Brief Course Description

Introduction:

Here's a cartoonist's version of what we it means to model the world with mathematics and what we will be studying this term:


Mathematical ideas underpin virtually all of the technology that keeps our civilization going. Fortunately, most of these ideas are, in their essentials, not very complex and there is no reason why the average citizen-in-the-street should not know about them and use this knowledge to have a better understanding of what are the potentials and limitations of this technology. To have a professional mastery of these mathematical ideas requires a lengthy apprenticeship. But the basic ideas are easily illustrated by simple and intuitive examples. In addition, some mathematical formulas are needed in order to give an precise expression to each idea and to make a bridge so one can convert the intuitive idea into an arithmetic calculation of how it will work in the world.

In this course, all these ideas will be presented in a dual way, using, on the one hand, readings from the primary historical sources where they were first discovered and, on the other hand, numerical calculations carried out with the simple yet powerful tool of a spreadsheet (e.g. Excel). We are blessed (and cursed) with computers, which can make manifest in a split second what a formula hides within itself. Many of the mathematicians of earlier centuries loved to calculate and would have been delighted to have such a tool, now available in the cheapest PC.

What we will not do is to present the formalism of mathematics for its own sake. This formalism is what you need to master to become a professional in one of the 'exact' sciences, notably pure or applied mathematics, physics, chemistry, engineering, economics or computer science. Many are in love with the elegance and precision of abstract definitions, with deceptively simple but truly deep questions about ordinary integers or the arcane skills employed in the manipulations of analysis: if so, take standard math courses and not this one.

Math is usually seen as divided into 3 areas: algebra, geometry and analysis. This course deals entirely with analysis, which is the area of mathematics which has grown almost entirely from its applications to the real world and which has in turn fed the technologies with which we master the world. Algebra and geometry, on the other hand, have developed to a large extent by pursuing their own internal logic, with occasional prods from reality. Algebra has had many applications in computer science and has hit the headlines recently due to its use in cryptography and secure internet transactions; and geometry has had major links with physics throughout the 20th century (esp. in the theory of gravity and in particle physics). But analysis applies to virtually every aspect of modeling the world.

Most of what we say here will relate directly to the 'real' world. We will need some half a dozen deeper math-facts, which are not at first obvious. But we will try to explain why they are natural and inevitable and give you a chance to flex your computational muscles with them to see that really are true. We will assume some acquaintance with calculus - e.g. one semester. But we will also review the basic definitions of calculus as we go along and, in fact, see them in their original form in the hands of the old masters Newton, Leibniz and Euler.

So exactly what are we going to study? The first topic is what started the Western World on the path of its technological success, whose fruits we enjoy today. I want to look briefly at Babylonian, Greek and Arabic mathematics, the two traditions, which started it all. The Babylonians really invented arithmetic, including fractions and decimals and converting between them - except that they used base 60 (sexagesimal) instead of base10 - and they reveled in calculating with these numbers. The Greeks, on the other hand, had less appetite for numbers but loved logic and discovered the curious fact that geometry could be studied on the basis of axioms without any measurement at all. Nonetheless, they sometimes put geometry and numbers together and did come up with a stunningly accurate estimate of the diameter of the earth. This is easy to mimic, as we will do in assignment #1. The Arabs did some great science (e.g. they understood the nature of light correctly) and put zero back in the number system, which had been foreshadowed with small place markers in Babylonian cuneiform but not properly appreciated. The Western World first stirred in the work of Nicole Oresme who introduced the idea of graphing a quantity, which changes in time. He introduced x- and y-coordinates and used them for making plots of space, time, velocity, even spiritual quantities like grace. In many ways, he was preaching the contemporary 'hot' topic of visualization and sounded much like its best-known apostle today, Edward Tufte. More or less at the same time, Europe fell in love with clocks and maps and began to appreciate all that can be done if you measure time and space accurately. The take-home message of the 14th century was that arithmetic could be used to measure and analyze and understand better the events taking place around us. The second topic is the work of Galileo, Newton and his immediate successors. It is usual to say that the key step was the discovery of calculus but I would describe the biggest impact of their work differently. They uncovered the fact that, for a large number of events in the world, if you measure (i) where everything is and (ii) how it is moving or changing at any point in time, then the laws of nature, if you know them, tell you (iii) how every object is accelerating and (iv) that knowing that this law will always hold, the entire future is determined. This is a stunning simplification of the ancient job of prophecy, a challenge that had pre-occupied every previous culture but which had met with quite limited success. Put mathematically, this discovery is that many aspects of the world are described by differential equations (to be technical, 2nd order ordinary and hyperbolic partial differential equations) and much of the next 3 centuries has been spent extending this model to ever-larger categories of events. I like to put it this way: God's laws for the universe are written in the language of differential equations. So it behooves everyone to have an inkling of what these are. We will read Galileo and Newton and try to see inside their minds. Folklore has it that it all started with Galileo's watching the swing of the censors and oil lamps in the cathedrals of Pisa. With the help of a spreadsheet and some digital movies of a swinging pendulum, we can readily check part of what he and Newton claimed. Ironically, science is quite circular here: an accurate clock needed to check this mathematical model and the mathematical model is needed to prove that a pendulum ticks like a first class clock.

