Transcription
Hello, my name is Sam Thompson, and today I'm going to present to you on the history of the function concept up until the 19th century.
Think back to when you first learned about functions in algebra. Do you remember how you first thought of them? Probably just thought of them as processes where you plugged in one number, an input, and then the function would spit out another number at the output, and then the two functions were somehow related to one another or associated. While this may seem like an oversimplification of what a function actually is, at its core, this is really just what a function is. It's just taking an input and associating it with exactly one output. Here are some examples of functions.
Now, that the Stewie and the Minion graph both have over 80 tall functions in them. Today, though, we're not going to be looking at exactly how those functions came to be on these graphs in particular, but rather instead, we'll be focusing on how the function concepts led to graphs like these being created. Compared to some of the topics we've studied in class, the concept of a function is a much more recent mathematical development. But just as Rome wasn't built in a day, neither was the function divined overnight. In fact, the concept of a function we know today really only first appeared a little under 300 years ago out of a famous controversy that was centered around determining the function of a vibrating string. And then, even after that, it would still continue to undergo profound changes during and after the debate that ensued afterwards. It wouldn't even take some more time for the functioning concept to evolve further within the 20th century as mathematicians continued to refine it over time.
One thing you might be asking yourself, though, is why didn't the function concept emerge earlier? I mean, after all, it seems pretty simple to us in today's world. And the reason that it didn't emerge earlier was because, quite simply, there wasn't enough necessary ingredients in place for it to occur. Like, for example, there wasn't any symbolic notation invented until roughly the late 16th century, and mathematicians weren't really comfortable with negative numbers yet, even in some cases just refused to outright consider or deal with them. Additionally, there was just a lack of motivation to consider a function concept because, well, without any abstract examples of a function, you don't really have a need to worry about defining one.
However, this would change over the course of two hundred years, from roughly 1450 to 1650, as many fundamental developments for the function concept occurred, such as the creation of symbolic algebra, where for the first time, symbols were being used to represent unknown quantities. And then, of course, the wedding vows draw and geometry, where the introduction of variables and the expression of the relationship between set variables by means of equations was first introduced. By expressing these relationships and by means of equations, this allowed a large number of examples of curves to be studied and would eventually lay the groundwork for the introduction of the function concept. However, what was missing was identifying the independent and deep and a variable within the equation, something that would come important later on during the vibrating string controversy.
Another thing that would become important for the development of a function concept would be calculus. But as we're well aware, the calculus developed by Newton and Leibniz did not originally have the same form that we now buy today. In particular, it was not a calculus of functions. Instead, the principal object of study in 17th century calculus was based around geometric curves. In fact, 17th century analysis originated as a collection of methods for solving problems about curves, such as finding tangents to curves, the lengths of curves, and velocity of points moving along curves. Since the problems that gave rise to calculus were geometric and kinematic in nature, and since mathematicians such as Newton and Leibniz were preoccupied with exploiting the wonderful tools they had created, it would be some time before calculus could be recast in its algebraic form.
The very first formal definition of a function came to be from Johann Bernoulli in 1718, out of correspondence between himself and Leibniz from 1694 to 1698. The definition came to be due to a lack of a general term for representing quantities dependent on other quantities in such formulas and equations. It stated, "One calls here function of a variable a quantity composed in any manner whatever of this variable and of constants." The only issue was that Bernoulli never really explained what "composed in a manner" meant. But overall, in the grand scheme of things, this would amount only being a small first step relative to Euler's definition that would arrive within the next 30 years.
Something that's worth keeping in mind, however, is the gradual change in focus for 18th century mathematics. What I mean is, from the 17th to 18th century, mathematics sees a gradual separation of some 17th century analysis from its geometric origin and background. This process saw the replacement of the concept of a variable applied to geometric with the concept of function as an algebraic formula. This change in the way mathematicians were thinking about solving curves helped generate an increased interest as wanting to find the formulas for said curves. As this increased emphasis was being placed on the formulas and equations relating the functions associated with a curve and on the role of symbols appearing in these formulas of equations, mathematicians were becoming more and more interested in the relations held among these symbols, independent of the original curve they came from.
This trend was embodied in Euler's "Introductio" published in 1748. It was originally intended as a survey of the concept and methods of analysis in analytic geometry, intended for a study of calculus. But Euler's "Introductio" was going to be the first work, in actuality, where the concept of function plays an explicit and central role. In fact, one can say that in 1748, the concept of a function left a prominence thanks to Euler's "Introductio." In it, Euler defined a function as, "a function of a variable quantity is an analytical expression composed in any manner from that variable quantity and numbers or constant quantities."
