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Overview
As the author shows, many significant scientific calculations are quite simple and fairly easy to understand, even for readers will little math background. But their implications can be surprising and profound.
For example, what links a famous comet and the cost of an annuity? Why do scientists claim there is “dark matter” in the universe if it can’t be observed? How does carbon-based life on Earth depend on a quirk of nuclear physics? The answer to each question is an illuminating calculation.
This accessible, engaging book will help you understand these breakthroughs and how they changed our view of life and the world.
Product Details
| ISBN-13: | 9781633880290 |
|---|---|
| Publisher: | Prometheus Books |
| Publication date: | 07/07/2015 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 414 |
| File size: | 6 MB |
About the Author
Read an Excerpt
Great Calculations
A Surprising Look Behind 50 Scientific Inquiries
By COLIN PASK
Prometheus Books
Copyright © 2015 Colin PaskAll rights reserved.
ISBN: 978-1-63388-029-0
CHAPTER 1
INTRODUCTION
in which I review the complementary roles of calculation and experiment in science; outline some general aspects of calculations and see them in operation in a first example; and explain the structure of the chapters to follow.
According to Einstein,
science is the attempt to make the chaotic diversity of our sense experience correspond to a logically uniform system of thought. In this system single experiences must be correlated with the theoretic structure in such a way that the resulting coordination is unique and convincing.
In science, we investigate sense experiences in a systematic way. This may involve planned suites of observations in astronomy or in zoology, or the arrangement of particular physical entities and conditions in experiments. The importance of the use of both theory and experiment was neatly summarized long ago by philosopher Immanuel Kant (1724–1804):
Experience [experiment] without theory is blind, but theory without experience is mere intellectual play.
(By the way, we must not be too disparaging about intellectual play because it might be said to cover pure mathematics, where the formalism itself rather than its applications is of interest; we shall see some famous calculations within this category, too, before we move on to applying mathematics in science.)
The development of this viewpoint by people like Sir Francis Bacon, Galileo, John Locke, Sir Isaac Newton, and Kant was central to the scientific revolution underway by the seventeenth century. Nevertheless, its recognition was still in question for many years as is apparent from this excerpt from an 1861 letter from Charles Darwin to Henry Fawcett:
How profoundly ignorant B. must be of the very soul of observation! About thirty years ago there was much talk that geologists ought only to observe and not theorize; and I well remember someone saying that at this rate a man might as well go into a gravel-pit and count the pebbles and describe the colors. How odd it is that anyone should not see that all observation must be for or against some view if it is to be of any service!
Today, it is generally accepted that we need theory to discover order in the mass of data revealed by experiments and also when planning those experiments (in Einstein's words, theory tells us what to measure). However, while there are many books about the outstanding, or most important, experiments (see those by Crease, Johnson, and Shamos for examples), very few deal with the actual calculations behind such experiments. There are books about the equations of physics (see Crease, for example), but equations are the tools of theory — much like the telescope, cyclotron, and oscilloscope are tools in experimental physics. Here I am concerned with the most influential results of using these theoretical tools.
One reason for the theory-experiment imbalance is that to treat the contribution of theory in the physical sciences often means coming to terms with a mathematical formalism. Mathematics has proved itself to be the tool required by scientists; for Galileo there is a mathematical languagefor the universe. Mathematics allows us to use and interpret observations to "go where we cannot go" — to explore the outer reaches of the universe, the interior of our planet, and the subatomic world — and even reveals what is inside our own heads.
Kant's idea that theory without experience is mere intellectual play is at the heart of a debate about the value of string theory in present-day fundamental theoretical physics. There is a danger that the search for ever more fundamental theories loses contact with experiment and the actual physical universe as we know it; for science to prosper there must be calculations and a direct comparison with observations.
I contend that many calculations in science may be appreciated without a deep knowledge of the mathematics involved, just as the output from an experiment may be appreciated without understanding the intricacies of apparatus design and manipulation. This book gives a discussion of calculations to go along with those books that describe the experimental side.
1.1 ABOUT CALCULATIONS
To put the mathematical aspects into perspective, I suggest that four questions should guide the discussions:
• Why was the calculation made?
• What was calculated?
• What was the result of the calculation?
• What impact did the calculation have?
The details of exactly how the calculation was made may be of secondary importance, although in some cases the approach and techniques used are so innovative or revolutionary that they do merit discussion. Mostly, if I do give technical details, I will put them in separate sections for those who wish to see them, and always in the bibliography I give references for readers who wish to study the original papers reporting the calculations.
