The comments indicating which drupal path is used for which function seem to be swapped.

In foursquare.module on line 16 it states "// initiate authorization of user" however upon further examination the function "fouresquare_authorize" states:
/**
* Callback when user is redirected from foursquare back to the Drupal site.
* Get the access keys and store them
*/

Line 24 states "// Callback endpoint to finish authentication of the user" and it appears to actually be the path to initiate authorization.