The third topic will be waves. The first intuition into wave phenomena was the analysis of musical chords by Pythagoras and his followers, noticing the strange fact that two strings plucked at he same time sound better together if the ratio of their lengths is a simple fraction like 3/2. We will look at some length at the vibrating string and the waves it produces, which will illustrate again how differential equations are the basis of all mechanical systems. With the help of strobe lights and digital speech recordings, we can easily 'see' both simple and complex waves. We will look at a human voice singing the major scale and check what underlies Pythagoras's beliefs and we will use a spreadsheet to solve the vibrating string equation. Finally, these two examples lead us to a very important piece of mathematics: Fourier Analysis. Rather like the familiar idea that a real number can be approximated by an infinite decimal to greater and greater accuracy, Fourier declared that every function could be approximated by an infinite superposition of simple waves of different frequencies. It is just the mathematical expression of a musical score. A quite unexpected twist to the story here is the amazing usefulness of a mathematical artifice of the Renaissance - introducing a new 'number' having nothing to do with lengths, the square root of -1. The fourth topic will be a whirlwind survey of how the same ideas have ramified in the last two centuries, enabling us to model the buckling of columns - we will look at some figures for the Twin Towers - waves in earth and water, and most importantly, electro-magnetic waves such as radio, TV and light. How can you tune one antenna or cable box to pick up so many different stations?: it's all Fourier Analysis.

The fifth topic will be chaos! The word has many meanings but the most important one for us is the idea, as Ed Lorenz put it, that the flapping of a butterflies' wings in Rio can set off a tornado in Oklahoma. This means certain phenomena of nature are fundamentally unpredictable. Understanding what is and what is not predictable has been one of the main problems driving applied mathematics in the last 50 years. Whereas the superposition of waves had been the key tool that unlocked the behavior of linear differential equations, the key challenge recently has been to understand all the complex things that happen with non-linear equations. Our focus will be Lorenz's equation for convection rolls in the atmosphere (the instability caused by the sun heating the earth hence the lowest layers of the atmosphere and this then rising). A new idea here will be that to visualize this and other equations, one makes plots in 'phase space', an idea that Oresme would have liked. We will draw for ourselves the 'strange attractor' that explains Lorenz's equation and this will lead us to the idea of fractals, shapes with convolutions on finer and finer scales, which model many of the more complex structures in nature. In the dialectic of mathematical models, the Newtonian thesis of predictable planets in smooth elliptical orbits has been replaced with the antithesis that chaos and fractals are ubiquitous. Finally, the last topic will be chance. Historically the most important observation of chance in Science was the discovery by Brown of the seemingly random motion of tiny particles in solution. As it turns out, this was the first empirical evidence of the atomic nature of matter! Individual molecules, moving randomly due their heat, collide with the microscopically visible particle and drive it this way and that. The most important thing to know about chance is that the size of its fluctuations is predictable. There is a formula here - another for your toolkit - and it applies to elections, medical trials and the reliability of fingerprints. Another fascinating aspect is how chance can be used to better compute things governed by immensely complex equations. The H-bomb was designed using so-called 'Monte Carlo' algorithms and we will see how it is easy to estimate the critical mass of uranium needed for a bomb, again with a spreadsheet!