In 1869, the Baptist fundamentalist Cyrus Reed Teed reported his divine revelation that the Earth was hollow. At first glance, nothing novel. Jules Verne had explored a similar concept five years prior in his science-fiction adventure Journey to the Centre of the Earth. But while Verne imagined a subterranean cavern of fantastic creatures, Teed declared in earnest that we were literally living inside the sphere. In this strange cosmology, the Sun, planets, stars and galaxies all occupy the Earth’s interior. The Earth’s crust is an infinitely thick layer of rock encasing the entire Universe.

Motivated by his new cosmology, Teed published a book, started a new religion, amassed disciples, and founded a new town in Florida. To many, Teed’s ideas sound like snake oil so thick only the most gullible could imbibe. Yet his influence was not limited to the United States, nor to the 19th century. Anti-intellectual sentiment within the Nazi party embraced concave hollow-Earth theory – or Hohlwelttheorie as it is called in German. According to the Dutch astronomer Gerard Kuiper, elements of the Nazi military might have even advocated looking up through the sky to spy on the Allies on the other side of the world. After all, there’s no place to hide inside a globe.

How did Teed’s ideas gain a foothold in the upper echelons of the Third Reich? A less cosmic, more conspiratorial notion of a hollow Earth was introduced in Europe by the English novelist Edward Bulwer-Lytton. In his novella Vril: The Power of the Coming Race (1871), Bulwer-Lytton depicts a master race living in the bowels of the Earth. In this cultural ambience, belief in a hollow Earth might have attracted German nationalists. Indeed, during the First World War, a German pilot named Peter Bender converted to Teed’s ideas as a prisoner of war. According to the geographer Duane A Griffin at Bucknell University in Pennsylvania, Bender introduced Hohlwelttheorie to the elite Nazi Hermann Göring, who oversaw the creation of the Gestapo. With this initial overture, alternative cosmologies might have slithered into Nazi thought. As Nicholas Goodrick-Clarke writes in The Occult Roots of Nazism (1992), ‘fantasies can achieve a causal status once they have been institutionalised in beliefs, values, and social groups’.

As the Nazi party rose to power in Germany, anti-Semitism drove the Jewish-German physicist Albert Einstein to take up residence in the US. Like Teed, Einstein had developed a deeply counterintuitive understanding of the Universe. To explain several curious observations about light, Einstein inferred that it must always have the same speed for all observers. Moreover, time and space change as one approaches light speed. Behold, very fast objects contract in length and experience a different passage of time! So twins might diverge in age if one starts travelling much faster than the other.

We take it for granted that time and distance are the same for everyone, just as we take it for granted that the cosmos contains Earth, and not vice versa. Yet both ideas have been challenged. How do we know that Einstein is right and Teed is wrong?

To get to the bottom of things, look to the construction of models – simplified descriptions of reality that explain how variables change over time. A red point that we call Mars moves across the starry firmament. How do we explain this nightly variance? A model is needed. From antiquity to the scientific revolution, humanity’s model of Mars has evolved. To the Ancient Greeks, Mars was the wandering star of Ares, god of war. From this early perspective, in which astronomy and astrology had yet to diverge, Mars was a heavenly wanderer embodying the traits of a warlord deity. This explanation for the planet’s nightly motion, while a start, is less than satisfactory. A robust model is not merely a hand-wavy explanation but a mathematical description that accounts for all variance in the data. Specifically, why does Mars move slower than some planets and faster than others? And why does it occasionally reverse direction, a phenomenon known as retrograde motion, for a few months, only to loop back on itself in the original direction?

An early attempt to model most features of planetary motion was developed by the Ancient Greek astronomer Eudoxus of Cnidus, and described by the philosopher Aristotle. In this model, an intricate system of 27 crystalline spheres encircling Earth explained the motions of the heavens, including the retrograde motions of the planets. Later, a different model by Hipparchus of Nicaea invoked a series of planetary orbits around Earth. The first orbit – called a deferent – was a perfect circle around the Earth, while the second orbit – called an epicycle – moved along the circumference of the deferent. Hipparchus’ contribution was immortalised for centuries in the work of the Egyptian astronomer, mathematician and geographer Ptolemy of Alexandria, who tweaked the concept into something so enduring that questioning it was nothing short of blasphemy. Just ask Galileo, who was persecuted for this heresy by the Inquisition in the 17th century.

