Quantum Leaps E-Book

Hugh Barker

• Herausgeber: Atlantic Books
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 336

Veröffentlichungsjahr: 2024

Beschreibung

From the author of Million Dollar Maths comes a fascinating and engaging look at the mathematics that lies behind our modern world. How does Google know what you want to type? How did humans first reach the moon? Could we ever have a supercomputer the size of a pinhead? In this thrilling numerical journey, Hugh Barker explores how mathematics has helped to build the technology of today, and the fascinating ways it is shaping the future. From green energy to 3-D printing and from quantum cryptography to machine learning, Quantum Leaps reveals the hidden mathematics in modern technology. Beautifully written and full of ingenious examples, this book will make you realise how the modern world would be impossible without our formidable mathematical armoury.

Leseprobe

Hugh Barker is a non-fiction author and editor; as the latter he has edited several successful popular maths books, including A Slice of Pi. Hugh is a keen amateur mathematician and was accepted to study maths at Cambridge University aged 16. His previous book wasMillion Dollar Maths: The Secret Maths of Becoming Rich (or Poor).

By the same authorMillion Dollar Maths

First published in hardback in Great Britain in 2024 by Atlantic Books, an imprint of Atlantic Books Ltd.

This paperback edition published in 2025 by Atlantic Books.

Copyright © Hugh Barker, 2024

The moral right of Hugh Barker to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act of 1988.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of both the copyright owner and the above publisher of this book.

No part of this book may be used in any manner in the learning, training or development of generative artificial intelligence technologies (including but not limited to machine learning models and large language models (LLMs)), whether by data scraping, data mining or use in any way to create or form a part of data sets or in any other way.

Every effort has been made to trace or contact all copyright-holders. The publishers will be pleased to make good any omissions or rectify any mistakes brought to their attention at the earliest opportunity.

A CIP catalogue record for this book is available from the British Library.

E-book ISBN: 978-1-78649-767-3

Internal illustrations © Diane Law Barker

Atlantic Books

An imprint of Atlantic Books Ltd

Ormond House

26–27 Boswell Street

London WC1N 3JZ

www.atlantic-books.co.uk

Contents

INTRODUCTION: Maths, Technology and the Imagination

CHAPTER 1: Where Are We Now?

The Maths of Location and Navigation

CHAPTER 2: Robots and AI

How Maths Underpins the Horizons and Limits of Silicon Valley

CHAPTER 3: Pattern Recognition

Why We All Need to Understand the Mathematics of Patterns and Algorithms

CHAPTER 4: Behind the Wheel

Who’s Going to Drive You Home?

CHAPTER 5: Speed and Flight

Getting from A to B

CHAPTER 6: Infinity and Beyond

The Maths of Space Travel

CHAPTER 7: Talk About the Weather

From Calendars to Sunny Spells

CHAPTER 8: Blah, Blah, Blah

The Maths of Global Warming and the Environment

CHAPTER 9: The Internet, Cryptography and Social Media

How Maths Deals with the Interactive Human Race

CHAPTER 10: Slices of Life

3D Printing and Medical Notes

CHAPTER 11: Taking the Red Pill

Computer Images, Games and Movies

CHAPTER 12: The Infinite Future

How High-Tech Maths Will Continue To Build the Future

INTRODUCTION

Maths, Technology and the Imagination

Since the dawn of human history, we have been trying to understand the universe we live in. Science is our attempt to decipher the rules that govern that universe, and to find out what it is ‘made’ of. By contrast, maths is an abstraction from those physical rules: it starts with the basic act of counting, then moves on to use numbers and operations that allow us to make calculations about attributes such as distance, volume, area, weight and so on. And this in turn makes it easier for us to record and evaluate our scientific observations.

So science often depends on maths. But maths is not always tied down to the real, observable world. As a theoretical discipline it can extend into strange realms such as imaginary numbers, levels of infinity, irrational and transcendental numbers and multidimensional objects. There are even many different possible ‘geometries’, depending on what assumptions you make about the world. But for the most part Euclidean geometry, our standard three-dimensional understanding of space, is the one that tends to describe our immediate surroundings most accurately.

