The Impossible Goal of Zeno’s Paradox

My struggles with time, academia, and existence (both mine and otherwise)

“You’ve got no idea what you’re talking about you fucking idiot,” one read. I had just gotten out of bed and checked my email. There were messages sharing similar observations. Others containing giddy superlatives and explanation points!!! People pleading for help with their theories of space, time, and the universe. An email written in bright multicolored font quoting long passages from scripture and postscripted with a personal message from God: “Peter, I am aware of the pain that you went through when you were three years old.” Another extolling the virtues of Ayn Rand. People with different takes than mine. Others enthusiastically agreeing but who I could tell hadn’t understood. Accusatory messages, some written in caps, telling me I was a FRAUD, and the editor of Hustler inviting me to write an article — about time, rather than likely sexier topics such as the origin of the universe or super massive black holes.

An internet search for my name returned thousands of results where there had previously been just one or two. The first was a message board discussion with the title “Peter Lynds is a hoax.” In addition to my possible nonexistence or hubris, much of the frenzied commentary revolved around Zeno’s motion paradoxes. These paradoxes, named after Zeno of Elea, the Greek philosopher who first expounded them some 2,500 years ago, purport to show that movement — and change in general — is impossible. Perhaps the most famous of the paradoxes, The Dichotomy, demonstrates this by showing one can never reach a said point or goal.

I have only read the first two sections as it is clear that the author’s arguments are based on profound ignorance or misunderstanding of basic analysis and calculus. I’m afraid I am unwilling to waste any time reading further, and recommend terminal rejection.

I find this paper very interesting and important to clarify some fundamental aspects of classical and quantum physical formalisms. I think that the author of the paper did a very important investigation of the role of continuity of time in the standard physical models of dynamical processes.

Author’s work resembles Einstein’s 1905 special theory of relativity, in that its validity is not destroyed by the petitio principii from which it is formally derived.

— Anonymous journal referee comments

In 2003, I published a paper in a U.S. physics journal that included what I saw to be the correct solution to Zeno’s paradoxes. The paper received a lot of media attention, and the public reaction was rather polarized and extreme — no doubt partly because I was a young nobody from New Zealand who didn’t have a degree. I was also naive and not expecting some of the vitriol that came my way from dementedly irritated internet defenders of physics orthodoxy, including one who paid a personal visit, and another who hacked into my voicemail and replaced my greeting with his own: “Peter Lynds is a fucking liar! Don’t believe him!”

There was also real hostility from physicists and philosophers. This backlash seems to have resulted in a previously supportive physicist — one I had been corresponding with days earlier and who had invited me to a conference — trying to distance himself by getting ontologically personal and claiming online that I didn’t exist. Then again, partly because I had to fight for several years to get the paper published, I didn’t care for aspects of academia and my actions showed it. This included the issue of a press release about the paper — something that is commonplace in physics, but with a university’s press office usually handling it instead of one’s girlfriend. Underlying my attitude was the radical belief that those with a university position and a Ph.D. don’t hold 100% ownership of good new ideas about how the world works.

That was more than 10 years ago now and things have settled down. I’ve published papers on other topics, gotten older and at least slightly less impetuous and more cynical, and carried on with my life — one that has little to do with academia. During this time I’ve also learned that trying to show how change is possible despite Zeno’s paradoxes is a lot easier than changing how people think about these things. Indeed, given that I’m a uncredentialed academic misfit — and notwithstanding my failings and foibles — I’ve come to realize that achieving such a goal is impossible. A sort of reverse Zeno’s paradox, in which one can see no practical reason why a said goal shouldn’t be reachable, but reality proves otherwise. Yet despite an awareness that the chase is futile, the first part also makes it impossible to just give up. Zeno’s masochistic paradox. Perhaps it sounds familiar. Perhaps not.

