I am struggling to parse no matter what. (There are a couple questions on the site about the phrase no matter what but they revolve around how to use it.)
Is matter here a verb? Is it a noun?
If it is a verb, why does it not take an auxiliary verb?
No matter appears in NOAD as its own unit:
[ with clause ] regardless of: no matter what the government calls them, they are cuts.
it is of no importance: “No matter, I'll go myself.”
In the briefer phrase no matter (e.g. "If you don't want to go, no matter, we will stay put") it reads easily enough as a noun: there is no problem or nothing to consider.
• an affair or situation under consideration
• a reason for distress or a problem (NOAD)
The what brings confusion, whether it is final or introduces the rest of a clause ("No matter what" vs. "No matter what comes next"). In "I won't believe you, no matter what you say" it doesn't really make sense to make matter a noun.
Perhaps a useless exercise, we can expand it with potential synonyms:
I won't believe you, no problem what you say. I won't believe you, no reason for distress what you say. I won't believe you, no substance what you say.
- I won't believe you, no consideration what you say.
I leave #4 unstruck because it makes the most sense but, I find, not enough. This exercise is perhaps unhelpful because maybe another synonym could be used, or a perhaps a preposition could save the day, but I hope it serves my point.
So is matter a verb?
be of importance; have significance (NOAD)
Before trying to defend matter's role as a verb we already see it functioning quite strangely: generally English requires an auxiliary verb for negating verbs. Is this my reason for confusion?
When what introduces the rest of a clause this more or less solves the matter:
⇒ "I won't believe you; it doesn't matter what you say"
but it remains murky when there is nothing to follow to the what:
⇒ "I won't believe you; it doesn't matter what
by no means provides the same sense of disregard and, besides, there isn't really a subject.