You are saying 2+2=4 is a "static timeless relationship that just exists", I am saying "summation" is an action. Similarly, "constructing a set" is an action even if "the set of all things contains the number 1" is a timeless relationship which just exists. The axiom addresses our capability to perform or not be able to perform the action of constructing a set from sets who have indistinguishable elements. If we take the axiom, I can choose elements from those sets despite their indistinguishability, if we do not take the axiom I cannot.


At this point, I think you are just misunderstanding my use of the AoC. Allow me to reintroduce the context of when I originally brought it up:
The AoC is not proof that we can make choices independent of cause and effect, and it is not a refutation that we can either. The point of this all in the greater context of our discussion, is that even in rigorously defined mathematics, some obvious "truths" in set theory are not possible to arrive at without taking choice as an axiom. In our discussion of free will, I find it similarly difficult to prove anything, but perhaps exploring the mathematical analog could yield some insight into our less rigorous exploration of free will.



There is no Motte and Bailey fallacy - what I said is still true and I offer no lesser idea. If the Axiom of Choice were somehow proven false, and became for instance "The Theorem of Non-Choosability without distinguishing characteristics", that would show, in my opinion of course, strong support for determinism. As to whether its possible to prove the AoC false, this bit "The Axiom of Choice will never become a theorem." is quite a strong statement mathematically speaking. If you were able to justify that in some rigorous way I would be very interested to hear it.



Yes I certainly admit I don't know how to prove the existence of an uncaused choice, but this is an incorrect use of a razor. If we had some decision to make here, we could use a razor to make it, but we're looking for rational proof of difficult things, and a razor is not a proof.

On a similar vein, I also don't know of any proof of causality ("caused" choices). We certainly have a lot of empirical evidence that cause-and-effect exists, but why not examine all cross-products of the following:


---------------------------Method
________________________
---------------------| empirical | rational |
---------------------________________________
Choice | causal | A | B |
-------------------------
| non-causal | C | D |
-------------------------
I hope you like my ASCII art.

I think A has a lot of empirical evidence, C perhaps has some, but B and D I've got nothing for. Can you rationally prove that choices are causal? (Yes yes shifting-the-burden-of-proof fallacy or whatever, but we're on the search for truth here!!!)


I would just like to point out I agree with this in general. Though I think for this to work you need a viewpoint which is able to balance determinism and free will - evaluating the various deterministic factors in society are important, but without free will I don't see a mechanism to assign responsibility (other than utility, but that brings us back to the moral questions and might makes right).

Again I generally agree with these courses of action but I don't see the connection to determinism. You're making a leap from a particular physical mechanism to a moral judgement that we should act a certain way in response to the physics. Insofar as I asked a question prior to this, and you tried to answer with the weather example, the answer I see in your explanation is just that we allow the "not-figure-out-able" part to be modeled indeterminately, aka allow for non-determinism until we can convert it to determinism (or indefinitely if we can't).

I suppose for more clarity, I was asking with respect to the idea of what do you do if your worldview is "purely" deterministic. If you allow for a mix of determinism and non-determinism these questions seem pretty easy to answer.

Summation is an action yes, you can write down the steps to add 2+2. You can program a computer to do it. But the AoC is explicitly invoked when a set is uncomputable. You cannot program a computer to pick from infinite pairs of identical socks.

This isn't right. Taking the axiom does not suddenly give you the ability to choose indistinguishable elements. You remain just as mathematically powerless as before. The axiom simply states that a theoretical set of choices exists in the abstract mathematical universe, completely independent of your ability to actually pick them.

So if I'm reading you right, you're saying 'hey don't take it so literally - let's just try and use AoC as a helpful analogy.' I'm still pushing back on this - it's a false equivalence.

In set theory, if we don't take AoC, we literally break entire fields of mathematics. We lose the ability to prove actual necessary theorems in functional analysis and topology. But in philosophy, what 'obvious truth' do we lose if we drop free will? We don't lose the ability to explain human behaviour. In fact, as I said above with the weather analogy, determinism actually explains 'obvious truths' of human behaviour BETTER than free will does. We don't need a magical axiom to explain why a man turns left at the crossroads; causality already explains it perfectly.

Furthermore, in math, axioms are structural necessitities. We literally have to start somewhere. But in the physical sciences (neurobiology, physics, human behaviour), declaring an uncaused event to be an 'axiom' just because the causal chain is too complex to map is intellectual surrender. It's just slapping the label' axiom over a gap in our knowledge / an area of our ignorance - god of the gaps fallacy striking again.

Okay, when I said earlier that AoC will never become a theorem - that wasn't my strongly held opinion. I was stating an established mathematical fact that was settled over 60 years ago.

