Wake me up when epistemology has sorted out anything of any use, then, rather than going round in circles about what anything means.
Don't mistake the tight little solipsistic circles that our OP is spinning in as an accurate representation of the philosophy of science. It's just coffeehouse navel-gazing served on a bed of AI slop. Be that as it may, all this rhetorical pearl-clutching serves philosophers a lot more than it serves scientists.
It seems our OP is still trying to pretend that anyone here believes in his "strong scientism" boogey-man.
I didn't say logic and maths were invalid, did I?
Nor have I.
Propositional logic is valid in that it describes what can be reliably deduced or reasonably inferred. As you pointed out, it is merely an aid to thinking. Thinking would still occur if we did not codify patterns of good thinking. The codification does not exist outside of human thought. We can apply those patterns to some real-world set of facts and evaluate the outcomes. We can, say, recognize the pattern of reasoning as a Barbara syllogism, and therefore of validating form, and further that the conclusion is verified by observation:
All science fiction fans are nerds.
Star Trek fans are science fiction fans.
Therefore, Star Trek fans are nerds.
The empirical proof of this is let as an exercise to the reader. But the reason why we know this pattern of thinking leads to reliable conclusions is not because we have exhaustively or statistically tested all possible examples. We know it works because we have found a way to express what it means to be a member of a category. We express the pattern verbally as a syllogism or graphically in a Venn diagram. But the formalization is simply
res mentis. The concept being formally described is self-existent.
Similarly, mathematics is valid because it expresses relationships among quantifiable concepts in ways that are sometimes self-evident. These relationships are expressed as rules, but the expressions themselves are not the rules. Because techniques like induction let us extend rules from things we can see to things we cannot see, we can continue to rely upon mathematics to provide knowledge beyond what we can verify directly.
One good example is the quaternion. It's a very abstract way of thinking about angles in three dimensions that works better than the familiar Euler angles (i.e., roll, pitch, and yaw). If you express directions as quaternions instead of Euler angles, you can more elegantly describe the angle between those directions. The observable effect is that your spaceship smoothly changes attitude using minimum fuel. But under the hood, quaternions let solutions sneak partway into four dimensions of space. A quaternion obeys all the rules of math. A quaternion lets you do things that you can observe in the real word to have the desired effect. But in an important way, they exist only as
res mentis. Something points naturally and self-evidently in a direction. Math gives us several formalisms to describe the directions; the formalisms do not separately or meaningfully exist outside the mind.
Statistics is also a good example. The normal distribution describes a real-world behavior. But it derives from an understanding of mathematical rules for how real-world events like coin tosses
must behave. We then rely upon the mathematical rules to draw conclusions about what we observe. The rules let us assign a probability to a predicted outcome. But a probability doesn't exist as a real-world thing. It exists only as an abstract way for us to reason about the prediction and its observed outcome.
There is no false dichotomy here. We're not trying to say that anything that can't be verified empirically can't be valid. In fact, science relies upon these human-contrived tools to accurately describe things we
cannot directly observe.
