And the trouble with the scientific method is that it's wedded to ceteris paribus, the "all things equal" assumption. (In philosophy, this is called the manipulationist theory of causality, e.g., [2].) This is the core reason that complex systems present such a problem for scientists---it is, if you will, the real reason economics is such a trainwreck. People want explanations to look like, "A causes B if, when you manipulate A, all other things equal, that drives changes in B." But if a system is fully contingent on itself, then that'll never work out.
That's the value of intuition: it's not that it's fast or feasible or individual, it's that the scientific method runs out of gas in highly-contingent situations. To put it in a computer science perspective, the clasically undecidable problems (the halting problem and the decision problem) will always stand in the way of robust automatic code generation, and yet we write code anyway. We can do that precisely because the formal methods we go on about don't describe how we actually are, and that's always going to make people both smarter and stupider than we seem.
[1] http://colinklein.org/papers/DualTrackWeb.pdf [2] https://plato.stanford.edu/entries/causation-mani/
Intuition is closer to recognition. Recognizing a person from the way they walk, recognizing a dangerous situation, recognizing a combination of clues whether to trust a person, recognizing a pattern to predict a future outcome.
Intuition is very powerful and obviously prone to bias[1], as predicted by the No Free Lunch theorem[2].
Kahneman describes rational thought as a crosscheck to intuition, and that seems highly reasonable. Emotion is a whole different category. You don't need an fMRI to experience these distinctions yourself. In fact I'm baffled if you don't.
[1] https://en.wikipedia.org/wiki/List_of_cognitive_biases
[2] https://en.wikipedia.org/wiki/No_free_lunch_theorem "if an algorithm performs well on a certain class of problems then it necessarily pays for that with degraded performance on the set of all remaining problems"
This is probably better seen from a couple other angles. Probably the most interesting thing, since you bring up cognitive biases, is that confirmation bias can be the only way we approach information. If not for confirmation bias, we'd be incapable of talking about "signal" and "noise," or else we'd wonder, how do you tell the difference?
Or to go deeper, it's known at this point that formal systems aren't decidable in their application (i.e., the entscheidungsproblem), so it's just not possible that rationality is like a formal system. But then, what is it? My view is that it's just the same thing as intuition (the sense we have of things) but more of it, which goes under the charitable name "abductive reasoning."
There are times when it's very important to distinguish between 0.999998 and 1, and times when it's very important to treat the difference as negligible.
Have you ever played with an analog computer? I think you'll find it interesting. Modular synthesizers are essentially lego for analog computation, and you can find completely analog sequencers and so on. It takes some time to get the hang of this, but learning to build self-sustaining chaotic attractors is both enjoyable and interesting. The demo version of teh software here will get you started if you are not near a place with a decent music store: http://www.nordkeyboards.com/downloads/legacy/nord-modular-g...
Your brain is not a perfectly precise or reliable organ, but it's taken you this far.
Analog computation is definitely interesting, but at least in terms of theory and standard practice, it's worlds apart from digital computation. Put another way, though this may be even more obscure, the problem with analog computers is that they're application-specific. "Digital" computers are still analog computers under the hood, and their application is simulating set theory (so the "universality" of computation is contingent on set theory being universal). My point here, really, is that, if you make that not the case, then you're just saying, what if we didn't use digital computers at all? An interesting enough question, but then you really do have to study analog computers, and whatever you come up with will be totally unrecognizable to current computer science.
In any case, my original point is that brains can't be formal systems, as a comment on the supposed "rational brain." Formal systems don't work out in practice for a lot of reasons, and many of them are exposed by the actual experience of computer science. Returning to analog computation would just mean agreeing with me.
...and it took only four billion years of evolution. (Or maybe 100 million years of mammalian evolution, or 20 of primate evolution, or 200 thousand years of human evolution. Whatever you pick, it's slow.)