1+1 Equals 2…OR DOES IT?

12 min readOct 8, 2023

The Paradox at the Heart of Mathematics

“The whole problem with the world is that fools and fanatics are always so certain of themselves, and wiser people so full of doubts,” Bertrand Russell

This statement is false.” Is that true? If so, that would make the statement false. But if it’s false, then the statement is true. This sentence creates an unsolvable paradox; if it’s not true or false, what is it?

Some things are just obvious. The sky is blue, the letter ‘A’ comes before ‘B’, and 1+1 equals 2 except that the latter isn’t as obvious as you might think. Think about it. What does it mean when we say that we’re adding stuff? What does it mean when we say two things are ‘equal’? What does it mean when we talk about the number 1? Here’s a fun little exercise for you: Try to describe what a number is without using the word ‘number’ or any other word/concept derived from the word ‘number’.

Now, if you really put all of your brain cells at work trying to accomplish the above task, it’s time to giddy your horses up. This is gonna be a wild ride as we venture deep into the world of pure logic and find an abyss that hides beneath the foundations of reality!

The Foundation of Mathematics: Logic

We learn about science by observing the world around us. We observe, theorize, and experiment to uncover natural phenomena. It seems pretty fascinating to imagine a universe where time moves backward, or where string theorists acknowledge experimentalists. However, when you try to think of a universe where 1+1 equals 3, it doesn’t feel right. It seems True by definition and even though math works exceptionally well when it comes to describing our universe, is it really about our universe? This begs the question,

Why is the Math the way it is?

Of course, we aren’t the only souls to have thought of it. This very question captured the attention of one of the greatest philosophers of all time, Plato. He thought of numbers, geometrical shapes, and the relations between them as objective truths. Regardless of what world they’re in, they are bound to be true. In Plato’s view, a ‘World of Forms’ existed independently where objects like numbers were present. However, one of his pupils, Aristotle, had contrasting views. He rather believed that numbers were properties of objects than numbers themselves being objects. For instance, if a human is said to have 2 eyes, it doesn’t mean that ‘2' and ‘eyes’ are separate objects. He thought of the number 2 as a property used to describe a pair of eyes. Numbers describe features of our world. Philosopher Immanuel Kant thought of them to be described by intuition and experience. WAIT, HOLD UP! You just used ‘2’ to describe ‘1’ pair of eyes!?

If numbers correspond to an object’s property, how can multiple numbers be used to describe the same object? It doesn’t make sense. This thought daunted German mathematician, logician, and philosopher Frege (Friedrich Ludwig Gottlob Frege). He reasoned that numbers don’t actually apply to objects but to concepts. He strongly believed that mathematics (especially arithmetic) can be reduced to pure logic i.e., you can understand math based on reason alone. Logic is a tool for reasoning about how different statements affect each other solely through deduction and inference. This idea is famously called Logicism.

Frege defined numbers by using concepts and extensions. A concept can be any idea you can think of- such as a square or a glass of water or perhaps this medium article; an extension is the set of all things that fall under that concept. Here are a few examples for a better understanding of concepts and extensions.

Concept: The Shape Square
Extension: The set of all square things
Concept: Round Squares
Extension: Empty set

In the same way, numbers are extensions of concepts.

The number 3 is the extension of the concept: **All things having a collection of these many objects.**

Frege stated that all concepts we can possibly think of would have a corresponding extension (set). This is known as the General Comprehension Principle.

Mathematics Cracked (LITERALLY)

Now, things start getting interesting. Frege was just about to have his work printed when one of his colleagues, Bertrand Russell, wrote him a letter. To understand the issue he raised, we need to travel back and pay a visit to ancient Greece (specifically the 4th century BC).

Finding the Right Angle with Euler

Back then, only 2 fields of mathematics were known: Geometry and Arithmetic. And where there’s geometry, there’s Euler. Euler was a Greek mathematician who believed math to be the language of the universe (debating on whether that is true or not deserves its own article :>). He laid the foundations of Euclidean geometry based on observable facts of the world. They are as follows:

A straight line segment can be formed by joining any two points in space.

Any straight line can be extended indefinitely on both sides.

Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as the center.

All right angles are congruent.

