Embracing infinity: could surreal numbers shape the future of physics?
Imagine Earth were to shrink to the size of a marble. We might be in trouble, but the planet would continue its smooth course around the sun while the moon would maintain its orbit, circling Earth once a month.
Isaac Newton
proved Earth's gravitational pull would be the same even if all the mass were concentrated in a single point. But the density at that point would be infinite, a condition physicists and mathematicians call a singularity.
Such singularities are found in black holes, stars that have collapsed under their own weight. According to general relativity, mass concentrations curve space-time, inducing the force of gravity. With enough matter in a small enough volume, gravity becomes infinitely strong.
In 1916, just months after
Albert Einstein's
general relativity appeared, Karl Schwarzschild discovered a solution of the equations with a singularity. Decades later, this idea led to the theory of black holes, crushed stars with spherical boundaries that trap anything falling inside, including light rays.
READ MORE
There is now abundant evidence that black holes exist, but do they really represent space-time singularities? Most physicists believe the singularities are mathematical artefacts, and would vanish in a more fundamental theory incorporating quantum effects.
Physical equations enable us to predict the future, but singularities imply a lack of predictability; theory just breaks down. It was hoped that quantum effects would eliminate infinities, but current versions of quantum gravity are plagued with singularities. It seems that infinite quantities are inherent and unavoidable.
[
Beyond the big bang: Irishman's universal evolution theory challenges accepted cosmology
Opens in new window
]
German physicist Hermann Weyl opened his essay, Levels of Infinity, with the statement 'mathematics is the science of the infinite'.
Infinity is at the core of mathematics. We can gain a first impression of it by placing all the counting numbers, 1, 2, 3 ... in a row stretching towards the right without end. Including the negative integers extends the row to the left. But there are gaps in the row, crying out to be filled. We can insert an infinity of fractions between any two whole numbers but, while the gaps become ever-smaller, their number grows without limit: they never go away.
Towards the end of the 19th century, two mathematicians, Richard Dedekind and Georg Cantor, found ways to define quantities known as real numbers, filling all the gaps and producing a mathematical continuum. But this may or may not correspond to the points on a physical line; we have no way of knowing whether we have too few or too many numbers for these points.
Cantor proved many startling results. There is not just one infinity, but an entire hierarchy of transfinite quantities, increasing without limit. Around 1970, John Conway discovered an entirely new way of defining numbers, which includes all the familiar numbers, all Cantor's transfinite numbers and a breathlessly vast universe of new numbers, both infinitely large and infinitesimally small. These are the surreal numbers.
[
Likely site of new 'gas giant' planet found by research team led by Galway scientists
Opens in new window
]
So far, the surreal numbers have not been used in physical theories. But this is typical; new mathematical developments often find applications only years or decades after their discovery.
Given that fundamental physical theories involve singularities, and infinite quantities are natural elements of the surreal numbers, these exotic numbers may prove valuable in future theories of quantum gravity. Perhaps physicists should embrace infinity rather than trying to banish it from their theories.
Peter Lynch is emeritus professor at the School of Mathematics & Statistics, University College Dublin. He blogs at
thatsmaths.com
Try Our AI Features
Explore what Daily8 AI can do for you:
Comments
No comments yet...
