Here is the full list of all 31 Legends of Mathmatica²: Under the Shadow of Certainty achievements.
Thousand step journey or something. Longer if you grind.
Added Maggie to the party! Get your healer early.
Added Dougal to the party! Time to cut onions!
Added Sigurd to the party! Apparently Hiro is not enough of a hero.
Reached High Noon. The goal is starting to come into sight!
Cleared the Cave of Friendship. Your bonds can overcome anything! Or, at least, Hiro thinks so.
Cleared the Totally Unexpected Fire Cave. Even a dragon is powerless against math.
Cleared the Ruins of Obligatory Sewers Inc. Don't blame me. It's in the contract.
Clear the Fabled Forest. So... now what?
Defeated Villain and saved the world! Congratulations!
Cleared ten Puzzle Dungeons. Getting used to them?
Cleared twenty-four Puzzle Dungeons. Only the hard ones remain.
Cleared all thirty-five Puzzle Dungeons. If you didn't use a guide, wow, congratulations! If you did, well, I can't blame you.
Got all of the Achievements. Truly, Congratulations and thanks.