History of the function concept facts for kids

A function in mathematics is like a special rule that takes an input and gives you exactly one output. Think of it like a vending machine: you press a button (your input), and you get one specific snack (your output). You won't get two different snacks from pressing the same button! Functions help us describe how one thing changes in relation to another. For example, how the distance you travel depends on how long you drive at a certain speed.

The idea of functions started to become clear in the 1600s, especially with the invention of calculus. At first, mathematicians thought of functions as formulas or equations. But over time, as new ideas like set theory came along, the definition became much broader. Today, a function is seen as a way to connect elements from one set (the inputs) to elements in another set (the outputs), where each input has only one output.

Early Ideas of Functions

Even before the 1600s, some thinkers had ideas that were similar to functions. In the 1100s, a mathematician named Sharaf al-Din al-Tusi in the Islamic world studied equations. He looked at how a certain mathematical expression changed its value. He even found the largest possible value for this expression to solve an equation. This showed an early way of thinking about how one value depends on another. However, his work wasn't continued much at the time.

Later, in the 1300s, a French philosopher and mathematician named Nicole Oresme also explored ideas about how quantities could be related. He thought about how some things might change independently, and others would change because of them. These early ideas were important steps towards understanding functions.

The development of analytic geometry around 1640 was a big step. This allowed mathematicians to connect shapes and curves (geometry) with equations and numbers (algebra). They started using "variables" like x and y to describe points on a graph. This made it easier to see how y changed when x changed. The word "function" itself started being used more often in the late 1600s by mathematicians like Leibniz and Bernoulli.

Functions in Logic (Before 1850)

In the early 1800s, logicians (people who study how we reason) were busy analyzing arguments and rules for thinking. While they didn't use the word "function" explicitly, their work laid some groundwork.

Thinkers like Augustus De Morgan and George Boole started to use symbols and variables in logic, much like mathematicians used them in algebra. De Morgan noted that a logical truth depends on the structure of a statement, not just the specific things being talked about. He would write statements like "X is Y," where X and Y were like variables. This showed an early form of abstraction, where general rules could apply to different specific cases.

Boole, known for Boolean algebra, also used variables to represent classes of things. For example, he might use x for "oxen" and y for "horses." He even defined a function in the context of calculus around 1849. He said that if one quantity changes steadily (the independent variable), then another quantity that changes because of it is a "function" of the first.

Functions in Logic (1850-1950)

As the 19th century progressed, logicians started to look deeper into the foundations of mathematics.

Boole and Venn's Ideas

In 1854, George Boole defined a function as "Any algebraic expression involving symbol x." He used algebraic expressions to describe both mathematical and logical ideas. For example, he showed how `1 - x` could mean "NOT x" in logic, and `xy` could mean "x AND y."

Later, in 1881, John Venn (famous for Venn diagrams) used the term "logical function." He connected this idea directly to "classes" (which are now called sets). He said that a logical function `f(x)` always represents a logical class of things.

Frege's New Way of Thinking

Gottlob Frege, in 1879, changed how people thought about functions. He moved away from the traditional "subject" and "predicate" in sentences. Instead, he used "argument" and "function."

Frege gave a simple example: Start with "Hydrogen is lighter than carbon dioxide." If you replace "Hydrogen" with "Oxygen" or "Nitrogen," the sentence changes, but the part "... is lighter than carbon dioxide" stays the same. Frege called this stable part the function. The changing part (like "Hydrogen" or "Oxygen") he called the argument. He also showed that functions could have more than one argument, like "... is lighter than ...".

Peano's Contributions

Around 1889, Giuseppe Peano also defined functions. He used the idea of a "class" (a group of objects). He said that if you have a rule (which he called a "function presign" or "function postsign"), it takes an object from a class and gives you a new object. For example, if the rule is "add a", then applying it to x gives you `x + a`.

Russell and Variables

Bertrand Russell was greatly influenced by Frege and Peano. In his 1903 book, The Principles of Mathematics, he emphasized the importance of variables in mathematics. He said that all mathematical statements contain variables, even if they're hidden. Words like "any" or "some" often point to a variable.

Russell believed that the core idea of a mathematical statement could be found by turning its specific parts into variables. He called these generalized statements propositional functions. For example, in the expression `2x³ + x`, the function is what's left when x is removed: `2( )³ + ( )`. Russell saw functions as more fundamental than just describing properties or relationships.

Russell further developed his ideas. By 1913, he considered propositional functions to be very important. A propositional function is like a statement with a blank space, such as "ŷ is hurt." When you fill the blank with a specific value (like "Bob"), it becomes a proposition ("Bob is hurt"). A proposition can be true or false. If it's true, the value you put in "satisfies" the function.

Functions in Set Theory

The idea of set theory (the study of collections of objects) grew from the work of logicians. It was pushed forward by Georg Cantor's work on infinity. However, some problems, called "paradoxes" or "antinomies," were discovered in early set theory, especially Russell's paradox. This paradox showed that if you weren't careful, you could create contradictions.

Russell's Paradox

In 1902, Russell wrote to Frege about a problem: Frege's system allowed a function to be an argument of itself. Russell then created a paradox: "Let w be the rule: to be a rule that cannot be applied to itself. Can w be applied to itself?" This question leads to a contradiction, shaking the foundations of mathematics at the time.

This paradox made mathematicians realize they needed a more careful way to define sets and functions.

Zermelo's Set Theory

To avoid these paradoxes, Ernst Zermelo proposed an axiomatic set theory in 1908. An axiom is like a basic rule that is accepted as true without proof. Zermelo's "Axiom of Separation" used a propositional function to define a subset. This meant you could only create a new set by picking elements from an existing set based on a clear rule. This helped prevent paradoxes like Russell's.

Ordered Pairs and Relations

Mathematicians also needed a precise way to define a "relation" and an "ordered pair." An ordered pair is a pair of objects where the order matters, like `(a, b)` is different from `(b, a)`.

In 1914, Norbert Wiener showed how to define a relation using ordered pairs. Later, in 1921, Kazimierz Kuratowski gave a definition of an ordered pair that is still widely used today. This was important because it allowed mathematicians to define relations and functions purely in terms of sets.

By the 1920s, the idea of a function as a "many-one correspondence" became common. This means that for every input, there is at most one output. This is a key part of the modern definition of a function.

Von Neumann's Approach

In 1925, John von Neumann proposed his own set theory. He chose to define "function" as a basic concept, rather than "set." He saw functions and sets as being equivalent: a function can be seen as a set of pairs, and a set can be seen as a function that gives two values. His theory, later refined, is known as von Neumann–Bernays–Gödel set theory.

Bourbaki's Definition

In 1939, a group of French mathematicians writing under the name Nicolas Bourbaki gave a very clear definition of a function. They defined it as a special kind of relation between two sets, E (inputs) and F (outputs). For every element x in E, there must be one and only one element y in F that is related to x. They also defined a function as a "functional graph," which is a set of ordered pairs where no two pairs have the same first element.

Functions Today

In modern set theory, a function is formally defined as a relation. A relation, in turn, is defined as a set of ordered pairs. And an ordered pair is defined as a set of two specific sets.

So, when you see a function like `f(x) = x + 1`, it's understood as a collection of ordered pairs like `(0, 1)`, `(1, 2)`, `(2, 3)`, and so on. Each pair shows an input and its unique output.

While older terms like "propositional function" were used, today mathematicians often use words like "formulae" or "predicates" when talking about expressions that contain variables and can become true or false statements. The concept of a function is now a fundamental building block in logic and mathematics, describing how things are related in a precise, single-valued way.