Total is the operative word. Total Air Temp is the total or the aggregate of the actual temp. (SAT) and the temperature rise (the increase due to compression).
So if the SAT is -40 and the TAT is - 30, you have a temp. rise of 10. TAT is the reference temp for many on board calculations both systems wise and by the pilot. For example, when you think about the need for engine anti-ice your definition of icing conditions is visual moisture at a temp of 10 deg. TAT or colder if you're airbourne. That's the temp. over the wing so it's more relevant than the actual SAT.