Zeno was born in southern Italy around 490 B.C. and was a proponent of the Eleatic school of philosophy founded by Parmenides, who held that contrary to the evidence of our senses, motion and change are illusions; the only thing that really exists is an eternal, unchanging whole called One. Parmenides contrasted the Way of Truth (Aletheia) with the Way of Opinion or Appearance (Doxa, which gives us orthodoxy), believing doxa led us astray. By introducing a logical technique we now call reductio ad absurdum, it is thought that with his paradoxes, Zeno was attempting to defend these views.

The first paradox asks us to imagine the athletic Achilles in a race with the cumbersome Tortoise. Achilles runs 10 times faster than the Tortoise, so Achilles gives the Tortoise a head start of 10 meters. After Achilles has run those 10 meters, the Tortoise has moved one. Achilles runs that meter, the Tortoise moves a decimeter, and so on. Although Achilles can get terribly close, there will always be some distance between the two, and despite being much faster, Achilles can never catch the Tortoise.

A second paradox, The Dichotomy, asks us to imagine trying to reach a said goal; say, a door. To get to the door, we must first travel halfway. Once there, we must travel half the remaining distance again, and once there, half again, and so on ad infinitum. We can never reach the door. Moreover, if we switch the paradox around, we can’t even get started: Before we can get halfway to the door, we must get a quarter of the way; before a quarter, one-eighth, and on and on until we can’t move at all.

Zeno’s third paradox, the Arrow or Fletcher, states that at any instant in time, a moving arrow is motionless. Movement, the paradox suggests, is impossible.

At first glance, Zeno’s paradoxes seem easy to dismiss. Motion and change are obvious features of the world. But it’s difficult to find the actual flaw in Zeno’s reasoning. Indeed, the paradoxes posed real headaches for philosophers and mathematicians over the centuries, leading Bertrand Russell to term them “immeasurably subtle and profound.” However, with the development of calculus, beginning with Isaac Newton and Gottfried Leibniz 300 years ago, the paradoxes were generally considered to be solved.

It is usually claimed that the Arrow paradox is resolved by the mathematical idea of limit. Consider a polygon inside a circle. If the number of the polygon’s sides is increased, the length of each side decreases, and the polygon becomes more and more like a circle. In reality, the polygon will always have a finite number of sides and will never achieve circledom, but it can get arbitrarily close, so for practical purposes, it is said that it might as well be a circle. In this example, the polygon is analogous to the arrow’s motion and velocity, and the circle, to the limit of the arrow’s velocity at an instant. In other words, although the limit of the arrow’s velocity at the instant is never reached, the arrow is said to have a nonzero velocity at the instant, and thus not to be motionless.

The paradoxes of Achilles and the Tortoise and the Dichotomy, on the other hand, are thought to be solved by infinite series. By summing an infinite series of progressively smaller time intervals and distances, the time taken for Achilles to overtake the Tortoise, or to traverse the distance to the door, is finite. The faulty logic in Zeno’s argument is seen to be the assumption that the sum of an infinite number of terms is always infinite, when in fact an infinite sum (such as 1 + 1/2 + 1/4 + 1/8 + 1/16 …) can be said to be equal to a finite number — in this case, 2.

Halfway through 1992, at age 16, I left high school to follow my father and his new wife to eastern Germany, where they had business interests. While I was already having troubles — some of which were probably mediated by feeling out of place, heavy drinking, and being involved in a fishing accident during which a close friend drowned — my experience in Germany brought the arrival of circling vultures. Upon returning to New Zealand three months later, I worked as a pool attendant for a time, in a sports store for a shorter period (before getting fired), tried my hand at sheep and cattle farming, pursued rugby, and ended up working for an insurance company, where I became deeply depressed about my life and the world (over and above the fact that I was working in insurance).

In the Ecclesiastes sense, it all seemed shallow, hollow, futile, and motivated by little more than ego and self-interest — chasing after the wind. I was also asking a number of questions and didn’t seem to be finding many answers. I ended up trying to kill myself, but didn’t do a very good job. When a large number of painkillers and tranquilizers failed to render me unconscious, I followed up with a bottle of aspirin and some whiskey, which just made me sick, emptying my stomach of the lethal levels of acetaminophen. Stupidity sometimes has its benefits.

