Even while considering the universe to be finite, one can do mathematics symbolically as a game with a system of rules. If the game doesn't have enough pieces we just add new pieces, with new properties or allowed moves as required. All that matters is that the enlarged system is compatible with the old system; that the smaller game is a subgame of the large one; that the smaller system can be embedded in the larger one.
When does something exist ? Well if there isn't something from amongst the objects under consideration that has the properties we want then we just create new symbols and define how they relate to the old ones.
If we were a pythagorean and the only numbers that exist are rational numbers, then we wouldn't call $\sqrt 2$ a number, but if we were also finitist symbolicists then we could embed any collection of numbers into a collection that contains not only numbers but also "splodges" which is what we're going to call $\sqrt 2$. It's important to always keep in mind that $\sqrt 2$ isn't a number - it's a splodge. In this new system of arithmetic we've invented we can add numbers to splodges to get new splodges like $1+\sqrt 2$. What a fun game. Let's add some more splodges. We're bored with algebraic splodges so let's add some non-algebraic splodges like the one in your question. Of course that expression is a bit cumbersome so we'll give it the shorthand symbol $\pi$ instead.
Given a splodge $x$, it would make calculus easier if there were a splodge $x+o$ that was nearer to $x$ than any other splodge, however that isn't possible so we embed the splodges in a larger system called the hypersplodges that contains not only splodges but also vapors, and contains not only the concept nearer but also the concept "nearer". Vapors like $x+o$ are "nearer" to $x$ than any splodge could ever be, and when you're finished using them they evaporate leaving just a splodge.
We want a splodge that satisfies $x^2+1=0$ however there isn't one, so we embed the splodges in a larger system called weirdums in which we've added a piece called $i$ with the rule that $i^2=-1$, and under the new system we can "add" splodges to weirdums to get new weirdums like $1+i$.
In solving differential equations we'd like a function which is zero everywhere except at a single point but which has a non-zero area under the curve. There is no function that behaves like this so we'll go to a larger system that contains not only functions but also spikes which do have the desired property because the larger system contains a rule about spikes which says they do. Conveniently certain calculations involving spikes cancel out leaving just functions.
A finitist or ultrafinitist shouldn't recognise the concept of infinite sets therefore the only sets are finite-sets and since all sets are finite the adjective finite is superfluous therefore from this point onwards we just use the term "set". Some people want to consider sets that contain things they haven't put in there themselves - which of course can't be done because a set only contains the items we've put there. So we embed the system of sets in a larger system that includes not only sets but also dafties. In this daft system the rules are that a daftie can have an "affinity" for things whether those things have been previously mentioned or not. Dafties have an affinity for things in the same way that sets contain things. A compatible embedding of a system of sets in the daft system means that a set has an affinity for the items it contains when the set is considered as part of the daft system, therefore by daft reasoning one can say things not only about dafties but also about sets. To each set one can attach a number. You can't do this with dafties so we embed the numbers in a larger system containing sinners and attach a sinner to each daftie. Sometimes there is a need for something that looks like a daftie but has no sinner - such things are called messes. A mess can have an affinity for collections of dafties that no daftie could have an affinity for. This could go on, but you need gobbledegook theory. The set system has zero gobbledegook. The daft system is level-1 gobbledegook. The messy system is level-2 gobbledegook. Finitists like to maintain a zero level of gobbledegook. Analysts are usually happy with one-level of gobbledegook and category-theorists are comfortable with any amount of gobbledegook.
There are two ways to compatibally extend a system:
1) a conservative extension adds new items but doesn't say anything new about the old items that couldn't be said before;
2) a progressive extension does say new things about the old items but only about things that were previously undecidable
P.S. We can combine the vapors, splodges and linedups in a system called the messysplodges but they haven't been studied much because they're a bit messy.