Does Pi exist?

If you listen to talks of Normal Wildberger on Youtube you might get confused about the foundations of mathematics. And indeed the topic is confusing. The reason for this is that the problems in the foundations are not really solved in some conclusive way, let alone elegantly. Our textbooks start at a point where the theory seems complete and indisputable. That is a practical standpoint. The fact that it works for practical problems justifies it. Yet, shouldn’t we at least glimpse under the hood?

To pinpoint the topic, let me ask you in what way the number Pi really exists? First of all, it must exist in some way. Otherwise, I could not talk about it here. If we take the geometric viewpoint we can define Pi as the area of the circle of radius 1. In analytic terms

\(\pi = 2 \int\limits_{-1}^1 \sqrt{1-x^2} \, dx\)

The integral on the right hand side can be written as a limit of Riemann sums, and it seems we have precisely defined Pi and proven its existence.

But wait! The limit of such a series does only exist if the real numbers do exist. Here, it is not clear what „exists“ means. We do at least want no contradictions in the theory. Now Gödel tells us that it is impossible to prove that there is no contradiction, even in the smaller field of arithmetic. So we are indeed on swampy grounds. To be fair, efforts have been made to base the real numbers on constructive methods. But that gets very complicated, and in any case the simple notion of Pi has to be replaced by some limit process.

Is it so bad to have a limit process? After all, we have computers and can compute the value of Pi to any desired number of places. Wildberger makes the case that „any“ is asking a bit too much since our world is finite. We probably need at least n computer steps to produce the n-th digit of Pi for all n (this is not known). So we will never know the n-th digit, if n is larger then the number of steps we can take in the space of our universe. But let us avoid such practical considerations and assume that we have potentially infinite time and space.

Then indeed, we can compute an interval as small as we want which contains Pi. Just take an upper and a lower Riemann sum of the integral. As a side remark, this does not mean that we can compute n digits of any well defined real number for any n. It could be that the decision if the number is rational or not needs infinite steps. For Pi, however, it can be done. Only that we might have to compute much more than n digits to be sure in which way to round for n digits. The devil is in the details.

The power of the field of Analysis as it stands today is that we are able to prove theorems about Pi. E.g., we can derive other limits, which are equal to Pi. Or we can prove

\(e^{i\pi} + 1 = 0,\)

with the meaning, that if we insert i times Pi into the complex series of the exponential function and add 1 to it, the series converges to 0. This can all be checked by computer programs. Or rather the other way around. It predicts the behavior of numerical computations.

And predicting what happens is the finest we can get from science.


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