I began to read a lot, and that reading soon brought me to physics. A light switched on, and I read whatever physics books I could get my hands on, especially those related to Albert Einstein and his work. While I had found the more earthly aspects of the story of Jesus’ life compelling, Einstein — a secular humanist, deeply kind and compassionate liberal, fiercely independent-minded rebel, music and tobacco lover, highly aware and full of penetrating, pertinent insight — seemed a much more convincing prophet. Furthermore, following the God he metaphorically spoke of — Nature — actually did offer a way to the Truth. His writings seemed to resonate strongly, and because I was a guy who had recently blown up his mother’s kitchen by throwing a jug of water into a pot of burning oil, I found this both surprising and encouraging.

Put another way — and although I had music and the likes of Beethoven and the Beatles — I was desperate for something, or someone, to believe in. With Einstein I got both. Something of an orgy of free thought followed, I taught myself the fundamentals of physics, and ideas started coming. First, an idea related to providing a logical foundation for Platonic existence, and shortly after, an idea about the origin and future of the universe, which I eventually published in 2006. Moreover, despite my family thinking I had now entirely lost my marbles, many pieces of the personal existential puzzle I had been having such difficulty fitting together fell firmly into place. Out of curiosity more than anything else, I decided to go to university, taking classes in physics and philosophy.

One night around this time I went to bed mulling over Zeno’s paradoxes and their said solutions by calculus. I had just watched the film IQ, which mentions the paradoxes and features a mechanic who, with the help of Einstein, fools everyone into believing he’s a physicist.

To my mind, there was an obvious problem with the accepted solutions. In the physical setting, both involved something akin to rounding up and didn’t address the actual problem posed by the paradoxes. That the paradoxes were treated as separate problems requiring separate solutions just further indicated something was amiss.

I realized that central to Zeno’s argument was the assumption of the existence of a durationless instant in time at which a moving object could be said to have an exactly determined or instantaneous position. But for something to be in motion, its position has to be constantly changing and undetermined; if it weren’t, the body couldn’t be in motion. By wrongly assuming one could freeze and dissect motion at an instant — thus assigning it an exact position — the paradoxes were created. In the real world, the object’s motion is continuous. We can’t freeze the world at an instant, because nature is forever changing.

Essentially I realized that instants in time and nows (the present tense version of an instant) couldn’t physically exist. If they did, any sense of motion and change would be impossible. The same applied to spatial points. This further meant that no instantaneous magnitude, such as instantaneous velocity and acceleration, could exist or have physical correspondence. Indeed, defined as it is, as a succession of instants, time itself couldn’t exist either, nor could space.

It seemed that nature had wisely traded time, stasis, and certainty for change and indeterminacy. I soon found that traces of this reasoning stretched all the way back to Aristotle and Saint Thomas Aquinas, so this wasn’t entirely novel. The idea that time doesn’t exist certainly wasn’t original, having been around for a long time in philosophy, and it was slowly gaining momentum in physics. But the assumption of the existence or correspondence of instants — the would-be building blocks of time — remained, even in the minds of physicists who had otherwise rejected time.

Problematically, much of this went against accepted physics back to Newton. But because the idea’s possible validity rested solely upon the simple question of whether a moving object could have a determined or instantaneous position, it was rather straightforward. Moreover, it didn’t mean calculus was wrong or that we should necessarily throw the idea of the instantaneous out the window. Rather, we needed to be careful with our assumptions regarding the use of calculus in a physical setting, and equally careful with instants.

A good example of this represents necessary caution when thinking about quantum gravity — the as-yet-undiscovered theory that would reconcile quantum theory with Einstein’s general theory of relativity. Many physicists expect that time and space, at nature’s smallest scales, are quantized — that is, inherently discrete and grainy — just as quantum theory tells us things such as the electromagnetic field, including light, are. But if time and space were literally quantized, this would require the physical existence of points of time and space to bound and determine their values as intervals, this in turn rendering motion and change impossible. However, being made up of particles and energy, the constitutes of clocks and rulers would be quantized, so their readings should be too. So while there is no sense in which time and space can physically be quantized, as a natural outflow of matter being quantized, the readings of clocks and rulers should be. This further means that a possible non-zero duration instant or “now” cannot exist, while — unless instants are first removed — the claim that Zeno’s paradoxes can be resolved by time and space being quantized also falls over. Once instants are removed, no paradox remains to be solved.

