I have to admit I'm not knowledgeable enough to appreciate everything you said.
I did some reading to have a clue, and I realize I'm a foundationalist. I don't get how it is a problem to assume some truths to be true and build the rest of the belief structure from these.
Yes, we all have to assume something to start off. That's undeniable. The problem, aside from assumptions ultimately being unjustified, is that there isn't always a "common assumption" that we all hold. If you and I disagree on a fundamental assumption, there's no evidence or argument to examine.
Some basic beliefs I have are: the external world exists, I exist, I have experiences,
There's no solution to hard solipsism, and further, simulation theory becomes more plausible every day. You might actually need to either demonstrate that artificial minds are different from ours, or else take that as one of your initial assumptions.
I'm not proposing either of these things, but rather stating that they are possibilities. They're unfalsifiable, but your assumption, just like any assumption, is unjustified.
the laws of logic are reliable,
Nope. Quantum mechanics.
moral values exist objectively,
Why? How? Would the world be any different whatsoever if moral values did not exist objectively? What a can of worms this is though.
propositions can be true or false depending on their correspondence with reality or logical consistency.
In logic, "true" is also another one of those primitive notions. That is, it is undefined.
Maybe electrons have weird behaviors, but how does that affect our book? At a given time it is or isn't on the desk.
Firstly, I don't need to address all possible scenarios; a single counter-example is sufficient. Secondly, Schrödinger's Cat is a thought experiment relating quantum mechanics to large objects and I don't think there's been a solution to it.
I'm not knowledgeable enough on Gödel's theorem to speak on it.
In any consistent, nontrivial logical system, there must be a statement which is either true or false but cannot be shown to be such. Effectively it is neither true nor false, but a third category: undecidable.
Concerning mathematics, you're saying it's just convention or some meaningless system?
Mathematics has no meaning. That's why we are able to assign meaning to it.
Imagine for a moment that we have only shown that that two firetrucks plus two firetrucks equals four firetrucks. We would then be unjustified in claiming that two apples plus two apples is four apples. It is precisely because mathematics is meaningless that we are able to assign to it any meaning we like. This is too abstract for most people and now you have Platonists running around claiming that the number 2 actually exists in some ethereal realm. It does not.
