Using personal preference or personal intuitions as priors instead of some objective measure along the lines of Solomonoff Induction

You can't do Solomonoff induction in real life. That by itself seems to be a good reason to look for something else.

As to objective/subjective priors, note that the classic Bayesian understanding of probability is as "subjective belief" which is what drives frequentists hopping mad. If you accept this concept of probability then there shouldn't be any problem with using subjective (="personal") priors.

Not looking at the world in a probabilistic way

Philosophy has long had the hope that eventually, somehow, it would find a set of elegant axioms from which the rest would regrow, like what happened in math. Several branches of philosophy think they did collapse it to a set of elegant axioms (though upon inspection, they actually let the complexity leak back in elsewhere). I think there's a fear, not entirely unjustified, that if you let probabilistic reasoning into two many places then this closes off the possibility of reaching an axiomatization or of ever reaching firm conclusions about interesting questions. Today, it's been long enough to know that the quest for axiomatization was doomed from the start - or at least, the quest for an axiomatization that wasn't itself a probabilistic thing. So allowing probabilistic reasoning shouldn't seem like a big scary concession anymore, but on the other hand, it's still difficult and most philosophers aren't dual-classed into maths.

Using personal preference or personal intuitions as priors instead of some objective measure along the lines of Solomonoff Induction

Unfortunately, Solomonoff Induction falls off the table as soon as the questions get interesting. As a next-best-thing, intuition is not all that bad. I'd criticize a lot of philosophy, not for grounding ideas in intuition, but for treating intuition as a black box rather than as something which can be studied and debugged and improved. Most LW-style philosophy does bottom out at intuition somewhere, it just does a better-than-usual job of patching over intuition's weaknesses.

Moral realism

When you're getting started on learning game theory, there is a point where it looks like it might be building towards an elegant theory of morality, something that would reproduce our moral intuitions and being a great Schelling point and ground morality really well. Then it runs into roadblocks and doesn't get there, so we're stuck with a hodgepodge metaethics where morality depends on an aggregation of many peoples' preferences but there are different ways to aggregate one persons' preferences and different ways to aggregate groups' preferences and some preferences don't count and it's all very unsatisfying. But if you haven't hit that wall yet or you're very optimistic or you're limiting yourself to sufficiently simple trolley problems, then moral realism seems like a thing.

Mathematical Platonism

This is a trap door into silly arguments about subtleties of the word "exist" which are cleanly and completely separated from all predictions. But if you want to engage with ideas like a mathematical multiverse, you do end up needing to think about subtleties of the word "exist", and math ends up looking more fundamental than physics.

Libertarian free will (I'm looking for arguments other than those from religion)

I'm not sure what libertarian free will is in relation to the rest of the ideas about free will, but I find thinking about free will gets a lot easier if you first acknowledge that our intuitions are guided by the idea of ordinary freedom (ie, whether there's a human around with a whip), and then go a step further and just think about ordinary freedom instead.

The view that there actually exist abstract "tables" and "chairs" and not just particles arranged into those forms

These ideas come back in slightly different forms when you start considering mathematical multiverses and low-fidelity simulations of the universe. For example, if you accept the simulation argument, and further suppose that the simulation would not be full-fidelity but would be designed to make this fact hard to notice, then you get the conclusion that certain abstract objects exist and their constituent particles don't.

The existence of non-physical minds (I'm looking for arguments other than the argument from the Hard Problem of Consciousness)

The idea of minds as cognitive algorithms leads to something sort-of like this; in that framing, minds are physical objects with a dual existence in platonic math-realm that diverges if physics causes a deviation from the algorithm.

Listen, what does "physical" even mean. Apparently, it's all made up of weird non-classical probabilities underneath. It doesn't seem like a very solid foundation to build a rock out of, and yet here we are.

My reading suggests the issue is this: You have some set of axioms, which you seem to be conflating to some extent with priors, because you hold the precepts of Bayesian reasoning as direct axioms, and those axioms are the logical constructs in which you run your brain. Other people have different axioms. You're asking for a set of logical steps, using your axiom set, by which you can "prove" their axioms, so that you can run theirs as well.

