Why are Proofs Important??

The only way to verify that a claim about mathematics is correct is to prove it.  Numerical evidence or the fact that the claim is true in some physical situation is suggestive but is not verification. We all know that 1+1=2 but think about why it is true. Because someone told us and we believed them? Take a look at its proof and think about how important they are to mathematics.

P1. 1 is in N.
P2. If x is in N, then its "successor" x' is in N.
P3. There is no x such that x' = 1.
P4. If x isn't 1, then there is a y in N such that y' = x.
P5. If S is a subset of N, 1 is in S, and the implication
(x in S => x' in S) holds, then S = N.

Then you have to define addition recursively:
Def: Let a and b be in N. If b = 1, then define a + b = a'
(using P1 and P2). If b isn't 1, then let c' = b, with c in N
(using P4), and define a + b = (a + c)'.

Then you have to define 2:
Def: 2 = 1'

2 is in N by P1, P2, and the definition of 2.

Theorem: 1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.
Then 1 + 1 = 1' = 2 Q.E.D.

Here’s a classic proof where 2=1. Can this be true? See if you can identify the fallacy in the link below?

http://www.math.toronto.edu/mathnet/falseProofs/first1eq2.html