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FQXI ARTICLE

February 6, 2023

True Lies: Why Mathematics is an Illusion

To find a theory of quantum gravity we may have to look through a different logical lens, abandoning conceptions of "truth" and "falsehood" and crossing over to a new "mathematical universe."

FQXi Awardees: Christopher Isham, Andreas Döring

November 30, 2009

CHRISTOPHER ISHAM

Imperial College London

Their journey together began five years ago. In September 2004, Isham was already a prominent name in theoretical physics, researching deep conceptual questions at Imperial College in London. He was wrestling with some of the most profound mysteries of modern physics, including the quest for a theory that unifies general relativity with quantum mechanics. His work caught the attention of Döring, then still a PhD student at the Johann Wolfgang Goethe-University in Frankfurt, Germany.

Specifically, Isham and a few like-minded colleagues were attempting to use

We need to change

mathematics itself,

the whole whack,

completely.

mathematics itself,

the whole whack,

completely.

- Christopher Isham

Isham remembers liking Döring right from the start. "I was very impressed with him, both as a person and a researcher," he says. "He’s obviously very bright, but he’s also a nice person. I put a lot of stock in a person’s personality."

The young scientist came to London and started working at Imperial a year later, in December 2005. The two began meeting weekly, and a collaboration into topos theory was underway.

Dynamic Duo

Isham and Döring’s working relationship was not without obstacles, however. Isham suffers from a neurological disease and needs a cane to walk, so he prefers to work from his home in London when possible. So Döring would swing by Isham’s apartment for intense brainstorming sessions that begin with tea and chatter before turning to physics.

TOUCHSTONE OF REALITY

Isham (left) and Döring (right)

In contrast to Isham and Döring’s dynamic partnership, the duo are working to bring together two of the biggest theories in physics that have been notoriously hard to unite. General relativity, which explains the interactions of cosmic bodies, and quantum mechanics, which defines interactions at the subatomic level, have both proven successful at their separate ends of the universe, the very large and the very small. But the two theories have so far refused to mesh with each other. A unified theory, the so-called Holy Grail of modern physics, would either meld the two together or replace them with something that works at both extremes.

For more than a dozen years now, Isham has been convinced that topos theory might provide the answer. Don’t confuse one type of theory with another, though. Unlike general relativity or quantum mechanics, topos theory isn’t a theory about

"What Andreas and I have been arguing is that we need to change mathematics itself to solve this problem," Isham says. "I don’t mean to find a different branch of mathematics. I mean actually change mathematics, the whole whack, completely."

True or False?

Reconstructing the rules of logic sounds mind-bending—and it is. But quantum theory wreaks havoc with logic at the best of times. For example, it suggests that the universe isn’t the solid construct it seems but actually an array of possibilities collapsed into a single reality by the act of observation. So maybe mind-bending concepts are exactly what’s called for.

BENDING THE TRUTH

Notions of "truth" depend on which topos you look through.

Credit: Stepan Popov

Topos theory may be able to incorporate this murky quantum logic. It underlies the laws of mathematics and logic that we use, and is more fundamental than either, the physicists argue. Each topos describes a different "mathematical universe," with its own mathematical constructions and logic. And each topos can be used as a different lens through which to view the world, depending on requirements.

Isham and colleagues have identified a topos in which quantum theory appears to make logical sense—as long as you embrace a new type of logic, in which "true" and "false" are no longer your only options. There are now multiple shades in between. A different topos recreates classical reality with its firm "yes" and "no" answers. Isham and Döring believe that every physical system, from atomic particles to the universe as a whole, can be viewed through different topoi.

"In a sense, using a topos

The work has won fans. "It’s really quite wonderful," says John Baez, a physicist at the University of California, Riverside.

So how do you keep grounded, when you spend your days rewriting reality? It helps that Isham’s background was originally in electrical engineering. "It’s very easy to get seduced," Isham says. "The mathematics is very intriguing, but we have to keep in mind that we’re working on theoretical physics."

Staying focused on the ultimate purpose of your intellectual endeavor is one of the many lessons that mathematician Döring has learned from working with Isham, who often reminds him to "think like a physicist."

Döring now lectures at Oxford University, UK, and shares these lessons with his students. His move makes those weekly meetings with Isham in London harder. "I strongly hope to keep up the collaboration with Chris in some form, even though the style will surely change," Döring says.

In other words, it’s not a breakup. It’s the beginning of a long-distance relationship.

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THOMAS HOWARD RAY wrote on August 15, 2014

Having just dropped 600 bucks on a new set of Firestone tires, I appreciate that more rubber is reaching the road than before. :-)

Having just dropped 600 bucks on a new set of Firestone tires, I appreciate that more rubber is reaching the road than before. :-)

THOMAS HOWARD RAY wrote on August 15, 2014

We agree, Rob.

Mathematical completeness is necessary, though not sufficient.

As you say, "It is necessary, that there also be a demonstrable, one-to-one correspondence, between 'every' physical entity/property, and each mathematical symbol, relevant to the physical phenomenon being described."

A mathematically complete theory, then, is one that guarantees measured correspondence between symbol and physical result.

"Demonstrating such a one-to-one correspondence...

We agree, Rob.

Mathematical completeness is necessary, though not sufficient.

As you say, "It is necessary, that there also be a demonstrable, one-to-one correspondence, between 'every' physical entity/property, and each mathematical symbol, relevant to the physical phenomenon being described."

A mathematically complete theory, then, is one that guarantees measured correspondence between symbol and physical result.

"Demonstrating such a one-to-one correspondence...

ROBERT H MCEACHERN wrote on August 15, 2014

Should have said "where the rubber meets the road"

It would be nice if one could edit.

Rob McEachern

Should have said "where the rubber meets the road"

It would be nice if one could edit.

Rob McEachern

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