We shall see how calculations help to turn data into information and thus fit them into the patterns of science. Sometimes single numbers are involved. Sometimes whole data sets, perhaps very extensive in nature, are to be treated. The calculation may result in a formula or parameters to be used in a descriptive formula. This often reduces experimental results to a form that requires explanation and lends itself to interpretation; one calculation may inspire another. Sometimes the formula is already suggested by theory, and then the calculation will be validating or discrediting that theory.
Remembering Einstein's theory tells us what to measure, we shall also see examples where the calculation produces a prediction for experimental verification. As mentioned above, this is the key step in real science: put the theory in a form that can be tested by experiment. When Einstein's general theory of relativity was published, it contained one calculation explaining a mysterious anomaly in planetary motion measurements and another predicting an optical effect. For many people, it was the prediction that was so impressive because there is always a lingering feeling that a theory may be manipulated to produce calculations of already-known effects. (Of course there is then the related question of whether data may be selected or doctored to fit a prediction.)
An example will help to clarify some of these matters.
1.2 EXAMPLE: MALTHUS GIVES US CALCULATION 1
I begin with a very simple calculation: generate two sequences of numbers, each starting with the number 1. In one case, go from one term to the next by doubling the term; in the other, just add 1 to the term. We obtain
1 2 4 8 16 32 64 ...
1 2 3 4 5 6 7 ...
The first one is an example of a geometric sequence (each term is just a multiple of the previous one and here the multiplying factor is 2). The second one is an example of an arithmetical sequence (each term is a sum of the previous one and a constant factor, and here that factor is 1). Not a very profound result, but we shall see more exciting numerical examples in the next chapter.
The interest in these two sequences increases dramatically when we note their use in Parson Thomas Malthus's Essay on the Principle of Population as It Affects the Future Improvement of Society(1798). Malthus begins with two postulata:
First, that food is necessary to the existence of man.
Second, that the passion between the sexes is necessary, and will remain nearly in its present state.
He goes on to deduce that population, when unchecked, increases in a geometrical ratio. Subsistence increases only in an arithmetical ratio. He then points out: a slight acquaintance with numbers will shew the immensity of the first power in comparison of the second. The numbers he uses as examples are just those given above. To make things clear, I define time periods (with zero indicating the initial situation), and the population and the available resources in those periods. Those resources can provide for a certain population, and I calculate that supportable population as three times the resources value. You might think of something like a number of years (Malthus assumed 25 years), millions of people, and thousands of acres of farming land. Malthus's results then appear as follows:
time period 0 1 2 3 4 5 6 ...
population 1 2 4 8 16 32 64 ...
resources 1 2 3 4 5 6 7 ...
supportable population 3 6 9 12 15 18 21 ...
Clearly life is good initially; the resources are more than adequate for supporting the population. But by period 4 the population is barely making ends meet. Life becomes very tough in period 5 (perhaps surviving by using stored food excesses from previous periods), and by period 6 there is complete disaster with the actual population more than three times the supportable one. In Malthus's own words:
This implies a strong and constantly operating check on population from the difficulty of subsistence. This difficulty must fall somewhere; and must necessarily be severely felt by a large portion of mankind.
I label this calculation 1, Malthus on population growth. By means of his simple calculation Malthus has identified a problem (the failure to match resources to increasing population) that has troubled mankind in almost all civilizations and continues to cause misery in much of the world today. The value of the calculation is in dramatically highlighting the problem.
1.2.1 Observations
Notice that we have now seen a common pathway for calculations with the following steps:
1. State the problem.
2. Identify the defining elements.
3. Translate the problem into a mathematical form.
4. Determine how to carry out the calculation.
5. Obtain the required result.
6. Analyze the result to see what it is telling us.
Malthus's example also illustrates another property of calculations: sometimes the objective is to identify a trend or type of behavior; in this case the way population growth outstrips the availability of resources needed to support it. The actual detailed numbers (population doubling in 25 years for Malthus) are not of particular interest; it is the pattern they reveal that is of importance.
Of course, some people will always want a more realistic assumption, as Charles Babbage amusingly demonstrated. After reading the couplet "Every minute dies a man, Every minute one is born," in Alfred Tennyson's Vision of Sin, Babbage wrote to Tennyson suggesting that it be changed to "Every minute dies a man, And one and a sixteenth is born." But even that was not enough as he continued "I may add that the exact figures are 1.167, but something must, of course, be conceded to the laws of metre." At least Babbage wished to build in the trend of an increasing population, rather than the static one as Tennyson would have it.
Malthus's calculation is a good example of one that produces a set of numbers. However, sometimes a calculation may result in a formula. For example, if we denote the population in time period n by pn, then Malthus tells us that pn = 2n. The formula may be one to be tested against experimental results, and its accuracy may be a test of the validity of the under lying theory. On other occasions, the formula may be deduced by finding the appropriate mathematical expression to describe data from natural observations or from carefully designed experiments.
