The pace of scientific discovery in the last few decades has been extraordinary. We’ve discovered new particles; seen habitable planets orbiting distant stars; detected gravitational waves; mapped the complete neuronal network of a C Elegans worm; and built new forms of carbon called graphene. In 2014 the science journal Nature reported that the number of scientific papers published has been doubling every 9 years since the end of World War Two.  So is there anything science cannot answer? Or could we possibly know it all?

Identifying the known unknowns was the task I set myself on the journey I’ve been on for the last three years writing my new book What We Cannot Know. Not just things we don’t know now. I wanted to identify whether there are any problems that are intrinsically unanswerable. One of the motivations for my journey comes from a proof that my own subject of mathematics has provable limits.

Called Gödel’s Incompleteness Theorem, the discovery made by Austrian logician Kurt Gödel in the 1930s rocked the mathematical community to its core. Ever since the ancient Greeks introduced the powerful tool of mathematical proof, mathematicians believed that, given any true statement about numbers, mathematicians should be able to produce a proof from the axioms of mathematics of the truth of that statement.

For example, Fermat famously claimed that the equations xⁿ+yⁿ=zⁿ don’t have whole number solutions if n>2. He actually believed he’d found a remarkable explanation but alas the margin of the book he was scribbling in was too small for the proof. This turned out to be the biggest mathematical tease in history. It took another 350 years before my colleague in Oxford Andrew Wiles finally came up with a proof that indeed you can’t solve these equations.

I think most mathematicians believed that if you’ve got a true statement about numbers then there should exist some way within mathematics to prove that the statement is indeed true. For example, we believe that it is true that every even number can be written as the sum of two prime numbers. (Prime numbers are those indivisible numbers like 7 and 17.) Called Goldbach’s conjecture, no one has yet come up with a proof that this is true of all even numbers.

What Gödel proved is that within our axiomatic system for mathematics it may be possible that Goldbach’s Conjecture is true but there doesn’t exist a proof that it is true. All those mathematicians chasing a proof might be chasing a shadow. The shocking revelation is the inspiration for a wonderful novel called Uncle Petros and the Goldbach Conjecture by Greek author Apostolos Doxiadis. In the novel on learning about Gödel’s Theorem, Petros suffers a complete meltdown at the revelation that his life’s work trying to prove Goldbach might be in vain.

Mathematics has been able to prove its own limits of knowledge. By turning mathematics in on itself, Gödel showed that any system of mathematics must contain true statements that are unprovable. So could we apply a similar self analysis to science to reveal the limits of what we can know about our universe?

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"The idea of repeating an experiment, something so dear to the way we do science, seems actually impossible"
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