# The Mathematical Truth Behind George Orwell's 2+2=4

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## Chapter 1: Understanding Orwell's Assertion

The statement “Freedom is the freedom to say that two plus two makes four” from George Orwell’s dystopian novel, *Nineteen Eighty-Four*, left a profound impression on me during my initial reading. My first thought was to question why Orwell chose 2+2=4 instead of a simpler equation like 1+1=2. This curiosity marked my journey towards a deeper understanding of mathematics, ultimately leading me to pursue a degree in the subject.

Before delving into the concept of why 2+2=4 holds true, it’s essential to lay out some foundational rules that underpin our daily calculations. This framework is known as the Peano axioms, a logical system concerning natural numbers proposed by the 19th-century mathematician Giuseppe Peano. These axioms form the basis of what is referred to as the Peano arithmetic system.

## Chapter 2: Proving 2+2=4

To demonstrate that 2+2=4, we can rely on several axioms and definitions derived from Peano’s framework:

**Axiom 1**

0 is classified as a natural number.

**Axiom 2**

For any natural number x, s(x), representing the successor of x, is also a natural number.

**Definition 1**

For ease of reference, we denote s(0) as 1, s(s(0)) as 2, s(s(s(0))) as 3, and s(s(s(s(0)))) as 4.

**Definition 2**

If x is a natural number, then a+0=a.

**Definition 3**

If x is a natural number, then a+s(b)=s(a+b).

At this point, I encourage you to pause and attempt to use this information to validate Orwell's assertion.

**The Final Proof:**

Using Definition 1, we can express 2+2 as s(s(0)).

Applying Definition 3, we rewrite 2+2 as s(s(s(0))) + s(0).

Using Definition 3 again, we can express this as s(s(s(s(0))) + 0).

Finally, through Definition 2, we simplify this to s(s(s(s(0)))), which by Definition 1 equals 4.

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