What then of time? What does it mean to say time doesn’t exist? Well, time is simply what a clock shows (not measures, as this would imply something being there to measure). Moreover, it is motion that enables the hands of a clock to rotate or the earth to rotate on its axis (which in turn gives rise to our notion of time); the existence of time doesn’t enable motion. As suggested by the block view of time and the lack of absolute simultaneity or a “preferred present moment” in Einstein’s relativity, the past and future do not exist either. All times, so-called past and future, are laid out together — analogous to viewing a sweeping sketch of the Rocky Mountains, as Kurt Vonnegut described in Slaughterhouse-Five. Yet we can still assign an order to events and times — in the same way we can say 2 follows 1, and 3 follows 2 — without referencing before and after, past or future. Finally, the present is also entirely subjective and, I think, inextricably linked to our conscious perceptions and ability to be self-aware. Indeed, I believe the present represents the key to unlocking the mystery of how consciousness is possible.

With time seemingly playing such an integral role in the way we view the world, it can be difficult to tell the real from the imagined, the actual from intuition. Indeed, I believe we consciously think within the context of static frames or instants in time, and that this is the root of much of our trouble with Zeno’s paradoxes and the instantaneous. For example, if asked whether a moving car has an instantaneous position relative to the road beneath, one can picture the situation only in the context of a frozen snapshot and not with the car’s position constantly changing. To mentally picture something, one has to hold that image in awareness for a certain period of time before we can picture a change to that image. A car seems to have an exact position in this static frame, so we assume it really has one, when in fact the act of imagining the situation has inadvertently frozen the car in its tracks. With time — whether instants, its apparent flow, and past, present, and future — being inextricably linked to how we think, we can’t help but have a strong anthropocentric bias to view the world a certain way, and it can be difficult to see beyond this. A similar thing could be said for humanity’s two greatest problems — self-interest and stupidity (if these often overlap).

During the 10 years since its publication, the paper has gained a certain traction and support, but not enough for it to stand a chance of gaining widespread acceptance. Most people believe such questions are long settled and beyond reproach. Given the amount of water that has flowed under the great bridges of physics and philosophy, the work also invites cynicism. Adding me to the picture doesn’t help. Of course, I could also just be wrong.

In trying to pursue my work, some very good things have come from the experience: meeting some leading lights of the physics world; being able to correspond with people I might not have otherwise (Noam Chomsky comes to mind, although I suspect he thoughtfully replies to everyone); having an office for the day at the Perimeter Institute for Theoretical Physics in Canada as the guest of a hero of mine, the physicist John Moffat; and becoming close friends with Fran Healy of the Scottish band Travis — who, in addition to a number of other incredibly kind acts, introduced me to another hero, Paul McCartney.

After reading about my work in the newspaper and then scolding an angry mob brandishing tuning forks on an internet physics forum, Fran first emailed in 2003, and as a postscript, casually mentioned he wanted to dedicate his next album to Kurt Vonnegut and me. I was already a fan of his music, so this was really something. My friendship with Fran has easily been the best thing to come from my work.

There have also been a number of not-so-good experiences, including numerous fights (generally of the written variety and usually involving someone purposely standing in the way); a continual string of frustrations and disappointments at not being able to move by; encountering a level of politics, self-interest, and fear of nonconformity I didn’t think would or should exist in science; actions by singular academics that might have impressed Machiavelli or Mussolini; through to running out of money and sleeping on the street in New York City and Toronto while trying to pursue my work (although this was my fault, and part of me enjoyed it).

