The rationalist descartian position states that everything can be doubted except doubt itself. With the phrase "cogito ergo sum" it claims to establish the existence of one ones existence, if i did not get it wrong, through exerting doubt which is supposedly undoubtable.
So this (that you exist therefore you doubt, and vice versa) are in their consideration the only two fundamental axioms existing. Now i have some questions, i couldn't up with an answer for:
if these two assumptions or truth are 'just' a set of axioms, does that imply that a different set of fundamental axioms can not be excluded?
if so, could we potentially establish these or similar two truths also with another set of axioms? could we try to construct a theory not necessarily based on the need of an entity such as existence (so basically a thought school based on non-existence) or is that self-contradicting?
cogito ergo sum is a circular implication, does that mean that it excludes provability?
being a circular pair of axioms can it still be possible to use these as premisses for more complex syllogism deriving more complex knowledge from these two fundamentals by using methods like descartian circle, or is that a not completely "kosher" method in logic?
If we couldn't derive komplex knowledge like language from these two axioms, how could they provide the tools (language, logic...) to analyze themselves and therefore be proven as true facts? Or is there always a degree of uncertainty involved because the axioms cannot prove themselves but need a "homunculus", something outside the system, to prove them?
If methods to produce more complex knowledge from these two axioms fails and being a cirular pair of axioms can we safely apply Wittgensteins expression "sinnlos"? Though they may or may not have some degree of true-ness to them, they might not have any "Bedeutung" attached to them, therefore making no statement about anything substantial.
I finally have one last question, which isn't really related to this directly: If -as i read in other posts- logic is not provable by logic itself, therefore downgrading logic from a tool of absolute verity to a "mere" model to acquire close-to-truth information, wouldn't that make any logically proven theorem, or in general every axiom system a theory rather that a fact? wouldn't that make axioms themselves a matter of debate?
sorry for my bad english, and if any of these questions are too naive...but i would appreciate any clarification