Shankaracharya (788-820 CE), great philosopher and promoter of Avaita Vedanta, traveled India in his day winning debates with non-dualist arguments. He beat great debaters of Sankya and Buddhism - and some even attribute the decline of Buddhism in India to Shankaracharya himself.
However, while Shankaracharya promoted a non-dual approach, one must ask - who was he promoting it to? If we are all, indeed, one - our separate lives a mere allusion - then would it not follow that the knowledge of oneness be self-evident? If our separateness is an illusion, and our senses are merely perceiving themselves, would not all knowledge derived from these senses be false? Proponents of non-dualism must admit that the human being (and our earthly counterparts) rely on some level of dualism to interact in this world, on this plane of existence.
Think of the universe as the human body, and the philosophy we adopt our lense. When observing the human body with the naked eye from afar, imperfections are hard to see. Come within one foot of that same body, and we see more closely the subtleties of the skin and pores. Now imagine observing that same body under a microscope! How many more worlds open up to us? And go even deeper now to the cellular level - or even farther, to the individual cell. Worlds within worlds spinning upon each other. Would it be fair to call any level of reality we just explored false? No! The dualism of our world is necessary for us to come to know the non-dual reality of eternity. Our human existence mandates our learning through the senses and living within time and space.
Where does this lead? Who wins the dual of dualism? Neither can fully answer the question "what is this" or "who am I" without the other. Essentially, we must begin our inquiry grounded in the reality of our existence - and then as we dive deeply into that reality, it will flip inside out and expose the beautiful absurdity of life.