Russell’s Paradox – A Lesson On How To Crash Mathematics

Russell’s paradox is one of the most famous paradoxes in all of mathematics. The paradox is named after Nobel laureate and polymath Bertrand Russell. It has gotten famous over time for the technical challenges it poses to the field of mathematics.

However, the fascinating and gripping story behind this paradox is not as well-known as its technical details. This essay will focus on the story behind Russell’s paradox rather than the mathematical intricacies. We will still cover the scientific essence of the paradox. However, we will skip the rigorous mathematical details that are irrelevant to the story.

You will go through a thrilling and gut-wrenching series of events that almost crashed mathematics. You might even be surprised to know that Bertrand Russell himself is not the protagonist in this story.

Are you not intrigued? Without any further delay, let us jump directly into the thick of the plot.

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The Great Mathematical Debate

When we consider sciences such as physics, chemistry, and biology, we see that these sciences are subject to experimentation with real-world phenomena. However, with mathematics, this is not the case.

Sure, we use mathematics to make sense of real-world phenomena, but apparently not the other way around. In this sense, mathematics starkly differentiates itself from other sciences. This difference has caused many to question if mathematics is a science in the first place.

Whilst this is still an ongoing debate today, back in the 19th century, this uniqueness of mathematics triggered an even more fundamental set of questions amongst deep thinkers. Such questions included the following:

1. Is mathematics true by definition?

2. Why is mathematics apparently not based in reality?

Around the same time such questions were popping up, mathematicians noticed that the different branches of mathematics such as algebra and number theory were fragmented. So, they were looking for a unified foundational theory that describes mathematics at the most fundamental level.

The Fundamental Theory of Mathematics

To be clear, mathematicians had been trying to establish a fundamental theory of mathematics for centuries before the 19th century. Historically speaking, the world of mathematics and the world of philosophy often intersected in this context.

Plato considered numbers, variables, and their mathematical relationships as objective truths. He even invented a theory of forms for these special mathematical creatures. Aristotle, who followed after Plato defined numbers as properties of objects, and not mathematical objects themselves.

For instance, if you have 3 candy bars on a table, according to Plato, the number ‘3’ was an object and the candy bars themselves were also separate objects. The number ‘3’ (as an object) is then related to the candy bars (as a set of objects) through a mathematical relationship. But according to Aristotle, the number ‘3’ was just a property of the set of candy bars. His theory deviated from the theory of forms that Plato had developed.

Over the next centuries, deep thinkers and philosophers deviated from the view that mathematical objects were objective truths. For instance, Immanuel Kant believed that mathematics could not be understood without developing an intuition for it.

With this situation as the backdrop, the protagonist of our story makes his appearance.

Friedrich Ludwig Gottlob Frege

Gottlob Frege was a German philosopher, logician, and mathematician. Born to a family that valued education, Frege encountered philosophies in his early childhood that would guide his scientific career. His father had written a “help book” for children aged 9 to 13 that taught the German language. This book was Frege’s first exposure to the logic of language.

He matriculated at the University of Jena, where he focused on mathematics and physics. Following Jena, he continued his studies at the University of Göttingen, which was the leading university for mathematics in the German-speaking world at that time.

Here, he had access to some of the world’s best thinkers and innovators of the time such as Wilhelm Eduard Weber (co-inventor of the electromagnetic telegraph with Carl Friedrich Gauss). He eventually got a doctorate with a dissertation on “A Geometrical Representation of Imaginary Forms in a Plane.”

Even though his early academic career was focused on geometry, Frege’s focus quickly turned towards the world of logic.

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Frege and Logic

Frege fundamentally disagreed with Aristotle’s view that numbers were properties of objects. According to Frege, if this were the case, then each object should have correspondence with one number only. But in reality, this is not the case.

For instance, if you have rice grains in a bowl in front of you, depending upon how you look at them, the number changes. If you look at them as individual rice grains, then there are a lot of them that you will need to count. But if you look at the bowl as the measure, it is just one bowl of rice. Other examples for this phenomenon could be a pair of shoes (or two shoes), a deck of cards (or 52 cards), etc.

Based on this, Frege developed the notion that numbers are related to concepts and not objects. He also disagreed with the notion that one needs to develop intuition first before one can understand or work with mathematics. He famously set forth on a journey to prove that the laws of arithmetic could be derived from the basis of reason (logic) alone.

His main goal was to reduce mathematics to logic, and prove that logic is the foundation of mathematics and all of its branches (today, this idea is known as Logicism). Frege described numbers using the notion of concepts and extensions.

Concepts and Extensions

According to Frege, a concept could literally be anything that could be described. Examples of concepts are cars, ships, dogs, dogs with one ear, etc.

An extension, on the other hand, is the set of all things that fall under a particular concept. For instance, if we consider dogs with one ear as the concept, then its extension would be the set of all dogs with one ear (time-independent: past, present, and future).

Here is an interesting concept: circular triangle. Based on the definitions of circles and triangles, we can safely say there exists no triangle that is a circle. So, the extension to this concept would be an empty set (we will see the relevance of an empty set later).

Frege built this thought process as an axiom into his theory in “Begriffsschrift”. Begriffsschrift was a turning point in the history of logic. It solved (among others) the problem of multiple generality. Previously, logical systems had difficulties understanding logical constants such as ‘some’ and ‘all’.

For example, it was not easy for older logical systems to differentiate between the sentences: “every boy loves some girl” and “some girl is loved by every boy.” But the formalism stated in Begriffsschrift could handle it easily.

