At the end of last semester we talked
about the beginnings of modern science in Europe, in the 16th
and 17th centuries, and we saw that they can be characterized
as a separation of physics from metaphysics. Remember that metaphysics
meant those doctrines which dealt with the inner, active principles
in things, principles which are usually hidden. Thus, for example,
it was thought that celestial bodies had an inner, active principle
in them, which made those bodies move in circles, epicycles, etc.,
whereas the bodies down here, on earth, have an inner, active
principle (gravity), which makes them move down towards the center
of the earth. The principle of light is to illuminate, to make
things visible. Modern physics tries to do away with all those
inner, active principles, and tries to explain everything, every
phenomenon, whether on earth or in the heavens, by an analysis
of a single phenomenon: motion. Furthermore, modern physics seeks
to explain all phenomena on the basis of the motions of atomic
particles: like geometry, it starts from something called points,
whose only inner principle is a non-active one: to be indivisible,
that is, to be as small as possible.
The purpose of modern physics is to find the laws of all motions, and these laws must be written in the language of mathematics. As Galileo put it, "The book of Nature is written in mathematical language." This was as true in the 17th century as it is today. But to find the laws of motion we have to make sure that we can define the basic numerical properties of motion in a meaningful way. These properties are:
Let us look closely at these properties one by one.
Regarding (1), the position of a particle, we saw last semester that Descartes provided us with a method to specify it, the so-called Cartesian coordinates. Let's start with the simple situation where a particle moves along a straight line, which we call the x axis. We pick a point on it and call it 0, "the origin," then pick a unit of measurement and mark it on the axis as the segment from 0 to 1. Thus, as Descartes observed, each point on our straight line will correspond to a "real number": those points on the side of 0 where our 1 is will be positive numbers, those on the opposite side will be negative. Now suppose that a particle is moving along this line: at a given time (call it t) it will be located on a certain point corresponding to a number x, and to make sure we understand this, we'll say that at that time t the particle is located at the point x(t). What we get here is a function, a correspondence: to each value of the variable t, which represents time, there corresponds a number x(t) which tells us where on the line the particle is located. It all seems very simple (although it really isn't, but we'll ignore the complications involved in concepts such as "real number," "instant of time," etc.) If our particle doesn't move on a straight line, but rather on a plane, then we will need a Cartesian system of two coordinates, two axes, x and y, instead of just one, and so we will need two functions of time, x(t) and y(t), to describe the motion of the particle. Similarly, in three-dimensional space we will need three functions, x(t), y(t) and z(t).
The problem starts when we try to define (2), the velocity of the moving particle at a given time. Because most of us drive cars and there's a speedometer in our car, we think we know what velocity means. The problem, though, is not that simple. Imagine you are driving and a cop stops you: "You were going at 80 miles an hour!" he tells you. Well, you might reply, "How can it be? I left home just 15 minutes ago; I haven't been driving for an hour, so how could I be going at 80 miles an hour?" At this point the cop may become quite mad, even though your question was a good one. What does it mean that you were going, when the cop spotted you, at 80 miles and hour? One can say: it means that if you were to drive for an hour at that same speed, you would cover 80 miles. Yes, but you weren't going at the same speed all the time; you were not going at a uniform speed; you went slow, then you went faster, then... So, how are we going to define that statement: "At 12:35 (say) you were going at 80 miles per hour"? This needs some Calculus, which was one of the great inventions of modern science.
We might start by saying: that you were going at 80 miles/hour at 12:35 means that between 12:30 and 12:35 you covered 80 miles divided by 12 (because 5 minutes is 1/12 of and hour), that is, 6.66 miles. But this is not good enough, for in those five minutes, between 12:30 and the time the cop spotted you at 12:35, you didn't go at a uniform speed; maybe you were stepping on the gas, going faster and faster. So we might say, okay, let's take the time interval between 12:34 and 12:35: what the cop claims is that in that one minute you covered 80/60 = 1.33 miles. Still, this is not good enough, for, again, during that minute you were not going at a uniform speed. What we need is a definition of instantaneous speed, the speed you were going at exactly 12:35. And the way to define it correctly is as follows: we take the distance covered in a small time interval, call it h, before 12:35; this distance will be x(12:35), your location at time 12:35, minus x(12:35-h), your location at time 12:35-h. This distance, then, is
x(12:35) - x(12:35-h).