I sometimes wonder whether I was right the person for my ideas. But this is at least partly offset by the doubt that many others would be idiotic enough to keep running into a brick wall for them. And yet, on top of bricks of my own making, as I see much of this wall as simply being composed from a juxtaposition of the behind-the-scenes reality of academia and human nature, I’ve also come to question the bravado of doing this too. It’s all well and good to dramatically clench one’s fist and vow to keep fighting and never give up, but a person is generally going to be wasting their time if fighting not against established ideas but something like human psychology. It’s difficult to punch for one.

Several people have suggested I get a Ph.D. if I want my work to be taken more seriously. There have also been instances where I’ve thought it likely my work has influenced a particular physicist or a philosopher, but, presumably because I don’t count, they haven’t acknowledged it. But as I don’t need a Ph.D. to do that work, I would be getting a degree only for the benefit of other people’s perceptions — something that strikes me as absurd. A good idea is a good idea regardless of whether its author has a Prof. or Dr. before their name. That academia and its funding bodies — the people who ultimately decide what constitutes a good idea — think otherwise ought to be its own reductio ad absurdum.

Until the mid-20th century, it was not uncommon for formally unqualified “hobby” or “gentleman” scholars to make real contributions to science. This included William Herschel, an 18th-century German-born British musician who discovered the planet Uranus, several moons, and infrared radiation, and Michael Faraday, a lowly 19th-century apprentice bookbinder who, despite being entirely self-taught and lacking a knowledge of higher mathematics, became one of history’s most influential physicists for his work on electromagnetism. More recently, there is the science Cinderella story of John Moffat, an iconoclastic physicist, now in his 80s, who has made notable contributions to cosmology, gravitation, and particle physics.

In Moffat’s story, Einstein played the role of a fairy godfather after a teenage Moffat — a struggling artist turned rapidly self-taught physicist — wrote to his idol pointing out what he saw to be a mistaken assumption in the great theorist’s work. To Moffat’s surprise, Einstein, then age 73, wrote back, acknowledging the unemployed college dropout had a point. This led to an ongoing correspondence, with Einstein treating the young upstart as a comrade and equal, which not only inspired Moffat to become a scientist but also led to his entering the doctoral program at Cambridge University’s Trinity College, where he became the first person in the school’s 500-year history to be awarded a Ph.D. without first obtaining an undergraduate degree.

Such unqualified outsiders and their ideas don’t seem to be making it to the Scientific Ball these days, and in physics this is partly explainable by the complexity of the subject growing to such an extent that it can take several years of study to master a single small area of the field. This isn’t always the case, however, and somewhat like stepsisters blocking their lowly sibling from attending the party, in recent decades physics has become much more insular and exclusive. Indeed, such circumscription hasn’t just applied to individuals, but to a whole academic field — philosophy.

Until around 1950, philosophy and physics often went hand in hand, with many physicists being deeply concerned with the foundations and assumptions underpinning their theories. For such physicists, science was about discerning objective reality and truth. Einstein was the epitome of this, and combined with the rare coupling of a strong grip of mathematics and an acute physical intuition, it no doubt helped make him the special talent he was. Today, many physicists view philosophy with disdain, preferring to concentrate on solving equations rather than consider questions of reality (given the obscurist and redundant nature of too much of philosophy, I don’t think this is entirely without good reason.) This has become known as the shut-up-and-calculate approach, and with theories of extra dimensions gaining traction and thousands upon thousands of papers on the (unobservable) interior of black holes, it doesn’t seem unreasonable to link this paradigm with theoretical physics also becoming much more detached from the empirically verifiable, and arguably from reality itself.

As a recent, public example of this contempt for philosophy, on the first page of his 2011 book, The Grand Design (co-authored by Leonard Mlodinow), Stephen Hawking saw fit to proclaim “Philosophy is dead” — an action that, if not for publicity reasons, I suspect had more to do with philosophers sometimes having been critical of Hawking’s theories. Meanwhile, the physicist Lawrence Krauss, reacting to a highly critical New York Times review of his 2012 book, A Universe from Nothing: Why There is Something Rather than Nothing, called the reviewer, David Albert, a “moronic philosopher,” decrying his right to knowledgeably comment. Albert is a highly regarded philosopher of physics from Columbia University, well-published in the relevant area of quantum field theory, and he was simply pointing out what many with a concern for the foundations of physical theory were thinking.

