Why does the time/space tradeoff exist
It doesn't. You're biased to results you find interesting.

If we measure a particular algorithm's space $s$ and time $t$ complexity, and then improve the algorithm. One of the following things can and does happen:


We reduce $s$ and leave $t$ unchanged.
We reduce $t$ and leave $s$ unchanged.
We reduce both $s$ and $t$.
We reduce $s$ and $t$ increases. WOW!
We reduce $t$ and $s$ increases. WOW!


Now 1, 2 and 3 happen all the time. Why are you not asking for a "mathematical principle" for them?

Because they are boring. Since we strictly improved the algorithm it must mean the previous algorithm was suboptimal, and is immediately forgotten. Only in cases 4 and 5 we suddenly have two algorithms which do not have a clear winner, and only then do you start comparing them and wondering about a "mathematical principle" behind this difference.



What's happening is that given two objectives, you get a plot of possible solutions:



Anything above and to the right of the critical red line is immediately forgotten as non-interesting, as there are algorithms that are strictly better in both $s$ and $t$. Your mind only looks at the red line (also known as the Pareto frontier), ignoring the rest of the plot.