According to [constructivism](http://en.wikipedia.org/wiki/Constructivism_(mathematics)), "it is necessary to find (or "construct") a mathematical object to prove that it exists". There are several formulas to calculate `pi`, such as: 

    
![formula for pi](https://i.sstatic.net/SI96R.gif) [(image source)](http://mathworld.wolfram.com/images/equations/PiFormulas/NumberedEquation3.gif)

so I take it `pi` exists according to constructivism.

According to [finitism](http://en.wikipedia.org/wiki/Finitism), which is a form of constructivism, "a mathematical object does not exist unless it can be constructed from natural numbers in a finite number of steps."

Where does that leave irrational numbers, such as `pi`? Do they simply not exist according to finitism? How does one reason about the ratio between a circle's circumference and its diameter, if one is working within a finitistic/ultrafinitistic framework?