The cogito 'I think therefore I am', is when you think about it, not quite what it means. Consider a fox, one rather suspects that it has some inner consciousness that it is, and this without enunciating to itself the cogito (what is the private language of a fox, for it assuredly has not a public one, at least one not reducible to a language of signs and gestures, which in its width is no longer a language in the usual sense; and Wittgenstein assures us that there can be no private language with a public counterpart). In this direction, one can reduce this statement to 'I am'; or just 'am'.
So, what then, Descartes cogito? It has the form of an axiom, a la Euclid; but the similarity is spurious as the ambit of Euclid - plane geometry - is tightly controlled, with specific and formal rules of organisation. One should see it as an 'inspiration'; and thus it is neither an axiom, in the mathematical sense, and thus nor a proof.
In mathematics one does not prove obvious things; if one does it is usually in terms of a larger project. For example, the natural numbers were axiomatised only in the early twentieth century. Given that plane geometry was axiomatised, why then the long wait to axiomatise the natural numbers? This is simply because they were seen as obvious (hence natural); and therefore not in need of any kind of formal machinery - that it was done was due to the emergence of a larger programme - to reduce mathematics to logic; hence the neccessity to find a formalisation that would fit into this framework.
Descartes axiomatic way of considering philosophy inspired Spinozas 'geometrisation' of neo-Platonism, and Wittgensteins 'logical' method of reducing facts (statements about the world) to propositions (logical statements) and also Leibniz Monadology which applied the same systematic approach to a synthesis of Greek philosophy (atoms and substance) to Christian theology.
What this then brings us to is a consciousness of what it means to prove; and this should, as it does, mean to convince someone who is aware of the larger aims of that area of knowledge; hence one finds differing kinds of arguments in differing domains of knowledge that are taken as valid.
To put this in context, one often hears that in principle one can prove the notions in physics - for example that fibre bundles naturally interpret physical theories in a geometric manner. This is a platitude, as it would take on average six years of serious study to be able to do this. It is not as simple as showing that 1 and 1 is 2, which is a monents work; and because it's a moments work is probably not seen as worthwhile; that it is is due to the presence of a tradition.
Proof in mathematics, in one sense, means that someone can prove it, and usually not yourself; that we are convinced by these proofs is more often than not due to other factors: the presence of tradition, and therefore of authority; it's utility, and therefore the presence of practical knowledge. It's after these factors that has driven the mass education of mathematical thinking. Still, there are a few, and they are usually only a few, who do appreciate mathematics as a subject, for what it is.
All this doesn't answer your question 'what are the fundamental truths of philosophy'; and this because it is too wide and too broad to admit any easy answer, or even a difficult and long one; which doesn't mean that it isn't an important one.
One might say the fundamental truth of philosophy is the verb in Descartes axiom: to think.