I believe we were talking about "is it possible to make a choice when there are no factors to base that choice on" and this is the exact same topic which the axiom addresses.
We're you're going wrong there is you're still equivocating "choice". (equivocation fallacy) In philosophy, a choice is an
action occuring in time. A brain processes data and outputs a decision. In set theory, a
choice function is not an action. Nothing is "happening". It is a static, timeless relationship that just exists.
A
category error is assigning a property to a thing that cannot possibly possess it, eg "what colour is the number 4?". Taking a timeless, non-physical property of abstract set theory and using it as mechanical proof for how a biological brain operates in linear time is the textbook definition of a categoery error.
I would dispute this - the axiom resolves the ability or inability of the mathematician to choose a set in the absence of determining factors. The axiom specifically addresses the mathematician's ability to choose in the absence of any possible choice function.
This is a fundamental misunderstanding of the axiom. The Axiom of Choice is the
exact opposite of what you are saying. In set theory, if the mathematician can actually define a rule to pick elements from sets,
you do not need the axiom of choice. In math, there are two ways to prove something exists: Constructive Proofs and Non-Constructive Proofs, and the Axiom of Choice belongs entirely to the latter. Example:
Infinite Pairs of Shoes (famous Betrand Russel analogy)
Imagine you have an infinite number of pairs of shoes. You are asked to create a new set containing exactly one shoe from every pair. You do not need the axiom of choice for this - you just invent a new rule "Always pick the left shoe." Because you (the mathematician) are able to physically define a rule that does the
choosing, you have provided a
constructive proof. You did the choosing. The math didn't need to do anything magical.
Now imagine instead of being asked for exactly one shoe from every pair you are asked to create a set containing one sock for every pair of an inifinite number of pairs of
identical socks. Now you are screwed. There is no left or right sock. You cannot write a rule like pick the heavier one or pick the left one. Because the set is infinite, you cannot go through and pick them by hand one by hand. You are entirely powerless to make this choice. You cannot construct a rule.
THIS is where the axiom of choice steps in. It's a mathematical decree that states: "Even though it is impossible for a rule to be written, and impossible for a human to do the picking, a choice function still logically exists in the abstract." The AoC is a non-constructive proof.
Q.E.D.: you have it entirely backward. The AoC doesn't give you, the mathematician, the ability to choose. It explicitly steps in when the mathematician
lacks the ability to choose. It is a declaration that a mathematical structure exists independently of any human cognitive agent's ability to construct it.
You trying to explore AoC to prove a human brain can make a choice without a rule fails - because AoC literally represents a choice that no human COULD ever make or define. It is the ultimate proof that set theory is completely divorced from human agency and physical causality, and thus not useful in our discussion.
There is a simple rebuttal to this - if an axiom were proven to be true or false, it would no longer be an axiom, it would just be a proven theorem.
Sure axioms have sometimes become theorems, but this is Motte and Bailey fallacy. This was not your original position. You originally challenged me: "if you were able to prove it (Axiom of Choice) false, that would probably show very strong support for determinism". I've hopefully explained by now that what you're asking is nonsensical in the context of ZF theory. The Axiom of Choice will never become a theorem.
Very well - a man is out for a walk. He encounters a fork in the road. He turns left instead of right.
Like you say below, this is begging the question.
I don't know that I can provide a better description of an uncaused choice without first having proof that parts of the universe are not deterministic. Obviously what we would like is to examine a choice and from its consequences be able to eliminate all deterministic factors that went into making it, showing that both the choice has occurred and there were 0 deterministic factors in making it, therefore "a non-causal choice has been made". Frankly I don't know how to do that though, given that the list of prior deterministic factors is practically infinite...
Sounds to me what you're saying is "I can't prove the choice was uncaused, because there are too many causes." It's god of the gaps/argument from ignorance. You're trying to sift through an impossibly large sea of deterministic factors to find this hypothetical uncaused event. Occams razor - there is no uncaused event.
This leads me to ask, can we choose a choice to examine with a small/known/finite number of deterministic factors - but that seems like begging the question to me.
