Abstract
The study shall seek to explore the deep, underlying correspondence between the mathematical world of pure numbers and our physical reality. The study begins by pointing out that while the familiar, one-dimensional real numbers quantify many aspects of our day-to-day reality, complex numbers provide the mathematical foundations of quantum mechanics and also describe the behavior of more complicated quantum networks and multi-party correlations, and quaternions underlie Einsteinian special theory of relativity, and then poses the question whether the octonions could play a similar role in constructing a grander theory of our universe. The study then points out that by increasing the level of abstraction and generalization of axiomatic assumptions, we could construct a more powerful number system based on octonions, the seditions, or even other hypercomplex numbers so that we may more accurately describe the universe in its totality.
Highlights
What is it that breathes fire into the equations and makes a universe for them to describe? —Stephen Hawking As this article attempts to take a closer look at the various ways in which the macroscopic and microscopic world of physical reality and the mathematical world of pure numbers interact, influence and inspire each other, it is of great importance that we try to understand the intrinsically mathematical nature of our own universe
The four number systems can form a 64-dimensional space and the components of the subspace when multiplied by other components remain fixed in the subspace which in turn allows the particles to stay particles even when they are rerating, interacting, moving and transforming. This is the ‘R⊗C⊗H⊗O combination’ of four number system where R stands for reals, C is for complex numbers, H represents quaternions and O the octonions
Yanofsky explains the analogy using the following diagram: in both mathematics and physics, when we tend to focus our attention to the description of subsets of phenomena rather than the entire phenomena, we discover structures, symmetries and pattern while if we shift our focus towards the vaster scales, we find mathematics becoming more complex than before and physical theories failing to grasp the crux of the problem
Summary
What is it that breathes fire into the equations and makes a universe for them to describe? —Stephen Hawking As this article attempts to take a closer look at the various ways in which the macroscopic and microscopic world of physical reality and the mathematical world of pure numbers interact, influence and inspire each other, it is of great importance that we try to understand the intrinsically mathematical nature of our own universe. Many fundamental constants of nature and initial conditions of the universe do appear as being exquisitely fine-tuned to an unimaginably high degree of precision. Max Tegmark, in his Our Mathematical Universe, explores the controversial idea that “the ultimate reality is purely mathematical, demoting familiar notions such as randomness, complexity, and even change to the status of illusions, and implying that there’s a fourth and ultimate level of parallel universes” (13). Barrow and Tipler (1986), Leslie (1989), Davies (2006), Ellis (2007), and Barnes (2012) have pointed out this fine-tuning aspect of cosmos. The smallness of cosmological constant compared to the enormous value of Planck energy is another example in fine-tuning. There are as many as 26 dimensionless constants which further point towards the extremely fine-tuned, and intrinsically mathematical nature of our universe. The ideas of dark energy and quintessence are invoked to describe the nature of this cosmological constant
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