I don't know if this question even makes sense, but let's say we have a group of people debating a set of ideas. Over the course of their discussion one idea starts to win out. Not because of a material base, but like the philosophical rigor of the argument is the best. That wouldn't be dialectical materialism, the ideas would be resolved by an idealist dialectic. Assuming they're talking about pure math or something like that. Or is the Marxist idea that even mathematics is a result of materialism?
This debate would still be "material" in at least three ways.
The debaters themselves are still material, and a product of a certain kind of society. The debate about abstract math would probably not happen in a hunter-gatherer society, or a society that does not spend resources on education, or a society that does not create an intellectual class.
Let's assume the object of the debate is something that directly or indirectly matters to the debaters and to people of the society they live in. Then, the new ideas or understandings created in this debate do not end there, they extend to a larger community or even the whole society. What kind of ideas can spread in a society is influenced by material factors, for example, mathematics that can be used in complex lens systems would be more likely to spread in a society that creates those systems.
The ideas themselves are based in the material human brains. Furthermore, however abstract and separate from the material world the ideas appear to be, they do not exist discretely from other ideas, and these other ideas are related to the material reality.
I'm asking about the artificially narrowed scope of the dialectical interchange between the ideas mid debate. Just the changing conceptualizations and support for the ideas changing based on the interplay between them. What is that called?
This question doesn't actually make sense because the mathematical ideas are either right or wrong, inherently. There would be no dialectical process of development, or if there is, it would exist in the material minds of the people debating