Credits: https://www.slideserve.com/elsu/euclid-s-five-postulates

For at least a thousand years, geometers were troubled by the disparate complexity of the fifth postulate. It seemed less evident, and mathematicians thought it could be derived from the first four. 2000 years later, in the 19th century, Nikolai Lobachevsky and Friedrich Gauss experimented with what would happen if they entirely discarded the 5th postulate. Voila, Geometry was still shockingly consistent! This geometry (without the 5th postulate) was very creatively named ‘Non-Euclidean Geometry.’ This further branched into hyperbolic and elliptic geometry.

Consider two straight lines indefinitely extended in a two-dimensional plane that are both perpendicular to a third line (in the same plane):

In hyperbolic geometry, they “curve away” from each other, increasing in distance as one moves further from the points of intersection with the common perpendicular; these lines are often called ultraparallels.
In elliptic geometry, the lines “curve toward” each other and intersect. By formulating the geometry in terms of a curvature tensor, Bernhard Riemann (German Mathematician) allowed non-Euclidean geometry to apply to higher dimensions.

Systems existed where angles of a triangle added to a sum of greater than 180 degrees, surfaces had a single side, objects existed in fractional dimensions, and all of them were CONSISTENT. This was a big BIG blow to mathematics, and it wasn’t the only one.

Cantor’s Infinity

Non-Euclidean geometry was supplemented by new mathematical fields such as complex numbers, calculus, and set theory each with their own paradoxes involving abstract concepts like infinity. Now, set theory was a very controversial field and wasn’t much welcomed by mathematicians. Set theory is the mathematical theory of well-determined collections, called sets, of objects that are called members, or elements, of the set. Georg Cantor, the founder of set theory, rifted mathematicians into 2 teams: Intuitionists and Formalists. I mean, wasn’t math the purest form of knowledge and just speaks the truth about our universe? How can math fracture? … THAT DOESN’T SOUND POSSIBLE! Well, not so fast. The reason revolves around infinity.

Now, Cantor wanted to know if 2 infinite sets, say a set of natural numbers and a set of real numbers between 0 and 1 were equal. By equal, he meant that there is a one-to-one correspondence between the elements of the two sets i.e for one element of the first set, there is an available element in the second set to pair up with.

Initially, you might assume that in a similar fashion, the set of all natural numbers and the set of real numbers are equal (I mean both are infinite, right? RIGHT?). Cantor showed a different picture. He showed that the set of real numbers is infinitely larger than the set of natural numbers. What? How?

Let me explain. On the left, you can see the set of natural numbers going all the way to infinity. On the right, are placed random real numbers to correspond to the set of natural numbers. Both seem to go on forever; However, there is a catch. Let’s assume that both the sets have a one-to-one correspondence, and all natural numbers match up with a real number. Now, I want you to consider the first decimal place of the first real number and add one to it. Consider the 2nd decimal place of the second real number and add one to it. If you go on forever using the same trick, well congratulations! You have successfully found a real number that is different from every other real number that was present in the infinite set. In fact, you can perform the same steps for an infinite amount of time and find infinitely more real numbers than natural numbers. Cantor called these Countable and Uncountable infinities. The fact that some infinities were larger than others is what gave birth to the intuitionists and formalists. The intuitionists believed that math was a pure creation of the human mind and infinities like Cantor’s weren’t real. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true. The formalists, on the other hand, believed that math could be entirely built on pure logical foundations like Cantor’s set theory. This group was informally led by David Hilbert (again, a German mathematician).

All in all, a secure mathematical foundation had to be built to fix the cracks that were arising, and this is where we go back to the letter Bertrand Russell sent to Frege.

Shaving Russell’s Paradox: Tackling the Barber’s Dilemma

Russell, being a formalist, pretty much acknowledged what Frege claimed about numbers and their extension sets. However, there was one problem and it turned out to be a fundamental flaw in set theory and later, in math as a whole. To understand the flaw, let’s first consider a set that is not a member of itself.
For example: The set of all letters of the alphabet- X1= {A, B, C, D…}
Clearly, X1 is not a member of itself. Now, Russell asked Frege to consider the set of all sets that are not members of themselves. He then proceeds to ask if that set is a member of itself or not. If it is, then it is not a member of itself. However, if it isn’t, it actually is a member of itself. This is called a self-referencing problem.

Credits: https://www.abdn.ac.uk/law/blog/russell-revisited-a-return-to-russells-paradox-and-the-principle-of-parliamentary-sovereignty-/

Confused? Let’s get a better understanding through a Hairy Analogy.
Consider a village populated entirely by grown men with beards. There exists a strange law in the village. The law states that the barber of the village must shave all and only those men who don’t shave themselves. Now, the barber lives in the village too and he’s obviously a bearded man. Who shaves the barber? If he doesn’t shave himself, then the barber has to shave him but he himself is the barber. If he does shave himself, then he (the barber) doesn’t shave him. So, he can’t shave himself, because the barber only shaves those who don’t shave themselves!

