A big part of mathematical logic is getting the correct set of axioms to begin with. Euclid's issues in doing this led to non-Euclidean geometries being developed a couple thousand years later. It might be interesting to have students in a geometry class develope (and critique) a set of axioms just as Descartes did. He struggled to come up with what anyone absolutely knew about our world. He came up with the fact that he knew his own existence - "I think, therefore I am." That sentence, in isolation, is usually all we hear. I came across his quote in his writings. It is contained in Discourse on the Method of Rightly Conducting the Reason and Seeking for Truth in the Sciences which is often shortened to Discourse on the Method.
What is the context for his famous line? In Part IV Descartes begins with, "I do not know that I ought to tell you of the first meditations there made by me, for they are so metaphysical and so unusual that they may perhaps not be acceptable to everyone." Amen to that.
Halfway through the next paragraph he states, "Thus, because our senses sometimes deceive us, I wished to suppose that nothing is just as they cause us to imagine it to be; and because there are men who deceive themselves in their reasoning and fall into paralogisms, even concerning the simplest matters of geometry, and judging that I was as subject to error as was any other, I rejected as false all the reasons formerly accepted by me as demonstrations. And since all the same thoughts and conceptions which we have while awake may also come to us in sleep, without any of them being at that time true, I resolved to assume that everything that ever entered into my mind was no more true than the illusions of my dreams. But immediately afterwards I noticed that whilst I thus wished to think all things false, it was absolutely essential that the "I" who thought this should be somewhat, and remarking that this truth "I think , therefore I am" was so certain and so assured that all the most extravagant suppositions brought forward by the skeptics were incapable of shaking it, I came to the conclusion that I could receive it without scruple as the first principle of the Philosophy for which I was seeking.
