What is metaphysics and why is it important?

Is the difference among QM interpretations related to disagreement about what probability means? Are the two main interpretations that probability is either a measure of reality or a measure of one's knowledge of reality?

I don't think so. Every interpretation of probability is compatible with Kolmogorov's formalism (the current mainstream mathematics of probability). I don't think we can say the same thing about the various interpretations of QM. They are not all compatible with Kolmogorov.

Great topic. I understand OP is looking for something beyond the basic definitions but it might be worth having them in front of us anyway - see below. I'm wondering what the OP is looking for in the word "useful". Useful in guiding science toward better theories? What does OP think of the new stochastic process model of quantum theory by Jacob Barandes? Is it metaphysics when he discards the Markov assumption? A "sins of the fathers" reality. What happens on the journey really does matter at the destination. Spooky action from the past. See videos at bottom, both a short intro and full interview.

Metaphysics:

Google:
"the branch of philosophy that deals with the first principles of things, including abstract concepts such as being, knowing, substance, cause, identity, time, and space."

Wiki:
"Metaphysics is the branch of philosophy that examines the basic structure of reality. It is traditionally seen as the study of mind-independent features of the world, but some theorists view it as an inquiry into the conceptual framework of human understanding."
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This Simple Change Makes Quantum Theory (Finally) Make Sense

The Physicist Who Found Quantum Theory's Unnoticed Assumption

PairTheBoard

Everything is pure math. Or at least it could if you apply symbols to reality. I'm stuck on that intuition statement. I'm imaging the game Go and what's happening when the best human player is
up against alphaGo. The human isn't thinking in equations, sure. Neither is the model.
https://www.youtube.com/watch?v=E-v6Y4Gc...

"The mathematics of QM is physics" makes no sense.

Competing interpretations of QM arise because of bias. The hard materialist leans towards many-worlds, "everything is real". It conveniently fits the goldilocks universe we are in, etc. Mathematically and experimentally there is nothing against it. But it is not required. Copenhagen is consistent with deeper much older philosophy, physics should be consistent with well developed philosophy.

Meta-physics is above physics, above nature, i.e. supernatural. If you baulk at that then have super-sensible, the transcendent. That which cannot be reconciled exclusively by the senses, separate from the phemomenal plane. Knowledge that is impossible to attain sensibly. That's quite an important discussion i think.

It would not seem as important if i had just come off a 16 hour shift and had to go back to work in 8 hours, and i was say 10 years old. There are other more important things to address.

The formalism is not flouted. It is simply ignored. Non-mathematicians often ignore higher mathematical formalism. I would guess that many physicists do not even realize that the "probability" they are doing is not compatible with mainstream probability theory.

The short explanation for the incompatibility is this. If one assumes that the universe is "local" and one interprets this locality as probabilistic independence, then there are experiments that cannot be formalized within probability theory. I don't know how simple it is, but John Bell gave an example of this involving spin measurements on three particles. See this article.

I watched the video you linked to. I also skimmed the paper,

https://arxiv.org/abs/2302.10778

arXiv.org

The Stochastic-Quantum Correspondence

Now, it seems plausible that Jacob, at this point in his career, knows the definition of a Markov process. But his paper is about something other than that. The language he uses makes it sound like it's probability. But the mathematics reveals that it is not. Or at the very least, it does not show that it is.

Quantum theory is not formulated using probability theory. It uses Hilbert spaces and operators. So there are no stochastic processes. Which means there is no Markov property. You cannot discard something that is not there. It feels like he has appropriated the language of probability theory and made a word salad out of it.

Jacob gives no rigorous formulation of his so-called processes. Is a rigorous formulation even possible? It could be. Jacob might have omitted it because he wanted to reach a larger audience. Bayes rule, though, would suggest that this is unlikely. If he had a rigorous formulation, he would have at least mentioned it. No, the most likely explanation is that Jacob is not using actual probability theory. He is not using actual stochastic processes. He is only using the language of our discipline to describe his work. Nonetheless, I find it interesting and it is definitely worth further investigation.

