# What makes a mathematical proof?

A mathematical proof is an argument that shows, using logical reasoning, that a mathematical statement is true.

A mathematical proof is an essential element of mathematics, and its importance cannot be overemphasized. It is a logical argument that demonstrates the truth of a mathematical statement based on established facts, axioms, and theorems. Proving a mathematical statement requires logical reasoning and a systematic approach to get from the initial assumptions to the desired conclusion. To quote mathematician and philosopher Alfred North Whitehead, “The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful; the ideas, like the colors or the words, must fit together in a harmonious way.”

Before we delve deeper into what makes a mathematical proof, let us look at some interesting facts about the topic. Did you know that:

• The concept of proof in mathematics dates back to ancient Greek philosophy. Mathematicians like Pythagoras, Euclid, and Aristotle formulated some of the foundational rules of proof that are still in use today.
• In 1936, mathematician Alan Turing proved that there are some mathematical problems that are unsolvable. This led to the development of computer science and the concept of computability.
• The Four Color Theorem, which states that any map can be colored with just four colors such that no adjacent regions have the same color, was only proved in 1976 by Kenneth Appel and Wolfgang Haken and required the use of computers to check over 1,800 cases.

Now back to the question at hand, what makes a mathematical proof? A proof must satisfy two fundamental criteria: validity and soundness. Validity means that the proof is logically correct and follows the rules of logical inference. Soundness means that not only is the proof valid, but it also has to be based on true and established facts, axioms, and theorems. In other words, the argument must be both valid and true to be considered a proof.

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However, there is no single method for constructing a mathematical proof. Some may use direct proofs, where the conclusion is verified based on established facts and theorems, while others may use proof by contradiction, where the negation of the desired conclusion is proved to be false. There are also other methods like proof by induction, proof by exhaustion, and so on.

To summarize, a mathematical proof is a convincing logical argument that demonstrates the truth of a mathematical statement based on established facts and theorems. As Richard Feynman, the famous physicist and mathematician, once said, “Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.” But with the right tools and a systematic approach, even the most complex mathematical statements can be proved.

Criteria for a Mathematical Proof Explanation
Validity The proof must be logically correct and follow the rules of logical inference.
Soundness The proof must be based on factual and established facts, axioms, and theorems.

A visual response to the word “What makes a mathematical proof?”

The video discusses the importance of going back and identifying key results and techniques to solve math proofs. The speaker advises spending time on a problem, but if it becomes too challenging, one should analyze the solution step by step and rewrite it on their own to understand every step. By doing so, individuals can learn more techniques for future problems and understand the significance of each step in the solution.

## I found further information on the Internet

A proof in mathematics is a convincing argument that some mathematical statement is true. A proof should contain enough mathematical detail to be convincing to the person(s) to whom the proof is addressed.

Thematic fact: The idea and demonstration of mathematical proof were first presented in ancient Greek mathematics. Thales and Hippocrates gave the first proofs of the fundamental theorems in geometry. The axiomatic method given by Euclid revolutionized mathematical proof.
Thematic fact: Mathematicians are proud that their deductive proofs are irrefutable. Assuming that the proof is correct, this is true. However, note that this is in spite of the mathematician. It is in the nature of the deductive method – from the general to the particular.

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