Related Articles
Irish Times
17 hours ago
- Irish Times
SpaceX Starship rocket explodes during testing in setback to Musk's Mars mission
SpaceX's massive Starship spacecraft exploded into a dramatic fireball during testing in Texas late on Wednesday, the latest in a series of setbacks for billionaire Elon Musk's Mars rocket programme. The explosion occurred at about 11pm local time while Starship was on a test stand at its Brownsville, Texas Starbase while preparing for the 10th test flight, SpaceX said in a post on Mr Musk's social media platform X. The company attributed it to a 'major anomaly', and said all personnel were safe. 'Preliminary data suggests that a nitrogen COPV in the payload bay failed below its proof pressure,' Mr Musk said in a post on X, in a reference to a nitrogen gas storage unit known as a Composite Overwrapped Pressure Vessel. READ MORE 'If further investigation confirms that this is what happened, it is the first time ever for this design,' he continued. SpaceX didn't immediately respond to a request for further comment. The Starship rocket appeared to experience at least two explosions in quick succession, lighting up the night sky and sending debris flying, according to video capturing the moment it exploded. The 122m (400ft) tall Starship rocket system is at the core of Mr Musk's goal of sending humans to Mars. But it has been beset by a string of failures this year. In late May, SpaceX's Starship rocket spun out of control about halfway through a flight without achieving some of its most important testing goals. The Starship lifted off from SpaceX's Starbase, Texas, launch site, flying beyond the point of two previous explosive attempts earlier this year that sent debris streaking over Caribbean islands and forced dozens of airliners to divert course. Two months earlier, the spacecraft exploded in space minutes after lifting off from Texas, prompting the US Federal Aviation Administration (FAA) to halt air traffic in parts of Florida. Videos on social media showed fiery debris streaking through the dusk skies near south Florida and the Bahamas after Starship broke up in space shortly after it began to spin uncontrollably with its engines cut off, a SpaceX live stream of the mission showed. Mr Musk called that explosion 'a minor setback'. The FAA said earlier this month it had closed an agency-required investigation into the mishap, citing the probable cause as a hardware failure in one of the engines. SpaceX identified eight corrective actions to prevent a recurrence and the FAA said it verified SpaceX implemented those before the late May Starship mission. In January, a Starship rocket broke up in space minutes after launching from Texas, raining debris over Caribbean islands and causing minor damage to a car in the Turks and Caicos Islands. – Reuters
Irish Times
a day ago
- Irish Times
Embracing infinity: could surreal numbers shape the future of physics?
Imagine Earth were to shrink to the size of a marble. We might be in trouble, but the planet would continue its smooth course around the sun while the moon would maintain its orbit, circling Earth once a month. Isaac Newton proved Earth's gravitational pull would be the same even if all the mass were concentrated in a single point. But the density at that point would be infinite, a condition physicists and mathematicians call a singularity. Such singularities are found in black holes, stars that have collapsed under their own weight. According to general relativity, mass concentrations curve space-time, inducing the force of gravity. With enough matter in a small enough volume, gravity becomes infinitely strong. In 1916, just months after Albert Einstein's general relativity appeared, Karl Schwarzschild discovered a solution of the equations with a singularity. Decades later, this idea led to the theory of black holes, crushed stars with spherical boundaries that trap anything falling inside, including light rays. READ MORE There is now abundant evidence that black holes exist, but do they really represent space-time singularities? Most physicists believe the singularities are mathematical artefacts, and would vanish in a more fundamental theory incorporating quantum effects. Physical equations enable us to predict the future, but singularities imply a lack of predictability; theory just breaks down. It was hoped that quantum effects would eliminate infinities, but current versions of quantum gravity are plagued with singularities. It seems that infinite quantities are inherent and unavoidable. [ Beyond the big bang: Irishman's universal evolution theory challenges accepted cosmology Opens in new window ] German physicist Hermann Weyl opened his essay, Levels of Infinity, with the statement 'mathematics is the science of the infinite'. Infinity is at the core of mathematics. We can gain a first impression of it by placing all the counting numbers, 1, 2, 3 ... in a row stretching towards the right without end. Including