2 Induction on edge
All night long, Wang Miao dreamed of pool balls. They flew in the dark without any patterns. He woke up in a sweat and thought of the shooter hypothesis and the farmer hypothesis, something that the members of the Frontiers of Science often discussed. Both involve the fundamental nature of the laws of the universe.
In the shooter hypothesis, a good marksman shoots at a target, creating a hole every ten centimeters. Now suppose the surface of the target is inhabited by intelligent, two-dimensional creatures. Their scientists, after observing the universe, discover a great law: “There exists a hole in the universe every ten centimeters.” They have mistaken the result of the marksman’s momentary whim for an unalterable law of the universe.
The farmer hypothesis, on the other hand, has the flavor of a horror story: Every morning on a turkey farm, the farmer comes to feed the turkeys. A scientist turkey, having observed his pattern to hold without change for almost a year, makes the following discovery: “Every morning at eleven, food arrives.” On the morning of Thanksgiving, the scientist announces this law to the other turkeys. But that morning at eleven, food doesn’t arrive; instead, the farmer comes and kills the entire flock.
Wang felt the ground beneath his feet shift like quicksand. Everything seemed to wobble and sway. What was it about these two hypotheses that made Wang faint?
2.1 Inductivist Turkey
The farmer hypothesis borrows from Bertrand Russell’s cruel parable of chicken—fed every day until the morning it becomes dinner. The turkey scientist did nothing “wrong.” He observed, and he observed diligently. Each morning’s meal seemed to confirm the law: at eleven, food arrives. The more he recorded, the stronger his confidence grew. Until confidence itself was his undoing.
This is induction in its rawest form: the animal instinct that tomorrow will echo today. The more we see, the stronger the habit grips us. A thousand sunrises persuade us of the next. A thousand billiard shots reassure us of Newton’s mechanics. A thousand experiments across every branch of physics harden our trust in Noether’s theorem—that symmetry begets conservation. All of these, fundamentally, is no different from “at eleven, food arrives.” As Russell quipped, “Our instincts certainly cause us to believe that the sun will rise tomorrow, but we may be in no better a position than the chicken which unexpectedly has its neck wrung.”
At the shallowest level, the story teaches that induction is fallible: expectation built from repetition can betray. To rely on induction blindly is to live as animals do—entranced by habit, unarmed against surprise. The cruelty of the farmer is philosophy’s reminder: what feels like law may only be routine, one axe-blow away from collapse.
Induction can go wrong—not a revelation that philosophy is proud to offer. The real lesson cuts deeper. Hume drew a brutal conclusion: reason has no authority here. Habit does. We are animals of expectation, not creatures of proof. We are not guided by reason but by psychological inertia. And here is why.
2.2 Hume’s Problem
Induction is the act of reasoning from repeated past observations to expectations about the future—to expect tomorrow will echo today. Philosophy calls it a form of logic, yet there is no proof—it is muscle memory dressed as credence.
David Hume stripped induction to its bare bones. Why trust induction? He asked. One might say, “because it has worked in the past.” But this answer relies on induction again—the very act of reasoning on trial. A circular argument, like a snake swallowing its own tail.
Another answer, “because nature resembles itself.” But how do we know? Because it always has. Once more, the circle closes.
To break the circle, we have to stop somewhere. Either we fully embrace induction, no question asked; or we accept, without further doubt, that nature resembles itself—the Principle of the Uniformity of Nature. Either way, we are placing a bet. We wager that the world is consistent across space and time so that induction can help us learn something useful from this consistent world.
To believe in science is to place such a bet. That symmetry is not a trick and gives us conservation—Noether’s Theorem. That invariance will endure beneath noise—Uniformity Principle. That regularity outruns randomness—the law of large numbers. It is faith disguised as laws. Philosopher C. D. Broad concludes, induction is “the glory of science and the scandal of philosophy.”
Uniformity of nature is the invisible credit card science swipes every day, with no guarantee the bank won’t suddenly decline. In the end, we are no different from the turkey scientist, and we might very well get our necks wrung when nature ceases behaving uniformly. This is why Wang Miao staggered—he glimpsed that science rests not on proof, but on habit, or worse, on faith, as fragile as a turkey’s routine.
