(by this month’s guest-columnist, Jeff Glibb)
An esoteric branch of math called fraction theory
may hold the answers to science’s deepest mysteries
You may think you know what numbers are. Chances are, you learned to count before you entered kindergarten, and number-names like “one”, “two”, and “three” were among the first words you learned.
But what if I told you that, lurking in the spaces between the counting numbers, is a vast swirling sea of number-ish things, things that many mathematicians and physicists insist are every bit as real as the counting numbers you know and love? What if I said that “one, two, three,…” is just the tip of an iceberg—a mere sliver of a far deeper, more thrilling reality?
Welcome to our fractional universe. It’s where you’ve always lived; it’s just that, up till now, you didn’t know it.
Let’s start our journey into the in-between with something simple: a four-inch-long popsicle stick and the seemingly ridiculous question “How many feet long is it?” You might be tempted to shrug and say “None”, or to dismissively reply “That’s an absurd thing to ask; a popsicle stick is too short to be measured in feet!” But the revolutionary doctrines of fraction theory urge us to put common sense aside and to say that the popsicle stick is “reciprocal-3” feet long, corresponding to the fact that if we switch the roles of the measuring stick and the popsicle stick, a foot is 3 popsicle sticks long. Nonsense? Perhaps; but this kind of paradigm-shattering “nonsense” is ushering in nothing less than a new era in science.
Reciprocal-3, more compactly expressed as 1/3, is an example of a new kind of number: a fraction. Like all world-changing ideas, the idea of fractions is at bottom a mind-bogglingly simple one—so simple that with the benefit of hindsight it can seem surprising that fractions weren’t invented much earlier. Indeed, Princeton physicist Edmund Wissenschaft has called fraction theory “a piece of fifth-century mathematics that fell into the twentieth century”. Indian scientists came heartbreakingly close to the modern concept of fractions fifteen hundred years ago, very nearly stumbling upon the in-between world of fractional numbers that had been hiding in plain sight. But it wasn’t until the twentieth century that humanity finally unlocked the full, breathtaking power of fractions, setting the stage for an era of scientific progress the likes of which the world has never seen.
“We humans, in our eagerness to get from one to two,
from two to three, from three to four, and so on,
overlooked the fractions. But they were there the
whole time, waiting for us to notice them.”
— Gaston Matzobrei
In his groundbreaking 1970 manifesto “The Fractional Arithmetic of Nature”, mathematician Gaston Matzobrei showed how “numbers” like 1/3 appear throughout the natural world if only we would open our eyes to them. For instance, when a river splits into three tributaries, we can consider each tributary as 1/3 of the river that gave rise to it. Learning to think this way takes time and practice, but once one has mastered the math, it can be applied not just to water systems but to diverse natural phenomena, from the infinitesimally rapid jitterbugging of atoms to the majestic, eons-long waltzes of swirling intergalactic dust clouds. “Numbers” like 1/3 provide a universal language for describing those structures. Indeed, the deeper one goes into the theory of such numbers, the more one comes to realize that in some as-yet-dimly-understood way they are woven into the very fabric of the cosmos. They aren’t just language-tools that help us unravel the world’s secrets; if we listen closely, sometimes the universe itself whispers “Reciprocal-three … Reciprocal-three …”
Matzobrei called his new numbers “fractions”, from the Latin verb meaning “to break” (the English word “fracture” comes from the same root). And fractions indeed break many of the rules that ordinary numbers follow. For instance, if you multiply two ordinary numbers together, the product is usually bigger than either of them. But when you multiply 1/3 by itself, you get a fraction that’s smaller than 1/3, not bigger!
Particle physicists were quick to realize that fractions provided a way for them to resolve the infamous Quark Charge Problem that had stymied the brightest minds in physics for nearly a decade. Experimentalists had found indirect evidence of particles, dubbed quarks, that carried less electrical charge than protons or electrons, even though protons and electrons had hitherto been regarded as possessing the smallest possible amounts of charge. The problem shook the foundation of physics to its very core.
Physicists could have dealt with the problem by simply admitting that the unit of electric charge they had chosen as fundamental was wrong; that, more specifically, the unit they’d chosen was three times as large as it should have been. However, this would have necessitated a huge change in the way electricians measure voltage, current, and a host of other electrical quantities, and the process of transitioning to a new system could have led to massive cost and world-wide confusion. (Think Y2K squared.) Instead, physicists realized that quarks could be assigned fractional charges, retaining mathematical consistency while avoiding global disruption of electrical services. Order was restored. A literal Dark Age was averted. Physics textbooks did not need to be pulped and rewritten. But at what cost?
