nicholas Clairmont] wrote my dissertation a few years ago arguing for this idea, which is called “semicompatibilism.” It’s gaining ground in philosophy circles due largely to its greatest champion, a California philosopher named John Martin Fischer... The notion that we are morally responsible for something because we could have done otherwise even though we didn’t is actually really weird, isn’t it?

I'm in complete agreement with him. Anyway, your question suggests the complete opposite. That, without the "forthcoming findings of physics" supplying a reason to believe we could have acted differently, no-one can lay claim to some "unearthly power" needed for 'moral responsibility'.

I have no idea what that quality could really amount to, besides the belief that we will have acted in such a way, and should not.

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