My intuition tells me that any theory, whether expressed using mathematics(and therefore more precise and structured) or argued for using natural languages have to involve blind faith in certain propositions (or statements). In science (except mathematics) these propositions appear as postulates, whose truth value and validity are ascertained by observation of the natural world. But this becomes tricky when you argue in mathematics, where these propositions may not have their validity based in observations of the natural world. Such statements then become the axioms of that theory. My first question is
- Can every mathematical theory be proven to have axioms which are blindly believed in as a necessity?
Next, when we enter the realm of the internal world of thoughts and feelings to form logical theories of how to conduct ourselves in the world, which is the task of philosophy, one can't objectively prove the validity of many statements. My next question therefore is
- Can philosophical theories of how best to live life also be shown to adopt propositions which are blindly believed to be true?
I haven't been able to find answers to these questions anywhere online. Could you please answer while also citing the sources for them so I may read further?