Wednesday Wisdom

WHO?

Euclid was a famous ancient Greek mathematician, often referred to as the "Father of Geometry." While much of his personal life remains shrouded in mystery, his mathematical contributions and teachings have had a profound and lasting impact on the field of mathematics. Euclid is believed to have been born around 300 BCE, in Alexandria, Egypt, in the Ptolemaic Kingdom which was part of Greek (Hellenistic) Alexander the Great’s empire. Little is known about Euclid’s early life, family, or education, but it is likely that he studied mathematics in Athens, Greece, which was a center for intellectual and philosophical pursuits during his time.

What he produced

Euclid is best known for his work Elements, a comprehensive compilation of knowledge in geometry and number theory. Elements is considered one of the most influential and enduring works in the history of mathematics and is believed to be the second most printed book after the Bible.  Euclid presented a systematic and rigorous approach to geometry, organizing knowledge into a series of propositions and theorems, built upon a small set of axioms which are universal truths. This approach formed the foundation of deductive reasoning in mathematics. He also made significant contributions to number theory, including the study of prime numbers and the famous algorithm for finding the greatest common divisor known as the Euclidean algorithm. Euclid's work laid the groundwork for the study of geometry and mathematics for centuries to come, and it was widely used as a textbook well into the modern era. Building upon Greek philosophers like Socrates, Aristotle and Plato, Euclid applied the philosophical logic technique known as “Reductio ad Absurdum” (Reduction to the absurd) to mathematics.

Euclid used “reductio ad absurdum” as a method of proof by contradiction to establish his mathematic theorems. By assuming the opposite of what he wanted to prove and demonstrating that this assumption led to a logical contradiction or absurdity, Euclid ensured that his conclusions were based on rigorous and indisputable logical foundations. This method was crucial in "Elements" as it allowed Euclid to demonstrate the validity of his geometric propositions and theorems, laying the groundwork for modern mathematics by emphasizing the importance of rigorous deductive reasoning. Euclid's use of “reductio ad absurdum” set a standard for mathematical rigor that has influenced mathematicians for centuries, shaping the way mathematical proofs are constructed and ensuring the reliability of mathematical knowledge.

The earliest example of a “reductio ad absurdum” argument can be found in a satirical work attributed to Xenophanes of Colophon criticizing a poem by Homer about the attribution of human faults to the gods. Xenophanes states that humans also believe that the gods' bodies have human form. But if horses and oxen could draw, they would draw the gods with horse and ox bodies. The gods cannot have both forms, so this is a contradiction. Therefore, the attribution of other human characteristics to the gods, such as human faults, is also false.

This technique is commonly used in mathematics, philosophy, and logic to challenge arguments and beliefs. It is a powerful tool for exposing inconsistencies or contradictions in reasoning and can be an effective way to critically analyze and evaluate arguments.

2023 why do we care?

One argument in favor of gun control might use “reductio ad absurdum” by considering a scenario where there are no restrictions on firearm ownership because of the word “infringed” in the second amendment to the US Constitution. The argument might proceed as follows: Assume that there are no restrictions on firearm ownership. Consequently, anyone can own any type of firearm, including dangerous military-grade weapons. This leads to a potential situation where individuals with malicious intent can easily bazookas, missiles or tanks. This would lead to misuse these weapons, resulting in increased violence and deaths. This outcome is clearly undesirable and contradictory to the goal of public safety. Therefore, the initial assumption that there should be no restrictions on firearm ownership leads to an absurd or contradictory conclusion. As a result, the argument supports the need for reasonable gun control measures.

In practice, reductio ad absurdum involves assuming the truth of a proposition or assumption and following its logical implications to a point where the conclusion becomes absurd or contradictory. By demonstrating the absurdity of the conclusion, one can infer that the original proposition or assumption must be false.

Philosophers and scientists have employed and popularized the method of “reductio ad absurdum” as a valuable tool in logical reasoning, mathematics, and discourse. It appears that sometimes it takes the absurd to bring about the rational.

And now you know...

Philosophy is the art of thinking, the building block of progress that shapes critical thinking across economics, ethics, religion, and science.

METAPHYSICS: Literally, the term metaphysics means ‘beyond the physical.’ Typically, this is the branch that most people think of when they picture philosophy. In metaphysics, the goal is to answer the what and how questions in life. Who are we, and what are time and space?

LOGIC: The study of reasoning. Much like metaphysics, understanding logic helps to understand and appreciate how we perceive the rest of our world. More than that, it provides a foundation for which to build and interpret arguments and analyses.

ETHICS: The study of morality, right and wrong, good and evil. Ethics tackles difficult conversations by adding weight to actions and decisions. Politics takes ethics to a larger scale, applying it to a group (or groups) of people. Political philosophers study political governments, laws, justice, authority, rights, liberty, ethics, and much more.

AESTHETICS: What is beautiful? Philosophers try to understand, qualify, and quantify what makes art what it is. Aesthetics also takes a deeper look at the artwork itself, trying to understand the meaning behind it, both art as a whole and art on an individual level. A question an aesthetics philosopher would seek to address is whether or not beauty truly is in the eye of the beholder.

EPISTEMOLOGY: This is the study and understanding of knowledge. The main question is how do we know? We can question the limitations of logic, how comprehension works, and the ability (or perception) to be certain.