The key difference between Euler and Bernoulli's definition of a function was Euler's use of the term "analytical expression." Why Euler never does to find the term "analytical expression," he instead attempts to give it meaning by explaining that an analytical expression can consist of any combination of the four algebraic operations, roots, exponentials, logarithms, trigonometric functions, derivatives, and integrals. He classifies functions as being either algebraic, which meant any function that included the four algebraic operations and roots, or as transcendental, which was defined as any operation that could not be expressed in terms of a finite sequence of break operations, such as that of exponentials, logarithms, or trig functions. Even further, he classified functions as being either an explicit function, meaning the function can be expressed as an equation in terms of one variable, or an implicit function, which are functions that originate by a solution of equations. Although the concept of a function did not originate with Euler, it was he who first gave it prominence by treating the calculus as a formal theory of functions.
To give a brief forward, the vibrating string problem was of extreme importance for the subsequent evolution of the concept of a function. I'm going to try and do my best to cover the main points of the vibrating string controversy, so I do apologize if I leave out any details that may be to your interest. The controversy was based around determining the function that describes the shape of an elastic string with fixed ends that's deformed into some initial shape. It then released, vibrating at its core. The controversy was centered around the meaning of a function, and several mathematicians, including Euler, Dalembert, Daniel Bernoulli, and Lagrange, attempted to claim their interpretation as being the correct one.
Before that, there's one thing we need to understand about 18th century mathematics, which is the "articles of faith." This states that if any two analytical expressions, that is to say, two formulas, agree on an interval, that they may agree everywhere else. This assumes that the entirety of a curve given by an analytical expression is determined by any small part of the curve. For example, in 1744, Euler wrote the Goldbach of pi minus x divided by 2 is equal to the summation of sine in x divided by n from n equals 1 to infinity. As we can see from the graph below, these two expressions only agree from 0 to pi and nowhere else. Why, to us, this may seem like a big issue in today's world, at the time, Euler believed examples like the one below or an insignificant exception to the function concept that I'd need not be worried about. At the time, this wasn't a stranger, uncommon assumption, but would eventually lead to the need for later revaluation of the function concept.
Viewpoints during the vibrating string controversy: First off, Dalembert, after determining the partial differential equation that governed the motion of the string, believed that the most general solution of the string depended on the initial form of the string and insisted that the function which described the initial velocities of each point on the string had to be expressed by a single analytical function. On the other hand, Euler, after working and solving the problem himself, found the same solution as Dalembert and agreed with them, but disagreed with him on the solution's interpretation. In his own experiments, Euler claimed the day on ABARES solution gives the shape of the string for different values of T, even if the initial shape isn't given by a single formula. Euler claimed that for physical reasons, more general expressions for the initial form of the string had to be allowed. Day on bear, of course, disagreed with them, and the two battled back and forth. But according to the articles of faith at the time, neither of these two types of initial shapes given by Dalembert could be given by a single analytical expression. Therefore, Dalembert's solution was deemed false by Euler.
Eventually, Daniel Bernoulli stepped in and gave his take on the problem, which he based off of physics and his knowledge of musical vibrations. Both Euler and Dalembert found Bernoulli's solution to be absurd, as relying on the articles of faith, they argued that since f(x) and the corresponding sine series agree on 0 to L, they must agree everywhere else. But of course, this led to the absurd conclusion that an arbitrary function f(x) is odd and periodic. The debate would go on for several more years, later being joined by Lagrange, and then would eventually die down without ever actually being resolved. Essentially, the debate could be characterized as one between Dalembert's mathematical world, Bernoulli's physical world, and Euler's no-man's land between the two. The debate did extend the function concept in major ways to now include functions defined piecewise by analytic expressions in different intervals, and also can now include functions drawn freehand and possibly not even given by any combination of analytical expressions.
Sometime after the vibrating string controversy, Fourier would come along and give his definition of the function concept to the mathematical world. Initially, this was met with much skepticism by mathematicians, as one of the big things that he did away with was the articles of faith that were held so closely by 18th century mathematicians. Another thing that he did was he sparked a renewed emphasis in analytical expressions, which in turn would force another revaluation of the function concept. One of the mathematicians to do so in revaluing the function concept would be Dirichlet. What he did was he undertook a careful analysis of Fourier's original theorems and work on the function concept in order to make it more mathematically respectable. As originally, Fourier's proof of his theorems was loose, even by early 19th century standards. This led to Dirichlet's concept of a function, which would end up being the first to seriously take the notion, excuse me, take the seriously the notion of a function as an arbitrary correspondence between the two values x and y. Another thing to know is that in Dirichlet's definition, he was among the first to explicitly restrict the domain of the function to an interval.
This covers the history of the function concept up to the 19th century. I hope you enjoyed my presentation, and thank you for listening.