Frege continued developing his ideas and was getting closer and closer towards isolating logical principles. This would mean proof that there would be no intuition required for understanding or working with mathematics. He published part of his work in Vol. 1 of Grundgesetze der Arithmetik (Basic Laws of Arithmetic) in 1893. A decade later, ever closer to his goal, Frege was preparing to publish Vol. 2 of Grundgesetze der Arithmetik.

It was at this critical moment that he heard from a certain Bertrand Russell, the antagonist of our story!

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Bertrand Arthur William Russell

Bertrand Russell was a Welsh polymath who was born into royalty. His paternal grandfather, Earl Russel, had twice been Prime Minister in the 1840s and 1860s. Russell had a lot of misfortune as family members around him kept dying due to illness when he was a child. Eventually, he was transferred to the care of his grandmother, Countess Russell. She turned out to be the dominant family figure for the rest of his youth.

In his autobiography, Russell recalls the following about his youth:

“nature and books and (later) mathematics saved me from complete despondency..”

Apparently, he had considered committing suicide several times. His curiosity to learn more mathematics was the only wish that had kept him from doing so (committing suicide). Needless to say, any such deep desire typical bears strong fruits. Russell turned out to be a prolific mathematician and a deep thinker.

He went on to become an academic and focused on a slew of topics in mathematics and philosophy (among other interests). He was deeply impressed by Frege’s work but hit upon a worrying “difficulty”. As Frege was preparing to publish Vol. 2 of Grundgesetze der Arithmetik, Russell wrote the following in a letter to Frege (don’t be alarmed if the text does not make sense immediately):

“Let w be the predicate: to be a predicate that cannot be predicated of itself.

Can w be predicated of itself? From each answer, its opposite follows.

Therefore, we conclude that w is not a predicate.”

Again, if this sounds confusing to you, worry not. We will look at what Russell means here momentarily. Just note that Russel’s original letter was in German (the quote is translated). Furthermore, the term ‘predicate’ is confusing. Instead of going into the details of Russell’s letter, let’s look at an intuitive example that explains Russell’s “difficulty”.

The Barber Paradox

The barber paradox is essentially a puzzle that Russell derived in order to intuitively describe his “difficulty”. It helps understand the problem at hand without going through unnecessary technical jargon.

Imagine a town that has only one barber. The barber is the “one who shaves ALL those, and those ONLY, who do not shave themselves”. If this is true, then the question posed is:

“Does the barber shave himself?”

If the answer is yes, then he is a town denizen who shaves himself and therefore, is not allowed to shave himself (as per the rule). If the answer is no, then the barber is not allowed to shave ALL of those who did not shave themselves (a violation of the rule).

This is essentially Russel’s paradox. Russel had questioned a notion of a set that is a set of itself. Consider a set of all shoes. Such a set is not a member of itself.

On the other hand, consider a set of all sets that are not shoes. Such a set is a member of itself. So far, so good. But Russel’s “difficulty” was:

Is the set of all sets that are not members of themselves, a member of itself?

You are welcome to take your time and re-read this sentence (several times if need be) to appreciate the value of Russel’s paradox. It knocks at a fundamental problem of set theory logic and mathematics.

If the set of all sets that are not members of themselves is a member of itself, then it is NOT. If it is not a member of itself, then it IS! This is essentially the paradox.

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Frege’s World Crashes Down

This seemingly simple question posed by Russell had exposed a fundamental flaw in one of the axioms assumed by Frege. In turn, Frege was so distraught by this event that he went through a mental breakdown and had to be treated medically.

Regardless of his situation, he responded with earnest respect and professionalism to Russell and tried to resolve the issue. Russell, in turn, marveled at how humbly and courageously Frege handled the whole situation.

“As I think about acts of integrity and grace, I realise that there is nothing in my knowledge to compare with Frege’s dedication to truth. His entire life’s work was on the verge of completion. Much of his work had been ignored to the benefit of men infinitely less capable.

His second volume was about to be published. And upon finding that his fundamental assumption was in error, he responded with intellectual pleasure clearly submerging any feelings of personal disappointment.

It was almost superhuman and a telling indication of that of which men are capable if their dedication is to creative work and knowledge instead of cruder efforts to dominate and be known.”

— Bertrand Russell

Frege continued to toil in an attempt to solve the issues, but unfortunately, had to abandon a lot of his work because the issue was at such a fundamental level.

The sad part is that Frege was never really recognized during his time, and had to live with that painful burden all along his difficult journey.

It was only after his life that his work gained popularity when Russell and other philosopher mathematicians along this path gave (due) credit to Frege and built upon his work.

Final Remarks

The intriguing point about Russell’s paradox is that Russell was not even the first person to discover it. Apparently, a German mathematician named Ernst Zermelo had discovered it independently before Russell.

However, Zermello had not published his findings, and the knowledge remained known only to David Hilbert and a few other academics at the University of Göttingen. Russell was simply the first person to express/publish the paradox.

As I write this essay in an attempt to do justice to everyone who was involved in this beautiful story, I cannot help but be fascinated by how many different people’s lives were involved in recognizing and tackling Russel’s paradox.

Fast forward to the present, I’m happy to share that Frege’s efforts did not go in vain. The Zermelo-Frankel set theory successfully tackled Russel’s paradox (I’ll cover how it did so in a future essay) and is widely accepted as the most common foundation of mathematics.

To be fair, the Zermelo-Frankel set theory is not perfect by any means. But it carries a lot of Frege’s original work with it and showcases the grit of generations of mathematicians who did not give up in the face of adversity.

If Frege were to be shown the mathematics of the Zermelo-Frankel set theory, I am sure that it would bring a smile to his face!

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Credits and References: Bernard Linsky (research article) and Jade Tan-Holmes (presentation).

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Further reading that might interest you: The Thrilling Story Of Calculus and Why Do You See Mirrors Flipping Words?