I have been recently introduced to a book about Geometry’s presence in nature. I was brought back to my days at Columbia University when I worked with my brilliant math tutor from the Republic of Georgia-Alex- who helped me get through my beast of a calculus course. We are again working together on more abstract and philosophical mathematical concepts in 2024.
The Greek Philosophers of old saw numbers and shapes and math in everything they experienced in life. It was their way of trying to understand their existence. But also led them to understand that so much was so far beyond their knowledge, comprehension, and grasp.
For example- the Monad- or number 1- was represented by them as a circle. The study of circles led Greek mathematicians like Archimedes to explore the properties of the circle's circumference and diameter, which brought them to the concept of pi (π), an irrational number.
The fact that pi cannot be expressed as a simple fraction, and that it has infinite decimal places, was a profound discovery and showed that circles were mathematically unique and mysterious.
This infinite nature added to the circle’s allure and the idea that it was linked to deeper truths about the universe.
Lastly ponder this fact: A perfect circle does not exist, nor can it be constructed, by human instruments.
1. Pythagoras (c. 570–495 BCE) – Numbers and the Nature of Reality
- Concept: Pythagoras and the Pythagoreans believed that numbers were the fundamental building blocks of reality. They saw mathematical relationships everywhere, from the harmony of musical scales to the motion of celestial bodies.
- Universe Connection: Pythagoras held that reality is deeply mathematical and that understanding the relationships between numbers can reveal truths about the cosmos. For example, the concept of harmony in music was expressed through mathematical ratios, which he believed mirrored the harmony of the universe.
- Philosophical Impact: This view laid the groundwork for a worldview in which mathematics and structure are intrinsic to the cosmos, influencing later scientific and philosophical thought.
2. Plato (c. 428–348 BCE) – Mathematical Forms and Eternal Truth
- Concept: Plato introduced the idea of the "world of forms"—an abstract, eternal realm where perfect mathematical objects like triangles and circles exist. He believed that the material world is an imperfect reflection of these ideal forms.
- Universe Connection: According to Plato, the forms, including mathematical objects, represent the most real aspects of the universe, as they are unchanging and perfect. Geometry, for instance, connects us to this higher reality, since geometric shapes in the physical world are imperfect imitations of the ideal forms.
- Philosophical Impact: Plato’s philosophy of mathematics suggests that mathematical knowledge is a way to access eternal truths about the universe, influencing thinkers like René Descartes and Immanuel Kant.
3. René Descartes (1596–1650) – Mathematical Certainty and Knowledge
- Concept: Descartes believed that mathematics, particularly geometry, provided a model of certainty for human knowledge. In his method of systematic doubt, Descartes sought to find knowledge that could not be doubted, and mathematical truths became foundational for his philosophy.
- Universe Connection: Descartes viewed the physical universe as fundamentally mathematical and mechanistic, where space and matter could be understood through geometry and laws of motion. His development of Cartesian coordinates, which allowed for the representation of geometric shapes through algebra, emphasized the interconnectedness of the physical and abstract worlds.
- Philosophical Impact: Descartes’ mathematical approach to philosophy helped establish the idea that reality can be understood through clear, rational structures, like those found in mathematics.
4. Gottfried Wilhelm Leibniz (1646–1716) – Infinite and Infinitesimal Calculus
- Concept: Leibniz developed calculus, which deals with change and motion, and introduced the concept of the infinitesimal—an infinitely small quantity. His work helped to describe continuous processes, like the motion of planets or the growth of organisms.
- Universe Connection: Leibniz believed that the universe could be understood in terms of mathematical principles, particularly through his work on infinitesimals. His calculus allowed for a better understanding of how objects in nature move and change over time, capturing the continuous flux of the universe.
- Philosophical Impact: His view that the universe operates through mathematical laws, and that these laws can be described using concepts like infinity and continuity, influenced later scientists and mathematicians. Leibniz also speculated on the idea of a "pre-established harmony" in the universe, rooted in mathematical order.
5. Bertrand Russell (1872–1970) – Logical Foundations of Mathematics
- Concept: Russell aimed to reduce all of mathematics to logic. He believed that mathematical truths could be derived purely from logical principles, and this led him to work on the foundations of mathematics.
