There’s no way to predict what the next unsolved pi digit will be just by looking at what came before it. It’s neither predictable nor deterministic. The very existence of calculations to get the next digit supports that.

Note: I’m not saying Pi is random. Again, the calculations support the general non-randomness of it. It is possible to be unpredictable, undeterministic, and completely logical.

Note Note: I don’t know everything. For all I know, we’re in a simulation and we’ll eventually hit the floating point limit of pi and underflow the universe. I just wanted to point out that your example doesn’t quite fit with pi.

I mean, unironically yes. It seems the most popular stance is that all math regardless of how weird is Platonically real, although that causes some real bad problems when put down rigorously. Personally I’m more of an Aristotelian.

In the case of things like rational or real numbers, they have a counterpart that’s weirder (irrational and imaginary numbers). For the rest I’m not sure, but it’s pretty common to just pick an adjective for a new concept. There’s even situations where the same term gets used more than once in different subfields, and then they collide so you have to add another one to clarify.

For example, one open interval in the context of a small set of open intervals isn’t closed analytically under limits, or algebraically closed, but is topologically closed (and also topologically open, as the name suggests).

We’re so used to dealing with real numbers it’s easy to forget they’re terrible. These puppies are a particularly egregious example I like to point to - functions that preserve addition but literally black out the entire x-y plane when plotted. On rational numbers all additive functions are automatically linear, of the form mx+n. There’s no nice in-between on the reals, either; it’s the “curve” from hell or a line.

Hot take, but I really hope physics will turn out to work without them.