The Foundations of Mathematics (Part 1)

Hello Reader,

In this post I would like to show you what mathematics actually is. Mathematicians on the whole do try to avoid philosophical questions, but in this particular case, the question required answering for mathematics to progress. To understand why, we must return to the mathematical world of the 20^th century and a legendary German mathematician called David Hilbert.

In the 20^th century, thanks in part to the work of Hilbert, mathematicians were also beginning to ask the same question we are: What is mathematics? How do we know that anything we prove in mathematics is true?

When we prove something in mathematics, we inevitably make some assumptions. For example, in Euclid’s proof of the existence of infinite primes, he assumed the fundamental theorem of arithmetic. Of course, we can then prove the fundamental theorem of arithmetic, but then we make some assumptions about other, even more basic things. As we go further and further down the rabbit hole of provable statements, we eventually come to statements called axioms, which are essentially unprovable statements that are so obviously true that we just assume they are. For example, the axiom of equality states that x = x.

So, in essence, mathematics is the search for logical conclusions from a set of presumed truths called axioms....but, unfortunately, it gets more complicated than that.

It wasn’t until the early 20^th century that mathematicians really began to question the foundations of mathematics. After all, based on the above definition of maths, the entire subject is like a tower of cards, each card depending on others for support all the way down to the base cards (the axioms). But what if these base cards were shaky...the entire tower could come crumbling down. This period became known as the foundational crisis of mathematics.  Mathematicians desperately began to try to clarify the foundations of mathematics, but each attempt seemed to find paradoxes and inconsistencies. Many feared the end of mathematics was nigh.

David Hilbert, who by this time was a renowned and highly respected mathematician, set up the Hilbert Program. The aim of this program to strengthen the foundations of mathematics by creating a finite system of axioms that could be shown to be consistent (they would never contradict on another) and complete (all statements about mathematics could be shown to be true from this set of axioms). Hilbert essentially wanted to create a foundation for the tower of cards that would never crumble or have to be re-laid. Wouldn’t that have been nice...

Unfortunately for Hilbert, another German mathematician by the name of Kurt Godel was about to utterly destroy the Hilbert Program by showing that what it hoped to achieve was in fact impossible. In 1931, Godel proved two fundamental truths about mathematics:

1)      Within any axiomatic system with a finite set of axioms, there exist statements which cannot be proved true or false within the system (ie. No matter how many axioms we have, there will always be unsolvable problems in mathematics)

2)      For any axiomatic system, if the system includes a statement of the system’s own consistency, the system is inconsistent (ie. The axioms of maths cannot be proved to be consistent using maths)

To say this shook the very core of mathematicians around the world would be an understatement. Many mathematicians’ worst fears had been realised, that mathematics could all be somehow untrue. Thankfully, as is often the case with revelations as large as this, the reaction was a little overstated.

The next post will explain what followed Godel’s revelations, how the crisis was solved (actually leading to mathematics rising from the ashes stronger than ever before) and maybe some notes on a truly mind-boggling but relatively uncomplicated, personal favourite topic of mine.

Thanks for reading J