“Fraction physics is less a theory than a communicable illness.”
— Sabrina Hassenpfeffer
Have physicists been seduced by the hermetic tidiness of the mathematicians’ fractional universe? Physicist and physics critic Sabrina Hassenpfeffer thinks so, and has taken sharp aim at the intellectual juggernaut that is fraction theory. With acerbic wit and uncompromising clarity, she has argued that the wholesale adoption of fraction theory in physics is nothing less than a fad, kept afloat by a few taste-makers at top universities. “It’s a tulip mania for the modern era,” she has gibed, famously quipping that fraction theory is “not even stupid”.
Few in the physics community are willing to agree with her publicly (though some of the younger ones that I interviewed for this story, speaking off the record, confessed in whispers that they share her qualms but fear that open dissent from fraction orthodoxy could harm them professionally). Yet many philosophers have taken an undisguisedly dim view of fractions for decades. “You simply cannot have it both ways,” writes one philosopher; “either ‘reciprocal-2 feet’ is just a pretentious way of saying ‘6 inches’, in which case we do not require the fanciful jargon of fractions, or else it means something entirely different, something untethered from reality and entirely without purpose.”
Modern philosophers have identified no fewer than four major problems with fraction theory, each one as mind-bending as it is intractable: The Hamburger Enigma, The Dirty Room Paradox, the Puzzle of the Barber, and the Midpoint Indeterminacy Conundrum. Even the most ardent fans of fraction theory admit that “The Big Four” pose a serious challenge that physicists have yet to fully grapple with.
The first of these problems is the easiest to grasp, though no less disturbing for its accessibility. In ordinary math, 3 is less than 4, but in the upside-down world of fractions, a hamburger that weighs 1/3 pounds is bigger than a hamburger that weighs 1/4 pounds. Counterintuitive? Certainly. But buckle your brain’s seat-belt, because the ride is about to get even bumpier.
Next up: The Dirty Room Paradox. This paradox concerns a floor that needs to be swept. Easy, right? “Just hand me a broom,” you say. But in the fractional universe, things are not always what they seem. There’s a catch in the sweeping job: before you can finish the job you must finish the first 1/2 of the job, and then you must finish the first 1/2 of the last 1/2 of the job, and then the first 1/2 of the last 1/2 of the last 1/2 of the job, and so on. The task of sweeping up the dirt becomes an infinite desert you cannot cross, an infinite mountain you cannot climb, an infinite ocean you cannot swim, an infinite string of metaphors that, unlike this one, which only seems to go on forever, actually does.
Similar in spirit but even more dizzying is the (admittedly androcentric) Puzzle of the Barber, which goes like this: Before a barber can finish shaving your beard, he must shave the first 1/2 of your beard; but alas, the poor barber cannot even accomplish that much, because before he can finish shaving the first 1/2 of your beard he must shave the first 1/2 of the first 1/2. But even that task cannot be completed, because … you get the idea. The task of shaving off your beard isn’t just one that can never be completed; it’s one that can never be started. We find ourselves in a labyrinth of infinitely many infinitesimal tasks, a labyrinth that renders the act of shaving not only unfinishable but unstartable, nay, unthinkable.
Even the many philosophers who have never wielded a broom or a razor agree that something is wrong here. A few think that the problem comes from our understanding of the concept of Beards, or the concept of Dirt, but the majority agree that the concept of 1/2 lies at the root of the problem.
The most technical of the Big Four quandaries, the Midpoint Intedeterminacy Conundrum, reared its ugly head when fraction pioneers tried to build a bridge between their revolutionary work and the “classical” way of understanding the world with counting numbers. They called this bridge “rounding”. In this view, the fractional numbers are organized into “clouds” surrounding the ordinary numbers, with each fractional number belonging to the cloud that surrounds the ordinary number that it is “closest” to. At first, rounding seemed like a brilliant way to tame the wild zoo of new numbers fraction theorists had unleashed, the answer to all their fervent prayers. But rounding gave rise to an impossibly difficult dilemma: Which ordinary number is the fractional number 1/2 closest to? To this day no resolution is in sight. Some cling to the answer “0”; others stubbornly insist on “1”; and a third, more mystical camp contends that 1/2 simultaneously rounds to both and neither, trapped in an eternal tension of irresolution.
