How do you become an Atheist?

[Anomalocaris]

Also, science (probably) can't establish values, so philosophical theories of value (moral and common) are likely to be irreplaceable by any empirical methods, you can determine the relationships between sets of values in competition a priori.
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But if there is no free will, then at least in principle an advanced version of neuroscience that incorporate models of the effects of the interaction between changes in neurological state and the physiology and physical environment can establish a priori how the philosophical theories of value will be formulated and how the competition amongst the values will be resolved. So the philosophy of value is illusory to exactly the same degree as free will is illusory. If will is in principle externally determinable by empirical means then values too can be determined the same way.

The whole concept of a non-contingent and necessary thing is that it isn't dependent on perception and cognition, so improving our understanding of those things won't help in the slightest. They aren't in any way 'artifacts' of somewhat flawed cognition either, we could be cognitively perfect (whatever that would mean) and what is necessary is still necessary and what is not dependent on physical things is still not dependent on physical things. We can possibly improve the methods by which we arrive at these necessary truths, but to say that they will 'lose their appeal' is like saying the law of non-contradiction will lose it's appeal. Really, it's not going to happen.

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My approach to this issue is based not on concept of non-contingent truth but on neurological effect of the grasp of the non-contingent truth. With sufficiently detailed understanding of how neurological artifacts influence interaction with the physical body and between the physical body and the environment, one could at least in principle establish, based on the interaction of their respective neurological effects and physiological and physical environment, which version of necessary truth as comprehended by the human mind would statistically be most unlikely to be overturned by subsequent philosophizing, or will make the most headway towards the objectives for which these concepts are toyed with in the first place.