It is a feature of many good calculations that they lead us to ask questions and to build in other characteristics of the problem. In chapter 12 we will see how Malthus's work leads into whole new areas of both population modeling and mathematics.
It may be that an original calculation fails to take account of some essential aspects of the situation under investigation, perhaps something unknown at the time when it was made. This would invalidate any conclusions to be drawn from the calculation. We will see some famous examples in later chapters.
The results of a calculation might also lead us to different but related questions; in Malthus's case: How many people can the earth support? In fact, this question had been considered by Antonie van Leeuwenhoek (1632–1723), a man better known as a pioneer in microscopy. (Details are given by Cohen, who explains how Leeuwenhoek came up with a maximum population of 13,385,000,000.)
Finally, we should note that Darwin was aware of the conclusions to be drawn from Malthus's work when he considered how the essential competition for scarce or limited resources might be incorporated into his theory of evolution. Calculations may clearly have an influence far beyond their immediate context.
The actual calculations made by Malthus may have been extremely simple, but they have led to some vitally important developments in science and in the affairs of mankind.
1.3 OUTLINE FOR THE BOOK
I have grouped my chosen calculations into broad subject areas, and within each group, I use chronological ordering so we can see how calculations sometimes mark a major milestone in the history of science. Sometimes I will refer to a group of calculations. Emphasis is placed on the physical sciences, since it is there that mathematical descriptions and calculations have long been accepted as part of the subject, but examples from the biological sciences are also given. However, first, in chapters 2 and 3, I present some examples of calculations that are of purely mathematical interest, although in almost every case there is also a link to the applied mathematical world.
Chapter 4 shows how calculations have helped us to get to know our home, the earth. Calculations are becoming ever more important (think weather forecasting and climate change), and I will touch on some issues involved in a discussion section.
Chapters 5, 6, and 7 examine the part played by calculations when we consider our earth as a planet in a larger framework, first in the solar system and then in the universe as a whole. It is here that calculations become of necessity and exceptional importance.
Chapter 8 moves from the physical to the biological (having already seen Malthus's calculations earlier in this chapter). The physical world seems to demand the use of mathematics when we study it, but the triumphs of such an approach are fewer — but still vitally important — in the biological domain, and I will speculate a little about that.
The next three chapters deal with the basic and most fundamental parts of our physical world: light and the building blocks of matter. Discussion of the latter subject is divided into two chapters since nuclear physics provides some of the most spectacular (and perhaps disturbing) examples of the impact of calculations.
Chapter 12 examines how calculations have allowed us to discover the nature of motion in its various forms and to see how general mathematical descriptions were developed. It is here that we shall see the impact of modern computers as a vital ingredient in the business of calculation.
Finally comes a chapter of summary and evaluation. Can we say how, and to what extent, calculations have molded and supported the progress of science? Even more contentiously, can we pick out those calculations that are of supreme importance, those worthy of the label "great calculations"? It is an evaluation of their history, context, and impact that contributes to an assessment of a particular calculation, and here opinions are bound to differ. Please note that the title of this book is Great Calculations and not THE Great Calculations, or some other title suggesting that there is a definitive list. This is surely a personal matter, and many readers may have a quite different list. That is part of the fun, and I ask you not to be too irate if your personal favorite is missing.
References for all chapters are given in the notes section and also in the bibliography, which contains details of all mentioned books and papers. I have given examples of suitable references so that the interested reader may further explore each calculation.
If you cannot wait for the story to unfold, but want to see my list of important calculations right now, please go to the beginning of chapter 13!
(Continues...)
Excerpted from Great Calculations by COLIN PASK. Copyright © 2015 Colin Pask. Excerpted by permission of Prometheus Books.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Contents
Acknowledgments, 9,Preface, 11,
Fifty Questions for the Calculations to Answer, 13,
Chapter 1: Introduction, 15,
Chapter 2: Ancient Mathematics, 25,
Chapter 3: Steps into Modern Mathematics, 43,
Chapter 4: Our World, 67,
Chapter 5: The Solar System: The First Mathematical Models, 99,
Chapter 6: The Solar System: Into the Modern Era, 117,
Chapter 7: The Universe, 149,
Chapter 8: About Us, 181,
Chapter 9: Light, 213,
Chapter 10: Building Blocks, 251,
Chapter 11: Nuclear and Particle Physics, 273,
Chapter 12: Methods and Motion, 315,
Chapter 13: Evaluation, 353,
Notes, 371,
Bibliography, 385,
Index of Names, 407,
Index of Subjects and Terms, 411,