O.K. Luis, I see where you're coming from- but consider this:
Leading physicist Paul Davies wrote a book "The Mind Of God. Science And The Search For Ultimate Meaning."
In chapter 7 he quotes Roger Penrose admitting that rigorous argument is usually the last step- and that much of his mathematical work is guided by beauty (i.e. he has to make many guesses, and aesthetic convictions are very important in this.)(see p.176)
Paul Davies acknowledges that the concept of mathematical elegance is highly subjective. Yet it was a sense of aesthetics that led both Dirac and Einstein to their major breakthroughs.
For me, it was in the defense of introspection that I made (to respond to your arguments); that delivered what may yet prove to be a breakthrough in physics. So you might argue that, just as "rigorous argument is usually the last step'; so it has proven with me. You provided the context for rigorous argument; thus I extracted some hard-science elements from my introspective journey. Even Yanniru phD did not challenge a piece I wrote; but just asked if it was intuitive or mathematical.
It is not yet proven that I made a significant discovery for physics- time will tell. But even John Cramer's transactional analysis interpretation of quantum mechanics seems to derive from what I came up with. And Dick phD appears to have fallen silent, unable to counter the theory. (That everything a physicist does involves contemplating Existence directly, or comparing and matching patterns. And that a three way match between 'jump oscillation (quantum) fields' (musical chairs) gives you anything in physics.
Knowing the difference (relativity) between a 'musical chairs' field and the at 'right angles' 'join the dots' field is critical. Alex agreed with 'the three compensating accelerations to give stationary object' model; but hasn't commented on the "musical chairs/ join the dots/ Know the difference" model.
But in due course I think I can show how any problem in physics and well beyond physics can be 'dissolved' by this 'comparing and matching patterns' system. Can the amateur angler catch the big fish? Who knows. If I'm up the creek without a paddle; I sure learned a lot on the way.
So we might agree on a less extreme postulate: that introspection PLUS logical analysis might be effective in what are traditionally termed 'scientific matters'. I would go further than that though.
A philosophy text acknowledges that there is a class of knowledge involving 'having the experience' (e.g. pain); that this is real knowledge. Sure, I look like a cognitive miser when I propose a data-bit (or quanta) view of 'knowledge"- that knowledge is made up of quanta and to have one quanta is to have one bit of knowledge.
I wrote the following once elsewhere:
In "An Introduction To Philosophical Analysis", by John Hospers, many pages are given on the subjects of "meaning of words", "definition", and "concepts". Of note (p56) is "OSTENSIVE DEFINITION"- this avoids the problem of words defined by yet more words. It is the most fundamental kind of definition- necessary to even get started.
Ostensive definitions are learnt when we are young from usage and context (which I would call "pattern matching").
Example: someone uses the word "wood" in the context of pointing to a chair, a table, a fence-post, a piece of timber, a tree- you figure out the common pattern "wood"
relates to the sound pattern "wood".
(My view: knowledge is thus: relationships between patterns. All three: (a) pattern of real wood in different situations; plus (b) sound pattern "wood", plus (c) comparison of the two; gives the knowledge of the meaning of the word "wood")
Ramachandran, a neurologist, tells us that there are about 30 different areas of the brain involved in "seeing"- shape, movement,........emotional connections....etc. So there is a lot of pattern matching going on. What Ramachandran may have overlooked is the voluntariness of the strategy one adopts when patterns appear to match.
In the "wood" example; the word "brown" might have been used at all the listed occasions wood was used. One might make a mistake and think "brown" meant "wood"; and call a yellowish wood desk a "brown desk"
meaning a "wood" desk.
Then someone points to brown eyes and says "brown". A rule of philosophy is: when you meet a contradiction, make a distinction.
You may realise that "eyes" contradicts the "wood" pattern that you thought the sound "brown" meant; so you solve the contradiction by making the distinction that the "wood" sound is distinguished now only by the wood-ness of the yellowish desk, and of the other wood patterns that happened to be brown as well.
There is maths that matches nature-patterns, and maths that doesn't. Some maths you get a number of solutions to an equation; and some are absurd and discarded while others fit nature. Hence maths is not reality; reality delivers some maths matches. If maths was reality; what of the discarded solutions? Where are those objects? Reality is more basic than maths. This doesn't prevent reality patterns matching maths patterns.