The typographical question may prove to be as thorny as the philosophical...
Its inventor John Wallis, well-versed in the Classics, is assumed to have derived the design from lower-case omega (the last letter of the Greek alphabet, metaphorically the end), possibly with a bit of Ouroboros (the serpent biting its tail) thrown in. So that makes it somewhat of a character, and the corresponding type design treatment would have it match the lower case.
But it has also been described as "indefinitely more" -- making it akin to a mathematical operator. So the corresponding treatment would be a thin monoline, even in a contrasting stress face.
Or is it a transcendental value? -- like pi, in which case it would be more appropriately treated as a Greek script character.
And how big/wide should it be?
Mathmaticians, typographers, which font has your favourite infinity symbol?