- Universe Connection: By demonstrating how mathematics arises from logic, Russell suggested that the universe itself might be structured according to logical and mathematical principles. The regularities and laws of the universe, from the orbits of planets to the behavior of particles, could be understood as expressions of logical-mathematical truths.
- Philosophical Impact: Russell's work deeply influenced the development of analytic philosophy and the philosophy of mathematics, suggesting that mathematics not only describes the universe but also reflects the underlying logical structure of reality.
6. Immanuel Kant (1724–1804) – Space and Time as A Priori Knowledge
- Concept: Kant argued that space and time are not empirical concepts derived from experience but are instead the a priori structures that shape all of our experiences. In his view, the human mind imposes a mathematical structure on reality.
- Universe Connection: Kant’s view suggests that our understanding of the physical universe is inherently mathematical because the way we perceive space and time is structured by mathematical intuitions. Geometry, for instance, is a tool that our minds use to navigate the spatial world, and arithmetic structures our understanding of time.
- Philosophical Impact: Kant’s work laid the groundwork for later philosophical inquiries into the relationship between mathematics and the physical universe, particularly in the realms of physics and cosmology. His ideas influenced Einstein’s theory of relativity, which also redefines the nature of space and time.
These philosophers' explorations of mathematics and the universe helped shape both scientific and philosophical thought, contributing to our understanding of reality.
Can a perfect circle be drawn?
In practical terms, a perfect circle cannot be drawn by hand or even by machine, at least not in the physical world. Here are several reasons why achieving a truly perfect circle is impossible, along with some philosophical and mathematical perspectives on the idea of a "perfect" circle:
1. Physical Limitations
- Imperfections in Tools: Whether you are using a pencil and compass, or even advanced machinery, there are always slight imperfections. The surface of the paper or material you're drawing on, the precision of the instrument, and the physical limitations of the tools will result in deviations from a mathematically perfect circle.
- Atomic Structure: On a microscopic level, the materials we use (such as paper, metal, etc.) are composed of atoms, which are not continuous. This atomic "graininess" introduces minute irregularities, meaning no physical circle can ever achieve the true mathematical definition.
- Environmental Factors: External factors like air resistance, gravity, or even the motion of the Earth can subtly affect any attempt to draw a perfect circle.
2. Mathematical Definition
- In mathematics, a perfect circle is defined as the set of all points in a plane that are equidistant from a center point. This means that, conceptually, a perfect circle exists only in abstract, mathematical space. It is a purely theoretical construct that exists within geometry, where there are no physical imperfections.
- The concept of pi (π), an irrational number, further underscores the impossibility of creating a perfect circle. Pi represents the ratio of the circumference to the diameter of a circle, and because pi has an infinite number of non-repeating decimal places, it is impossible to express the circumference or diameter of a perfect circle with absolute precision.
3. Plato’s Realm of Forms
- From a philosophical perspective, especially drawing on Plato's theory of forms, the perfect circle belongs to the realm of abstract, ideal forms. According to Plato, no perfect object (including circles) exists in the material world. Instead, physical circles are mere approximations of the true, perfect circle that exists only in the world of forms or ideas.
- The circles we see or draw are always flawed versions of the ideal, which we can only conceive of in our minds.
4. Technological Advances
- While modern technology can create extremely precise circles, even the most advanced machinery or software cannot produce a perfect circle. High-precision devices, such as computer-assisted drawing (CAD) systems or 3D printing, can come remarkably close, but they are always limited by pixelation, rounding errors, or material imperfections.
- On a digital screen, for example, what appears to be a circle is actually composed of pixels, which are tiny squares or dots, creating only an approximation of the continuous curve of a true circle.
5. Perfect Circles in Nature?
- Some circles in nature, like the shape of planets, raindrops, or bubbles, can seem remarkably circular. However, even these are not perfectly circular due to forces like gravity, rotation, and imperfections in the environment.
- Planets, for instance, are not perfect spheres or circles; they are slightly oblate due to their rotation, which causes them to bulge at the equator.