2.3 The Flatland Perspective
If uniformity is not a law but a bet, what abyss does that open? The farmer hypothesis includes a farmer, a powerful being who can change any laws in turkey science at a whim. But at least, turkey scientists, if they survive, can observe the farmer’s axe and learn that their “laws of nature” were conditional all along.
The shooter hypothesis suggests a worse possibility—the imperceptibility of such an omnipotent being.
When the shooter casually changes his shots, all that 2D scientists can observe is that the distance between “holes” changes. What are these observations like?
Now try it yourself. What do you see? A line? An ellipse? Science is built on such shadows.
Edwin Abbott made the point in his Flatland (1885): a coin becomes a circle from above, an oval at an angle, a line at the edge. Flatlanders see only the line, never the circle.
Flatland scientists lack words for circles or holes. Those are three‑dimensional luxuries. To them, a shooter’s hole is not a circle but an edge—an irregular boundary where things vanish. A line that swallows matter and light.
Observation is tethered to perspectives.
2.4 Plane vs. Sphere
For the flatlanders, geometry isn’t a system of axioms—it’s instinct. They live in the flat, breathe it, never question it. Every triangle they draw closes perfectly; every straight line feels honest. To them, flatness isn’t an assumption—it’s the air they breathe, the rhythm of their universe. Only from above would anyone suspect the ground itself might be curved.
They might picture themselves living in what we call a Euclidean plane, an infinite marble floor—smooth, predictable, obedient. Draw two straight lines and they will never meet, no matter how far they run. Triangles close neatly—their angles always summing to 180°; the world behaves. It’s the geometry we inherit without knowing, the quiet default of our thinking—even in middle school, we learned it without question: parallel lines never meet, and space feels obediently flat, both on paper and in our three‑dimensional imagination.
From the shooter’s perch, things might look completely different. What the flatlander calls straight, the shooter sees as curved; what they call certainty, he calls projection. The surface itself might swell and bow, following Riemannian geometry, a curved world where straight lines bend and the rules of triangles change. On a sphere, for instance, the shortest paths (“geodesics”) curve along the surface, and the angles of a triangle can add up to more than 180°. To a flatlander, these deviations are invisible; every triangle still looks straight. To the shooter, the distortion is obvious.
In the widget below, you can toggle between these two geometries—Euclidean and Riemannian—and witness how the same world appears under different assumptions. Draw a line. Make a triangle. Send Johnny the Walker on this flatland to keep going straight. See what happens from these two views respectively.
What looks law‑like and straight from within may curve and warp from above.
2.5 Geometry as Convention
Wang’s vertigo deepens: if geometry itself can change with perspective, what else can? Henri Poincaré once asked the same question and reached a scandalous answer—that geometry is not discovered but chosen. The shape of space is a matter of convenience, not truth.
We measure, compare, and claim that rulers stay constant, yet the decision to treat space as Euclidean or curved is a convention—a bet on simplicity and convenience. The difference between straight and curved is not in nature but in our bookkeeping.
Now try another widget. This was Poincaré’s thought experiment. Flatlanders now live on a disk, but with a special caveat. As one moves away from the center towards the periphery, everything—including one’s own size and the ruler in hand—shrinks.
In our view, flatlanders live on a disk, flat, bounded, except that things shrink as they move away from the center; In the flatlander’s view, they live in an infinite and unbounded world. Both systems work, each internally consistent. The only test of geometry is usefulness.
This was Poincaré’s quiet revolution. The laws we take as fundamental are often the shorthand of creatures limited by their vantage point. Experiment does not reveal which geometry is true, only which is comfortable. As he wrote, “[Experiment] tells us not what is the truest, but what is the most convenient geometry.”
Try toggling the above widgets again. What you see is not two worlds, but two grammars for describing one. Euclidean geometry tells a simple story of flat obedience; Riemannian, a more complex tale of curvature and connection. Neither owns reality—they are translations. And every science, whether physics or cosmology, writes its truths in one dialect or another.
To the shooter, the world curves; to the flatlander, it doesn’t. Both are right, and both are wrong. That is the quiet horror Poincaré handed down to modern science: even the ground beneath reason can be merely a matter of convention.