Of every hundred facts upon which to reason, ninety-nine depend on authority...I am told that there are still Logical Positivists out there, who believe if a thing cannot be measured, it cannot be discussed intelligently. That formulation is much better than the "if you can't prove it, I don't believe it" that it grew out of. In the early part of the 20th C it was a popular philosophy, especially among those in the hard sciences, and I suspect that is where it may still hang on. It has strengths. It does promote a certain clarity in argument and reasoning.
For example, few of us have followed the reasoning on which even ten percent of the truths we believe are based. We accept them on authority from the experts and are wise to do so; for though we are thereby sometimes deceived, yet we would have to live like savages if we did not. C.S.Lewis "Why I Am Not A Pacifist" 1940.
My objection to it is not so much that I don't believe it, but that I suspect its adherents do not. They are like all of us, not only believing a thousand things we have never quantified and subjected to hard thinking, but a hundred things which we would continue to believe even if we had. Not all things admit of easy measurement, fewer still admit of proof even in theory, but we often have to believe some thing to cross the street, do laundry, sing a song. My suspicion is that the Logical Positivism is sometimes convenient, to get one off the hook having to think about something. I am hinting at religious questions here, but there are many other things in the mix: some questions are too painful, but most are simply too inconvenient to reexamine. So we go on believing what we did before.
Similarly, there is a belief gaining traction from neurological research which is based on older ideas questioning Free Will - that our reasoning is only approximately correct, useful rather than reliable. (Philosopher Eddy Nahmias challenges that, as I previously linked.) It may be a true idea. Our reasoning, pace Thomas Aquinas, may not only not be divine, but merely cleverly animal. It may be good enough to get us by, no more. Yet I suspect that the proponents of the idea don't believe that particular bit of reasoning is only approximate. They believe it is true. Lewis wrote frequently of the self-refuting aspect of that strict naturalism. It saws off the branch it is sitting on, for if our thoughts are not reliable, neither is that thought reliable.
That is usually waved off with a smile that yes, technically, it is an inconsistency, but what can one expect with such flawed reasoning as ours? So it may be. But I would be much more sympathetic to the idea if it were not only trotted out at convenient times, rather than when it is inconvenient specifically to the Approximate Believer.
Accusations of bad faith are not the most powerful of arguments, I admit, and that is what I have done here. Nonetheless I do not rescind it. I think it is what is happening, whether it is possible to measure or possible to rigidly prove or not. My painfully inexact animal cleverness encounters it too often to ignore, if that suits you better.