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You're probably wrong

About axioms and how they relate to life

Ugochukwu Chukwuma

Ugochukwu Chukwuma

08/07/2025 · 9 min read


Intro

Last year, while taking an MIT OCW math course, I came across the concept of axioms in Mathematics. On the surface, axioms are simple, but the most interesting thing about axioms is the staggering implication they carry: a single overturned axiom can collapse everything we think we know about a topic, it can turn a whole world of knowledge on its head and as God might have it, this upheaval has happened before, actually, more than once. We all live with our own set of axioms, this article will help you understand them a little bit more and try to help you navigate the unsettling thought that everything you think you know might be wrong.

Definition

An axiom (or postulate) is simply a statement we accept as true without proof, and it’s so intuitive that we hardly notice it. For instance, in everyday geometry we take for granted that “through any two points there is exactly one straight line.” It feels obvious, so we treat it as a given, but it has to be so for geometry to work.

Mathematics (and much of science) is built on these bedrock assumptions. Axioms serve as the starting points for all our reasoning: from them, entire systems, models and even worldviews spring forth via deductive logic.

Think of axioms like the rules of a game. Once everyone agrees on them, they define exactly what moves are possible, and what isn’t.

Consider chess:

  • The board is an 8 × 8 grid.

  • Each player starts with 16 pieces arranged in a fixed pattern.

  • Pawns may advance one square (or two on their first move) and capture diagonally.

  • Knights move in an L-shape, two squares in one direction, then one square perpendicular.

  • The game ends only by checkmate, stalemate, or resignation.

From these simple, unproven rules, an astonishingly rich world of strategies, tactics, and manoeuvres emerges. In exactly the same way, mathematics unfolds: complex theorems and entire branches of study grow out of a handful of seemingly self-evident axioms.

Let’s bring it back home. For example, the ancient people thought the earth was flat and they built whole systems out of that or that the sun moves because they could clearly see it rise from the east and set in the west, nothing made more sense. They also thought that earthquakes and other natural disasters happened when the gods were angry. Eventually we figured all this was wrong but it served its purpose, we built entire systems on it.

Axioms are so fundamental, they are the foundation of all systems, models and worldviews, and considering how important foundations are, if you shake, shift, or replace one, the entire “structure” built on top of it can collapse.

Now let’s go back to our chess example, if you change just one tiny little axiom, say how the knight moves, every opening, tactic, and defence involving knights must be rethought because the foundational principle underpinning its existence is no longer what it used to be.

That’s exactly what happened in mathematics and science and your life when certain axioms are challenged: everything relating to that axiom thought true heretofore will have to be re-evaluated, maybe even abandoned.

Examples

Next, let’s explore some of the most dramatic examples of this phenomenon, how key axioms were redefined and what new, self-consistent systems emerged as a result. Follow closely: it’s a journey into the very roots of how we know anything.

  1. Axiom of Choice → Multiple Mathematical Realities

The Axiom of Choice (AC) is one of the strangest turning points in modern mathematics, not because it’s obviously true or false, but because it can be both, depending on the world you choose to work in.

At its core, the Axiom of Choice says:

if you have a collection of non-empty sets, it’s possible to pick one item from each, even if the collection is infinitely large and there’s no rule telling you how to choose.

In simple English, if you have an infinite set of sets, say for example infinite teams of football teams, you can go to each team and pick a player, even if the number of teams are infinite and there is no rule telling you how to choose a player. Simple idea. But when mathematicians ran with it, things got weird.

In one world, where this axiom is accepted, you get results like the Banach–Tarski Paradox, where a solid sphere can be sliced into a few pieces and reassembled into two identical spheres. It’s mathematically sound, but physically absurd.

In another world, where this same axiom is rejected, you avoid these paradoxes, but lose access to many powerful results in algebra, topology, and analysis.

And here’s the twist: both versions of set theory turn out to be perfectly logical. In 1938, Kurt Gödel showed that if Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC) is consistent, then adding the Axiom of Choice doesn’t introduce any contradictions. Then in 1963, Paul Cohen proved that you can’t derive the Axiom of Choice from the other ZF axioms, it’s independent.

