We can treat π purely symbolically if we like, but as soon as we want to do anything useful with it we need some kind of approximation or algorithm.
For example, if someone asks you to explain Euclid's proof of the infinitude of the primes, and you say that Euclid did not provide any such proof and nothing more, I think it's quite disingenuous. It would be more proper to say, from a constructivist view, the argument Euclid made isn't a valid proof, and then either explain the proof in the logical context in which it was made or decline to.
In this case, the point of discussion was separating the definition of multiplication from an algorithm implementing it. It's quite unfair to silently take a position that a mathematical definition without an algorithm isn't valid or meaningful and then on that basis argue that only numerical approximations to transcendentals have meaning.
So many common mathematical concepts such as "the integers" have no meaning in a constructivist approach that it's not sensible to engage in mathematical discussion without establishing that one's fundamental basis of approach varies so widely from the common one.
For example if we want to know π^2 to about 3 decimal digits (~9.87), we can start by using an approximation of π to about 4 decimal places (~3.142) and then multiplying the two decimals.
For rational numbers, multiplication algorithms are usually built on breaking a number down into constituent pieces, multiplying every pair of pieces from the two multiplicands, and then adding up all of the partial products.
Matrix multiplication has the additional complication that the elementary terms involved (entries in different places in the matrix) cannot be added to each-other. But the basic procedure is still the same: break the two multiplicands down into basic units which we already know a multiplication table for, compute all of the partial products, then sum them up.