I believe I have a good answer to this question, and would love to get feedback on my answer as well as see other people's answers. Here is my answer:
The following is my sketch of a proof of why 1) theoretical philosophy/spirituality/religion and 2) science are not only compatible, but how both are actually necessary if you seek to understand universal truth.
When I say seek universal truth, what I mean is to better understand the world we live in and how it works -- the reality we observe... Throughout our lives we have had a single person's perspective on the world we live in, and we've seen through the lenses of other people who see the world slightly or very differently than us... My belief is that there is an objective reality out there that each of us is observing, and that if we strive to understand it, describe it, and know how it works, that we're doing our best to understand the world we were born into and how to live best within it.
All the great religions have lessons on how to understand universal truth, they also have major contradictions with themselves and each other. Most of them say if you don't believe in their flavor of religion then you will be eternally damned, but by choosing any one religion all the others damn you. To me none of those answers sounds very good to me, because I'm just betting on the fact that I happened to have chosen the right religion. Instead I trust in myself to listen to the wisdom from all religions, spiritualities, and laypeople, and decide for myself what I believe to be the wisest way to live life based on the great moral and physical lessons learned from religion science and spirituality alike.
First, some definitions:
Science - Science is defined as a knowledge base of assumptions and logical rules of inference along with experiments that validate these assumptions and rules of inference.
The assumptions are so basic we assume them to be true (for example, mass energy either exists or does not exist at a point in time in 3-dimensional space).
The rules of inference are provable ways to get from A to B consistently
Experiments are conducted using the scientific method and validate/refute hypothesis made from assumptions and rules of inference to consistently expand knowledge and validate that predictions of scientific theories are accurate.
Theoretical Philosophy - This is philosophy that uses reason and rational thought to make conjectures about things that are not provable / verifiable through Science.
Spirituality / Religion - This is the belief in a knowledge base which is inconsistent but complete -- Eg it tries to answer all questions, but does not have consistency in its answers… Meaning there can be contradictions found from the knowledge base. It has an answer to all questions, but sometimes can be wrong.
Accepted science - This is scientific hypotheses that have been validated through experimentation and commonly accepted as scientific fact. For example, that the world is not flat.
Gödel’s incompleteness theorem states that any knowledge base is either incomplete and consistent or complete and inconsistent. I’ll paste a little more detail from another paper to support this claim:
Gödel's Incompleteness Theorem Overview
Godel’s theorem proves that logic can’t be both consistent and accurate for all truths
While many are familiar with the theorem, I will provide a brief overview for those that are not. I will not re-prove the theorem here((, but will leave it to readers to convince themselves – I encourage them to do so as when I first read the theorem I didn’t believe it, until I read the proof several times and thought through it carefully.))
Gödel’s incompleteness theorem refers to any logical system ((you can think of it as the most powerful supercomputer in the world that understands the world as perfectly as is possible… which Godel proved is imperfectly)), which I term here a “black box” which proves things true or false. The “Perfect” black box can tell you exactly what is true or false, see appendix item 1.
However Gödel’s Incompleteness Theorem proves that the “perfect” black box doesn’t exist: instead if you are using a black box to prove things true and false, then you will be in one of two situations:
Everything you prove true is true, however there are some things that are true that your black box will say are false (consistent but incomplete – See appendix item 2)
You can prove true everything that is actually true, however there are some things that you prove true that are actually false (complete but inaccurate – See appendix item 3)
Said another way, no black box can be both complete and accurate. Gödel’s incompleteness theorem proves that any black box will either be consistent but incomplete, or complete but inaccurate.
((While this has many broad implications, I will dig into just one for the purpose of this paper))
Note that the black box I will be referring to is the scientific method – Formal logic leveraging postulates and rules of inference to prove things about the world that are true or false. ((may need to add detail about the fact that the postulates and rules of inference proving things true assumes that the laws of physics remain relatively consistent or constant throughout time so we can actually learn something about them.))
Where does science fit in?
Science is consistent, by nature… Therefore science is also incomplete… There are things that are true that science can’t prove true. Scientific knowledge is limited today, but even taking into account all that science can ever learn, there are things that science will never be able to prove true, that are in fact true. This shows that science, while it can explain MANY things and explain them consistently and usefully, is limited in what it can prove… this leaves a gap for how you answer the questions that science not only can’t answer now, but that science can never answer. Such questions are things like “What’s the meaning of life?” and “What is moral?”
That said, accepted science’s scope is limited now and can grow much vaster. The discipline of making up theories about how science can grow can be termed a part of scientific discovery itself, or we can also call it Scientific Philosophy, in that its philosophy that aims to expand the knowledge base of current scientific knowledge with theories that are either provable or disprovable.
It may be worth noting that the difference between something science could eventually be able to prove true and something it can’t, in general, is that something science can prove to be true is something that is provable through experimentation. That is, one can theorize a result of an experiment, perform the experiment, and see if the results are consistent with the theory. There is no experiment one can come up with to prove a universal “meaning of life” and therefore this question is one for theoretical philosophy, spirituality, or religion.
Where do theoretical philosophy, spirituality, and religion fit in?
Theoretical philosophy, spirituality, and religion are inconsistent… Even the most stalwart zealots of these disciplines will admit that beliefs can be inconsistent depending on how people interpret them. They’ll say there’s a right and wrong way and theirs is right and others are wrong but that itself admits it’s inconsistent (since they have interpreted their belief system differently than other people who share their belief system). While these disciplines are inconsistent, they can be complete (they can prove true everything that is true) in that it tries to explain everything, including things science can’t yet prove, and never will be able to prove.
The right path forward
Therefore the safest path (and the one closest to universal truth) is to use accepted whenever you believe that science is consistent (accepted science at a minimum), and use theoretical philosophy, spirituality, and/or religion when science doesn’t have an explanation for the question you’re asking.
Science and theoretical philosophy/spirituality/religion are NOT at odds or uncomplimentary… In fact, leveraging both belief systems is the only provable way to ALWAYS be right… Though you have to keep in mind that even if you use both science and religion you will be right a much higher percentage of the time, but there’s no way to prove that you will always be right. And SO… we should recognize that the chances are good that we’re wrong about some things and therefore should be willing to accept that we may be wrong, whatever our beliefs are. A healthy skepticism for all beliefs, including one’s own, is the best way to grow and develop your knowledge of the world. Do so leveraging science when it has clear answers, and theoretical philosophy, spirituality, and religion when science doesn’t have clear answers.