Everything we know about maths could be wrong

Figure it out: A single proof might contain 1,000 pages of dense workings-out. © Alamy

Is maths best described as an art or a science? A top academic fears that modern theorems are riddled with errors. Computers could fix them. But if they did, would the result still be maths?

When Professor Kevin Buzzard opened the 10th Interactive Theorem Proving Conference before an audience of expert mathematicians the other day, little did they know that he was about to tell them something that would turn their world upside down.

“I’m suddenly concerned that all of published maths is wrong,” explained the number theorist. “Because mathematicians are not checking the details.”

The problem, he said, lies in mathematical proofs. A proof is a mathematical statement which logically demonstrates that a theorem is true.

However, Buzzard believes that proofs have become so complex that no one can understand them — or really knows if they are correct.

A senior mathematical expert might cite 20 established proofs in one paper. When the paper is published, other mathematicians take it for granted that these proofs are correct without really checking, particularly when a single proof might contain 1,000 pages of dense reasoning.

Buzzard uses the example of Fermat’s Last Theorem, once listed by the Guinness Book of World Records as the world’s “most difficult maths problem”.

"I believe that no human, alive or dead, knows all the details of the proof of Fermat’s Last Theorem. But the community accepts the proof nonetheless,” he says.

Because theorems depend upon each other like building blocks, one tiny mistake in the foundations could bring the whole edifice crashing down.

So, how could we know if there was a mistake?

Scientists have developed a proof verification software called Lean. The computer programme can follow the logical steps in a proof conclusively to determine if it is true.

Computer scientists are working on “a general automated theorem prover”, which could create its own proofs and develop its own theorems. In the future, maths could be entirely automated.

But Professor Michael Harris of Columbia University worries that such a machine would take the creativity out of his field. And without creativity, it would not really count as maths.

Mathematics is “wildly creative”, says Professor Anthony Bonato of Ryerson University in Toronto. “There are elements of both art and science in the field, but it isn’t a subset of either.”

Granted it isn’t either. But which best describes the essential process: art or science?

Doesn’t add up?

Sciences use logic and rationality to tell us about the world around us, and this is exactly what maths does. While numbers do not physically exist, we use them to describe the natural world and the intricate patters found in it. This is why many mathematicians view the field as a discovery, rather than a human invention. Like all science, mathematics is a body of provable knowledge.

But as G.H. Hardy wrote in his Apology, maths — like the greatest art — is concerned with symmetry and beauty. “A mathematician, like a painter or a poet, is a maker of patterns […]. Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere […], sublimely pure, and capable of a stern perfection such as only the greatest art can show.” A mere computer would not get far with this.

You Decide

  1. Is maths an art or a science?
  2. Is maths a discovery or an invention?

Activities

  1. Draw a picture using a mathematical pattern. Read the piece from The Guardian in Become An Expert for inspiration.
  2. Research a famous mathematician. Make a one-page profile with details about their life, and sum up their area of study in a few sentences.

Some People Say...

“Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

Bertrand Russell (1872-1970), British philosopher and mathematician

What do you think?

Q & A

What do we know?
The 10th Interactive Theorem Proving conference was held in Portland, Oregan last month. Kevin Buzzard, Professor of Pure Mathematics at Imperial College London, delivered the opening talk. In an interview at the conference, the algebraic number theory specialist argued that some of the mathematical proofs that mathematicians take for granted, and base their theorems on, may be wrong.
What do we not know?
If maths is really an art or a science: it is open to interpretation. In the Merriam-Webster dictionary, mathematics is defined as “the science of numbers and their operations, interrelations, combinations, generalisations, and abstractions”.

Word Watch

Theorem
A statement or hypothesis that is proved through a chain of reasoning.
Fermat’s Last Theorem
According to the Encyclopedia Britannica, the theorem states that “there are no natural numbers (1, 2, 3,…), x, y, and z such that xn + yn = zn, in which n is a natural number greater than 2.”
Edifice
A large, imposing building; a complex system of beliefs.
Automated
Operated by automatic equipment; without human involvement.
G.H. Hardy
An English mathematician known for his work on number theory. This quote is taken from his 1940 essay, A Mathematician’s Apology.

Subjects

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