Go was the inspiration behind Conway's invention of the surreal numbers. (A good reference for Go on this is Mathematical Endgames, by Berkelamp and Wolf). https://en.m.wikipedia.org/wiki/Surreal_...

So whilst the players may not realise it, one can view the game in terms of surreal numbers and the stronger player is better at assigning themselves the higher number. It strikes me this may be the metaphysics of Go. Calculating the appropriate surreal number for a typical middle game is too non-trivial to be practical.

Bohm without the physical pilot wave?

Friggin realists. Always trying to put light back in the dark. Visual consciousness is the luminous ether.

For a realist, the world is pitch black, with light meaning e/m fields in space. So if a tree falls in a mind-independent world…

Realists can’t even claim there’s a tree, only what they’d call a tree-zone. Luminosity is what draws the demarcation line in a continuum of radiation. i.e., there are no particles without a luminescent mind to perceive them.

Same with the double slit. For a realist, electrons move through slits in a dark void their interference just probability waves over e/m fields. But realists can’t even claim there are “slits” but only "slit-zones." Luminosity is what carves them out and makes the pattern appear.

Same with Bell. For a realist, particles separated across space somehow 'signal' to each other in the dark. But realists can’t even claim there are two particles, only two correlated zones in a continuum. Luminosity is what makes them appear separate and what makes their correlation show up at all.

Even entropy. For a realist, the second law is just counting microstates in the dark. But realists can’t even claim there are “states” without luminosity only undifferentiated flux. It’s appearance that gives us distinct configurations and the arrow of time.

For me, that's the kind of stuff metaphysics is for.

"Physics is a kind of Metaphysics"
- Einstein

PairTheBoard

Simplest answer I can think of.

Physics asks "What are quarks?"
Metaphysics asks "Why are quarks?"

Examples:
Materialist metaphysics effectively claims "Because there's nothing else than quarks. The physical world is the ground and only state of reality."
Christian/certain deist metaphysics claims "Because God created the world; a higher and ultimate reality exists above us."
Simulation theory metaphysics claims "Because we're inside a computer and nothing is real; a higher (possibly not ultimate) reality exists above us."

There are probably thousands of distinct views.

These are all ultra-dumbed down one-line summaries but they give you a feel for it. For what it's worth, I'm a Catholic so I align with the metaphysics appropriate to that belief. I think God created the world.

Are metaphysical claims falsifiable? I am inclined to favor a definition of metaphysics that says no. That is, if a claim is provable or falsifiable (using empirical observation and logic), then it is, by definition, not a metaphysical claim.

I would say pretty much all metaphysical claims are unfalsifiable, though that's not the same as "they have no supporting evidence".

Cogito ergo sum is a great example of this, and I would say a kind of metaphysical claim.

"I think, therefore I am."

To yourself, it is self-evident that you exist.
But you are unable to prove to others that you do or do not exist.

You can pinch them as hard as you like; from the perspective of others, solipsism is always a valid retreat. They might be hallucinating you. You could be an NPC in the simulation. Or, you might actually be real. This is widely regarded as the most likely answer, if only because it's more convenient than assuming you're the only real person and going all Grand Theft Auto on the world.

Same goes for Christianity - there's a great deal of supporting, historical evidence regarding the life of Jesus Christ. But for me, it was a personal experience (much like cogito ergo sum is) that got me to believe that Christianity is true. I can't perfectly prove it to you but I can make a case.

Or take simulation theory - imagine one day you find a giant floating error box in the sky. "404, sky not found".
That would be evidence for simulation theory, but it's not conclusive - the true metaphysics could be that you're in a kind of hell, and a malevolent, higher being has constructed your world and is tormenting you with such confusions.

I asked ChatGPT if mathematical axioms are metaphysical. It said only if you're a Platonist.

