The man who ruined mathematics

Kurt Gödel, the man who ruined mathematics, was one of the most important thinkers of the 20th century. He was born in 1906, smack-bang in the middle of the greatest crisis that maths has ever known. Just a few decades later, he would help resolve this turmoil, but in doing so doom mathematicians to a smaller world than the one that came before.

Mathematics, as an intellectual framework, is incredibly powerful. The entire point is taking one set of logical ideas and using them to build another, making maths the closest thing we have to a cognitive perpetual-motion machine – there is always a new mathematical idea lurking across the horizon, and we just need to assemble the steps to get there. Or so it might seem. But in reality, there is a dark fundamental truth at the heart of mathematics that places limits on our intellectual exploration. It is called Gödel’s incompleteness theorem.

The story of this theorem begins in the late 19th century, when mathematicians had begun to unpick the foundations of their subject and quickly discovered that the intellectual edifice of the past 3000 years or so was built on quicksand. Unruly paradoxes began to crop up, and mathematicians were plunged into panic.

As the century turned, one man decided to fight the chaos. Taking the stage at a Paris conference in 1900, mathematician David Hilbert presented a list of 23 unsolved problems in mathematics, laying the groundwork for a research programme that would occupy mathematicians for much of the 20th century. “As long as a branch of science offers an abundance of problems, so long is it alive,” he told the assembled crowd.

The task that would later come to occupy Gödel was Hilbert’s second problem. It concerns the axioms of a given mathematical arena, essentially the assumptions that serve as the rules of the game and allow you to make logical derivations from them. Hilbert’s challenge to his fellow mathematicians was to prove that the axioms of arithmetic, specifically, “are not contradictory, that is, that a finite number of logical steps based upon them can never lead to contradictory results.”

This is a very desirable thing to prove. Imagine playing a board game where one interpretation of the rules gains you points, while another, equally valid interpretation of the same rules can lose you points. Such a game would become, well, pointless.

For the next few decades, Hilbert and close colleagues attempted to chip away at his second problem, developing what was known as Beweistheorie, or “proof theory”, a way of turning proofs into mathematical objects. While a proof is normally a collection of natural language words and mathematical symbols, this transformation into more abstract mathematical concepts allowed Hilbert and his associates to study proofs themselves with the tools of mathematics – a bit like a recipe book that contains a recipe for making recipes. In 1928, he gave a lecture entitled Die Grundlagen Der Mathematik (“The Foundations of Mathematics”), explaining that this new method would allow him “to definitively resolve the fundamental questions in mathematics by transforming every mathematical statement into a concretely demonstrable and rigorously derivable formula”, though he admitted that “a great deal of work will still be necessary”.

By this point, Gödel was a 22-year-old PhD student at the University of Vienna, Austria. He was working under the supervision of mathematicians who followed Hilbert’s programme, although we have no historical evidence showing that the two men ever met or directly corresponded. A year later, as part of his PhD thesis, Gödel published his completeness theorem – a good step towards Hilbert’s goals.

The completeness theorem concerns models of sets of axioms, with these models being the mathematical understanding that links a collection of symbols like “2”, “+” or “=” to the actual mathematical objects they describe. This is quite abstract, so it is worth running through a small example. Imagine our axioms are “there are two things” and “things are different”. These are not very powerful axioms – you can’t prove much with just these two alone – but they are perfectly valid. We can apply many different models to these axioms, such as the faces of a coin (heads or tails), your hands (left or right) or even just numbers (0 and 1). Although these models appear different, they describe the same mathematical object – a collection of two distinct things.

What is important is that you can apply many different models to the same set of axioms, and what Gödel proved is that any statement that is true in all possible models of a set of axioms must therefore be provable from those axioms. That might sound slightly circular, as things often are when we delve into the bowels of mathematical definitions, but it was promising for Hilbert’s effort to firm up the foundations of mathematics.

Not that Hilbert seemed to notice, mind you. Gödel presented his completeness theorem on 6 September 1930 at conference in Königsberg (today known as Kaliningrad in Russia). Hilbert was at a different conference in Königsberg and gave a grand speech on 8 September, in which he famously rejected the idea that there are limits to human knowledge. “We must know. We will know,” he said – words that were eventually engraved on his tombstone.

There is just one problem with Hilbert’s rallying cry to mathematicians – Gödel had already destroyed all hope of it the day before. Not on 6 September, when he presented his completeness theorem, but on 7 September. During a discussion with fellow logicians that day, Gödel let slip that he had identified the possibility of “undecidable” statements – ones that cannot be proven true given a certain set of axioms, but crucially cannot be proven false either. This was the genesis of an idea that would limit the horizons of mathematics forever.

It is tempting to imagine Gödel in the audience of Hilbert’s talk, silently chuckling to himself, though we have no evidence this ever happened – the two conferences were in different parts of the city. What we do know is that Gödel published his incompleteness theorem – a dark mirror to his PhD thesis – a few months later in January 1931.

This theorem makes two important claims, worth examining separately. The first is exactly what Gödel came out with in the 7 September discussion – that whatever your axioms, there will always be problems that are undecidable within those axioms. These are a bit like a mathematical version of the paradoxical phrase “this sentence is false”, a statement that renders itself neither true nor false. Mathematicians now call this finding about undecidable problems Gödel’s first incompleteness theorem, and it is still relevant nearly a century later – here’s a fun story I wrote about computer programs with the theoretical potential to break mathematics, all because of undecidable problems.

The first incompleteness theorem is a major rewriting of our understanding of what mathematics can do, but it was what we now call Gödel’s second incompleteness theorem that really threw Hilbert on the ropes. That’s because Gödel showed that any sufficiently powerful set of axioms (essentially, the ones mathematicians care about) can never be used to prove that those same axioms won’t produce inconsistencies.

To go back to the board game analogy, you can read the rules all you want, but you can never be sure they won’t produce contradictory results. An assurance against contradiction is exactly what Hilbert was seeking for the axioms of arithmetic – and Gödel showed that exactly this problem is undecidable. There is a get-out clause: if you shift to another set of axioms, you can potentially prove the consistency of your previous axioms. But this doesn’t solve the problem, because there will now be other inconsistencies in your new axioms. Instead of chasing infinite mathematical horizons, mathematicians must be content with the unknowable.

So how did Hilbert react to this earth-shattering news? Incredibly, he didn’t, at least not publicly. According to Gödel’s biographer, John Dawson, we know that Gödel sent a draft of his paper to Hilbert’s assistant and close collaborator Paul Bernays, who acknowledged receipt, and later sent copies of the final published paper.

Dawson says Gödel’s results “provoked Hilbert’s anger”, but the one and only time Hilbert ever put pen to paper in response to Gödel didn’t come until 1934. “The view, which temporarily arose and which maintained that certain recent results of Gödel show that my proof theory can’t be carried out, has been shown to be erroneous,” Hilbert wrote in a book co-authored with Bernays.

In other words, poor Gödel never got a proper response from Hilbert after essentially destroying the latter’s vision of mathematics as an infinite engine for knowledge. Perhaps Hilbert simply couldn’t bring himself to accept it. Gödel won in the end – incompleteness is accepted as part of the mathematical canon, with the resulting limits to mathematics leaving us both richer and poorer for it. Despite that, I can’t help but wonder whether being spurned by Hilbert left Gödel himself feeling incomplete.