Central to his main question is the idea that questions that depend on moral determinations of judgement to be answered are by definition "moral" ones and in so being are out of the analysis scope of being probed or described by the methods of Science. I makes this statement plain when he writes:
"I claim that whether you want something to happen or not – what value there is in the result, and how you judge the value of the result (which is the other end of the question: Should I do this?) – must lie outside of science because it is not a question that you can answer only by knowing what happens; you still have to judge what happens – in a moral way. So, for this theoretical reason I think that there is a complete consistency between the moral view – or the ethical aspect of religion – and scientific information." -- The Relation of Science and Religion
This is what I disagree with and I find interesting that Feynman held this view (at least at this time in his life, he lived on for another 3 decades so it is possible he changed or refined his view I've not investigated that fact as of yet) given that he knew some of the people that worked on ways inadvertently, that science and mathematics specifically indeed does explain moral questions. In particular, I refer to the game theorists...people like Von Neumann and others that studied the economics of resource based decisions in the creation of the new branch of mathematics and economics dubbed game theory starting with papers in the 30's.
Judgement, moral or otherwise is not done in a vacuum, we judge based on our own visceral metrics for value on the concepts under judgement. We assign these values by our innate emotional connection to the outcome, if it is something we are apathetic to we disregard it. You don't spend your days wondering how badly a rock on the beach in Peru is being weathered, it is being weathered but it is of no visceral import to you. Moral judgements emerge from visceral imports..or things we can associate with as being personally and physically important to our survival or by proxy of empathy to the survival of other personable agents at some time.
As those determinations are encoded in our brain body association between autonomic/emotional and higher cognitive sensory experience they are, if distally modulated by Scientifically describable relationships. A definition of morality, are the social rules of behavior that emerge when individuals bound by the resource constraints of a given environment become known to them.
I assert moral conflicts do not exist where the dual characteristics of environmental constraint and resource constraint are not present. In either case, the formation of those rules of behavior emerge from the mathematical realities of the constraints (resource and environment) as modulated by the individual autonomic and emotional internal modulating factors that lead individuals in the groups to "judge"; emerging a possible moral tapestry from which judgements are made.
The idea of mutual cooperative altruism among social animals is intimately tied to this idea of moral emergence. We cooperate because the value for doing so exceeds the drawback of not doing so, a thought experiment reveals that there must be an association to the presence of limited resources within the orthogonal constraint of a limited environment of interaction. All we need do is change the examined group size of individuals in the environment.
If a particular resource or resources is required for our survival and we are the only agent in the environment that can gain value from the resource we have no inherent desire to be altruistic, this may seem trivial but it is not...our altruism is not an absolute characteristic of our make up but rather an emergent quality of moral expression that ONLY becomes extant when others that also value that resource are present and subject to the same constraints.
If within the experiment the other individuals are able to range away from our resource cache to find their own there is a desire to do so rather than attempt to engage in moral interaction with others. This is because moral interactions for resources have more stakes than just the securing of the value of the resource they also include the retention of the value of ones life when those resources are viscerally important as defined previously.
If however the resource restriction is such that despite the open nature of the environment (ability for other individuals seeking that resource to go find it else where) the resource is only found in one area, then again moral conflict and the need to engage in altruistic negotiations presents but again ONLY because the group size of the agents involved has gone from n=1 to n=k where k is any number above 1.
Thus if moral decisions can be so readily modulated by a mathematical concept as fundamental as group size and since the restrictions of resource and environment modulate those decisions, the emergent web of possible options all mathematically described are entirely within the description, if not in the absolute than in the probable of Scientific methods.
As aside, these issues are not ones that only humans possess, all animals that interact in groups either temporarily or continually via social interaction must be constrained by these realities.
Apparent difficulties in doing (describing the "moral" by Science) so are only failings of applying sufficient resolution to the problem of describing the dynamics of group size, visceral resource requirement within the individuals in the group, availability of resources in the local environment and finally openness of the environment to exploration for additional caches of desired resources. They make identification of the choices that emerge in the altruistic and moral interactions that result when those attributes are variably restricted more difficult but they remain solidly within the realm of Scientific questions even when the judgements being made seem to slip beyond the empirically testable and quantifiable realm.
So that settled what of Feynman's question? He put it as:
"I don't know the answer to this central problem – the problem of maintaining the real value of religion, as a source of strength and of courage to most men, while, at the same time, not requiring an absolute faith in the metaphysical aspects." -- The Relation of Science and Religion
Given the results of the previous exposition the answer is it doesn't matter, the dynamism of possibilities involve all manner of personal explorations of the interactions. That some people can come to maintain high value derived from their religious views while simultaneously holding scientific views should not be surprising as we shouldn't be examining the problem as a static entity. People's views evolve over time, their moral fabric (as proven clearly in the thought experiment above) appears and disappears, expands and contracts in quantifiable if numerically exhaustive variation but the nuance of particular views based on modulation of the recognized factors of group size, resource need, resource availability and environmental openness enable parallel existence of seemingly contradictory positions realized as these needs shift.