>The way I see it, if we don't verify a thing, we should not accept it. We simply say we don't know, that we are working on a hypothesis. We should explicitly state our assumptions and gratefully receive correction. We can say that 'such and such is the common consensus but that I have not proven this to myself'. We should let a modicum of doubt be expressed in what we say, if there is a modicum of doubt in what we are stating.
While I appreciate your point of view and, especially in terms of policy and governance, it's a good idea to at least try to verify the veracity of the claims made by those seeking to implement policy, I think you go too far in dumping science and mathematics into that bucket.
For example, do you find it necessary to prove the commutative property of addition, or the existence of rational numbers before you can compute the sales tax due on a purchase?
Do you need to prove General Relativity (one of the most precise theories we've ever come up with) in order to trust your GPS?
You can "verify" this by punching in an address in your phone's directions app and use the GPS receiver to confirm your location as you move. When you arrive at your destination, you'll find that you're within a few meters of where you wanted to go.
That's only possible for GPS because the clocks on GPS devices and the satellites providing GPS signals are synchronized pretty precisely.
Without General Relativity providing a framework for accurately measuring time for objects with vastly different velocities, this would be impossible.
Or Quantum Mechanics, whose predictions have been shown to be more precise than any other scientific theory ever developed.
If we were to take your advice and distrust those theories because we can't personally verify them (whether that be a lack of mathematical knowledge or a lack of equipment and methodologies to do so), then we should reject the idea that GPS devices can give is accurate directions and claim that lasers (or any sort of coherent EM emissions) don't exist.
Which is objectively false.
It's a good idea to question what others assert, but (as another poster put it), having to prove everything from first principles before accepting its validity is a huge waste of time and effort.
That's where critical thinking, assessment of sources and the application of Occam's (and Hanlon's) razor really shines.
I completely agree that we shouldn't trust everything we hear, read or even see.
At the same time, rejecting everything we can't personally verify seems both counterproductive and deeply destructive of our science, our technology and our societies.