A Collection of thoughts on Mathematical Languages:
Natural Reflections on Algebra & Geometry
This post is inspired by an intriguing thought that disturbs me often when I try to relate the beauty of algebraic notions and their syntactical complexities with the physical dimensions and measures of geometric nature. Geometrical shapes of vivid complexity is all over us. Algebraic knowledge becomes richer and deeper day by day.
At the outset, algebra appears to me as the attempts of human mind to find an order within the physical and its own humane cognitive nature. Hence at times algebra becomes associative and at times abstract. Geometry is mostly occupied with unique physical operators for measuring nature and natural objects. As human mind and its boundless imagination works in both the streams of knowledge, they always explore anew territories and often produce complex functions and abstractions. Humankind crafts nature in their own reflection.
As we know, our innate tendency to imitate nature has been one of the fundamental driving force in all our scientific, artistic and engineering initiatives. This tendency gets transformed into further advanced forms of knowledge as we apply our reflections of nature into our labor process. This tendency can be seen in the mathematical languages as well. Our mathematical observations about geometry has helped our investigations to find patterns in the field of numbers and algebraic symbols. Is there a difference in the nature of measurement in Algebra and Geometry? Though the approach remains the same the challenges differ widely.
Some metric level contradictions in the nature of Algebra and Geometry
I was always perplexed about the physical differences between various elementary mathematical operators like addition, subtraction, multiplication and division. When I add entities of different properties, the sum never accumulates in nature. It occupies a physical space less than or equal to the nature of individual entities.
Taking an example, if we add one mango and an orange in a physical box they take space as equal to their physical dimensions. In the representational mathematical space also they occupy distinct locations. When we take two mangoes here they occupy same space in the physical box whereas they add on to the same quantity in the representational space.
The question is whether the mathematical symbolic space is as accurate as the physical geometrical space. For convenience we may denote this representational space as algebraic. The algebraic space is often limited by the limits of our abstractions and the capacity to associate them whereas the geometric space is challenged by the scope of observation and the limits on the minutest quantum that we can reach.
Numbers are the base constructs of the symbolic space. They become complex when the physical world in the scope becomes wider and deeper. We think of complex numbers and complex plane when we reach a limit to measuring the diagonal of a triangle made of single unit edges. Then the new symbol enters our mathematical vocabulary. This is just an example. Similarly when we think of trigonometric measurements, we find new sinusoidal and cosine patterns emerging in symbolic space. It is a new synthesis of physical and symbolic space.
Next question is can symbolic space travel ahead of the physical space. In other words, can we dream a physical entity before experiencing its real physical nature. I must say that at times yes. It is not magic or miracle. It is because of the intellectual capacities of human cognition and the power of knowledge to expand the horizons of our recognition. Each strides in the cognitive skills of human mind and our social experiences enrich our mathematical languages, whether they reflect our knowledge about symbolic space or physical space.
