You would think that 1+1=2 is simple: one tennis ball plus one tennis ball equals two tennis balls. What is there more to say? But that is not how a pure mathematician thinks of 1+1=2. From a pure mathematician’s point of view the statement 1+1=2 is true because of a combination of definitions, axioms, proofs, and theorems. In mathematics, definition is an exact statement of the meaning, nature, and/or limits of a mathematical object. An axiom is a proposition regarded as self-evidently true without proof. A proof is a rigorous mathematical argument which unequivocally demonstrates the truth of a given hypothesis. A hypothesis that has been proven is called a theorem.
You can think of mathematics as building a brick wall. The axioms are the foundation of the wall. The theorems are the bricks in the wall. And the proofs are the cement that attached the theorems to the wall. The wall started in approximately 300 BC with Euclid who stated axioms and then proved theorems. With Euclid’s definitions, axioms, proofs and theorems, he created the geometry we call Euclidean Geometry. Euclid’s book the Elements was used as a mathematical text book for over two thousand year.
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Year after year, century after century, millennium after millennium, mathematicians have been building the wall higher and higher.
Around 1900 AD mathematicians noted a problem; inconsistencies were being introduced in the wall particularly in “naïve” set theory. The Barber Paradox illustrated one of the inconsistencies. Here is the Barber Paradox: “A town only has one male barber. The barber only shaves all the people who do not shave themselves. So, who shaves the barber?”
- If you say the barber shaves himself, you arrive at a contradiction; the barber only shares people that don’t shave themselves.
- If you say the barber is shaved by someone else, you arrive at a contradiction; by the barber being shaved by someone else, all the people that don’t shave themselves are shaved by the barber, that is to say, he shaves himself.
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The inconsistencies unleashed a crisis in mathematics which was solved by careful consideration of the axioms used. Several axiom systems were proposed: the Zermelo-Fraenkel axioms, the Dedekind-Peano axioms and others to solve this crisis. We’ll use the Dedekind-Peano (Peano pronounced like “piano”) axioms to prove that 1+1=2.
The first axiom (A0, zero exists) is that zero, written 0, is a number and an element of the set of Natural Numbers. A set is a collection of objects. The Natural Numbers are written: ℕ. The phrase “is an element of” is written: ∈. The phase “equals” is written: =. So, the first axiom, in mathematical notation is written: 0∈ℕ. Intuitively, ℕ is the set of numbers: 0,1,2,3,4,… and so on to infinity. We have not defined the Natural Numbers; we will do that from the axioms. We do not know what is in the set ℕ yet, except for the number 0.
The second axiom is that there exists a successor function, called s, which takes as input a Natural Number “a” and returns another Natural Number “b”. A function is a rule that takes an argument “a” and returns a result “b”. We write A1 in mathematical notation: if a ∈ℕ and s(a) = b, then b ∈ℕ. Intuitively, s(a) = a+1, this is the “rule”. But we have not defined addition yet, so we do not know this.
There are three other axioms that define the set of Natural Numbers, but we don’t need them to show 1+1 = 2. But we do need to define numbers other than zero and we also need to define addition. Let us start by defining numbers other than zero. The number zero exists from A0. And we define the following:
- 1 is defined to be s(0).
- 2 is defined to be s(s(0)).
- 3 is defined to be s(s(s(0))).
- 4 is defined to be s(s(s(s(0)))).
- and so on to infinity.
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We define the Natural Numbers to be the sequence: 0, s(0), s(s(0)), s(s(s(0))),…
We define addition with two rules:
- Addition1: if a ∈ℕ, a + 0 = a. In English, if “a” is a Natural Number, then “a” added to zero is “a”.
- Addition2: if a, b ∈ℕ, a + s(b) = s(a + b). In English, if we have “a” added to “s(b)”, this is equivalent to “s(a+b)”.
Here is the proof that 1+1=2:
- start with 1 + 1
- 1 + s(0): by definition of 1
- s(1 + 0): by Addition2
- s(1): by Addition1
- s(s(0)): by definition of 1
- 2: by the definition of 2
So from the point of view of a pure mathematician, 1+1 = 2 is true because of definitions, axioms, and a proof. Note that this independent from any physical or real-world definition; sorry tennis balls. Next, we’ll prove that 2+2=4. Just kidding.
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Category: WHY
