The debate of realism and anti-realism

Gottlob Frege

The secular debate between realism and nominalism (or anti-realism, its now preferred name), has been expressed in a few new theories of the so-called analytical philosophy, whose origin dates from the early twentieth century, with Gottlob Frege, Bertrand Russell, Ludwig Wittgenstein, the Circle of Vienna and several philosophers of the last fifty years, especially in the Anglo-Saxon world.

Currently, the two camps, realist and anti-realist, agree on one thing: science works. But although this is considered an incontrovertible fact, very divergent positions are posed to explain it.

As it has always happened throughout history, neither of the two fields is united. Both realism and anti-realism are divided into two branches, at the least.

Let us start by describing the realist position:

Gödel and realism

Kurt Gödel

Kurt Gödel (1906-1978) was one of the most important mathematicians of the 20th century. In 1931, when he was 25, he rose to fame with his mathematical proof that the attempt to build a complete axiomatic system, from which one can deduce all the arithmetic of natural numbers or any equivalent system, is doomed to failure.

His first incompleteness theorem says the following:

Every consistent formal system as powerful as elementary arithmetic is not complete (it contains true undecidable propositions).

Let us look at a simplified informal demonstration:

Let theorem G say the following: This theorem G cannot be proved from the axioms and rules of system S.

• If we assume that Theorem G is false, system S is inconsistent, since a false theorem can be proved from the axioms and rules of S.
• Then if S is consistent, G must be true, and therefore cannot be proved from the axioms of S.
  

Gödel’s theorem shows that every axiomatic formalization of arithmetic is either inconsistent (it allows false theorems to be proved), or incomplete (it contains true theorems that cannot be proved).

The mystery of the Great Pyramid

The Great Pyramid of Giza, also called Pyramid of Cheops or Pyramid of Jufu, was built to be the tomb of the pharaoh Jufu (called by the Greeks Cheops), of the fourth dynasty, the high point of the Ancient Egyptian Empire. The reign of Jufu is usually dated in the 26th century before Christ, over 4500 years ago.

The current height of the Great Pyramid is 138.8 meters, but the pyramid is truncated, having lost its top. It is easy to calculate that its original height was about 8 meters higher: 146.7 meters, or 280 Egyptian cubits. The base of the pyramid is a square with a side of 230.34 meters, or 440 Egyptian cubits.

Observe a curious point: the semi-perimeter of the pyramid (twice the side of the base) is equal to 880 cubits. If we divide it by the height of the pyramid, we get the following:

(880/220) = (22/7) = 3.142857...

Are the digits of Pi real?

Martin Gardner

In an article published in Discover magazine in 1985, Martin Gardner wrote this:

As it happens, the thousandth decimal of pi is 9... The question: Was [this assertion] true before the 1949 calculation? To those of the realist school, the sentence expresses a timeless truth whether anyone knows it or not... [Others] prefer to think of mathematical objects as having no reality independent of the human mind.

This problem is quite old, as we have been discussing it for over two thousand years. The question about whether mathematical objects really exist or are a pure creation of our mind is a particular case of another problem, much more general, that debates whether ideas and concepts (like the dog species) really exist, or just this dog and that dog exist. This is the problem of universals, famous in the Middle Ages, which has not yet been solved to everyone’s satisfaction. In fact, at present, this debate is more virulent than ever.