How Can You Learn and Understand the Philosophy and Mindset of Mathematics?
Embarking on the journey to learn and understand the philosophy and mindset of mathematics requires a deep dive into the nature of mathematics itself. Central to this exploration is the fundamental question of what mathematics is. This question, seemingly simple, leads us to delve into two major philosophical perspectives: the Platonic view and the cognitive representation view. Let’s explore each in detail.
Understanding Mathematics: The Platonic View
The Platonic View:
According to the Platonic view of mathematics, the mathematical concepts and theorems that we encounter do not originate from the physical world but pre-exist it. In this view, mathematics is a realm of eternal and immutable truths that exists independently of human thought and external reality. This realm, often referred to as the universe of ideas, is accessible only through the workings of human consciousness. The laws of mathematics, therefore, are not discovered but rather are discovered and applied by the human mind.
One might wonder, where do the laws of mathematics come from? According to this perspective, these laws are eternal and exist before the physical universe came into existence. They are considered as intrinsic to the fabric of reality. However, it is only through the human mind that these mathematical truths can be perceived and utilized.
The Platonic ideal of mathematics emphasizes the idea that mathematical truths are objective, eternal, and universal. These truths exist independently of human knowledge, and it is our role as mathematicians to uncover and understand these truths through rigorous logical reasoning and abstract thought.
Understanding Mathematics: The Cognitive Representation View
The Cognitive Representation View:
In contrast to the Platonic view, the cognitive representation view posits that mathematical concepts and theories are products of the human mind and have not always existed. This perspective suggests that our understanding of mathematical concepts is derived from how our mind represents and perceives the world around us. This view is best articulated in the works of George Lakoff and Rafael Nú?ez in their book Where Mathematics Comes From.
The cognitive representation view argues that mathematics is not a pre-existing entity but an artifact of the human cognitive system. It suggests that our mathematical understanding has evolved from the way our mind processes sensory input and perceives reality.
For example, consider the recognition of symmetry in a kaleidoscope. This ability to recognize symmetry and other perceptual features is immediate and comes from parallel processing in the visual system. This indicates that our perception of symmetry and other mathematical concepts is deeply rooted in our sensory and cognitive systems.
The ability to perceive and understand mathematical concepts like edges, surfaces, and volumes stems from how our perceptual system is designed to detect patterns and make simplifying judgments about the world. Our minds evolved to perceive and categorize the world in a way that is useful for survival and understanding the natural world.
Connecting Abstract Math to Perceptual Concepts
Understanding the relationship between abstract mathematical concepts and our perceptual system is crucial. For instance, the concepts of edges, surfaces, and volumes are primal concepts that our minds can grasp instinctively. These concepts form the building blocks of mathematical thinking.
Our perceptual system is designed to detect simplicity in the form of stark contrasts and symmetries. This is evident in our ability to recognize the symmetry and periodicity of ornamental patterns in cathedrals like Notre Dame or the Al Hambra. These patterns reflect the underlying mathematical principles of symmetry, geometry, and transformation.
Moreover, our love for simplicity and order, such as the distinction between dark and light, and the shapes of spheres, cubes, and Platonic solids, reveals the deep perceptual roots of mathematics. These perceptual sensations and simplifications underpin our understanding of mathematical concepts and help us make quantitative judgments about the world.
In summary, the philosophy and mindset of mathematics are complex and multifaceted. Whether you subscribe to the Platonic view or the cognitive representation view, understanding the nature of mathematics requires a deep dive into the origin and development of mathematical concepts. By recognizing the connection between abstract math and perceptual concepts, we can better grasp the underlying principles that govern our understanding of the physical world and mathematical truth.
Key Takeaways:
The Platonic view suggests that mathematics is a pre-existing realm of eternal truths. The cognitive representation view argues that mathematics is a product of the human cognitive system. Understanding the relationship between abstract math and perceptual concepts is crucial for a comprehensive understanding of mathematics.Further Reading:
Lakoff, G., Nú?ez, R. E. (2000). Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being. Plato's dialogues, particularly Timaeus, provide a significant philosophical foundation for the Platonic view of mathematics. Studies in cognitive science and psychology that explore the roots of mathematical understanding in perception and cognition.