This particular problem was later solved by the Zermelo–Fraenkel set theory (ZFC). However, self-reference didn’t give up that easily. It popped up in art, literature, language, computing, biology, quantum physics, and most evidently in logic and mathematics. David Hilbert, along with other formalists like Russell, wanted to build a complete formal system of mathematics: A symbolic and logical language with a rigid set of manipulation rules that are free of paradoxes.

Math from Scratch: Principia Mathematica

David Hilbert idealized math to be complete, consistent, and deterministic. To prevent mathematics from collapsing, two tragic heroes, Alfred North Whitehead and Bertrand Russell decided to take charge. They decided to start from scratch; and provide an entirely new mathematical framework. They published Principia Mathematica, a 3-part volume, in 1910,1911, and 1912 respectively after a decade’s worth of struggle. This is the book that contains the infamous 379-page proof of 1+1=2. Given that you’ve read right to this point, it might be understandable enough. They weren’t trying to prove 1+1=2 but rebuilding the entire foundation of mathematics.

How? Using logical axioms and inferences. Logical axioms form the base on which the pillars of mathematics are built on. They are statements that are so self-evident and ‘obvious’ that they do not need proof for their validity. ‘1+1 equals 2' might seem very self-evident to an average person. That is because statements like these are taught to us like blatant facts and we’ve known such information since we were toddlers. We’re taught numbers and addition, and we never seem to question why it is the way it is. In fact, 1+1 equals 2 has an umpteen number of assumptions built into it. So, what are these logical axioms? For example, If ‘p is true’ is a valid statement, then ‘p is false’ is a valid statement as well, and a valid statement can be either true or false.

Next, we need inferences. Inferences are conclusions that can be drawn from axioms based on reasoning and evidence. Examples include:

If ‘p is true’ is true, then ‘p is false’ is false.
If ‘p is true’ is true, then ‘p is not true’ is false.
If ‘p and r are true’ is true, then ‘r and p are true’ is also true.

Finally, we also need logical symbols (they’re really not that logical, trust me). They helped to move from axioms to inferences and deduce further. You can check the below link to see what the notation looked like.
The Notation in Principia Mathematica
Great! Now we’re completely ready to understand why 1+1 actually equals 2!

1+1=2

Let’s break the above proof down (I promise you don’t have to gauge out your eyeballs trying to comprehend this book and its proof. They’re really not human-readable). First, we must understand what the symbols 1 and 2 mean. 1 is not just- 1 ARTICLE or- 1 FINGER. As Frege had stated, 1 is an extension of a concept. It is essentially the extension of the concept of all sets with exactly 1 element in it. What does ‘=’ mean? We had talked about this in Cantor’s set theory and how he used one-to-one correspondence to prove if 2 sets are equal to each other. So far so good. Now, it’s time we proceed to the much-awaited proof.

The first statement (54.43) states what the below section is going to prove.
54.26 references to a section that proves if set A has 1 element in it, and set B also has 1 element in it, then the set of numbers having all numbers from set A and B (their union) will have 2 elements given that both the numbers aren’t equal. The next two statements are simple as well. They state that if the 2 sets have distinct numbers, then their intersection (common elements) has no elements in it i.e they have no overlap and the sets are different. 11.54 talks about how if 2 things exist, one of those things exists, and also how the other thing exists as well. They, then proceed to conclude that if set A and B both have 1 element in it and their intersection has no element in it, the set containing both set A and B has exactly 2 elements in it, or how the very cool Gen Z would put it as 1+1=2
One interesting thing to note is that they haven’t defined ‘Arithmetic Addition’ yet (Hopefully it’s in the later sections) … so the proof is essentially incomplete HAHA!

Anyway, it seems like significant progress was made. We’ve finally laid the foundations of mathematics free of any paradoxes. It left no room for errors or flawed logic. Did Hilbert’s dream come true? Math was finally-

Complete (Every true statement can be proven true)
Consistent (Free of contradictions)
Decidable (There is always an algorithm to show that a statement follows from the axioms)

A decade’s worth of struggle was worth it! Or WAS IT?
That is to be found out in the next article (welp this article has already been obnoxiously long)

Till then,
Stay Curious!