In order to prove that we have no knowledge of the external world one must first define the two. I will use Descartes definition of knowledge as a conclusively justified, true belief. For the purpose of this essay I will define the external world as all a priori and a posteriori knowledge. As defined in Philosophical Problems by Laurence BonJour and Ann Baker, a priori are reasons independent from experience and a posteriori are reasons based on or derived from experience. So the two types of knowledge are that which you’ve experienced (a posteriori) and that which is independent of experience but you have prior knowledge of (a priori). Descartes examines both types of knowledge in Mediation I in order to determine whether either can be certain or if they have doubts.
According to Descartes’ Dream Argument in Mediation 1 we cannot trust our beliefs based on sense experience. Since any experiences you can have while awake can also be dreamt, you can never know for sure whether they happened or not. You must know you are not dreaming in order to trust your experiences. However, it is impossible to know that you are not dreaming. Because of the possibility that our sense experience actually occurred in a dream and thus cannot be justified by experience, it is impossible to trust any of our sense experience. Therefore, we cannot know anything about the external world on the basis of our sensory experience.
As seen in Descartes’ Evil Genius Argument