Incorrigible knowledge

For Descartes, what form of knowledge can be proven as incorrigible through methodical doubt? Why is it incorrigible? What is the relevance of such proof in establishing the existence of the mind as something that exists independently of the body? How can we use his proof of its incorrigibility as an illustration for the rationalist claim that reason (not the senses) is the sufficient guide to truth? How can the foundations of scientific and mathematical knowledge be considered corrigible through methodical doubt? (30 points)