It’s as if science stands up and says: ‘Dammit, there’s got to be a simpler explanation!’

Because established models become entangled with our sense of reality, doubting them is often an act of defiance, if not utter sacrilege. Today, of course, we know that Ptolemy was wrong. The heliocentric model introduced by the Polish astronomer Nicolaus Copernicus in 1543 places the planets in elliptical orbits around the Sun, and best explains the data (including retrograde motion). Moreover, we now know that planets are neither stars nor even spheres. The English physicist and mathematician Sir Isaac Newton was the first to realise that Earth and other planets are actually ‘oblate spheroids’, globes that are squished a bit at each pole due to their rotation.

Why did Copernicus triumph over Ptolemy? Selecting the simplest, most parsimonious model is a criterion for deciding the truth called Occam’s razor. It’s as if science stands up and says: ‘Dammit, there’s got to be a simpler explanation!’ If someone must weave a complicated tale where a simple story will suffice, stay with the simple story. Ptolemy’s epicycles gave his model just enough rope to hang itself. As the popular saying goes: ‘Everything should be made as simple as possible, but not simpler.’

Yet alongside the elegance of Copernicus come the reactionaries, such as Teed.

The epitome of pseudoscience fantasy, Hohlwelttheorie shrinks billions of light years of mostly empty space – strewn with 100 billion galaxies and 1 million billion billion stars – down to a tiny point at the centre of the hollow-Earth Universe. While Hohlwelttheorie might sound utterly indefensible, a second quixotic mathematician from Alexandria – Mostafa A Abdelkader – has risen to the challenge. In the 1980s, Abdelkader described the mathematical gymnastics needed to conceive an inside-out cosmos. Among other assumptions, this geometric inversion trades the centre of the Earth with infinity. Imagine, if you will, cutting a seam in a basketball. As you turn the rubber inside-out, everything outside the ball – you, the room you’re in, the entire Universe – is sucked inside. The air that was previously inside the ball now forms an atmosphere outside the ball extending towards infinity. Et voilà! A once-humble basketball now contains the Universe. While this verbal analogy is imprecise, the abstract mathematics of Abdelkader’s paper accomplishes this transformation accurately. To embrace it, convince yourself that ‘inside’ and ‘outside’ are as arbitrary as left and right, or up and down.

As a consequence of Abdelkader’s physics, most of the Universe is mapped to a tiny point at the hollow Earth’s centre or origin. Describing this unthinkable transformation in 2012, Griffin writes:

Pluto shrinks to the size of a single bacterium floating seven metres from the origin, while Alpha Centauri, the star closest to our own Sun, becomes an infinitesimally small speck situated a mere millimetre from the origin. Every other star and object in the cosmos, therefore, is contained in a sphere less than two millimetres across that hovers 6,371 kilometres above our heads.

The late American science writer Martin Gardner brilliantly interprets Abdelkader’s mathematics in his skeptical tour of fringe science On the Wild Side (1992). In the strange new physics of Abdelkader’s world, light rays travel not in straight lines but rather in curved arcs. Like a pinwheel grazing the inside of a hollow globe, arcs of sunlight illuminate the near inner surface of the Earth, but curve around and miss the far nighttime surface. For this reason, the Sun appears to set, even though the Earth’s curvature is concave.

Santa Claus, like Mars, is just a model – but one that cannot be refuted or retained

But if other planets are closer to us than Spain is to New Zealand, why does it take so much longer to reach them? Fair question. In the mathematics of this cosmology, the speeds and sizes of moving objects approach zero as one nears the centre of the Universe – located at the centre of the Earth. According to Gardner, Abdelkader’s mental acrobatics create ‘a consistent physics that cannot be falsified by any conceivable observation or experiment!’

