The Essence of Numbers: Invention or Discovery?
Have we invented or discovered numbers? This question has puzzled philosophers and mathematicians for centuries. To better understand this, we need to differentiate between the concepts of invention and discovery.
Understanding Invention and Discovery
Invention refers to the creation of something new, such as the Wright Brothers inventing the airplane. On the other hand, discovery involves identifying something that already exists, like Sir Isaac Newton discovering gravity.
When it comes to numbers, the line between invention and discovery becomes blurred. Before the advent of numbers like 1, 2, 3, and so on, people used pebbles to count objects. This means that the concept of counting was possible without a formal numerical system. However, to simplify and organize counting, we invented number systems.
Numbers as an Abstraction
Some theories suggest that numbers are simply abstractions created within our minds. According to David Bohm, the very concept of objects as separate and defined is a limitation of our perception.
Bohm believed that everything is interconnected and represents the whole universe. It is our senses that make us perceive certain parts as isolated entities, even though they are deeply interconnected, such as our fingers or the waves in the sea.
If there were no other minds in the world but our own, numbers, even as abstract ideas, would not have existed before we invented them.
Existence of Abstract Concepts
Unlike physical objects which exist in space and time, abstract concepts like numbers do not have the same criteria for existence. It is nonsensical to ask when a number like π first existed, as it is not bound by space or time.
However, this does not mean that numbers lack meaning. For example, there exists a number whose square is 4 and whose cube is 8, but no number exists whose square is 4 and whose cube is 9. This highlights the intrinsic logic and structure inherent in numbers.
Might we need to use different terms for these concepts? Perhaps we should use exists_physical for the existence of physical objects and exists_abstract for the existence of abstract objects like numbers.