the negative integers extends the row to the left. But there are gaps in the row, crying out to be filled. We can insert an infinity of fractions between any two whole numbers but, while the gaps become ever-smaller, their number grows without limit: they never go away. Towards the end of the 19th century, two mathematicians, Richard Dedekind and Georg Cantor, found ways to define quantities known as real numbers, filling all the gaps and producing a mathematical continuum. But this may or may not correspond to the points on a physical line; we have no way of knowing whether we have too few or too many numbers for these points. Cantor proved many startling results. There is not just one infinity, but an entire hierarchy of transfinite quantities, increasing without limit. Around 1970, John Conway discovered an entirely new way of defining numbers, which includes all the familiar numbers, all Cantor's transfinite numbers and a breathlessly vast universe of new numbers, both infinitely large and infinitesimally small. These are the surreal numbers. [ Likely site of new 'gas giant' planet found by research team led by Galway scientists Opens in new window ] So far, the surreal numbers have not been used in physical theories. But this is typical; new mathematical developments often find applications only years or decades after their discovery. Given that fundamental physical theories involve singularities, and infinite quantities are natural elements of the surreal numbers, these exotic numbers may prove valuable in future theories of quantum gravity. Perhaps physicists should embrace infinity rather than trying to banish it from their theories. Peter Lynch is emeritus professor at the School of Mathematics & Statistics, University College Dublin. He blogs at
Irish Times
a day ago
- Irish Times
Elio star Yonas Kibreab: ‘I saw my first Pixar movies when I was four, so to be in one is surreal'
Is there life on Mars? Or anywhere besides our pale blue dot? In April, astronomers using the James Webb Space Telescope detected dimethyl sulfide and dimethyl disulfide in the atmosphere of an exoplanet 124 light years from Earth. On our own planet these compounds are produced by marine micro-organisms such as phytoplankton. The findings are by no means conclusive, but they are considered the strongest evidence yet for extraterrestrial life. Or at least extraterrestrial plankton. Maybe it's these headline-making biosignatures. Or maybe it's a way to escape the trials of contemporary life on Earth. But aliens are experiencing a moment in the movieverse. Steven Spielberg is beavering away on a top-secret ET-themed science-fiction adventure, his first consideration of outer planets since he made War of the Worlds, in 2005. Alien: Earth, a new series serving as a prequel to Alien, Ridley Scott's 1979 film, will premiere in the US in August. READ MORE Younger sky-watchers can head to see Elio , Pixar 's new alien-populated movie, in which a young orphan – voiced by the 15-year-old Filipino-American actor Yonas Kibreab – struggles with grief, an overly vivid imagination and a deep sense of otherness. His guardian and aunt, a major in the US military – voiced by the Oscar-winner Zoë Saldaña – specialises in tracking space debris. That gives Elio the idea to use a ham radio. He's hoping to be abducted, but when an interplanetary misunderstanding leads a cosmic delegation to believe that Elio is Earth's official ambassador, he is teleported across the galaxy to represent humanity at the Communiverse, a sprawling congress of alien civilisations. When his good-natured hosts Questa (Jameela Jamil), Tegmen (Matthias Schweighöfer) and OOOOO (Shirley Henderson), a gelatinous supercomputer, draw the wrath of the warlord Grigon (Brad Garrett), it falls to Elio to use his nonexistent diplomatic skills to save the day. 'Elio's overall personality is like a superpower,' Yonas says. 'He doesn't care about what other people think. His personality is amazing. He's so cool. He wears capes. He's not worrying about what his classmates are going to say about him. And I think that's a very important message. Be yourself.' [ Elio review: Pixar's all-ages pleasures are in short supply in strangely half-formed animation Opens in new window ] In a welcome flourish, Elio is book-ended by references to the Voyager mission. Launched in 1977, Voyager I and II were sent hurtling billions of kilometres to the outer limits of our solar system to gather information about far-flung planets before sailing out into deep space. In 2012 Voyager I slipped through the heliopause and officially became the first human-made object to reach interstellar space. Both probes carry two golden records : 12-inch discs containing sounds and images selected to portray the diversity of life and culture on Earth, including greetings in 55 languages, birdsong, music and a message from the late Jimmy Carter, who was president of the United States at the time. 