“All physical theories should admit an explanation
so simple that even a child could understand them.”
— Albert Einstein
Some forward-thinking teachers, seeking to bring the physics curriculum up to date, have taken fractions into the classroom so that the next generation of students will be equipped to handle the cutting-edge science of our fractional universe. But the results have been nothing short of disastrous. Children, it seems, find fractions not merely difficult but utterly incomprehensible. Even seasoned teachers, veterans of the blackboard, have been left scratching their heads in confusion when not tearing out their hair in frustration, desperately trying to explain concepts that defy the limits of their pedagogical skill. Hassenpfeffer has urged her fellow physicists to take note of these struggles. She suggests that if fractions are too hard for children to understand, perhaps we should pause before basing our physics upon them.
Enter decimal theory. A sleek offshoot of fraction theory, decimal theory attempts to resolve some of the problems afflicting fractions by limiting attention to a special class of fractions called decimal fractions, such as 17/10 (also written as 1.7) and 17/100 (also written as 0.17). To see how decimal fractions can help us, consider the fractions 3/5 and 5/8. In ordinary fraction theory, deciding which of the two fractions is larger requires the laborious and baffling ritual of “cross-multiplication”, in which we first multiply 3 by 8 and then multiply 5 by 5 and lastly decide which product is larger, assuming we can still even remember what question we were originally trying to answer. But in decimal theory, comparisons are blissfully simple: it’s enough to inspect the numbers in their decimal forms. 3/5 is 6/10 or 0.600 while 5/8 is 625/1000 or 0.625, so it’s plain that the former is smaller. No mental gymnastics, no arcane calculations.
Yet decimal theory is plagued by a devil that theorists call the Infinity Problem, a problem that threatens to derail and unravel the entire decimal enterprise. Ironically, the Infinity Problem is exemplified by the very same fraction, 1/3, that physicists hailed as their savior in their solution to the Quark Charge Problem.
Here’s what we know now about 1/3 that we didn’t know back then: we can approximate 1/3 by the decimal fraction 3/10 (or 0.3), or by the decimal fraction 33/100 (or 0.33), but the chilling truth is, no decimal fraction that we can write down captures the infinite shagginess of the fractional number 1/3. To fit 1/3 into the world of decimal fractions, we would need a terrifying numerical monster with infinitely many 3’s. Needless to say, such unending expressions correspond to no process in nature, and so it has been justly asked whether they have any place in a physical theory. The dreams of mathematicians had become the nightmares of physicists.
“If we can truly understand decimals,
we’ll have a toolkit for understanding
every possible kind of computation.”
— Wilfrid Silicon
One scientist who was undaunted by these complexities was physics prodigy Wifrid Silicon, who in the 1980s turned his back on physics to explore the strange new world of in-between numbers. He had to reinvent himself as a software entrepreneur to do it, developing a now-widely-used system called Arithmetica™ that would be capable of handling decimals to the herculean precision his researches required. In a mere second, Arithmetica™ can check every calculation done by Archimedes; in a minute, Arithmetica™ can do the same for all of classical Greek mathematics; and in an hour, Arithmetica™ can do more calculating (and do it more reliably) that all of humanity had accomplished prior to the year 1900.
“Take a look at this,” Silicon said to me a few months ago, eagerly thrusting a laptop in my direction, his eyes gleaming with the special gleam seen only in the eyes of someone who has glimpsed the future. The screen showed a seemingly random parade of digits:
0.0140845070422535211…
“Do you know what that is? Don’t worry, I wouldn’t have known either. It’s the decimal fraction expansion of 1/71. Looks like gibberish, doesn’t it?” His hidden glee was evident. “But now look at this:”
0.0123456790123456790…
“It’s the decimal fraction expansion of 1/81. You see the pattern, I’m sure: those digits are actually counting from 0 to 9, over and over! So at one extreme you have decimal representations of fractions like 1/3 that are rather boring, and at the other extreme you have decimal representations of fractions like 1/71 that are chaotic, but in between there’s a fascinating regime of decimal fractions with a third sort of behavior, neither boringly predictable nor boringly random-looking, but genuinely complex, and these decimals are capable of embodying every form of computation!” Silicon leaned back in his chair. “I’ve devoted my entire career to classifying and understanding all the things decimal fractions can do. This isn’t just math—it’s a whole new way of thinking about computation and complexity.”
