The Discovery of Incommensurables: A Challenge to Pythagoras' Philosophy
Table of contents
The Discovery of Incommensurables: A Challenge to Pythagoras’ Philosophy
In mathematics, incommensurability refers to a situation where two lengths or quantities cannot be expressed as a simple ratio of integers. This concept was first encountered in the study of triangles and the length of their sides, particularly in the context of right-angled isosceles triangles.
Overview
The discovery of incommensurables challenged the fundamental principles of Pythagoras’ philosophy, which held that all numbers could be expressed as a ratio of integers. This idea was central to his concept of harmony, where mathematical relationships were seen as reflecting a deeper cosmic order. However, when mathematicians began to explore the properties of triangles and discovered incommensurables, they encountered a problem that seemed to contradict Pythagoras’ principles.
Context
The discovery of incommensurables occurred during the 5th century BCE, in ancient Greece, as part of the development of Euclidean geometry. Mathematicians were grappling with the properties of triangles and the relationships between their sides and angles. At this time, there was a growing interest in the application of mathematical principles to understand the natural world.
Timeline
- 500 BCE: Pythagoras develops his philosophy, which includes the idea that all numbers can be expressed as a ratio of integers.
- 450 BCE: The discovery of incommensurables is made by mathematicians working on the properties of triangles.
- 400 BCE: Euclid writes Book X of his Elements, which includes a proof of the existence of incommensurables.
- 350 BCE: The concept of incommensurability becomes widely accepted in mathematical circles.
- 300 BCE: Mathematicians begin to explore the implications of incommensurability for other areas of mathematics.
Key Terms and Concepts
Incommensurability
Incommensurability refers to a situation where two lengths or quantities cannot be expressed as a simple ratio of integers. This concept challenges the idea that all numbers can be expressed as a ratio of integers, which was central to Pythagoras’ philosophy.
A triangle is a polygon with three sides and three angles. The sides of a triangle are called sides, and the angles are called angles.
Pythagorean Numbers
Pythagorean numbers are positive integers that can be expressed as a ratio of two other positive integers, where both numbers have no common factors.
Harmony
In Pythagoras’ philosophy, harmony refers to the idea that mathematical relationships reflect a deeper cosmic order. He believed that all numbers could be expressed as a ratio of integers and that this reflected a fundamental harmony in the universe.
Euclidean Geometry
Euclidean geometry is a branch of mathematics that deals with the properties of triangles and other geometric shapes. It was developed by Euclid in his book Elements, which includes a proof of the existence of incommensurables.
Key Figures and Groups
Pythagoras
Pythagoras (c. 570-495 BCE) was an ancient Greek philosopher and mathematician who developed the concept of harmony and the idea that all numbers can be expressed as a ratio of integers.
Euclid
Euclid (fl. 300 BCE) was a Greek mathematician who wrote the book Elements, which includes a proof of the existence of incommensurables. His work became the foundation of Euclidean geometry.
Mechanisms and Processes
The discovery of incommensurability can be broken down into the following steps:
- Assume that each side of an isosceles triangle is 1 inch long.
- Suppose the length of the hypotenuse is m/n inches, where m and n are integers with no common factors.
- Show that if m and n have a common factor, then either m or n must be odd, which leads to a contradiction.
- Conclude that no fraction m/n will measure the hypotenuse.
Deep Background
The concept of incommensurability has its roots in ancient Greek mathematics, where mathematicians were exploring the properties of triangles and other geometric shapes. The discovery of incommensurables challenged the fundamental principles of Pythagoras’ philosophy, which held that all numbers could be expressed as a ratio of integers.
Explanation and Importance
The discovery of incommensurability was important because it challenged the idea that all numbers can be expressed as a ratio of integers. This concept has far-reaching implications for mathematics and our understanding of the natural world.
Objections to Pythagoras’ Philosophy
One objection to Pythagoras’ philosophy is that it is based on an overly simplistic view of number. The discovery of incommensurables shows that there are limits to what can be expressed as a ratio of integers.
Implications for Mathematics
The concept of incommensurability has implications for mathematics, particularly in the development of Euclidean geometry. It also highlights the importance of rigor and proof in mathematical inquiry.
Comparative Insight
In comparison with other philosophers, such as Aristotle, Pythagoras’ philosophy can be seen as overly simplistic. However, his emphasis on the importance of harmony and the interconnectedness of all things remains an important part of Western philosophical tradition.
Extended Analysis
The Significance of Incommensurability
Incommensurability is significant because it challenges our understanding of number and its relationship to the world. It highlights the limits of mathematical expression and encourages us to think more deeply about the nature of reality.
The Impact on Mathematics
The discovery of incommensurability had a profound impact on mathematics, particularly in the development of Euclidean geometry. It led to a greater emphasis on rigor and proof in mathematical inquiry.
Philosophical Implications
Incommensurability has important philosophical implications, particularly for our understanding of number and its relationship to the world. It challenges us to think more deeply about the nature of reality and encourages us to explore new ways of understanding the world.
Quiz
Open Thinking Questions
• What are the implications of incommensurability for our understanding of number and its relationship to the world? • How does the concept of incommensurability challenge Pythagoras’ philosophy? • What are the broader philosophical implications of incommensurability?
Conclusion
The discovery of incommensurability challenged the fundamental principles of Pythagoras’ philosophy, which held that all numbers could be expressed as a ratio of integers. This concept has far-reaching implications for mathematics and our understanding of the natural world.