Proofs

In many areas of life, we require proof. We want proof that a product will work before buying it. We need proof that a medical treatment will actually work and not severely harm individuals before it can be used. We need proof that someone is guilty before convicting him. No matter where you look, we are a civilization built upon reasoning and evidence, and math is certainly no exception.

Proofs are vital to the life of mathematics. They are what assure us that theorems, corollaries, properties, etc. are true. They are what allow us to build complex ideas from simpler ones. Proofs provide the logical structure of math.

But, what exactly is a proof? A proof is a logical argument that shows, or proves, that a conclusion is true when certain assumptions or criteria are met. They can also show that two ideas are equivalent through logic and reasoning. They can even lead to some surprising results that do not make sense intuitively. This is one of the main reasons why proofs are so important. Just as an attorney cannot convince a jury that a man is guilty solely on the basis of his intuition, a mathematician cannot convince other individuals of an idea solely on intuition. Mathematics is a field of discoveries made by hardworking individuals who worked from the ideas of other individuals over thousands of years. But this only worked because they were able to convince others of the truth and value of their discoveries. Intuition may start someone along the right path, but it is logic that convinces others to journey down the path with him.

Whether we are dealing with geometry, modern algebra, calculus, number theory, set theory, analysis, or any other area of math, we will see applications of proofs. Proofs are how we know what we know. In this section, we will start with the fundamentals of logic by discussing some terminology and notation. We will then discuss some different methods of how to prove ideas along with some examples to see how they are applied. These example proofs will come from areas such as real analysis and number theory and will even include ideas from calculus within them. The branches of mathematics truly are connected and intertwined, which is perhaps most clearly seen when the proof of an idea in one branch requires ideas from multiple other branches.

The importance of proofs and the ability to write solid ones cannot be overstated. Without proofs, we would not be able to discover many of the hidden truths and intricate designs that display the beauty of mathematics.