In 1938, Kurt Godel proved the standard axioms of set theory (ZF) cannot be used to prove the AoC false. He did this by constructing a mathematical model where ZF and AoC both hold true.

In 1963, Paul Cohen invented a mathematical technique called 'forcing' to rigorously prove that ZF cannot prove the AoC true either. He even won the nobel prize specifically for this proof. Together, they established that the Axiom of Choice is strictly independent from standard set theory. It is mathetmaically impossible to prove it true or false using the other rules of the system.

Occams Razor is an epistemic heuristic used specifically for evaluating competing hypotheses. It states entities should not be multiplied beyond necessity. If two models explain the same phenomenon, the one that introduces fewer untestable assumptions is the rational choice.

Yes, it's not a proof, and I never claimed it was a deductive mathematical proof. I'm using it to show the uncaused event "free will" hypothesis is irrational.

And dare I say I see a bit of hypocrisy here: demanding rational proof for determinism in the exact same sentence as conceding you don't know how to prove the existence of an uncaused choice. As you say below that is shifting the burden of proof indeed. I don't want to glaze over this either. If you're on team 'uncaused event', team 'free will' explicitly admitting you cannot prove your core premise is a massive concession.

Ah, this is David Hume's famous problem of induction. We can't logically prove that dropping an apple will make it fall. We can only empirically prove it by dropping it 10,000 times and observing gravity in effect. So I'd argue once again you're making a category error. Casuality is a property of the physical universe, which means it can only be established empirically. Pure rational proof belongs to mathematics and formal logic (eg 2 + 2 = 4). We know causality exists the same way we know gravity exists: through overwhelming, uninterrupted empirical observation.

As we're on the search for the truth, here is the state of the board based on your own table:

You admit box A (empirical causality) has a massive amount of evidence behind it (all of physics, neuroscience biology).

You admit you have absolutely zero evidence for box C (empirical non-cauasl) or box D (rational non-causal).

Since you have zero evidence for the non-causal model, and box A has the entire weight of the observable universe behind it, the only rational, truth seeking conclusion it to accept box A. You don't get to abandon empirical reality just because you can't deduce it with pure math (which as Hume demonstrated is an epistemological impossibility).

I would argue dropping free will doesn't destroy responsibility.

When a bridge collapses, we hold the engineering firm 'responsible'. We don't do this because we think the engineers have an uncaused magical soul that chose to build a bad bridge. We traced the causal chain, found what brought about the incident, and remove their engineering licence so they don't hurt anyone else.

I would argue trying to hang on to that last shred of free will is not being helpful and just allowing us to be cruel. It allows us to throw criminals into solitary confinement and say 'they deserve to suffer because they chose to be evil' instead of doing the actual hard deterministic work of rehabilitating them.

We don't need free will to build a moral society. We just need empathy, a desire to minimise harm, and the courage to actually troubleshoot the root causes of human behaviour.

Ok well first I want to make sure we don't confuse the word "indeterminable" (we can't calcuate it) with "indeterministic" (it has no cause).

A purely deterministic worldview is never modelling an unknown variable as non-deterministic. If I flip a coin and cover my hand, the result is indeterminable to me until I open my hand, but I haven't modelled the coin as 'non-deterministic magic'. I know it's determined I just am ignorant of the result.

So no I am not 'allowing for non-determinism'.

As for Hume's is-ought problem you elude to with the physics to moral judgment jump, and you're right, but .. gah how do I say this to you correctly. The is-ought thing is about context. Now we've been arguing a lot about axioms. So here's my moral axiom, here's my 'ought' that you have to accept or reject.

"We ought to reduce harm and make society safer."

Now I haven't gotten to that from determinism, you're right. It's the thing I'm asking you to start with me on. Now once you accept that, THEN we start talking about IS, WITHIN that framework of the baselessly asserted OUGHT.

And when you do that, determinism is a better model than free will for achieving that moral axiom. If I'm a doctor and my foundational goal is 'I ought to heal my patients' free will (demons) belief of "she freely accepted this demon, I need to prescribe an exorcism is physical punishment to drive the demon out to cure her" is going to not achieve that ought nearly as effectively as the deterministic model of germ theory - I need to prescribe antibiotics.

Am I making sense here? Within the subjective 'ought' context you have chosen, there are then definite right and wrong 'is' beliefs to achieve the 'ought' goal.

So I'm using determinism to explain we can throw out the false 'demon' belief the free will belief of retributive judgment and moral outrage, by saying no if our goal is [insert my ought goal here about reducing harm and make society safer] then determinisim is far more effective at achieving that goal.

