What does “equals” mean? For mathematicians, this simple question has more than one answer, which is causing issues when it comes to using computers to check proofs. The solution might be to tear up the foundations of maths.

When you see “2 + 2 = 4”, what does “=” mean? It turns out that’s a complicated question, because mathematicians can’t agree on the definition of what makes two things equal.

While this argument has been quietly simmering for decades, a recent push to make mathematical proofs checkable by computer programs, called formalisation, has given the argument new significance.

“Mathematicians use equality to mean two different things, and I was fine with that,” says Kevin Buzzard at Imperial College London. “Then I started doing maths on a computer.” Working with computer proof assistants made him realise that mathematicians must now confront what was, until recently, a useful ambiguity, he says – and it could force them to completely redefine the foundations of their subject.

The first definition of equality will be a familiar one. Most mathematicians take it to mean that each side of an equation represents the same mathematical object, which can be proven through a series of logical transformations from one side to the other. While “=”, the equals sign, only emerged in the 16th century, this concept of equality dates back to antiquity.

It was the late 19th century when things began to change, with the development of set theory, which provides the logical foundations for most modern mathematics. Set theory deals with collections, or sets, of mathematical objects, and introduced another definition of equality: if two sets contain the same elements, then they are equal, similar to the original mathematical definition. For example, the sets {1, 2, 3} and {3, 2, 1} are equal, because the order of the elements in a set doesn’t matter.

But as set theory developed, mathematicians started saying that two sets were equal if there was an obvious way to map between them, even if they didn’t contain exactly the same elements, says Buzzard.

To understand why, take the sets {1, 2, 3} and {a, b, c}. Clearly, the elements of each set are different, so the sets aren’t equal. But there are also ways of mapping between the two sets, by identifying each letter with a number. Mathematicians call this an isomorphism. In this case, there are multiple isomorphisms because you have a choice of which number to assign to each letter, but in many cases, there is only one clear choice, called the canonical isomorphism.

Because a canonical isomorphism of two sets is the only possible way to link them, many mathematicians now take this to mean they are equal, even though it isn’t technically the same concept of equality that most of us are used to. “These sets match up with each other in a completely natural way and mathematicians realised it would be really convenient if we just call those equal as well,” says Buzzard.

Having two definitions for equality is of no real concern to mathematicians when they write papers or give lectures, as the meaning is always clear from the context, but they present problems for computer programs that need strict, precise instructions, says Chris Birkbeck at the University of East Anglia, UK. “We’re finding that we were a little bit sloppy all along, and that maybe we should fix a few things.”

To address this, Buzzard has been investigating the way some mathematicians widely use canonical isomorphism as equality, and the problems this can cause with formal computer proof systems.

In particular, the work of Alexander Grothendieck, one of the leading mathematicians of the 20th century, is currently extremely difficult to formalise. “None of the systems that exist so far capture the way that mathematicians such as Grothendieck use the equal symbol,” says Buzzard.

The problem has its roots in the way mathematicians put together proofs. To begin proving anything, you must first make assumptions called axioms that are taken to be true without proof, providing a logical framework to build upon. Since the early 20^{th} century, mathematicians have settled on a collection of axioms within set theory that provide a firm foundation. This means they don’t generally have to use axioms directly in their day-to-day business, because common tools can be assumed to work correctly – in the same way you probably don’t worry about the inner workings of your kitchen before cooking a recipe.

“As a mathematician, you somehow know well enough what you’re doing that you don’t worry too much about it,” says Birkbeck. That falls down, however, when computers get involved, carrying out maths in a way that is similar to building a kitchen from scratch for every meal. “Once you have a computer checking everything you say, you can’t really be vague at all, you really have to be very precise,” says Birkbeck.

To solve the problem, some mathematicians argue we should just redefine the foundations of mathematics to make canonical isomorphisms and equality one and the same. Then, we can make computer programs work around that. “Isomorphism is equality,” says Thorsten Altenkirch at the University of Nottingham, UK. “I mean, what else? If you cannot distinguish two isomorphic objects, what else would it be? What else would you call this relationship?”

Efforts are already under way to do this in a mathematical field called homotopy type theory, in which traditional equality and canonical isomorphism are defined identically. Rather than trying to contort existing proof assistants to fit canonical isomorphism, says Altenkirch, mathematicians should adopt type theory and use alternative proof assistants that work with it directly.

Buzzard isn’t a fan of this suggestion, having already spent considerable effort using current tools to formalise mathematical proofs that are needed to check more advanced work, such as a proof of Fermat’s last theorem. The axioms of mathematics should be left as they are, rather than adopting type theory, and existing systems should be tweaked instead, he says. “Probably the way to fix it is just to leave mathematicians as they are,” says Buzzard. “It’s very difficult to change mathematicians. You have to make the computer systems better.”

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*Credit for article given to **Alex Wilkins***