Is there a reasonable argument against?

I suppose it depends on our definition of "evidence." But I am inclined to disagree. I'd say that metaphysical claims cannot have supporting evidence either.

Let us say that metaphysical claims are neither provable nor falsifiable. That is, we cannot determine their truth values using empirical observation and logic. There is a chasm of unknowability between the empirical world and these claims.

Your reply is that they are unfalsifiable, but have supporting evidence. The "but" clause suggests that the evidence mitigates the unfalsifiability. It somehow makes a partial bridge across the chasm. But for this to be the case, the supporting evidence must be empirical and/or logical. This is what I disagree with. Metaphysical claims cannot have supporting evidence of this type.

Let us use your example. I claim that you are not a philosophical zombie. Let us agree that this is a metaphysical claim. What empirical and/or logical evidence could support this claim? Granted, I may have philosophical reasons for accepting the claim. For instance, I could appeal to Occam's razor. Occam's razor, then, provides supporting evidence for the claim. But this is not the result of pure logical deduction on empirical observations. It is, for lack of a better term, "metaphysical evidence." Are we simply saying that metaphysical claims can have metaphysical evidence? If so, I can agree with that.

Physics explains how things behave.

Metaphysics asks what things are?

This may be of interest.

https://en.wikipedia.org/wiki/Identity_o...
Identity of indiscernibles

The identity of indiscernibles is an ontological principle that states that there cannot be separate objects or entities that have all their properties in common. That is, entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa. It states that no two distinct things (such as snowflakes) can be exactly alike, but this is intended as a metaphysical principle rather than one of natural science. A related principle is the indiscernibility of identicals, discussed below.

A form of the principle is attributed to the German philosopher Gottfried Wilhelm Leibniz. While some think that Leibniz's version of the principle is meant to be only the indiscernibility of identicals, others have interpreted it as the conjunction of the identity of indiscernibles and the indiscernibility of identicals (the converse principle). Because of its association with Leibniz, the indiscernibility of identicals is sometimes known as Leibniz's law. It is considered to be one of his great metaphysical principles, the other being the principle of noncontradiction and the principle of sufficient reason (famously used in his disputes with Newton and Clarke in the Leibniz–Clarke correspondence).
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https://en.wikipedia.org/wiki/Quantum_co...
Quantum contextuality

Quantum contextuality is a feature of the phenomenology of quantum mechanics whereby measurements of quantum observables cannot simply be thought of as revealing pre-existing values. Any attempt to do so in a realistic hidden-variable theory leads to values that are dependent upon the choice of the other (compatible) observables which are simultaneously measured (the measurement context). More formally, the measurement result (assumed pre-existing) of a quantum observable is dependent upon which other commuting observables are within the same measurement set.

Contextuality was first demonstrated to be a feature of quantum phenomenology by the Bell–Kochen–Specker theorem.[1][2] The study of contextuality has developed into a major topic of interest in quantum foundations as the phenomenon crystallises certain non-classical and counter-intuitive aspects of quantum theory. A number of powerful mathematical frameworks have been developed to study and better understand contextuality, from the perspective of sheaf theory,[3] graph theory,[4] hypergraphs,[5] algebraic topology,[6] and probabilistic couplings.[7]

Nonlocality, in the sense of Bell's theorem, may be viewed as a special case of the more general phenomenon of contextuality, in which measurement contexts contain measurements that are distributed over spacelike separated regions. This follows from Fine's theorem.[8][3]

Quantum contextuality has been identified as a source of quantum computational speedups and quantum advantage in quantum computing.[9][10][11][12] Contemporary research has increasingly focused on exploring its utility as a computational resource.
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PairTheBoard

I've added it to my watchlist, thanks. From the wiki quotes, I find it interesting that nonlocality is declared to be a kind of contextuality but, presumably, locality is not. I could be very easily mistaken, obviously, but I smell a potential True Scotsman lurking somewhere.