First off, yes you can, if you take the axiom of choice. Trivially speaking, you arbitrarily assign every sock an integer and then arbitrarily pick an integer and select that sock. If you don't take the axiom of choice, then you can't arbitrarily assign every sock an integer or arbitrarily pick an integer.

Secondly, the computability or "uncomputability" of a set is irrelevant to whether its possible to construct a particular set - every infinite set is "uncomputable" (meaning no computer can assign every element in that set some identifier in a finite amount of time) but we can still represent its construction in many, many cases.

Very well, I don't know what this abstract mathematical universe you've created is, or how its rules work or what it contains, but I journey joyously into the rabbit hole. Can you describe the construction of this abstract mathematical universe and most importantly, how it exists independent of any possible "actions" like summation or construction? If it is possible to describe this universe, can you also describe how it invalidates the application of axioms like AoC to the actions of those who do not exist merely within the bounds of its abstractions as well?



I would like to point out in mathematics ZF and ZFC are both functional fields of mathematics - one takes AoC and the other does not. We do not "lose" anything, we just arrive at different sets of results. But your question about what do we lose if we drop free will is still interesting, though I fear I'll just be re-iterating a number of things I've already touched on. Let me see if I can find a relevant previous post:


This is a good rebuttal to what I said about it becoming a theorem, good work. I would like to point out though that this still doesn't stop us from evaluating the implications of the AoC from actually being true or actually being false (or being some third non-binary state) - these mathematical results are about the provability of AoC, not the truth value of AoC. It may be mathematically impossible to prove (or disprove) the thing, but the thing may still evaluate to true, false, or neither, and we can still think about what happens in those cases.


I'm on team "lets see what happens" and am happy to demand anything that looks interesting. In truth, personally, my actual viewpoint is argument from utility - I don't see how removing agency from any model of the world is useful. Even if the world is actually deterministic, as you point out below the "indeterminability" or "uncalculability" of the complexities of the world requires some sort of simplification or compressed representation in order to model the world in a way that we can make decisions based on. I am happy to include deterministic factors in this "model" - presumably my model over time will rely less and less on this "Free Will" simplification and more and more on empirical evidence as both myself and humanity experiences more of the universe. I haven't been debating this with you for literally years at this point because I am afraid to find my "core premise" is unprovable, I've been doing it because the chance (however small) to find new thoughts to incorporate into a model like this is exciting. Sorry if I've pulled back the curtain only to reveal an empty podium.


Ok curtain closed, back in character! The use of Occams Razor is itself irrational - its an empirically founded rule of thumb, not a rational decision. You're begging the question here - showing something's irrational by using irrational tools. I can show you that 1 == 2 if you let me start from 1 == -1. Now, if you wanted to make a choice like say, which competing hypothesis should we evaluate first, then I would grant your use of this irrational tool.


If you'll notice, the table (uhg, my ASCII art did not turn out how I wanted!) has a column for empirical evidence as well, and I think of it more as "sharing" the burden of proof. I'm trying to be philosophically charitable with my argumentation and concessions. I have yet to hear any rational evidence for causal choices - perhaps you would concede that a causal choice is actually not a choice at all?


Ok Buster - throwing the philosophical charity out the window now - how does "C perhaps has some" equate to "You admit you have absolutely zero"


Agree
In the context of doing the actual real work of planning out a society or how to live a life, I'd be happy to start with you here. In our context of presumably trying to figure out whether free will is real or false or provable or unprovable or useful or not, it sounds like you're in the same boat as me - I don't know how to bridge the gap. Maybe (perhaps even very likely) there isn't any transcendental property of the universe which can tell us how to choose that starting statement. That is one thing I like about the idea of Free Will - if I assume its possible, and I have it, in order to create a consistent worldview I must conclude others have it, and as the Free Will property is itself immeasurable, the natural starting place is to give all agents with Free Will equal weight in the worldview. In determinism, or without Free Will, I don't know how to arrive at an analogous place. Free will has its own problems of course, like someone can just choose to be a murderer, or as you have pointed out many times we can over-value our weighting of free will versus known deterministic factors in making judgements. But I suppose my point is in a deterministic worldview there is no special emergent property which separates an agent from a non-agent, nothing which says society ought to be oriented towards freedom, which basically makes an insect-like hive-structure or a tyrannical society the same as a society of free agents. Thoughts?


Also on the idea of us not having Free Will, I would also happily concede the utility of that as a tool or thought experiment to help further scientific conclusions, or to better understand how to improve society and things of that nature.

Post has been edited 1 time(s), last time on Sep 11 2026, 8:23 pm by Vrael.