Beauty of Maths
Happy Pi day and Einstein’s birthday to boot.
As I tried to make the point last time, maths, like music should be enjoyed for its beauty. It provides a description of the universe and an insight into how it works. It is ever present in our daily lives in places like schedules, comparing value when shopping, risk assessment, dieting among many other activities. The universe is forever abiding by its rules.
Kids love maths as computer games, scoring sports and counting within games. They just don’t like worksheets and rote learning. Adults love maths when gambling, talking motor sports, ballsports, buying appliances, genetics, cooking and knitting (the ultimate in string theory). Just not when doing their tax or book keeping or otherwise being menaced by it.
So I’ll have a look at some of these after the next few blogs eg Gambling maths, including probability, risk analysis, game theory, data distributions; computer games using algorithms, neural networks, genetic algorithms; sports with laws of motion and entertainment with signal processing and time series.
Plus some more esoteric topics like limits, transforms, quantum mechanics, string theory and as always keeping the science and maths simple.
Continuing first with the theme of the last blog let’s have a quick look at our best descriptions of the universe. Again don’t get hung up on the detail or worrying that it is too hard to understand (no-one really understands it anyway).
These most modern descriptions suggest that the universe is just one huge mathematical entity where the smallest “particles” are units of information. There are currently two theories leading the way, General Theory of Relativity for large objects and Quantum Mechanics for small stuff. Both have proven invaluable in understanding the universe and both work brilliantly wherever they can be tested. Both have been invaluable in making our current lifestyle of smartphones, GPS, electronics, satellites etc. Mathematicians and Physicists are trying very hard to combine them into one Grand Unified Theory (GUT) with little luck to date.
This is the study of the real small stuff and, famously it’s a quite different universe to the big stuff we are used to. It is very difficult to describe, and to be honest no one really understands it. However maths is the process we use to describe it and Euler (and Schrodinger’s) equations are the basics to creating this knowledge.
Schrodinger built on Euler’s equation to come up with a method to find the probability of finding an electron at a particular point. Quantum mechanics tells us that very small particles (eg electrons and protons) have a dual nature. They are both particles and waves at the same time.
If an electron (or photon etc) is fired at a barrier with two closely spaced slits it will produce an interference pattern on a screen on the other side. It will also behave as a single particle travelling through just one of the slits (although we can’t tell which one). This was the first hint of weird behaviour at very small scales.
Quantum mechanics has proved weirder still over time. Quantum particles appear and disappear at random throughout the universe, even in a vacuum. Quantum particles can be entangled so that they know what the other particle is up to instantaneously over large distances. Einstein called this “spooky action at a distance”.
So our view of how the universe works at the small scale includes Eulers equation and these magic numbers e, i, . The wave equation describes this micro world. In the micro world things happen somewhat randomly and are described in terms of probability and uncertainty while in the macro world the probability distributions lead to certainty of events happening. A casino won’t know that a single event will go any particular way (eg a dice throw in a craps game) but they will know the certain outcome of millions of events that says they will be the winners. Similarly lots of seemingly random quantum events will create rock solid macro stuff (including us).
While it might all sound somewhat hypothetical it is the maths of quantum mechanics that gives us transistors, semiconductors, lasers, CD players, computers, smart phones etc. Playing with the maths will give us the new generation of these toys.
While understanding the wave equation will require a lot of detailed study, here it is in a simple form (time dependent equation). The symbol on the left is the Greek letter Phi, and while it describes “The Wave Function” it is not really a wave. We can think about it as the sum of all of the individual wave functions (small Phi) for the motion of say a photon (light particle). It looks a lot like Euler’s equation with some terms for Energy (E), Planck’s constant (h) and time (t). Planck’s constant comes into it via the Uncertainty Principle. It is the amount of energy of a photon of light. The quantum world is all about discrete energy packets (smallest possible amounts or quanta). Again it looks complicated, but just go with it.
So the quantum universe is all about energy travelling as waves and behaving like particles. Surfers are well aware of the energy contained in waves and of the interference patterns of these individual waves and reflected waves. To understand it all a little better, we will need to look at Probability theory, Distributions, Calculus, Time series, Fourier analysis and complex analysis.
