14 February 2020
Math is a powerful tool that drives innovation.
Making the simple complicated is commonplace; making the complicated simple, awesomely simple, that's creativity. (Charles Mingus)
Sadly, math is viewed by many as something we endured at school and hope to have never impinge on our lives again. And yet, the fruits of math surround us and exploring it reveals its true purpose - math is a tool to magnify the power of our minds.
Here are seven simple concepts to help you grasp the difference between math and arithmetic and understand the true value that it plays.
ONE: the difference between arithmetic and math
Fact. Many mathematicians aren’t much good at adding and subtracting. There are people who are said to be “super-numerate”; they can perform large number arithmetic calculations in their head. These same people aren’t necessarily great mathematicians.
Arithmetic (computation) is not mathematics.
Mathematics is about representing the real world using not only numbers but abstract rules that can be reused and applied to achieve insights into how the world operates.
For example, by representing the real-world using equations, Einstein mathematically derived E=mc^2 the famous equation that underpins relativity and the working of nuclear reactors.
The role that math played was to enable powerful manipulation using symbols to represent large and unwieldy concepts.
Math is a tool for simplifying the world.
TWO: Math builds like Lego
If you understand what a tool is then you get math.
A tool is something used by humans to magnify our capability. Knives allow us to cut materials that can’t be torn apart using just our hands. The knife applies the force of our arms and body weight to a very small surface area vastly increasing the ability to break atomic bonds.
The wheel reduces friction allowing massive objects to be moved. You get the idea.
Math dissects and shifts large concepts around by manipulating symbols that represent real-world things.
The math tool kit consists of a vast list of arcane symbols (sine. cos, tan, etc.) that are simply tools that define the rules of math.
The key concept is needing only to manipulate one set of tools at a time. Line by line the mathematician works through the calculations needing only to consider how to move from one line to the next until a new representation is achieved; one that better represents the real world.
This process is how Einstein arrived at E=mc^2
Understanding that math “builds” like Lego means that you only need to understand the building blocks, not the entire structure. This process is called ‘abstraction’.
THREE: understand the basic principle “if you can’t measure it you can’t manage it”
When we drive cars, we rely on the speedometer to know how fast we are travelling.
A speedometer is simply a tool for representing the real world (the speed of a car) in a simple abstract form. Mathematics build models of the real world to simplify information to produce decision making data.
We all understand the value of using stats to analyse the playing performance of sporting teams, and how the coach applies the insights from that data to make decisions about player selection, field positions, and applying skills coaching. This is a simple example of modelling using numbers (but perhaps not strictly math).
Representing the world in numbers allows us to see what is happening.
FOUR: The difference between “pure math” and “applied math”
When I studied math at University there were two streams “pure math” and “applied math”.
What’s the difference I asked? Applied math has application to the real world. Pure math has no application - it is math for math sake; instead of attempting to model the real world it simply models the world of math.
So, what’s the point? Simply put, all applied math started life as pure math.
Therefore, it is essential that we pay mathematicians to explore math because they are inventing powerful tools that will one day enable us to model the future.
The pure math toolkit may provide the tools to (for example) unlock time travel, defy gravity, or convert lead into gold.
FIVE: The power of modelling
A model is smaller, easier to handle and an easier to build representation of something much larger.
In mathematics, it’s the same thing. A model is a mathematical representation of a large real-world “thing”.
Engineers use mathematical models to predict the strength of bridges before they are built, rocket scientists predict the re-entry point of rockets before they fly, and manufacturing managers optimise production to maximise profits before they start the production line.
The numbers are simply the score, it’s the model that calculates the score.
SIX: Just because you don’t get math doesn’t reduce its value.
Not being able to build mathematical models doesn’t reduce the value of math; You may not know how to build a house, but you know what a house is. And it isn’t that hard to find a builder.
Similarly, with math. Mathematicians can be hired to model for you if you understand what it is that you are trying to achieve.
Wars have been won by hiring mathematicians to better understand the underlying principles of physical events, to model strategies, with the aim of optimising the outcomes.
Alan Turing used math to unlock the Enigma code that allowed the British to change the outcome of World War II.
SEVEN: Math mirrors the way that the human mind works
The human mind is not a very large machine and it doesn’t consume much power (comparatively) but punches above its weight in terms of information processing grunt.
It achieves this by abstracting information (simplifying it) to reduce the processing workload.
Even though we think we “see”, “hear”, “taste”, “smell” and “feel” the world clearly – what our brain is doing is converting the signals that it receives from our senses into simplified and abstract data.
The human mind is a device that models the real world.
An example of this is the way vision works. We only see detail at the very centre of our vision (a part of the eye called The Macula), the rest of the visual field is very low resolution (but processes visual information very quickly compared to the macula - a feature used to detect movement out of the corner of our eyes) .
Our brain assembles the two to create a picture that we perceive to be of uniform resolution across our entire field of view, but that’s an illusion. What we see in our brain is a model of what the world looks like not a pixel perfect image.
Our vision simplifies the amount of data that needs to be processed to both increase the capacity of visual memory and speed up image recognition; thus reducing the size of the brain needed.
Mathematics does the same thing.
Language is another example of abstraction. 26 characters in the English alphabet are assembled into words that are then assembled into language that describe the real world and provide an entire abstract thought process to conceptualise and understand how the world works.
Mathematics has a different alphabet, and different rules but applies the same principle.
If you understand these seven things about mathematics, you may never need to perform a mathematical calculation, but you may see the value of hiring someone who can.
Conversely, you may see the true value of Math and fall in love with its power and problem-solving capabilities; sadly, that is not how math is taught in schools; too many of us abandon math because when we are immature we write it off as an onerous and pointless annoyance.
Remember, the real gains in life are more often made accumulatively and incrementally at the margins, not in big leaps and bounds – a well-conceived mathematical model may assist you to optimise a process delivering the few per cent needed to turn a loss into a profit, to build a more competitive machine, or select a profitable investment.
The world needs more people skilled in math.
Further reading
The Emergence of Calculus: A Mathematical Journey of Human Thought