The irony of what you're asking is once we do this, create some super-specific artificial simple scenario - the determinism will just become incredibly obvious. Strip way all the confusing complexity of the human brain until you're left with let's say a thermostat, that 'chooses' to turn on the heat based on a very strictly finite number of deterministic factors (the ambient room temperature). It makes the choice, but no one would argue the thermostat has free will.
Human beings are just biological thermostats with a bazillion other functions. The only reason we feel like we have free will is because the math is too complicated for us to calculate in real-time.
So whether the list of causes if ininfitely large (the man at the crossroads) or your artificially finite scenario (the thermostat) the mechanics are exactly the same. The choice is always caused.
2. Perhaps this is overly pedantic, but for me at least, the important part of "free will" is "free" - that is the choices are made solely from the internal state of some unrestricted essence of self. This so-called self must be capable of making choices which contradict the influence of external factors. I would call this free will, if the internal essence was "determined" independently of the external environment. This of course, just brings us back to the question "can there be a choice made by an entity which is independent of external factors"
You're forgetting that determinism is not just external forces (like gun to your head, the weather). The internal 'essence' is also caused by deterministic causal chains. If your 'internal essence' dictates that you will choose A over B, then your choice was 100% determined by your essence. Did the self choose its own 'internal essence'? No. How can you choose your own essence before you have an essence to make choices with?? As Schopenhauer said, "A man can do what he wills, but he cannot will what he wills."
So our choices will fall into at least two categories - lets say "figure-out-able" choices and "not-figure-out-able" choices. What is the practical difference between having "not-figure-out-able" choices, and modelling part of the world as non-deterministic?
Here's the practical difference.
Let's look at a complex chaotic system: the weather.
It's a strictly deterministic system, however, because there are trillions of variables, predicting the weather perfectly a month in advance is entirely "not-figure-out-able".
So what's the practical difference between modelling the weather as deterministic vs non-deterministic?
If we model it is non-deterministic (the gods are angry, the clouds have free will) we stop trying to understand it. We do rain dances and just hope for the best.
If we model it as deterministic, we accept that it is caused by physics. Even if we can't trace the chain perfectly back to the beginning of time, we build Doppler radars, launch satellites, and trace the most immediate local causes to get highly accurate 7 day forecasts.
The practical difference is we replace judgment with empathy (They made a bad choice - vs - what environmental or biological inputs caused this behaviour and how can we prevent it in the future)
The practical difference is we replace arrogance with gratitude. (I authored my own greatness - vs - I am the lucky recipient of a causal chain)
Deterministic worldview forces a troubleshooting mentality on a societal level. I genuinely think the world would be massively more moral place if everyone accepted it.
if you have a deterministic worldview, what do you do when you encounter these "not-figure-out-able" choices?
Couple of options.
1 - just admit it. If we can't figure out the exact causal chain, we just we can't figure out the exact causal chain. We don't need to invent some magic uncaused ghost to fill the gap. Just accept with humility our own epistemic limitations
2 - inventions / technological progress - just because we cannot figure out the exact causal chain today, does not mean we can never figure it out. god of the gaps, as our knowledge grows, our technology develops, the gaps get smaller.
3 - probability and pattern recognition type calculations. I touched on this above with the weather analogy. If the precise perfect truth of a causal chain isn't possible, we can still use certain probabliatistic and pattern matching techniques to get closer to the truth than simply throwing up our hands and saying zeus did it. Clear cut physics example: we can't fully map out every possible trajcetory of every air molecule hitting a plane wing, but we know enough to shape a piece of metal that will reliably produce lift and keep 300 ppl safe in the air. Human behaviour example: when pt flips out at my hospital in mental health ward, wats going on in their brain is a black box, but we know enough about the employment of certain techniques, dimming lights, speaking in calm tone, removing the thing/person that set them off in the first place - to reliably de-escalate
if you want to give some precise examples of non-figure-out-able choices, instead of generic hypotheticals, we culd discuss that it may be interesting