I'll try to clarify, but I'm very pressed for time. Let me try this first: Does this comment help clarify what I was trying to claim in my previous post?

Your linked comment doesn't clarify for me what you meant by "mathematicians succeed and fail on this issue in a wide range of degrees." That said, the comment does give a meaning to the sentence "Conceptual analysis makes assumptions about how the mind works" that I can agree with, though it seems weird to me to express that meaning with that sentence.

The "assumption" is that our categories coincide with "tidy" lists of properties. First, I would call this a "hope" rather than an "assumption". Seeking a tidy definition is reasonable if there is a high-enough probability that such a definition exists, even if that probability is well below 50%. Second, it seems strange to me to characterize this as an issue of "how the mind works". That's kind of like saying that I can't give a short-yet-complete description of the human stomach because of "how epigenesis works".