Noting an increased insularity, as well as a lack of progress in the field, in his 2006 book, The Trouble with Physics, the physicist Lee Smolin argued that physics needs to be more inclusive of outsiders because of the novel ideas they can bring to the table. After all, although he had a Ph.D., Einstein was an outsider and remained so throughout his career. Many have questioned whether Einstein, if starting out today, could even get published. Smolin’s definition of an outsider, however, includes the stipulation that he or she must have, at the very least, a Ph.D. in physics — akin to defining “outside” as the covered porch area just beyond the front door.

Now I’m not suggesting the doors be flung open and everyone outside be invited in. There would be no place inside for the vast majority, myself included, and some would probably want to remain outside anyway — if not just for the freedom and view, so they could still throw stones at the windows. Moreover, not just anyone can play in the Berlin Philharmonic or Bob Dylan’s band, and academia is no different.

While popular culture has touted the value of the outsider, the day-to-day reality of the working scientist with an email address includes the far less romanticized outsiders who operate outside the bounds of reason and logic. Partly because of this, there will always be a need for filters such as peer review, the vetting of papers submitted to academic journals by anonymous experts in a field — even if this system is susceptible to personal bias and strongly disfavors novel ideas that go against the grain of consensus, which is normally a feature of the good ones.

But as an idea can speak for itself, such filters should pay little attention to anything else, nor should an academic balk if confronted with the otherwise competent work of an outsider. After all, the often implicit notion that good ideas about reality can come only from people in universities with a Ph.D. is like saying great songs can come only from people who formally studied music for several years. There are countless examples to counter the latter, yet despite both propositions appearing equally ridiculous, hardly any of the former.

John Moffat wrote, “To shift a paradigm long held to be true by the majority of physicists and astronomers demands persistence, thick skin, and a long life.” This is no doubt true; changing any generally accepted idea isn’t supposed to be easy. But like Achilles trying to catch the Tortoise despite instants and orthodoxy being against him, it shouldn’t be impossible for anyone either.

While writing this, I’ve been in Wisconsin helping my American girlfriend, a former New York City–based textile designer, organically farm an acre. After leaving university in 1999 (I was there for six months), I went to radio school, then ended up working as a tutor there, and via a number of different jobs have most recently been working out of Florida as a fishing guide for people with very large boats and far too much money. The plan is for us to move to New Zealand, where eventually we’d like to have a small farm where we can grow stuff. If all goes well, children, a cat, a dog, and music will figure, too. Financially it’s a long way off, and I really need to get moving if I don’t want to wind up alone in a gutter — although, despite the economy, the state of gutters has markedly improved in recent years.

Yet, and while I’m aware that continuing to pursue my work will largely result in frustration, I find it difficult to let go. I recently published a philosophy paper on the question of why something exists rather than not, and there are other papers I’d like to write too. As a result of being fed up or the need to make money, I’ll drop my work for extended periods, but I keep coming back to it. It would be nice to provide a more rounded end than this, but such is life, it’s unclear how things will pan out. This is a good thing, especially for people who deep down already know some struggle is doomed for failure or some goal isn’t reachable, but for any number of reasons from which they take real meaning, still want to try. With a person’s life most certainly doomed to fail, the time leading up to such a point constantly getting shorter and shorter, this applies to everyone.

This essay was provisionally accepted by Harper’s Magazine about five years agoa bit of a coup for someone like me. For various reasons, however, including my being something of a weirdo and somewhat stupid, I didn’t pursue its publication. As it was a time and place thing, and I’ve since moved on, aspects of it now make me cringe a little. Nonetheless, as I still feel the same way about the ideas in it, I thought I should at least free it from indefinite incarceration on my computer. A big thanks to the remarkably kind and talented Amanda Gefter for her help.

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