Technology is the means by which we use our knowledge of science and maths to achieve something useful in our everyday life. Early examples of technology came when we used a lever to lift a heavy weight, dropped a heated rock into a container of water to warm the liquid, or combined two metals to make a stronger alloy. Over time, humankind developed increasingly complex technology which allowed us to transport goods, build large structures, trigger chemical reactions, cultivate new crops, use steam turbines and much more besides.

We often celebrate the inventors of new devices and technologies as creative geniuses: the names of Thomas Edison, Galileo Galilei, the Wright brothers, George Stephenson, Alessandro Volta, Louis Braille and Marie Curie are immediately associated with the advances they made in science and technology. To make those advances, these inventors and scientists needed a creative vision: they had to imagine a possible application of science, then experiment until they worked out how to turn that vision into reality.

However, it’s easy to forget that maths is also a highly creative discipline, and it is the mathematical imagination that has driven progress in technology for millennia. It has always been one of the lynchpins of new and emerging technologies, from the pulley and the compass through to space travel, computers and future inventions. The Wright brothers needed mathematical equations to calculate the ratio of power, weight and lift that would allow an aeroplane to leave the ground. And, today, the developers of robots, self-driving cars and AI programs are dependent on thousands of different calculations, equations and algorithms that underpin every movement of a robotic hand, and every decision a program or car makes about its next move.

Mathematicians are constantly creating new ideas and fresh theories, and it is not always obvious which ones will end up having technological applications. Network theory was a fairly obscure discipline fifty years ago, but now it forms a key element of how the internet functions. Meanwhile, cryptography today is often based on the factorization of large semiprime numbers, a previously unappreciated part of number theory, while in future it may rest more frequently on the even more obscure maths of elliptic curves. Even those strange realms where maths deals with speculative concepts can end up having technological applications in the end. String theory, a purely theoretical attempt to explain the building blocks of the universe, only works if we assume higher dimensions of reality; in superstring theory, the universe is seen as ten-dimensional, in M-theory it is eleven-dimensional, while bosonic string theory requires twenty-six dimensions. Mathematicians have interacted with physicists to create some extraordinary interpretations of these multidimensional universes.

And, as we shall see, it turns out that the mathematical explanation of black holes used by some string theorists can also be used to describe particular kinds of quantum systems, which are called quantum bits or qubits. It has been speculated that information theory, when it is applied to qubits, could help with the future development of ultrafast computers and completely secure communication. So, whether or not string theory turns out to be a good description of the cosmos, its mathematical building blocks are already potentially useful in creating new technology. And they may have applications in the future that we can’t even guess at today.

Of course, when it comes to the most abstruse and difficult mathematical ideas, it can be difficult to give easy examples for the general reader, which is who this book is aimed at rather than at specialist mathematicians. So I hope it’s understood that at times the maths in this book will be described in general terms. On the other side of the coin, much of the maths can be understood by anyone who studied it at school; it is simply being applied in interesting ways. So where I have been able to find a way to bring particular problems to life with clear examples I have tried to do so.

In terms of organization, I’ve started with some general themes covering mathematical modelling, the development of AI and robots, and pattern recognition. Thereafter the chapters focus on different areas of life in which the interaction of maths and technology has changed, and will change, the way we do things.

We take the technological breakthroughs of the past for granted in our everyday lives whenever we use cars, mobile phones, GPS positioning, computers and many other devices. This book will take a look at how mathematics helped make some of those advances possible and the ways it is driving emerging technologies today, while speculating about the extraordinary devices and processes it may help us to create in the future.

