Chris Impey is a writer based in New York City. He's interested in all things tech, science, and photography related, and likes to yo-yo in his free time.


Follow him [on Twitter](https://twitter.com/chris-impey).

Chris Impey

Emma Hazel is a writer based in New York City. He's interested in all things tech, science, and photography related, and likes to yo-yo in his free time.


Follow him [on Twitter](https://twitter.com/emma-hazel).

Emma Hazel

Thomas Macaulay is a writer based in New York City. He's interested in all things tech, science, and photography related, and likes to yo-yo in his free time.


Follow him [on Twitter](https://twitter.com/thomas-macaulay).

Thomas Macaulay

In ancient Greece, by the sea, the followers of Pythagoras met in secret. They gathered in the dark, lit by flickering lamps, bound by a shared belief that numbers were sacred. Numbers, they thought, were the truth of the universe, the language of the gods. Above their school’s entrance, the words read: **"All is number."**

Among them was Hippasus, a young mathematician devoted to his craft. He, like the others, believed only in rational numbers. Mainly, every number can be written as a simple ratio of $\frac{a}{b}$. For example, 2/3 or 4/1, etc. Numbers were whole, clean, and perfect. But one night, Hippasus found something that broke this perfect world.

He discovered that some numbers, like the square root of 2, could not be expressed as fractions. No matter how he tried, they did not fit. He asked his teacher why would that be. The answer eluded his mentor as well, leaving him in the dark.  In a moment of profound despair, the teacher drowned his student.

### What Makes a Simple Number So Troubling?

Although the murder of Hippasus is  probably a matter of legend the secret was real, and certainly dangerous enough to the beliefs of those who knew about it—a warning whispered among those who knew. Why?

Imagine a right angle triangle with 2 sides equal to one. From Pythagoras’s theorem :

![](/images/blog/pythagorean.png)


We know that the hypotenuse's length squared is $1^2+1^2=2$, making the length of the hypotenuse itself $\sqrt2$ . The number which when multiplied by itself makes $2$. But what is that number? The square root of 2 isn't 1 because $1 \times 1 = 1$.  It’s not $2$  because $2 \times 2 = 4$. It's something in between but what is it?

At first, this wasn't a problem for the Pythagoreans. It was obviously a ratio of two whole numbers. They only had to figure out what that ratio was. After more and more frantic attempts, a horrible discovery was made. There is NO ratio that will produce $\sqrt{2}$. —it simply cannot be done. It's what we now call an irrational number. It can't be represented as a ratio of whole numbers.

This was a number that defied all their rules—a number we now call irrational. It can’t be expressed as a ratio of whole numbers, no matter how hard you try. For the Pythagoreans, this was not just a mathematical anomaly—it was a threat. They lived by the belief that numbers were the foundation of the universe, and every number fit into a neat, orderly system. But irrational numbers were a dark secret, a crack in their perfect world of order. To acknowledge these numbers was to admit that their entire understanding was flawed. Worse yet, it was to admit that the universe itself was not as perfect and orderly as they had believed.

Hippasus’s discovery was more than a flaw in their math; it was a challenge to their core beliefs. It was proof that the world was not built entirely on whole numbers and ratios. For the Pythagoreans, that was a truth too dangerous to accept. One that may have cost Hippasus, his life. 

### Troubles of Belief : Facing Our Fallibility

The case of Hippasus is far from unique. Throughout history, individuals—be they scientists, philosophers, or prophets—have encountered ideas that threatened the established belief systems of their time. These ideas were often met with ridicule, hostility, and at times, even violence.

In the “Myth of the framework”, Karl Popper makes an important remark that resonates with me. That is, 

> The history of intellectual progress is a history of prejudice and dogma, held with tenacity, often accompanied by intolerance and fanaticism. Yet, even amidst religious wars and persecutions, there is a brighter side. It is heartening to remember that countless individuals, from ancient to modern times, have been willing to live and die for their convictions—for ideas they believed to be true. Man, we may say, is not so much a rational animal as an ideological animal.
> 

“As humans, we are ready to die for our ideas!”

This noble tendency to cling to our beliefs is a double-edged sword. On one hand, it is honorable to defend and even die for one’s convictions. But on the other, what a waste, given the high likelihood that you, me and everyone is wrong. Is it worth it to die for a bad idea? What about living? 

Considering how little we know—both individually and collectively—it is crucial to acknowledge our limitation. The fallibility of human knowledge is an undeniable truth, one that we must confront if we are to engage with reality. 

This is the basis of my thesis : **As Human, we are fallible!** 

In the following paragraphs, I will explore the implications of this recognition—the necessity of being open to ideas that challenge our beliefs, and the importance of a method for determining when and how to abandon those beliefs.—Too ambitious? Probably!

I should perhaps clarify: when I refer to scientific theories, mathematics, or philosophy—fields I intend to borrow from—it is not to champion the conclusions of modern science or present it as an ultimate authority. I am not here to evangelize for science. My focus is on addressing the problems within my religious beliefs. I am interested in the methods and principles of these disciplines because they offer a framework for understanding how we arrive at knowledge—and, crucially, when and how we should be ready to let go of our beliefs when confronted with new evidence. 

Since I’m abandoning my beliefs, I find it fit to explore such a theory.

An essay about humanity's fallibility and the need for openness to ideas that challenge deeply held beliefs.

Dangerous Idea