User blog comment:TheValeyardOfGallifrey/Food for Thought./@comment-6414106-20120330000401/@comment-4674838-20120330013015

TL;DR: Go from today to the end of time, come back to today, and then go to tomorrow.

Without loss of generality, assume that the amount of time between today and the end of time is X days, where X is a positive number. Let A -> B denote the distance between two points in time, A and B, along the path from point A to point B.

Case 1: When traveling backwards in time, the time difference is of opposite sign as the time difference when traveling forwards in time.

today -> end of time = +X (positive number)

end of time -> today = -X (negative number)

today -> tomorrow = 1 day

today -> today = 0

today -> end of time -> today = X - X = 0

Since 1 > 0, tomorrow is a day which is further away from today than today. More generally, under this case, the difference between today and today along every path is 0, and the difference between today and tomorrow along every path is 1.

Case 2: When traveling backwards in time, the time difference is of the same sign as the time difference when traveling forwards in time.

today -> end of time = +X (positive number)

end of time -> today = +X (positive number)

today -> end of time -> today = X + X = 2X

today -> end of time -> today -> tomorrow = X + X + 1 = 2X + 1

Since 2X + 1 > 2X, tomorrow can be seen to be "further" away from today than today, given a certain choice of path. More generally, under this case, the difference in time between any two days is ill-defined, as it depends on the particular path chosen. For every path between today and today, there exists a path between today and tomorrow which is longer. Thus, the set of time differences between today and any other point in time along any path is unbounded, meaning that there does not exist a day which is "furthest" away from today.

Conclusion: even if time is permitted to flow in reverse, the statement "The farthest day away from today is today" is false, as, in case 1, tomorrow is a counterexample of day which is further away from today than today. In case 2, no day exists which is furthest away from today, since the notion of "time difference" is ill-defined.