How can something so stupid not be falsifiable? Such a question treats unfalsifiability as an alibi rather than a liability. It assumes, naively, that models are innocent until proven guilty.

Yet many arbitrary claims are unfalsifiable. A child’s claim that Santa Claus exists yet cannot be seen by any camera or scientific instrument is just that – unfalsifiable and thus untestable. Intellectually speaking, this is less like wearing a shield and more like wearing a millstone. Santa Claus, like Mars, is just a model – but one that cannot be refuted or retained.

The notion that models can be taken seriously only if falsifiable stems from the late philosopher of science Karl Popper. A model is built not on a concrete foundation but on stilts that can be quickly cast aside should new information arrive. As Popper wrote in The Logic of Scientific Discovery (1934), ‘no matter how many instances of white swans we may have observed, this does not justify the conclusion that all swans are white’. A good model creeps toward certainty; it chases the horizon of proof, yet never touches it.

Absolute certainty is never the benchmark for a scientific model. To be considered scientific, a model must make a prediction that can be supported by experiment later. To be successful, the experiment must verify this prediction. Only then does a scientist trust the model – but never fully. On this, Popper writes: ‘The game of science is, in principle, without end. He who decides one day that scientific statements do not call for any further test, and that they can be regarded as finally verified, retires from the game.’

Popper held that only testable models are scientific models. If no testable claim can be made, then the model is not falsifiable – and not science. The late British philosopher Bertrand Russell illustrated this point in 1952 by humorously declaring that ‘between the Earth and Mars there is a china teapot revolving around the Sun in an elliptical orbit’. Too small to be seen by telescopes, neither the existence nor non-existence of such a teapot can be tested by any reasonable experiment, argues Russell. ‘But if I were to go on to say that, since my assertion cannot be disproved, it is intolerable presupposition on the part of human reason to doubt it, I should rightly be thought to be talking nonsense.’

As Russell’s teapot shows, unfalsifiable claims or models cannot be taken seriously. They are not science. Otherwise, anything goes – teapots, inside-out Earths, you name it. Russell’s conclusion is mirrored by ‘Hitchens’s razor’, an adage of the late writer Christopher Hitchens: ‘What can be asserted without evidence can be dismissed without evidence.’ Hitchens’s razor, of course, belongs in the same toolkit as Occam’s razor, the contention that the simplest, most parsimonious model is a criterion for deciding truth, and the blade that slashes away at Teed. Simpler models are easier to falsify, and thus more amicable to science. A simple model requires fewer data to falsify it, while a complicated model requires more.

The probability that Earth is the Universe-container is equivalent to 80 coin-tosses coming up as heads

Though Hohlwelttheorie cannot be disproven or falsified, the majority of serious thinkers reject it because it is needlessly complicated (Occam’s razor), lacks evidence (Hitchens’s razor), and is unfalsifiable (shall we call this blade Popper’s razor?).

But our razors are not all perfect tools. When the number or specificity of assertions made by a model are unclear, Occam’s razor feels subjective. Is one scientist’s simplicity another scientist’s Gordian knot? Case in point, Abdelkader himself might have found simplicity in Hohlwelttheorie. By vanquishing those billions of light years of lifeless intergalactic void, Abdelkader saves us from the difficulty of believing in the fantastic vastness of the Universe and ‘the consequent reduction of the Earth to an infinitesimal’.

Though Occam’s razor is not an exact algorithm, Hohlwelttheorie’s arbitrary focus on Earth can indeed be quantified. Why Earth, in particular, as the Universe-container? Why not any of the other countless planets in the Universe? Why a planet, for that matter – could not a moon or a star be subject to the same sphere-inversion mathematics used by Abdelkader?

Because the mathematical inversion used by Abdelkader can be applied to any sphere, the Earth is a subset of all spherical objects in the Universe, and only one inverted sphere can logically contain the Universe and all its other spheres, the very specific claim that Earth is the Universe-container (and not the Moon, or Mars, or your favourite star) is even less likely to be true than the general premise that the Universe is contained inside a sphere. When we count the number n of all such bodies that might exist in the Universe, the probability that Earth is the Universe-container is 1/n. As there are at least 1024 such spherical bodies in the Universe, the probability that Earth fills the privileged role of Universe-container is less than 1/1024. This is like tossing 80 coins and having them all come up as heads.

