Here is an idea I have for solving the problem of induction, which draws upon the thoughts of Karl Popper but also manages to go a little further:
If I drop an object, it will fall. I can do it again and again, dropping the object a million times and watching it fall a million times. But how do I know that the object will drop every time just because it has dropped every time in my experience?
How do I know that the sun will rise tomorrow morning? I formulate a hypothesis that the sun will rise every morning, and I know that it is to be preferred because it is simpler. The reason it is simpler is because it employs fewer assumptions (assuming that the sun always rises is simpler than assuming that the sun occasionally rises and occasionally does not). The reason fewer assumptions are preferred is because, all things being equal, the more assumptions we make the more likely our hypothesis will be wrong (If the assumptions are equal to one another, and each one has, for example, a ten percent chance of being wrong, then obviously the hypothesis with fewer assumptions has a higher chance of being right). This brings us to mathematics and logic, which are self evident and must be accepted in order to even think.
Is my solution successful? You be the judge.