Why does reality have the appearance of obeying rules; rules which have the appearance of obeying mathematics?
Two questions are currently bothering me. The first is, what is mathematics? The second is why is there an arrow of time? This post will attempt to answer the former and will remain, due to the intractable nature of the question and my gross level of ignorance, just an attempt. Plus the second question is much too hard.
Why, you may ask, the question ‘what is mathematics?’ Having finished one opus: Bertrand Russell’s History of Western Philosophy, and attempted (but failed after only chapter 4) another: Roger Penrose’s The Road to Reality, of the ideas which we have, mathematics and numbers are very mysterious.
To highlight this mysteriousness I shall ask a question: do (Euclidean) triangles exist? This is a shape which is made up of three points connected by three lines whose internal angles add up to 180 degrees. Any triangle which you draw is not a perfect triangle; there is texture and imperfections in the paper, thickness of the ink and inaccuracy in human reproduction. Even if we use lasers at nano-technology scale, our triangles are still constrained by atomic particles. If we try and ‘prove’ a triangle using other means: algebra, set theory, axiomatic logic; then we are merely changing the language with which we choose to describe it. It is this seemingly metaphysical quality of things like the triangle example which led Plato to conceive of the theory of perfect forms, which have a level of existence separate to the physical reality we can touch and the mental world where we conceive ideas. These forms exist outside of time and our ability to understand them and are only revealed as we start to unravel the Universe. The perfect triangle existed since before the beginning of the Universe and physical reality, and it is only in our attempt to grasp at it mentally that we are alerted to its existence. It may seem absurd – disembodied triangles floating around the heavens – but the following question will highlight the importance of explaining what triangles and other mathematical concepts are.
“Why does reality have the appearance of obeying rules; rules which have the appearance of obeying mathematics?”
For example, in chemistry the periodic table, valence and thus chemical reaction, is based around the number of protons that reside in the nucleus of an atom. When an atom has eight protons it is ‘complete’ and doesn’t need to attract any electrons. There are elements with eight protons and they tend to be un-reactive – the ‘noble gases’ helium, neon and argon.Â If an atom has one proton, such as hydrogen, then it will tend to try and find other elements with which it can make up the magic eight, whether it is one chlorine (HCL) atom or two oxygen atoms (H_{2}0). Whilst valence is now seen as a simplistic way of describing atomic interaction, this example highlights how our understanding of the natural world appears to obey numbers.
The creation of complicated linguistic and mathematical explanations for things we perceive in the natural world can be explained by describing the way in which humans formulate ideas: through a continual reiterative process of experiencing and simulating the world into novel combinations. If you were to take a temporal (meaning right now) snapshot of your brains it would reveal neuronal processes which are simulating many things. As we think of a chair, and maintain as meditation attempts solely the idea of the chair and no other mental distractions or concepts, our simulation will become as close to the pure notion of a chair as our mental state can achieve. In the same way our snapshot of our brains when thinking of the number two will be neuronal processes performing a simulation. The two processes are equivalent, both are simulations which represent something else. The something else in the case of a chair and the number two are easy to understand as the relate to tangible objects which we can perceive through our senses; we can experience a chair which we can later simulate, we can experience two chairs and simulate the idea of the quantity of two.
The fact that these are just simulations and don’t prove their subjects existence is only self-evident when we use more abstract simulations, for example ‘justice’ and a ‘quadratic equation’ (ax^{2}+bx+ab=0). Both these simulations have little perceptible existence outside of other simulations. Justice is based on notions of how we feel when interacting with other human beings, past events and current circumstances and our other notions of good, fairness, right and wrong. The quadratic equation outlined here contains letters and numbers in algebra, which themselves are based on certain axioms which in turn are based on the rules such as those which underpin the number two (read Wikipedia for an extended explanation of what algebra is). The quadratic equation can represent a graph with x and y axes on which a curve sits. This curve in turn can represent something which we perceive and simulate, such as the trajectory of a chair being defenestrated. Neither of these representations are the perfect form of the quadratic equation; just like the perfect form of justice, the perfect quadratic equation does not exist outside of our simulation.
To finish and return to the earlier example of the science of the small the indeterministic quantum mechanics may be the science which is the harbinger of the destruction of physics as we know it. Whilst we still describe quantum mechanics in mathematical terms it would seem a logical step to take, given languages acknowledged separation from reality, for us to realise that the mathematics of physics is ultimately divorced from reality as well. A good description but never the same thing.