Occam’s razor tells us that Hohlwelttheorie is a bad model because there are simpler alternatives that explain the data just as well. But maybe we’re thinking too hard. Isn’t attacking Hohlwelttheorie with such a sophisticated thinking tool like swatting a mosquito with a sledgehammer? If we can reject Hohlwelttheorie with intuition alone, what need have we for the scientific toolkit?

And yet, intuition is a lousy filter for science. Many decades before Abdelkader, another individual also suggested that the fundamental geometry of the Universe had been misunderstood and that a journey through space changes an object’s size. Einstein’s theory of relativity probably sounded almost as laughable as Hohlwelttheorie when first introduced. With his new models of time, space and gravity, humanity exited the age of intuition.

Dizzy confusion was the feeling of late-19th century physics. Light from distant stars shows the same velocity relative to Earth, regardless of whether the Earth is moving towards or away from a particular star. To rescue physics, Einstein proposed that the speed of light is the same everywhere for all observers. Whether you move towards a laser beam or away from it, its light approaches you with the same relative speed.

And there’s more – time passes slower for an observer the faster he or she travels. Zooming towards a laser beam, our clock slows, limiting the relative velocity between us and the light. Yes, that’s right – a stationary clock on Earth ticks faster than a moving clock on a spaceship.

The hammer is raised, and with this first twist, Einstein chisels away at intuition. As we follow the story of history’s most celebrated physicist, things grow even stranger. Because time and distance are related by velocity, objects contract in length as they travel faster. Indeed, a lance thrown at nearly the speed of light would contract to a short stub. So goes Einstein’s theory of special relativity, the model that explains why the speed of light is constant in all frames of reference.

Using intuition as a handrail, one might behold Einstein’s model and see madness. Using science, one sees reason

Both special relativity and Hohlwelttheorie destroy basic human assumptions: one successfully, the other unsuccessfully. Special relativity tells us that time and space are relative. Hohlwelttheorie tells us that we live inside the Earth. The rope that rescued special relativity is also the rope that hanged Hohlwelttheorie: Popper’s falsifiability criterion. Special relativity offers many possible falsifications, all of which have thus far survived experimental tests.

Case in point: take two synchronised atomic clocks with nanosecond precision; keep one on the ground and fly the other on a jet around the world twice. Special relativity says their times will diverge – and indeed, they do. Any other outcome would have been game over for Einstein. Such accessibility to experimentation is precisely what keeps special relativity afloat. Using intuition as a handrail, one might behold Einstein’s model and see madness. Using science, one sees reason.

If Einstein and Abdelkader seem to converge, it is in the use of geometry to undo the world as we know it. Einstein’s special relativity uses a simple tool known as the Lorentz contraction to describe the manner in which objects contract in length as they approach light speed. Developed by the Dutch physicist Hendrik A Lorentz in 1892, this tool calculates the contraction of fast objects using nothing more than junior high-school algebra. Abdelkader might have been inspired by the Lorentz contraction when developing his mathematical framework for Hohlwelttheorie. Just as Einstein hypothesised that objects contract as they approach light speed, Abdelkader hypothesised that objects grow smaller as they approach the centre of the hollow-Earth Universe. The beauty of mathematics glows in both models. Yet the light of science does not shine on Abdelkader’s model.

Einstein offers an opportunity for his model to be proved wrong. The same cannot be said of Hohlwelttheorie. Whereas Abdelkader yearned to rescue Earth from the vast emptiness of space, Einstein sought to explain data showing that light travels at a constant speed in all frames of reference. And while Abdelkader’s motivation was anthropocentric, Einstein’s might have been shared by a Martian or any other sentient being. Abdelkader, of course, made no testable predictions, yet Einstein made many.