Laws of Motion
At the moment the biggest problem with fully understanding how the universe works is determining how the macro world relates to this micro world of quantum mechanics. The major theories behind both don’t gel particularly well and a Grand Unified Theory is some ways off.
While the micro world is described by quantum mechanics and deals with the nuclear forces and electromagnetic forces to a high degree of observed accuracy, the macro world is described by Newton’s Laws and Einstein’s theory of General Relativity and readily includes gravity (which does not come into quantum calculations). Einstein essentially expanded on Newton’s ideas.
A major difference between the two is that gravity and the macro world is continuous and deterministic, while the micro world is discrete and probabilistic. In a deterministic world we multiply things or add them to come to a specific answer. In a probabilistic world we look at the distribution of many events and calculate a most likely outcome.
Isaac Newton and Albert Einstein gave us a very deterministic universe and their laws work particularly well at macro scales. Newton’s Second Law for example, tells us that F=ma or Force is mass times acceleration. In the case of the apple, gravity provides the acceleration, creating a force from accelerating the mass of an apple downwards.
Acceleration is defined as the rate of change of velocity (delta V) with time (delta t) and is often written as delta V/ delta t or more succinctly dV/dt. The maths involved with rates of change is called calculus and was discovered by Newton (or some suggest by Gottfried Leibnitz). In this scheme, acceleration is called the derivative of velocity with respect to time where a = dV/dt and Force (F=ma = m dV/dt) is the derivative of Momentum (P=mV). Derivatives and their corresponding integrals are just another operator that we will look at later. Again don’t get too concerned about what exactly they are. Equations such as this, including derivatives (or rates of change) are called differential equations. In Hidden Figures, Katherine Johnson uses Eulers Method (another of his discoveries) to solve the differential equations to calculate the go / no go location.
Lots of fun maths will be had with Newton and calculus in other blogs, but first we need to look at how Einstein changed things. Well, a lot of people contributed, but Einstein put it all together to come up with his General Theory of Relativity (after first having a partial form of this known as Special Relativity). The General Theory described the concept of Spacetime replacing the two separate concepts of space and time. This led to Spacetime geometry defining how gravity worked and a limit to how fast things could travel (the speed of light or C). Gravity was viewed as a well in the fabric of Spacetime with objects having mass falling into this “hole”. (Einstein’s theories are even scarier than quantum mechanics!)
While Newtonian mechanics works well at speed way below C, Einstein showed how spacetime was altered at relativistic speeds and by large bodies. He showed that spacetime curved around large bodies and that time itself changed with near light speeds. He also showed that E=M , or the equivalence of Mass and Energy (speed of light C is a constant). Every experiment performed to either prove or disprove General Relativity has shown that it to be superbly accurate. We would not have GPS if we did not incorporate it.
Maths and the Universe
Experimental physics supplies the raw data about our universe and maths tells us how it fits together. The universe already knows and follows the maths, we just need to discover the secrets. Note the use of discover here. The universe already knows and uses the maths, it’s us that need to discover it. To some degree the ancient Vedic mystics of India discovered a lot of what we now call mathematics, including the number line, zero, negative numbers, power series etc. It is something that you can sit and meditate and think up. There are many versions of what you could think up and the endless experimentation is required to determine which solutions are right. Most of what is thought up proves to be nonsense and experimental proof is required.
While there is a lot of magic and beauty in numbers and patterns, it is easy to get blinded by easy answers. More people seem to enjoy numerology, astrology, tarot etc, even though these are made up mumbo jumbo from the ideas of the mathematical greats. They are earlier versions of the current “quantum fruitloopery” often being touted as cure-alls. Don’t get sucked into celebrity sales jobs – they do it to increase their own wealth and power (and reduce yours).
While terms and processes in mathematics need to be very well defined and proved in great detail, this thoroughness can be a major distraction to beginners. I have been very loose with terminology, proofs and referencing in order to just look at the simplicity and beauty (like listening to music without describing each chord). I recommend using the resources of the internet to follow up anything you may find interesting or that requires more detail. I will, of course also follow up many of these ideas.