“The ultimate version of F-theory will
bring the Infinity Problem to heel in
a harmonious unification of all the
D-theories into one grand synthesis.”
— Edmund Wissenschaft
While Silicon was unraveling the secrets of decimal complexity, another physicist, Edmund Wissenschaft, was busy creating a unified theory of in-between numbers. In Wissenschaft’s grand conceptual edifice-in-the-making, decimal theory is just one of infinitely many “D-theories”, each based on a different “denominator” or “base” (in decimal theory that base is the number ten). Wissenschaft’s upgrade of fraction theory, called “F-theory”, serves as a kind of map of the landscape of possible D-theories. Every D-theory is hobbled by its own version of the Infinity Problem, but each individual fraction avoids the Infinity Problem in infinitely many D-theories. For instance, 1/3 is problematic in 10-theory but non-problematic in 6-theory and 12-theory and infinitely many other D-theories. So even if each individual D-theory is at worst inconsistent and at best incomplete, the hope is that F-theory will knit together those individually incomplete views into a global picture that is both consistent and complete. This is Wissenschaft’s Holy Grail, his Moby Dick, his MacGuffin; and his audacious approach to it is one that only a true visionary like Wissenschaft could have conceived—using the very nature of infinity to escape infinity.
One famed champion of F-theory is Nobel Prize-winning physicist Roderick Pendragon. To understand Pendragon’s enthusiasm, return to our original example of fractions in which we used a ruler to measure a popsicle stick but then used the popsicle stick to measure the ruler. In his genre-busting best-seller “The Ruler’s New Mind”, Pendragon argues that the reversal of roles of the measurer and the measured—a cornerstone of fractional math—is integral to the phenomenon of human self-awareness wherein the mind that measures is also the mind that is measured. Pendragon contends that fractional math is the key to understanding the mystery of human consciousness, the Rosetta Stone that will help us hear not just what the universe whispers, but the things our brains whisper back.
“Nobody, not even Edmund, really understands this stuff.
Wait, are you writing this down? Please don’t quote me.”
— someone who didn’t want me to quote them
I spoke to a number of less-well-known physicists whose enthusiasm was more temperate than Pendragon’s and in some cases tinged with anguish. “We can see what the mathematics is telling us but we don’t really understand it,” one fractions expert bemoaned, speaking off the record.
Another physicist, visibly uncomfortable, offered a defense that sounded as much like an apology as an explanation: “It’s wrong to ask what fractions mean. We should just ask how they can help us figure things out about Nature.” There was a note of desperation in her voice, as if she were trying to convince herself more than me.
The most candid statement came from a weary postdoc, a rising star in the field of fractional physics, who confided: “The dirty secret of our profession is that nobody really understands why the rules for fractions work, especially the rule for dividing one fraction by another. Some people try to get to the bottom of it, but it’s career suicide. The advice I give grad students is, ‘It isn’t yours to reason why; shut up, invert, and multiply.’” His words hung in the air, a mocking echo of the founding precept of modern science, the article of faith that nothing need be taken on faith.
Shortly before this article went to press, I visited Wissenschaft, the man whose F-theory had transformed the face of physics. I had expected this scientific kingmaker to be an imposing and autocratic figure, but I found him to be tentative, gentle, and surprisingly undogmatic. I worked up the nerve to ask him whether he thought F-theory would prove to be the ultimate theory of reality.
We were walking towards his office on the Princeton campus. “Sometimes I think so, but then, …” he said, trailing off into deep thought. Students returning from summer break were beginning to throng the main avenues of the square central lawn, their laughter a sharp contrast to Wissenschaft’s thoughtful silences. He avoided the students, choosing to walk through the quad along a diagonal. “Sometimes I wonder … What if, scattered among all those infinitely many in-between numbers we know about, there are other, even crazier in-between numbers, right under our feet?”
Our diagonal path ended and we arrived at the physics department. “But then I put the fantasy aside and get back to work on fractions,” he said.
For more about fraction theory, see my new book “BETWEEN:
How the new science of fractions is remaking our view of reality”
by Jeff Glibb. 370 pp. Really Basic Books. $32
