Okay. But the Sages’ claim is that regardless of the truth or correctness of the argument, there is a principle of “incline after the majority.” So what was Rabbi Eliezer’s answer to that? Because it sounds like even if it is a mistake, you still have to follow the majority.
And also, if possible, I’d like some explanation of the meaning of the Sages’ radical response in burning the items he had declared pure.
Also, what the Rabbi wrote—that Rabbi Akiva came to him earlier too—where is that written? After all, the Talmud there explains that Rabbi Eliezer said to him: yours is harder for me than theirs, for if you had served me, you would have learned much Torah.

In my article on “On That Day” (which serves as a prologue to Moves Among the Standing), I addressed those points. There was a fundamental dispute there, not a dispute about an oven:

כל היכא דאמרינן ‘בו ביום’ ההוא יומא הוה – יום מכריע אחד בהשתלשלותה של תושבע"פ

By the way, I saw in the Jerusalem Talmud, Moed Katan 3:1, that it asks: did Rabbi Eliezer not know the law of “incline after the majority”? And it answers that Rabbi Eliezer objected only to the fact that they burned the items he had declared pure in front of him. And I wanted to understand: they burned those items at the end of the story, after the dispute had already concluded, but he disagreed with them even before that—so why did he disagree from the outset, before there was this reason of their burning his pure items?
B. How is his objection to what he saw as their improper behavior connected to his allowing himself to disagree with them and thereby violate the rule of “incline after the majority”?

It may be that in the Jerusalem Talmud the version of the story had a different order.
There is a situation in which a person does not accept the authority of the majority if they behaved toward him improperly. If the Sanhedrin were made up of Sadducees, for example, their rulings would not be accepted. Maybe this even fits into the view that the obligation to obey is when “they tell you about right that it is right,” but not when they tell you about right that it is left.

It seems to me that this is a bit different. In a Sanhedrin made up of Sadducees there is a substantive problem in their rulings, because the fact that they are Sadducees indicates that they are not ruling according to the Torah, so there is no obligation toward them at all; “after the majority” was not said about them. That is not the case with the Sages who disagreed with Rabbi Eliezer—there is no substantive flaw there, only perhaps a behavioral one at most. That certainly has nothing to do with the validity of the ruling or its correctness. (Somewhat like the Beit Yosef’s comments in Kelalei HaTalmud on Eruvin 13b, where he asks what difference it makes that Beit Hillel were gentle and humble, etc., and answers that this is an indication that they are correct—but in terms of character traits and personality, there is no connection to the correctness of the ruling. The Rabbi also discussed this at length in a lecture at Bar-Ilan on tolerant Jewish law and also in the appendix to Truth and Unstable.) If so, Rabbi Eliezer was required to be more rational and less emotional, and not disagree with them.

Obviously it is a bit different, and still I argue that perhaps Rabbi Eliezer thought it was similar enough not to obey the majority. When the ruling is correct but is accepted by improper people, there is room for the reasoning that one should not obey it. Certainly if, in my opinion (= Rabbi Eliezer’s), it is not correct.

By the way, Nachmanides raised the opposite possibility: that Rabbi Yehoshua said the oven is susceptible to impurity based on a tradition in his possession, and specifically Rabbi Eliezer challenged that on the basis of reasoning. “And it is possible that they were saying their words from received tradition, while he said: this is how it appears to me—and disagreed with them. Therefore they did not accept from him any proofs in the world, and had he issued a practical ruling in Temple times, he would have become a rebellious elder. Therefore they treated him stringently and excommunicated him,” end quote. This is brought in Nimukei Yosef.

Interesting. But it still requires a bit of examination in light of the Talmudic statement that Rabbi Eliezer never said anything he had not heard from his teacher. It is pretty clear that the straightforward reading is the opposite.

In any case, even according to Nimukei Yosef, you can see that they did not rely on their tradition but on the fact that they were the majority. Otherwise they would have accepted Rabbi Eliezer’s challenge.

If anything, he also said things that no ear had ever heard before. (And even according to Rabbi Kook’s explanation in the letter to the Sochatchover, those were still things the Sages who disagreed with him did not know…) so it seems to me hard to bring proof from that statement that he never said anything he had not heard from his teacher.