The idea of holding different axioms sounds insane to you, and you have trouble articulating why. It's because they're different axioms than those you think in. It sounds insane to you because it is; any attempt you make at it is going to be like somebody who only speaks English trying to imagine what it feels like to think in Japanese. You don't "think" in that language, and until you can, you're only aping it, using unintelligible nonsense where logic should be.

Is this a fair assessment?

Mathematical Platonism

Jimrandomh has addressed this well. I'll add that if you feel as confident of reductionism as the last two items on your list would suggest, it seems odd to reject out of hand the idea that we could reduce everything to math. (I don't believe in any version of Tegmark IV, but I'm open to arguments.)

For all I know, many who claim to believe in Platonism have something like this in mind, and it just isn't clear to you because philosophical terminology is not optimized for your understanding (or, indeed, for any individual's goals).

I intuitively feel that there really are objective morals (or: objective mathematics, actual free will, tables and chairs, minds).

Therefore, there really are objective morals (etc.).

"Morals" is just a word. But unlike some other words, it's not 100% clear to me what it means. There is no physical entity that "morals" clearly refers to. There is no agreed upon list of axioms that define what "morals" is. That's why, to me, "there are objective morals" doesn't feel entirely like a factual statement.

I might justify that there are objective morals by relying on my intuition. But that's not because I think intuitions are reliable sources of knowledge. That's because I think intuitions are the correct normative source of how we use words (together with common usage, I guess).

It's till possible that my intuitions contradict each other, or that they contradict facts. So they are not sufficient to say with confidence that objective morals exist. But they are relevant.

The existence of non-physical minds (I'm looking for arguments other than the argument from the Hard Problem of Consciousness)

There's actually some medical evidence for this, but it's considered to be outweighed by evidence for mental materialism (eg lesions, Phineas Gage). There are case studies of people diagnosed with hydrocephalus as adults, who had something like 25% of the brain mass of the average adult, and yet this didn't seem to affect their mental abilities at all. I've heard of people citing this as evidence of mental tasks being performed by an invisible soul, not by neurons.

EDIT: (In case you want more info, here's the blog post where I heard about this. This is a personal blog I follow, but it links to some more official articles: http://www.rifters.com/crawl/?p=6116)

"well, you're also ultimately basing yourself on intuitions for things like logic, existence of mind-independent objects, Occamian priors, and all the other viewpoints that you view as intuitively plausible, so I can jolly well use whatever intuitions I feel like too."

It's true that a priori using intuition is about as good as using an intuitive tool like inductive reasoning. However, induction has a very very strong track record. The entire history of science is one of humans starting out with certain intuitive priors, and huge numbers of them being challenged by experimental evidence. Babies learn about the world using induction, so each one of us has a mountain of evidence supporting its usefulness. Once you perform a Baysian update on that evidence, intuition looks like a much less useful predictor of future events.

Sure, you can deny the validity of induction. You can also claim that "I think therefor I am" is invalid because it isn't based on anything. (What is this "therefore" concept? What is "I"? What does it mean to "exist"?) You can even decide when to use induction versus intuition based on whim or based on which conclusion you want to “prove”. It’s just that doing so is incoherent.

Yes, I did just use an informal form of inductive reasoning, by using observations of evidence to demonstrate that induction seems valid upon reflection. Yes, that makes me feel dirty inside. But you have to start somewhere, and we don't really have any better options. It should be noted that anyone who says "intuition is a better option" learned those 5 words through induction, and probably uses induction to override intuitions every day. It is mathematically provable that no mathematical system can assert its own soundness without becoming inconsistent. We can't prove the validity of induction using induction. The best we can do is try to falsify the hypothesis "induction is valid". If looking at the track record of induction seemed to indicate that it wasn't valid, then we'd have an even worse mess on our hands, but fortunately that isn't the case. If someone can come up with a better alternative to induction, they'd better be able to demonstrate that it is better.

Some would call that faith. I'd counter that faith is belief regardless of evidence, and this is forming beliefs based on all available evidence. If that's faith, then so is every belief about anything.

"What could possibly be your motivation for sticking to your intuitions anyway?"

I have a theory which explains why you believe there was a past. It is that the world was created five minutes ago with false memories of a historical past. This theory shows why you would feel the way you do even if it didn't reflect the truth. So what could possibly be your motivation for sticking to your intuitions anyway?

Because it is also possible that your intuitions are right, even if there are theories that could explain them even if they were wrong.