Everything we know about maths could be wrong

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

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

Professor Kevin Buzzard, a mathematics expert from Imperial College London, has something playing on his mind.

“I’m suddenly concerned that all of published maths is wrong because mathematicians are not checking the details,” he explains.

The problem 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 on Earth knows if they are correct.

A senior mathematical expert might list 20 established proofs in one paper. When the paper is published, other mathematicians assume that these proofs are correct without checking.

One tiny mistake in a single proof could bring huge systems of advanced maths crashing down. But how could we check?

Scientists have developed a computer that can follow the logical steps in a proof to show if it is true.

However, Professor Michael Harris of Columbia University worries that the machine will take the joy and creativity out of his field.

Tech companies — motivated by profit — may want to use the software to ensure their programs are bug-free, but mathematicians like Harris do their work out of love. With the computer, there is little need for mathematicians.

Doesn’t add up?

Is maths an art or a science? Sciences use logic and reason to tell us about the world around us, and this is exactly what maths does. That is why many mathematicians view the subject 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.”

You Decide

  1. Is maths an art or a science?

Activities

  1. Draw a picture using a mathematical pattern. Read the piece from The Guardian in Become An Expert for inspiration.

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.
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”.

Word Watch

Theorem
A statement or idea that is proved through reasoning.
G.H. Hardy
An English mathematician known for his work on number theory.

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