1. I think that science (or more properly, the scientific method) is one specific, particular (and formalized) application of the broader concept/idea/philosophy/approach of rationality in general. I know you disagree, and I’d like to know why.
2. I think that both the world out there, as well as my mind, both of these comprise reality. That is why we have two modes, the subjective and the objective, of understanding reality (and also the in-between “inter-subjective”). I think you don’t quite agree with this either? I’d like to know more about why not.
1. first, in historical terms in western culture the idea of rationality goes back to the philosophers of the old Greeks. But there is a bias build into rationality, namely that you can explain reality using only reason and logic as opposed to emotions. Or in the weak sense that rationality is better than emotions.
So how is it, that I claim that general skepticism is not the same as science? Well, it ties into 2. Science or rather natural science, because in some other cultures science in the Anglo-Saxon sense is not science, science is the science of nature or natural science. You also have mathematics as a science, social science, human science (humanities) and philosophy. Another way to spot this is in the idea of a theory of everything. Reality is fundamentally physical and can be reduced to a physical theory of everything.
So the question is, can we explain reality using rationality and its formalized version natural science? Can we reduce reality down to physics using reason and logic? And the answer is neither yes nor no.

In philosophy there is a technical term: Necessary, but not sufficient. In the broad sense it means that e.g. gravity is necessary, but not sufficient in explaining reality. Even broader yet, natural science and rationality are necessary, but not sufficient in explaining reality. Now go back to the 5 versions of science and call them A to E. So can we add up A to E and explain reality using only reason, logic and evidence(natural science) and the answer is no.
In an everyday sense you are better off using reason, logic and evidence(natural science), because they are all necessary in understanding reality, but they are not sufficient. So how is that?
Well, it is contained in this quote by Protagoras:
"Man is the measure of all things: of the things that are, that they are, of the things that are not, that they are not."
We then add Nietzsche:
"Whoever fights monsters should see to it that in the process he does not become a monster. And if you gaze long enough into an abyss, the abyss will gaze back into you."
And finally Arthur Conan Doyle:
"Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth."
Protagoras is about cognitive relativism; i.e. how you make sense of reality is necessarily not the same as me.
Nietzsche in the broad sense is about morality and ethics (the monsters) and when you go deep and try to understand reality, cognitive dissonance (the abyss) is the result; i.e. reality does not add up, when only using rationality.
Doyle is a version of skepticism, i.e. when you realize that you can't use rationality alone, you realize that reality is not logical or governed by reason.
So how do I give evidence for that?
Well, it is simple. Remember subjective and objective. Now add a Boltzmann brain. It is the theory in theoretical physics that you could be in an universe, which only consists of your brain. So what is the probability of you being a Boltzmann brain versus you being in the universe you believe that you are in? It is unknown! How come?
Because the objective; i.e. the rest of the universe appear the same to you in both cases and in both cases are the cause of your mind. You are the effect of something else, the rest of the universe caused you to experience(your mind), but you can't know which universe you are in, because your experiences are the same in both universes. That is the limit of reason and logic. You can't using reason, logic and evidence to control the universe. The universe controls you in the objective sense and all you have is your subjective experience of it.
So the objective is unknown, but apparently the cause of your experiences. In the formal sense the objective is das Ding an sich(that is Immanuel Kant), i.e. the thing in itself independent of your experience of it. It is unknown what objective reality really is, other than it is independent of your experience of it, but the cause of your experience.
So now you have looked down the abyss long enough and the abyss has gazed back at you. Now take of a leap of faith and live the rest of your life in the belief that the rest of universe is as it appears to you in your experience of it. So now we are back in reality. But now we know, that there is a limit to rationality and evidence.
But what about God? Well, some humans are religious and others are not. But can't we decide that? No, because religion is all the way to the God of the gaps a part of das Ding an sich. Whether there is a God or not is as unknown as it is unknown whether you are a Boltzmann brain or not.
So enough about metaphysics, transcend beliefs and what reality really is.

For the rest of this text we are inside methodological naturalism; i.e. we treat reality is if it is as it appears in the mind and the objective matches the subjective experience of it.
Then we need ontology or rather the set of necessary, but not sufficient parts/explanations of how reality works in practice.
In practice in time and space for the arrow of time reality is connected as a strong philosophical principle; the physical gives rise to chemistry, which gives rise to biology, which gives rise to cognition in humans, which gives rise to philosophy, which gives rises methodological naturalism, which gives rise to the physical. Notice it is circular in one sense and dogmatic in another.
You don't question methodological naturalism, because that is what allows you to use reason, logic and evidence in practice. But you can't avoid emergent properties, which are not reducible to something else. I.e. the set of categories you use to describe reality include objective, inter-subjective and subjective terms, which are not reducible to each other. You need all 3 kinds and you hold them as interconnected, yet not reducible to only one category. In philosophical terms you describe reality as the human experience of the interconnected web of the objective, inter-subjective and subjective or if you like physical and mental terms or external sensational and intra-cognition as per epistemology.
So back to modern western natural science and rationality; i.e. the Enlightenment. You need a fundamental belief of what reality is (natural or super-natural; i.e.), you need to find your core values (morality, ethics, psychology/social science and aesthetics; i.e. subjective and inter-subjective) and you need to deal with reality is practical manner (rationality and natural science; i.e. objective). But you can't do it without all 3 if you look closer; i.e. skepticism.
So is reality in part objective? Yes, I believe so, but that belief is subjective. That is the joke or if you like the abyss.
I don't know what reality really is and I don't care, because it seems to work anyway.
With regards