Now we divide this distance by the time you took to travel it, namely h, and we get the average velocity of your car during that interval of time, between 12:35-h and 12:35. To get the instantaneous speed at exactly 12:35, we must look at that ratio,
{x(12:35) - x(12:35-h)} / h
and make h smaller and smaller, without however ever letting h
be 0. This operation is called taking the limit of our
quotient as h goes to 0. The result is the instantaneous speed
at 12:35, by definition.
To give a concrete example, suppose that instead of a car we consider a falling body, let's say a ball which we drop from a height of 5 feet. If we take convenient units of length (height) and time, the position of the ball at time t will be given by the function x(t) = (1/2)t2, one half of time squared. Now we want to compute the instantaneous speed of the falling ball at, say, time t = to. Proceeding as before, we look at the quotient
{x(to-h) - x(to)} / h = 1/2 {(to-h)2 - to2} / h
Using high-school algebra, and dividing by h throughout, this is equal to:
1/2 {to2 - 2toh + h2 - to2} / h = 1/2 {2to - h}
We said we would let h become smaller and smaller, and look at what happens to our function. Well, this one is quite easy: as h becomes closer and closer to 0, our function becomes closer and closer to to (after canceling the 2's). So that's the instantaneous speed of the falling body at time to: it is simply equal to to! Of course, this is what Galileo discovered: the speed of a falling body is proportional to the time elapsed since it was dropped.
This operation of looking at a quotient (it's called the incremental quotient of a function) and letting the quantity h approach 0, is called in Calculus taking the derivative of the function. What we just did is to prove that the derivative of the position function of the falling ball, x(t) = (1/2)t2, is the velocity function v(t) = t. This is always the case: velocity is the derivative of the position function. When we work in two (or three) dimensions, though, as I said earlier there are two (or three) position functions, x(t), y(t) and z(t). To get the velocity we take the derivative of each of them, thus obtaining two (or three) functions: x'(t), y'(t) and z'(t). These two or three functions are called the velocity vector. In more than one dimension, velocity is a vector!
Now it is easy to define property (3), the acceleration, or instantaneous rate of change of velocity. It is simply the derivative of the velocity. And again, in two or three dimensions the acceleration is a vector. Calculus doesn't only study derivatives, but also integrals. To integrate is the inverse operation of taking derivatives, much as subtraction is the inverse operation of adding. So by integrating the velocity function we can get the position function, and by integrating the acceleration we get the velocity.
We are in a position now to state Newton's laws of motion. They were discovered by Isaac Newton (1642-1727) in the years 1665-67, when he was about 24 years old, and they involve two other concepts, that of force and that of mass, which, as we'll discuss a little later, may look suspiciously like metaphysical entities. But let's us first state Newton's three laws of motion:
In those same 18 months, while Newton was in the countryside because the plague raged in the big cities, he made another historical discovery. As if discovering Calculus and the three laws of motion wasn't enough! This new discovery was the law of gravitation. It states that whenever we have two particles, with masses m1 and m2 respectively, and which are located at two points with a distance r in between, there is a force on each of the particles in the direction of the other (thus they are attracted), and whose intensity is proportional to the product of the two masses and inversely proportional to r squared. In math notation, magnitude of F = Gm1m2/r2, where G is a constant (the gravitational constant): its numerical value, of course, depends on the units of measurement that we choose.
