Euler diagram

An Euler diagram (/ˈɔɪlər/, OY-lər) is a diagrammatic way to show groups and how they relate to each other. These diagrams help make complicated ideas easier to understand by using simple shapes. They are like another tool called Venn diagrams, but they show only the important relationships between groups, not all possible ones.

The famous Swiss mathematician Leonhard Euler gave his name to these diagrams, although he did not invent them. People started using Euler diagrams for logic and later for grouping things together. In the United States, these diagrams became part of teaching set theory during the new math movement in the 1960s. Now, they are used in many different subjects and by many organizations.

Euler diagrams use closed shapes on a flat surface to represent groups. The way these shapes overlap shows how the groups are connected. If shapes do not overlap, it means the groups have nothing in common. If they overlap, the groups share some members. A shape that is completely inside another shape means it is a smaller group within the larger one.

History

Diagrams that look like Euler diagrams and serve similar purposes seem to have existed for a long time. However, we can only know exact dates for these diagrams after the invention of the printing press.

Before Euler

The first people to print an Euler-like diagram and talk about it in their writings were Juan Luis Vives (in 1531), Nicolaus Reimers (in 1589), Bartholomäus Keckermann (in 1601), and Johann Heinrich Alsted (in 1614). The first detailed explanation of these diagrams was by Erhard Weigel (1625–1699), who called them 'logometrum' (a measuring instrument for logic). Weigel was the first to prove all valid syllogisms using shapes in a flat plane. For general positive statements (all-sentences), the shape for the subject should be completely inside the shape for the predicate. For negative statements (no-sentences), it should be completely outside. For specific statements (sentences with 'some', 'some...not'), the shapes should partially overlap and not overlap. To prove a syllogism, one must first draw all possible figures for the premises and then see if the conclusion can also be read from them. If this is possible, the syllogism is valid; otherwise, it is not.

Euler and the time after

In his Letters to a German Princess, Euler focused only on traditional syllogistics. He further developed Weigel's approach and not only tested the validity of syllogisms but also developed a method for drawing conclusions from premises. At the same time as Euler, Gottfried Ploucquet and Johann Heinrich Lambert also used similar diagrams. However, the diagrams only became widely known in the 1790s through Immanuel Kant (1724–1804), who used them in his lectures on logic, and his students then spread knowledge of the diagrams throughout Europe. In the 19th century, Euler diagrams became the most widely used form of representation in logic, especially by 'Kantians' such as Arthur Schopenhauer, Karl Christian Friedrich Krause, or Sir William Hamilton.

Since the history of the diagrams was only partially researched in the 19th century, most logicians attributed the diagrams to Euler, leading to many misunderstandings, some of which continue to this day.

Euler diagrams in the era of Venn

John Venn (1834–1923) commented on the widespread use of Euler diagrams:

"... of the first sixty logical treatises, published during the last century or so, which were consulted for this purpose–somewhat at random, as they happened to be most accessible–it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of the Eulerian scheme."

But he argued that the Euler diagram scheme was not suitable for general logic and noted that it fits poorly even with the four propositions of common logic to which it is normally applied.

Venn ended his chapter with the observation that the use of Euler diagrams is based on practice and intuition, not on a strict algorithmic practice.

Finally, Venn criticized Euler diagrams for not fitting well with the propositions of common logic and for demanding the creation of a new group of appropriate elementary propositions.

Modern use of Euler diagrams

In the 1990s, Euler diagrams were developed as a logical system. The cognitive benefits of the diagrams soon became clear. The diagrams were therefore not only used as set diagrams but have since been used in many different ways and functions in computer science including artificial intelligence and software engineering, information technology, bioscience, medicine, economics, statistics and many other fields, and their philosophy and history have been discussed. In 2000, the conference series The Theory and Application on Diagrams: An International Conference Series began, which regularly addresses current research on Euler diagrams, among other topics.

Relation between Euler and Venn diagrams

Venn diagrams are a special type of Euler diagrams. A Venn diagram must show all possible ways sets can overlap, using circles or other shapes. In Euler diagrams, only the actual overlaps are shown, which can make them simpler, especially when there are many sets.

For example, with three sets, a Venn diagram has many sections, but an Euler diagram might just show a few shapes that represent the real relationships between the sets. This can make Euler diagrams easier to understand for complex ideas.

The text also discusses how Euler diagrams and Venn diagrams can be used in logic and mathematics, showing relationships between sets like "Animals" and "Minerals." Venn diagrams might not always show these relationships clearly, while Euler diagrams can sometimes make them easier to see.