I'm sorry if this sounds like a stupid question, but how can we know that our logical approach to ideas is not in itself based on assumptions. For example, how can we know that the workings of the entire universe are consistent with what we observe? Do we know that the same laws of physics apply in a distant galaxy, or have we simply decided to take the leap of faith because finding any explanations would otherwise be impossible.
Perhaps a clearer example is addition. We have defined addition to simply represent the counting of the number of elements within a two distinct groups. For example, in order to recognize that 3 + 8 = 11, we can draw the following diagram:
| | | + | | | | | | | |
and count the total number of lines. Since nobody has "counted the number of lines" in an addition problem like 5893 + 2485, how can we be certain that our result will be 8378? How do know that consistency does not break down in addition specifically for these two numbers? Would it be fair to say that "addition works" is more of a conjecture than a proven axiom?
Ultimately, what I'm asking is how we can claim that consistency in the patterns we observe extends beyond what we observe. Is it possible to know that a tree will make a sound when it falls unheard or is that merely an attempt to bring continuity into a world that may or may not be? Is it just a practical, intuitive, unprovable assumption that is extremely convenient?