In plain terms, the Axiom of Choice is like a fork in the road: you can take it (working in ZFC) or leave it (working in ZF alone), and either way, you end up with a self-consistent mathematical universe. There’s no hidden “truer” system, just two equally valid frameworks.

As the Stanford Encyclopedia of Philosophy puts it:

“The Axiom of Choice does not lead to contradiction when added to ZF set theory, nor does its negation, meaning there is no ultimate ‘correct’ position. It’s a matter of which universe you wish to live in.”
[source: Stanford Encyclopedia of Philosophy]

So what’s fascinating isn’t just the axiom itself, it’s the implication that there is no one ‘true’ mathematical universe. We get to choose the rules, and different sets of rules lead to different, yet internally coherent, realities, we have two different truths about the same thing happily co-existing.

By the way this is the mathematical equivalent of parallel universes. All from one axiom.

  1. Euclid → Non-Euclid (Geometry as a Multiverse)

For over 2,000 years, Euclidean geometry was considered absolute truth. It was built on five postulates (axioms), and the most infamous of these was the fifth postulate, the “parallel postulate”, which essentially says that through any point not on a given line, there is exactly one parallel line. That is if you have a straight line (A) and you have a point outside the line there is exactly one line (B) you can draw that will be parallel to the first line A. Pretty intuitive right?

This axiom felt true. Nobody questioned it seriously, until the 19th century.

That’s when Lobachevsky and Bolyai asked: what if we just tweak that fifth postulate? Instead of insisting on one and only one parallel, they imagined a world where multiple lines could pass through that point and never intersect the original line.

Surprisingly, the math still worked. They built what we now call hyperbolic geometry, and it was just as logically valid as Euclid’s. Soon after, Riemann creates elliptic geometry, where no parallels exist.

So now, we had multiple geometries, each one self-consistent, but built on a different version of the same axiom.

Again, there is no contradiction. Each system is internally consistent, you just need to choose the geometry that fits your needs, whether you’re designing a floorplan (Euclid) or mapping the curvature of space-time (non-Euclid).

  1. Newton → Einstein (Physics is Relative)

For centuries, Newtonian physics explained everything from falling apples to planetary motion. It was elegant, intuitive, and it worked, assuming space and time were absolute. That was Newton’s hidden axiom, it was intuitive so again no one questioned it: time ticks the same for everyone, and space is a fixed stage.

But in the early 20th century, Einstein challenged this. What if space and time were not absolute? What if they were relative, fluid, flexible, and intertwined?

This shift gave us Special and later General Relativity, where space and time are warped by mass and energy. Time moves slower near massive objects. Light bends around stars. Everything Newton couldn’t explain suddenly made sense, but only by changing the axioms.

And again, here’s the key idea: Newtonian physics isn’t “wrong”. It’s still valid within its domain (low speeds, weak gravity). It’s a special case of a more general framework. But Einstein’s model is a new universe of thinking, built on a different assumption about the nature of time and space.

Why is understanding this important to me?

Good question. By now, you’ve seen that tweaking a single axiom, whether in geometry, physics, or set theory, opens up entire, self-consistent worlds. Here’s what I want you to know: all knowledge is tentative, especially yours, built on starting assumptions we might one day overturn. As Karl Popper reminds us,

science is not a quest for certain knowledge, but an evolutionary process in which even our most basic statements remain corrigible and open to review

Each of us lives by countless axioms, your religious and political beliefs, personal worldviews, or even conspiracy theories. None enjoys a monopoly on truth. Just as non-Euclidean geometry and relativity coexist with their classical cousins, so too can opposing beliefs be internally coherent, it is no wonder Christ said “Give to Ceaser what belongs to Ceaser”. The key is humility: recognise your “axioms” and understand that they make sense within your chosen framework and that is totally fine, but others’ frameworks may be equally valid.

So don’t waste time proving you’re “truer” than anyone else or looking for whose axiom is the “truest”. Instead, I suggest you take some time to think about your own axioms, remember they have this characteristic of being so intuitive we hardly recognise them, so that is your first job, to recognise yours, then understand that as much as you think it is true, it might be wrong or not totally correct, as a matter of fact it most likely is.