CHAPTER 2

Robots and AI

How Maths Underpins the Horizons and Limits of Silicon Valley

I’m completely operational and all my circuits are functioning normally…Hal (2001: A Space Odyssey)

Let’s face it, we are already living in the future. We may not have our jetpacks and flying cars yet, but we do have extraordinary things like robots, tasers, mobile phones and self-driving cars. The path to such inventions is a fascinating one that often starts with the fantastical imaginations of science fiction writers: think how the stun guns and handheld communicators of Star Trek have effectively become reality over the years since the show first aired. Essentially, the first element of an invention is the pure idea, and writers and artists are as good as anyone at having ideas. But it takes the laborious work of generations of mathematicians and engineers to turn that fantastical vision into reality.

We Are the Robots

One of the most significant historical changes we are going through is the increasing use of AI and robotics (in the broader sense rather than narrowly meaning a humanoid figure that walks and talks) in the workplace; it is unclear at this stage whether we are heading for a future in which the human workforce is replaced by actual robots (in which case we may need a new economic model) or, as encapsulated in the idea of ‘multiplicity’, a future in which robots will be supplementing the human workforce, meaning we are working alongside them, just as AI is already handling everyday tasks such as dealing unhelpfully with customer complaints over the internet.

The word ‘robot’ was first used by the Czech playwright Karel Capek in his play R.U.R. (Rossumovi Univerzální Roboti or Rossum’s Universal Robots) which featured androids made of organic material. It’s intriguing that, from the start, Capek introduced two fears that have accompanied the development of both real and imaginary robots: firstly, his robots turn out to be more efficient than the humans, and secondly, they end up setting off on a killing spree. Long before The Matrix and The Terminator we were already starting to wonder whether these robot things were such a great idea after all.

Bear in mind that a robot need not be built in humanoid form, although there are situations in which there might be reasons to do that. All it needs to be is a machine that can automatically carry out a series of complex interactions with its environment. Think of the difference between a remote-controlled toy car and the Mars rovers, which have to make some decisions and take actions autonomously. The latter can reasonably be called a robot, the former cannot.

The first device that could truly be called a robot is generally said to be Shakey, which was built in California by SRI International in the 1960s. It was an awkward, slow moving tower on wheels, which could navigate its way through a fairly simple environment, and which was equipped with a camera and sensors to help avoid collisions. In the same period, the first robotic arms were being used in factories; the forerunners of the machines that are so common in car factories today.

The first humanoid robots followed in the 1980s, when Honda developed P3, which could walk, wave and shake hands. More recent humanoid robots included Honda’s ASIMO (Advanced Step in Innovative Mobility), well known for having a soccer kickabout with President Obama, and TOPIO (the TOSY Ping Pong Playing Robot). And the robots are getting smarter. Take the Berkeley Robot for the Elimination of Tedious Tasks (Brett, for short), a small robot that can manipulate things in its environment. Brett can learn simple tasks on its own. You can see videos of it working out how to put a square peg through a square hole. This is due to reinforcement learning in which the robot is ‘rewarded’ for getting a task right. In one fascinating experiment, a robot that had been given the instruction to move forward as rapidly as possible made the imaginative leap from walking to running, without having been specifically programmed to do so.

Now we have astonishing specialized robots like SMURF, the soft miniaturized underground robotic finder, which is a very small robot on flexible wheels designed to search disaster sites and find survivors under the rubble.

And, of course, some people worry about the so-called singularity, the point at which machines become so advanced that they surpass the humans who built them.

So, you might ask, what is the maths that underlies all of these advances?

The first thing a robot needs to be able to do is to sense its environment. For this reason it will be equipped with sensors. These can be basic, like Shakey’s bump sensors, or much more advanced. For instance, there are now sensors that can perceive ultraviolet light, air pressure and even smells and convert these into mathematical data. The most fundamental element in most modern robots’ visual armoury is lidar (Light Detection and Ranging), which sends out laser pulses to sense the distance between the sensor and the object it hits. The robot knows the speed of the laser pulse, and when this is pinged back from an obstacle the robot can calculate distance by multiplying speed by time. The data from the pulses is converted into a number stream which gives the robot detailed information about the obstacles around it. This in turn allows the robot to build up a 3D model of its environment, which can be combined with cameras that create an even more detailed image. There is a similar process going on with any other kind of sensor used by the robot: the information is converted into a data stream, which in turn is turned into a model of particular aspects of the environment.

