“How do we know what we claim to know?” is quite easily the most important question in science. In fact, the scientific method is designed precisely to answer that question. Through a process of careful observation, hypothesizing, and tightly controlled experimentation, scientists have been able to explain why they know what they claim to know for hundreds of years. Rigorously following this procedure is what separates science from all other disciplines.
Now here is where the wheels came off:
Most aspects of our lives cannot be scrutinized with the scientific method. But what we can do is apply the underlying philosophy of the scientific method to our daily lives. And that philosophy is one of skepticism. More.
Actually, the philosophy isn’t skepticism as such. One could end up doubting that two and two make four.
Rather, the philosophy is to look for evidence rather than rely on authority. But we cannot look for evidence if we doubt that we are capable of evaluating it.
See also: The illusion of consciousness sees through itself.