The Infamous Proof of 1+1=2
Introduction
In the realm of mathematics, there exist certain statements that are considered to be universally true. One such statement is the equation 1+1=2. While it may seem trivial, this equation has been the subject of much debate and scrutiny among mathematicians and philosophers alike. In this article, we will delve into the fascinating world of mathematical proofs and explore the rigorous demonstration of the equation 1+1=2.
The Proof
The proof of 1+1=2 is often attributed to the German mathematician and philosopher, Gottlob Frege. In his seminal work, "Begriffsschrift" (1879), Frege presented a formal system of arithmetic, which included a proof of the equation in question.
The proof proceeds as follows:
Step 1: Definition of Equality
To begin with, we define equality as a binary relation, denoted by the symbol "=". This relation satisfies the following properties:
- Reflexivity: a = a (every object is equal to itself)
- Symmetry: if a = b, then b = a (equality is symmetric)
- Transitivity: if a = b and b = c, then a = c (equality is transitive)
Step 2: Definition of Successor
Next, we define the successor function, denoted by S(x). The successor of a natural number x is the number that comes immediately after x.
Step 3: Definition of Addition
We define addition as a binary operation, denoted by +. For natural numbers x and y, x + y is defined as the result of applying the successor function S(x) to y.
Step 4: Proof of 1+1=2
Now, we are ready to prove that 1+1=2. Let's start with the definition of 1 as the successor of 0, denoted by S(0). We can then rewrite the equation as:
1 + 1 = S(0) + S(0)
Using the definition of addition, we can expand the right-hand side as:
S(0) + S(0) = S(S(0) + 0)
By the definition of successor, we know that S(S(0) + 0) = S(S(0)) = S(1).
Therefore, we can conclude that:
1 + 1 = S(1) = 2
Conclusion
In conclusion, we have rigorously demonstrated the equation 1+1=2 using a formal system of arithmetic. This proof, while seemingly trivial, highlights the importance of mathematical rigor and the need for precise definitions and logical reasoning in mathematics.
As the great mathematician, David Hilbert, once said, "Mathematics is a game played according to certain simple rules with meaningless marks on paper." While the game of mathematics may seem simple, the rules that govern it are far from trivial.