Two remarks on this: first, it's not at all clear that the masses we are using when we state the law of gravitation are the same as those which appear in the statement of Newton's second law; still, experimental evidence shows that they are the same. Second, when we attribute this strange attractive property to massive particles, aren't we indulging in metaphysics? For we are saying, indeed, that matter has a inner, active principle: matter attracts matter. At the time, physicists (who called themselves "natural philosophers") accused Newton of doing exactly that, indulging in metaphysics, and the followers of Descartes (mostly in France) couldn't stomach the law of gravitation. What can we say in Newton's defense? Well, surely he was indulging in metaphysics, but with a difference: he wasn't just saying, like others had been doing for centuries, that things have an inner, active principle and leaving it at that; he gave a mathematical law for that inner, active principle. That made a lot of difference. He abstained from answering the metaphysical question, "What is this attractive force?" Rather, he just gave a mathematical formula for it. Still, the main reason for the acceptance of Newton's gravitation was its tremendous success. As the saying goes, nothing succeeds like success.
Let's explain the success of Newton's gravitation. First, he succeeded in going from particles or mass points attracting each other to big bodies such as the earth and the moon. To do this he used integration, the Calculus operation mentioned before, which allowed him to add up the contributions of very small (infinitesimal) portions of matter; the net result is that a body like the earth exerts a gravitational attraction on other outside bodies which is as if all the mass of the earth were concentrated at its center of gravity (in a spherical and homogeneous body that would be its geometrical center). The details of this proof are too technical for this lecture. Then, again using Calculus, he derived all of Kepler's laws for planetary motion (which we saw last semester) from the law of gravitation: so he proved that planets move in ellipses (or, more generally, conic sections) with the sun at one focus, that the area velocity is constant, and that the square of the periods of revolution of the planets are proportional to the cube of their mean distance from the sun. Newton also managed to explain the tides as a consequence of the gravitational pull of the moon on the waters of the earth oceans. This is not as simple as it seems: you should try to explain, yourselves, why there's a high tide not only on that portion of the earth facing the moon, but also on the opposite side. In any case, all kinds of phenomena, including the pendulum studied earlier by Galileo, were for the first time derived mathematically from simple formulas: the law of gravitation plus Newton's three laws of motion. That was no mean achievement, and the world couldn't help but take notice. Many thinkers, including Descartes, had maintained it is thanks to God that the universe keeps going; after Newton, the idea became more and more popular that if God was necessary, it was only to give the universe its initial impulse; after that, it kept on going according to the laws of gravitation and motion. The most usual metaphor was a clock: God was the clockmaker, and the one who wound up the mechanism, but his further intervention was not needed. By around 1800, however, even the initial divine intervention had been abandoned: when Laplace, a great French astronomer and mathematician, presented his Celestial Mechanics to Napoleon, the ruler asked, "Where is God in all this?", and the scientist replied, "Sire, I had no need of that hypothesis."
For reasons we cannot get into, Newton did not publish his discoveries until much later, in 1687, in a book titled (in Latin) Philosophia Naturalis Principia Mathematica, or Mathematical Principles of Natural Philosophy, probably the greatest book in the history of science. Meanwhile, another great scientist, the German Gottfried W. Leibniz (1646-1716) had (whether independently from Newton or not is still a matter of bitter contention) discovered the essentials of Calculus. Leibniz' Calculus had one advantage over Newton's: notation. This may seem a small matter, but giving good names and symbols to mathematical concepts can be of great help in thought and computations. In any case, the fight over the priority in the discovery embittered both men, and had a great influence over the subsequent history of math. There is another aspect of the difference between Newton's and Leibniz' thought which left deep traces, becoming an important difference between English and Continental (European) science and philosophy. Newton was content to demonstrate that phenomena occur according to laws formulated in math language, without inquiring too much into what is force, what is mass, and especially how can two masses attract each other at a distance, without contact. Leibniz, instead, was one of the greatest metaphysicians, who could not accept an explanation unless he understood those thorny questions: "What is it?" and "How?" So, for example, he maintained that gravitation is caused by the motions of a very subtle substance, rather like light, which he called "ether." This ether was to have an agitated life among physicists, until, by the end of the 19th century, it was discovered experimentally that we cannot say much about it; we cannot even measure its velocity. And so ether was dropped, probably for good. We will return to that by the end of the semester.
