To deny the material world is silly.
To say that ideas like solipsism or existential skepticism hold merit is to sacrifice the scientific value of thought because if you deny the physical world, you can also deny any other observable phenomenon. Therefore Moore argues that people should trust their common sense. That's an assumption, but it's the main idea of his writing.
Philosophy should correspond to real experience.
One main assumption that Moore asserts is that philosophy should be tethered to real human experiences, instead of British idealism in that day, which was becoming something of a creative endeavor. He is asking his philosophical community to get their feet back on the ground and their heads out of the clouds.
We are obviously co-experiencing reality.
The key word in this theme is "obviously." That word means that experience is observable and unmistakable. If a man punches another man in the face, both men feel that impact. Therefore, the physical material world is observable, obviously. This is Moore's famous "argument from common sense." He is asking his peers to be a little more honest about their creative ideas.