'I've always found it fascinating,' Garrett – a towering presence even sitting down – says. 'It can't just be us in the universe. That's just man's vanity, right?' Garrett's career began in the 1980s, when, as a young comedian, he became a grand champion on the American talent show Star Search; he subsequently appeared on The Tonight Show Starring Johnny Carson and performed alongside Frank Sinatra and Sammy Davis jnr. His breakout acting role came in 1996, when he played the bumbling, henpecked cop Robert Barone in the TV comedy Everybody Loves Raymond, a part that earned him three Emmy awards. A Pixar regular, he has voiced characters in Finding Nemo and Ratatouille. Elio, he says, is a bit different. 'This is the first time I played a villain,' he says. 'What I liked about it is that he is one of the few villains where you get to see him evolve in an emotional way. He's a dad and he gets to show a parental side that he's never had before. That happens just in time when his son really needs it.' Yonas Kibreab attends the gala screening of Elio at Vue West End in London on June 15th. Photograph:Yonas is also a voice-acting veteran, following his portrayal of Phinny in the Disney Junior series Pupstruction and Damian Wayne/Little Batman in Merry Little Batman. He has also appeared in Silicon Valley, Blumhouse's Blood Moon and the Disney+ series Obi-Wan Kenobi. 'With voice work you have to communicate emotions through the mic,' he says. 'It's hard to do that at times, because other actors don't really see your face and you don't see theirs. But what I love about it is just how free you are to do whatever. You can go in the studio and, because there's no cameras on you, you don't have to perform with your face.' Despite the teenager's extensive work across various franchises, it was the Pixar film – the 29th animated feature to emerge from the studio – that won him over to both science fiction and watching the skies. 'When I started Elio I did a lot of research on extraterrestrial life, aliens and sci-fi,' Yonas says. 'That gave me an excuse to watch a lot more sci-fi movies. I think it definitely got me into anything that has to do with space. I really enjoy all that now.' Elio is part of Pixar's renewed push for theatrical dominance. Last summer its animated feature Inside Out 2 took a staggering $1.7 billion at the box office, to become the highest-grossing film of 2024 and the eighth-highest-grossing film of all time. The numbers are especially promising following the direct-to-streaming releases of the Pixar films Soul (2020), Luca (2021) and Turning Red (2022), on Disney+, and the poor theatrical showing of Lightyear , Pixar's underwhelming Toy Story spin-off. Since its founding, in 1986, and breakthrough with Toy Story, in 1995 – that film was the first fully computer-animated feature – Pixar has consistently combined technical innovation with emotionally impactful storytelling. But recent box-office wows, notably the Spider-Verse sequence and Teenage Mutant Ninja Turtles: Mutant Mayhem , trumpet the return of traditional hand-drawn animation. Elio is part of Pixar's renewed push for theatrical dominance. Photograph: Disney/Pixar Pixar films are changing. The current American aversion to the values of diversity, equity and inclusion have made a dent in Pixar's commitment to culturally sensitive storytelling. In response to external pressures, Disney removed a transgender storyline from the Pixar series Win or Lose in advance of its debut on Disney+ last February. Behind the scenes, the company is restructuring. In May 2024, Pixar laid off 175 employees – about one in seven of its workforce – as part of the broader cost-cutting programme at Disney , its parent company. Under its chief executive, Jim Morris, the animation studio is prioritising films with 'clear mass appeal', moving away from director-driven, autobiographical narratives such as Turning Red and towards existing intellectual property: Toy Story 5, Incredibles 3 and Coco 2 are all in development; Toy Story 5 will premiere on March 6th, 2026. Brad Garrett, for one, is not worried. 'I started working with them early on,' he says. 'A Bug's Life was my first Pixar film, which is remarkable. I've been doing cartoons since the 1980s. But when Pixar came along, me and everyone else thought, wow, this is the new frontier. They do it like no one else. They're so incredibly collaborative. The work the animators do is unprecedented. It's an honour to be part of it.' [ Dismayed by pop culture's shift towards Trump? Then you might be one of the people to blame Opens in new window ] 'I've been telling everyone how surreal it is for me,' Yonas adds. 'Because, when I was four or five, I saw my first Pixar movies, Toy Story and Up. And those are still two of my favourite movies to this day. So to be in one, especially an original Pixar film, and to be the lead, is a big, big deal for me, and I'm very grateful for it.' Elio is in cinemas from Friday, June 20th