Newton studied the phenomenon of light, and published his Optics, at about the same time he was occupied with the invention of Calculus and the laws of motion. Among other experiments, he decomposed a ray of white light into the colors of the rainbow by making it pass through a glass prism, then he recomposed the different colors back into white light. In relation to light, we must add that before Newton, the French mathematician Pierre de Fermat (1601-1665) had stated the principle that bears his name: light travels always on a path which minimizes the time it takes to go from a point P to a point Q. Naturally, if light travels on a homogeneous medium without any obstacles, it will go in a straight line (assuming space is Euclidean); but when we have different media, for example first air then glass, then air or another type of glass, etc., Fermat's principle lets us derive the laws of reflection and refraction about which we talked in the first semester. This principle of Fermat is quite puzzling if we put ourselves in a metaphysical frame of mind: when light starts from P, how does it know which of the infinitely many paths will take it to Q in the minimum possible time? Yet that's the way things happen.
Finally, let's go back to the motion of an object on a straight line. As we saw last semester, Zeno of Elea formulated the first paradoxes about motion. One of them is the so-called Achilles and the turtle. Remember that Achilles runs a foot race with the turtle, but he of course runs much faster than the animal, so he gives it a headstart. Lets us say, to simplify matters, that Achilles starts one mile behind the turtle, and that his speed is constant, 2 miles per hour, while the turtle runs at a constant speed of 1 mile per hour. Let us compute the times. It takes Achilles 1/2 hour to get to the point where the turtle started, meanwhile the turtle will have advanced 1/2 mile and is now located at the point T1, 1/2 mile ahead of Achilles. Then it takes Achilles 1/4 of an hour to get to that point T1, but the turtle will then be located at a point 1/4 of a mile ahead of him, at the point T2. It takes Achilles 1/8 of an hour to get to that point, meanwhile the turtle... and so on. If we add up all those times we get the following: 1/2 + 1/4 + 1/8 + ... The problem is that the sum goes on and on: although the terms becomes smaller and smaller, there are infinitely many of them. One of the great achievements of the Calculus invented by Newton and Leibniz is that it allows us to compute such an infinite sum, which is called an infinite series.
How do we do it? First, observe that each term is obtained from the previous one by multiplying by 1/2: this is called a geometric series. Now, one learns in high school (we won't do it here, but you may look it up or ask) that when we have a finite geometric series such as: c + c2 + c3 + ... + cn, this sum is equal to: c(1 - cn)/(1 -c). In the case we are considering c is 1/2, so if we add the first n items we have, that is, if we add our terms beginning with 1/2 and stopping at 1/2 to the power n, we get: 1/2(1 - 1/2n)/(1 -1/2). Since 1 - 1/2 = 1/2, simplifying we obtain simply: 1 - 1/2n. But remember, we are only adding n terms, and n may be very big, but this is still a finite sum. How do we get the infinite sum? Again, the solution is what's called taking the limit. We ask, what happens to our expression, 1 - 1/2n, as n gets bigger and bigger? And the answer is: as n gets bigger and bigger, 1/2 to that power gets smaller and smaller, that is, closer and closer to 0. The limit of our expression is therefore equal to 1. In other words, our infinite sum, starting with 1/2 and adding all the powers of 1/2, is equal to 1! Or, yet in other words, it takes Achilles exactly one hour to overtake the turtle. Leibniz was a master at adding infinite series quite harder than the one we just did. Next lecture we will have more to say about such infinite series.
Chapters 7, 8 and 9 of Richard Feynman's Lectures on Physics, Volume 1.