Theories and physical laws are not mere equations. They really mean something about the material world we live in. The difference between Einstein and his predecessor, Newton, is a case in point. Newton described gravity as a force, while Einstein described it as curvature. Einstein’s general theory of relativity asserts that space and time are on a four-dimensional continuum known as spacetime. Mass warps the very fabric of spacetime like a basketball warping a rubber sheet. That’s all gravity is – warped space-time. As the late American theoretical physicist John Archibald Wheeler put it in 1990: ‘Spacetime tells matter how to move; matter tells spacetime how to curve.’

So gravity is not the familiar beast you think it. Because spacetime is warped (or curved) by mass, objects do not fall in the sense we usually think they do. Contrary to Newton, gravity is not even a force. Nothing ‘pulls’ a falling object down towards the Earth. The ‘falling’ object follows a straight line in curved spacetime.

Indeed, the flavour of Einstein’s Universe is utterly different from that of Newton’s. Yet, aerospace engineers might shrug at the choice between Newton’s physics and Einstein’s physics. The choice is not between the correct solution and the incorrect solution but, rather, the appropriate solution and the inappropriate solution. If your spaceship isn’t travelling close to light speed or near an extremely large mass, the appropriate solution is the simpler solution: Newtonian mechanics. Likely for this reason, the engineers behind NASA’s New Horizons mission to Pluto steered through space not by Einstein’s physics, but by Newton’s. The results are incredibly accurate. After travelling across the Solar System for nine and a half years, the New Horizons spacecraft’s flyby of Pluto in 2015 was off by a mere 72 seconds.

Newton’s world is a billiard-ball Universe, while Einstein’s world is a hall of mirrors

By the same token, Griffin has noted that accepting Abdelkader’s physics would make no noticeable difference to day-to-day life: ‘From a practical standpoint … we experience the Universe as Euclidean space with Earth’s surface or (occasionally) the Sun as our reference framework, and we can pass our entire lives without ever having to take an Archemedian perspective that views the framework itself.’ Like Newton and Einstein, Copernicus and Abdelkader strangely converge in the realm of the mundane.

In many cases, Newton’s physics and Einstein’s physics tell a space probe to do virtually the same thing. While convergent in this sense, they utterly diverge in metaphysics. Newton’s world is a billiard-ball Universe. Moving objects have no speed limit. The rules of the game are clear: forces act instantaneously, taking effect immediately from any distance. Einstein’s world, on the other hand, is a hall of mirrors. Spacetime – the very fabric of reality – is bent. Time and space are relative. Forces are limited by the speed of light.

The late American physicist Thomas Kuhn noted that successive scientific theories often give utterly different accounts of reality. Just as Newton’s house of billiards and Einstein’s hall of mirrors are two utterly different venues, one can envision a third.

This alternative arena for reality comes from the Dutch theoretical physicist Erik Verlinde who, in 2011, derived Newton’s model of gravitation from other first principles or fundamental truths in physics. So why is Verlinde’s rediscovery of a three-centuries-old model a big deal? Because gravitation itself is a first principle, and thus should not be derivable from other laws. Whether we think of gravitation as mass attracting mass, or as mass warping spacetime, the fact of gravitation cannot be reduced to anything simpler. As such, it should not be possible to rediscover it from other, disparate areas of physics. This would be like deducing the US Constitution from the Swiss Federal Constitution.

Unless, of course, gravitation is not a first principle. In this spirit, Verlinde frames gravity as an emergent phenomenon. Emergent phenomena appear when interactions on a small scale give rise to new laws, principles and structures on a larger scale. Consider the beautiful ice crystals we call snowflakes. The formation of snowflakes is driven by thermodynamics, the laws that govern the transfer of heat energy between molecules. And yet, crystals do not exist on the scale of individual molecules. They appear only on a larger scale, when many molecules exchange energy in a particular manner.

Much as we can obtain snowflakes from thermodynamics, Verlinde argues that we can obtain gravitation from thermodynamics. If the Universe were a computer program, there would be no line for gravitation in the code. In this view, gravitation is less like a constitutional article and more like a side effect.

