What are mathematical proofs, really?

Introduction

In an elementary class, the teacher asked the pupils to do an experiment. He told them to plant flowers and group the flowers into two.  The first group was to be placed outdoors where there is sunlight. The other group was to be kept in a dim room. After a while, the pupils observed that the flowers outdoors grew healthy, while those kept indoors either died or were not as healthy. The pupils concluded that plants need sunlight.

flowers and mathematical proofs

In the experiment, the pupils observed that flowers with enough sunlight are healthier compared to those placed indoor. Some pupils may have an idea (a hypothesis) about this concept based on prior observation. The conclusion that plants need sunlight was based on the observation from the experiment.

In mathematics, similar situations occur. The hypotheses in mathematics are based on observed patterns. For instance, we can “conclude” from a few examples that the sum of two even integers is even.

What’s the difference? 

The difference is that in mathematics, we do not conclude after seeing a finite number of patterns or cases. In fact, the reason why I placed quotes in the word conclusion in the previous paragraph is because it is not really a conclusion per se. In mathematics, it is still a hypothesis or a conjecture. We can only conclude if we have proved it mathematically. Conjectures that are proven are called theorems.

A mathematical proof is a series of statements or logical deductions based on assumptions and theorems. The assumptions which are also called axioms or postulates are statements that are accepted without proof. Euclidean Geometry, for example, was built on five postulates. From these postulates, theorems and theorems are built on. Aside from the postulates, there is also a need for definitions to proceed with further theorems (see Euclid’s Elements for details).

Mathematical proofs is all encompassing. If a theorem is proven, it is true and will always be true for all possible cases. We have proved that the sum of two even numbers is even, so we are sure that it is true even though we have not paired and added all even integers. This is contrary to the experiment above. We cannot be sure that all plants need sunlight because it is impossible to experiment using all plants (of course, we know that not all plants need sunlight). That is what separates mathematics from other sciences — the beauty of its mathematical proofs.\square

Photo Credit (Creative Commons): Per Ola Wiberg via Flickr

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