Next the robot will need to be able to interact with its environment. This involves ‘actuators’, which are the combined motor and gearbox that can be found in moving parts of a robot, such as the joint of a humanoid arm. The strength and manoeuvrability of the actuators will define how strong and dextrous the robot is.

This is where some really basic maths comes in handy. If a robot is reaching out to pick up a tool, then it needs to find ways to convert its 3D map of its environment into action. As it reaches out its arm, it has a constant feedback loop which will tell it, among other things, what the angle at the joint of the arm is. And the same information will be flowing from the shoulder joint (if there is one) and the joints of the hand or whatever gripping device it has been equipped with.

Given the lengths of each body part and all of these angles, simple trigonometry can be used to ascertain the location of the hand and fingers. Obviously, this calculation gets more complex as robots get more complicated (and as they have more joints) but at heart it remains a problem that can be solved using the same maths that helped us to understand Pythagoras’s theorem at school.

The last fundamental feature of the latest generation of robots is the ability to learn. This can be achieved through reinforcement learning, demonstrations, or through giving the robot goals to achieve. Humans are natural learners: our brains demonstrate neuroplasticity, in which each new experience creates new connections in our brain, and we can gradually improve our skills. Since computer programs are not as flexible, the challenge is to create an emergent system in which the lower level of inflexible programs can support a higher emergent level which ‘learns’. (In the theory of science, the phenomena of ‘emergence’ occurs when an entity has properties that its parts do not have on their own.)

The essence of machine learning is the way that a robot can recognize and learn about patterns: imagine we show a robot many pictures of an apple, in order to teach it to recognize pictures of an apple that it hasn’t previously seen. After giving the robot the training data, we show it previously unseen pictures and feed back information about when it correctly identifies a previously unseen apple. All the time, the robot is abstracting and refining a model from the images in order to become better at discarding non-apple pictures in future. In order to do this, we need an algorithm which can apply optimization methods to its own performance, tweaking the parameters it uses to make the right choice, so that it improves through experience.

The areas of maths involved in this task are wide and varied. As well as optimization theory and programming languages, some of the key areas are linear algebra, calculus, matrices and probability; essentially, the computer will receive data and store it in ways that are usable and retrievable. The algebra and calculus are all part of the machine’s ability to model the world and to make predictions using probability. Once the algorithm can generate predictions, there then needs to be a way of giving the machine feedback on when it gets things right and wrong, in order for it to adjust its parameters.

Computing Prehistory

All robots rely on the technology of computing. We’re all used to having a computer in our pocket these days (a modern mobile phone would be seen as an incredibly sophisticated miniature computer just a few decades ago). In order to understand how artificial intelligence works, it may be useful to take a brief look back at the development of devices that allow for rapid calculations of various sorts.

One of the earliest was the abacus. It isn’t known where this originated, but it was widely in use in the Mediterranean area in antiquity. One of the first things we know about it comes from the Mesopotamian period (during the third millennium BCE): the abacus involved a series of columns representing figures, with beads on bars representing unit numbers. However, it was probably limited in its mathematical power, as the Mesopotamians used a sexagesimal (base 60) number system, which is unwieldy when it comes to calculations such as multiplication or division.

However, by the end of the first millennium, far more efficient abacuses were in use: we know them from Roman examples, and also from the Chinese ‘Suan Pan’, a Chinese abacus first described in writing in the Supplementary Notes on the Art of Figures by Xu Yue in the second century CE. This was a highly efficient counting device, but it could also be used to do multiplication, division, and even square root and cube root operations, using highly efficient methods.

One fascinating artefact from the second century BCE (or possibly a couple of centuries earlier) is the Antikythera mechanism (named after the Greek island near to which it was recovered from a Roman era shipwreck). It was a small, damaged wooden box which enclosed thirty interlinked gears and levers, making it look a bit like a modern wind-up alarm clock.