This is a bit of a long story. I am a graduate student working on mathematics and the topic I have been working on is not directly within my advisor's expertise (pretty closely related to though). My thesis basically consists of two parts which deal with two related but independent problems. At the beginning when I chose these two problems to work on, I was kind of discouraged by my advisor as he thought these problems despite interesting did not seem reachable. But actually, when I decided to work on these problems, I already had some rough ideas on tackling the problems.
Anyway, the first part was basically done in last summer/fall which I also presented my proof orally to him. For the second part, he was once kind of skeptical of my proofs, however, his confusions and doubts seemed to be resolved through our conversations, email exchanges and my improvement of the exposition. I sent him my first draft in April (I was supposed to submit my thesis in June). In late June, when I had to submit my thesis, he said he had not really finished reading my thesis.
After my thesis was submitted, he told me that my thesis was very badly written which made him not even able to follow my work. But wait, he had two months to take a look at my thesis but he had not said a word about that every time we met. Anyway, my thesis defense was thus delayed for two months and in these two months I was constantly asked to make changes big or small on the thesis. At certain point, my advisor seemed to feel my thesis was in good shape and I should arrange for the defense.
Alright, I did the defense in August and it went well. But recently I received my thesis feedback report in which my advisor suggested that I should remove the whole second part from my thesis because he thinks that my proofs/arguments presented there are not solid enough. I always appreciated when he could point out specific mistakes/confusions I made in my thesis. But it seems that after all of these have been resolved, he is still not convinced that there is no gap in my proofs. I really do not know what to do now.
Should I argue with my advisor or should I simply follow his request to remove that chapter and get my degree safely?