Having caught the ball, Verlinde keeps running with it. After deriving Newton’s model of gravitation, he continues in the same 2011 paper to derive important pieces of Einstein’s model. But what does this change? Isn’t gravity still gravity? Perhaps not: Verlinde sees room for improvement.

Dark matter might be a desperate attempt to reconcile a failing theory with observation

Like many other physicists, Verlinde is troubled by an apparent shortcoming of Einstein’s model. Despite its many triumphs, general relativity fails to predict the manner in which galaxies rotate. To rescue Einstein, physicists have simply hypothesised that there is much more mass around galaxies than what we can actually see. The invisible mass, called dark matter, outnumbers regular matter more than five to one! Without its gravitational influence, there is no way to reconcile astronomical data with general relativity.

To Verlinde, the dark-matter story sounds familiar. In 1859, the French astronomer Urbain Le Verrier described an anomaly in the orbit of the planet Mercury. Newton’s laws do not entirely explain the gradual repositioning (called ‘precession’) of the planet’s egg-shaped orbit. To make sense of the situation, Le Verrier hypothesised the existence of an unseen planet orbiting close to the Sun. Dubbed Vulcan, the gravitational influence of this extra mass would explain Le Verrier’s anomaly by perturbing Mercury’s orbit, thus reconciling Mercury’s orbital precession with Newton’s laws. Vulcan, of course, was never discovered. General relativity explained Mercury’s orbital precession in 1915, eliminating the need for any such model.

Sound familiar? Verlinde thinks so. Like Ptolemy’s epicycles and Le Verrier’s Vulcan, dark matter might be a desperate attempt to reconcile a failing theory with observation. Perhaps it’s time to start a new page? Verlinde did just that in late 2016. In the paper ‘Emergent Gravity and the Dark Universe’, he gives new credence to an idea that originated in 1983 with the Israeli physicist Mordehai Milgrom: when gravity grows weak enough, the idea goes, its influence declines less with distance. With this modification alone, the Universe is no longer filled with invisible matter. Dark matter goes poof! Indeed, with parsimony on his side, Verlinde might ultimately wield Occam’s razor as his sword.

Science is a shark tank. It’s easy to mock someone who thinks the Earth is flat. But after mulling over special relativity, the Flat-Earther might have one last laugh. A 2014 video by the American educator and internet personality Michael Stevens demonstrates that, in some frames of reference, Earth is actually a flat disc. No, really. In the frame of reference of a cosmic ray moving towards our planet at nearly the speed of light, the Earth is literally a flat disk 17 metres thick, oriented towards the particle.

Remember, objects moving near the speed of light contract in their direction of movement. And who’s to say that the particle is not actually at rest while the Earth moves towards it? Yes, it’s all there in the physics. Earth moves towards the particle at nearly the speed of light and is immensely contracted in its direction of movement. As Stevens puts it in the video:

Science, of course, rejects a theory if a better one fits more of our observations, but why the egoistical obsession with our observations? A cosmic ray particle could use the very same scientific method we use and conclude that the Earth was, in fact, flat.

The German philosopher Friedrich Nietzsche, a 19th-century forerunner of postmodernism, wrote: ‘There are no facts, only interpretations.’ But even this is an interpretation. What Nietzsche really wrote was: ‘Es gibt keine Tatsachen, sondern nur Interpretationen.’ And this, still, is an interpretation of ink patterns on a piece of paper.

And perhaps Nietzsche was wrong – there could be something more fundamental than interpretations. After all, for theories of gravitation to exist in the first place, there must be something that needs explaining. Even a complete and utter denial of gravitation in any of its various formulations – Newtonian, relativistic, emergent – would admit that there is something we observe that might only be an illusion. Nothing can undermine the fundamental fact that there is a basic observation called gravity that needs explaining. All interpretations diverge from this observation, rooted in the shared experience of human beings.

As the late science writer Isaac Asimov once wrote in an exasperated letter to a patronising student:

John, when people thought the Earth was flat, they were wrong. When people thought the Earth was spherical, they were wrong. But if you think that thinking the Earth is spherical is just as wrong as thinking the Earth is flat, then your view is wronger than both of them put together.