Embeddings: Meaning as Coordinates

How would you write code that knows "log in" and "sign in" mean the same thing?

A list of synonyms is the usual first attempt. Each entry handles one pair, but language has thousands of such pairs, and no list covers them all. A help center search shows the problem. A user types "I can't log in," and the search returns nothing. Yet two articles answer the question exactly: "How to reset your password" and "Fixing sign-in problems." The search matched words, and the user's keywords appear in neither title.

A better approach compares meaning instead of spelling