Is this even remotely acceptable? Thank you.
Background: I am a 30 year old undergraduate at a college majoring in mathematics with goal of specializing in philosophical aspects of it. There are certain books and theories that has affected my personal philosophy which made me a skeptic (mathematical atheist if you will). One such book was Keith Devlin's "Goodbye Descartes: End of Logic" along with Wittgenstein final nail that "Whereof one cannot speak, thereof one must be silent." This was 8 years ago and then I came across Kurt Godel's proof and Raymond Smullyan's books. Latter's Taoist philosophy attracted me greatly. Also I delved into Zen Buddhist literature and Hofstadter. Zen encourages to maintain a beginner's mind and to question everything. So when I meditate on logic itself and dig further 'down the rabbit hole' it appears that mathematics is actually based on assumptions or axioms or formally ZF or ZFC. Why is 2+2=4? One can show Russell and Whitehead's proof which further begs what is the foundational 'glue' of logical connectives or what is the formal definition of entailment? As I studied further I came upon Non-well founded set theory, concept of predicativity, relativity of structures, randomness, algorithmic information theory, Chaitin's conclusion mathematics is random which further ossified my beliefs that mathematics may be based on 'beliefs' or 'assumptions'. In passing I allude that I enrolled back in college and registered in a Contemporary Philosophy class where (controversial according to some) "What the BLEEP do we know?" was shown.
Personal philosophy: I apologize if I am asking for a silver bullet for answers of all the philoso-mathematical problems. I am actually lost. I do not know where to begin. When I was a child I used to labor over problems, solve them and check the answers to the 'correct' solution. However at this stage of my life I find no motivation because I do not know what I know or what we can know or where to even start. Naively after inner meditations I come to the conclusion that mathematics must be a quasi-empirical science or subject based upon assumptions. Further to confuse and muddle my thinking, I read Field's fictionalism, Banach-Tarski paradox, Brouwer's intuitionist ideas, anthropological beliefs, smattering of New Age quantum mechanical popular books, idea of digital physics and all paths seem to inevitable point to the a conclusion that it could be based upon 'thin air'. It appears that what appeared to be air tight logic is actually full of paradoxes, circularity and relativity.
Question: Is the aforementioned conclusion or observation valid or sound?