There is a Mímir related to the myth of the well where Odin offered his eye to gain knowledge. There is myth of Mímir post-Aesir-Vanir war where his head is chopped by the Vanir. Are the two considered to be the same? Is there any evidence that the two are related or to think of them as different beings? Or is it very vague?
1 Answer
Let me start by saying that most of the available evidence supports the idea of a single Mímir, and scholars seem to unanimously agree on this.
However, as it is often the case with Norse mythology, there is a lot of confusion surrounding this topic, fueled by the fragmentary and obscure nature of the available sources.
Part of the confusion stems from the fact that the character appears with two similar, but distinct names: Mímir and Mímr1.
For example in the Völuspá, which is one of our main sources of information on this character, we have at stanza 29 (all translations are by Larrington):
‘Why do you question me? Why do you test me?
I know all about it, Odin, where you hid your eye
in Mímir’s famous well.’
Mímir drinks mead every morning
from Father of the Slain’s pledge—do you want to know more: and what?
Here the reference to the well where Odin offered his eye is clear, and it is exactly this passage that Snorri quotes in chapter 15 of his Gylfaginning. It is important to note that both the drinking of mead and Odin's eye sacrifice are associated with the acquisition great wisdom and numinous knowledge.
Then, at stanza 45 we have:
The sons of Mímr2 are at play and the Measuring-Tree is kindled
at the resounding Giallar-horn;
Heimdall blows loudly, his horn is in the air.
Odin speaks with Mímr’s head.
Apart from the obscure kenning "sons of Mímr", which is typically rendered as 'giants' but whose meaning is still very much open to debate, here we have a character called Mímr, whose head is strongly implied by the wording to be separated from the body, and who Odin consults just before Ragnarok. Similarly, in stanza 14 of Sigrdrífumál we find:
then Mímr’s head spoke
wisely the first word
and told the true letters.
We again have the motif of detached head featuring extraordinary knowledge/wisdom. Given the similarity of the names, and the association with great wisdom and knowledge, modern scholarship tends to equate the 'Mímir of the well' and the 'wise head of Mímr'.
Snorri was certainly of the same opinion: he used only the name Mímir and conflated the two both in the Prose Edda and in the Ynglinga Saga. It is in the latter that he tells the story of his decapitation after the Æsir–Vanir war.
Indeed
1 There are also a couple of instances of a Mími, which again probably indicates the same character.
2 Larrington anglicizes it as Mim, but for the sake of this answer I prefer to keep the Old Norse form Mímr.