Mathematicians Complete the Classification of All Finite Simple Groups

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Mathematicians Complete the Classification of All Finite Simple Groups

Overview

After 150 years of work capped with several decades of intense effort, mathematicians were able to demonstrate that they had classified all mathematical entities known as finite simple groups. This proof was the longest ever completed, consisting of more than 500 scientific papers that filled over 15,000 pages in mathematical journals. The proof, in final form, was over 5,000 pages in length. Mathematics' acceptance of this proof, which will likely never be reviewed in its entirety by a single person, marked an interesting change in the manner in which increasingly complex research is performed and accepted by the scientific community.

Background

In 1980 Ohio State University mathematician Ronald Solomon completed work on a mathematics problem that had begun in 1832 with a mathematics paper that was rejected by the French Academy of Sciences. The author of this paper, 20-year-old Evariste Galois (1811-1832), died in a duel just a few months later. His paper went unremarked for 12 years, until the great mathematician Joseph Liouville (1809-1882) personally presented it to the French Academy as a worthy achievement.

Galois's paper, which explored methods for solving the quintic equations (in which one of the terms is raised to the fifth power), introduced the mathematical concept of a group as a tool for attacking these equations. What Galois did not realize was that group theory would prove enormously useful in many areas of mathematics and physics, culminating in Solomon's paper.

Mathematically speaking, a group consists of five elements:

A set, usually designated G.
An operation * performed on group elements x and y such that x * y is also a member of G.
This operation must be associative, i.e. (x * y) * z = x * (y * z).
The group must have an "identity element."
Every member in the group must have an inverse.

There are an infinite number of groups. Most of these groups can have an infinite number of members and are quite complex, some with a daunting array of operations or complex operations. However, these groups are entirely composed of a class of groups called finite simple groups, which can be thought of as the building blocks of group theory. In a way, they are analogues to the prime numbers, which can be multiplied and added to form every other number.

Because of the fundamental nature of the finite simple groups, mathematicians made many efforts to understand them. This work proceeded sporadically until the 1950s, when efforts and progress picked up significantly. Around this time, it was realized that the finite simple groups fell into certain categories, with all of the members of each category sharing some fundamental properties with each other. The last of these groups, called the monster group, was described mathematically in 1980. This was followed by Solomon's mathematical proof stating that, with the description of the monster, all the finite simple groups had been discovered and described. When all the published papers in this area were compiled, it was found that more than 100 mathematicians had published over 500 papers with a total of 15,000 pages of text over more than a century to bring this problem to its conclusion.

Solomon's proof was a mathematical argument, designed to convince other mathematicians that a certain premise or statement was true. His specific proof was not long and was essentially intended to tie up the remaining loose ends in this long quest. The entire proof for this work will eventually be published in nearly a dozen volumes, reflecting the contributions of many mathematicians. But this raises the question of what a mathematical proof really is.

As mentioned above, a mathematical proof is an argument, couched in the language of mathematics, that is designed to show that something either is true or cannot be true. Mathematical proof dates back to the ancient Greeks. The first recorded proofs were generated in the sixth century B.C. when Thales (c. 624-547 B.C.) was able to use basic facts about the nature of length and area to prove the truth of common geometrical observations. Not much later, Pythagoras (c. 580-500 B.C.) developed a proof of the theorem that bears his name. Since that time, mathematical proof has evolved considerably in both length and complexity, but it retains its original character: a series of logical steps, based on mathematical concepts that are basic, that leads inexorably to the conclusion.

Over the course of centuries, mathematics has grown immeasurably more complex than in Pythagoras's time, and proofs have become correspondingly longer, more complex, and less accessible to the nonspecialist. The proof of the finite simple groups is one of the longest mathematical proofs ever generated, and it is doubtful that any single person could competently review or critique it. In addition, we entered the era of computer-generated proof in 1976, when a proof of the four-color map problem was generated, relying heavily on computer programming. In this case, the calculations required are too complex and lengthy for a person to perform, necessitating the use of computer power. This reliance on computers for many high-level problems in mathematics is increasingly common but makes many mathematicians uneasy. However, it is likely to continue and, however limited people may be with respect to performing calculations by hand, it should be some solace to remember that nobody has yet programmed human intuition or insight into a computer.

Impact

The proof that all finite simple groups had been accurately classified raises some interesting questions about the nature of mathematical proof, questions that are as much philosophical as mathematical in nature. These questions will be discussed following a short summary of the impact of group theory in mathematics and physics.

Group theory is one of the more fruitful areas of mathematics. Galois used it to help answer questions regarding quintic equations. Chemists and crystallographers use it to better describe and understand a variety of crystal structures. Group theory has also proved useful in particle physics and atomic physics. This widespread use of group theory utilizes all groups, finite and infinite, simple and complex. However, the finite simple groups are the fundamental units of which all groups are composed; understanding and classifying these fundamental units leads to a better understanding of all groups.

These aspects of group theory notwithstanding, this particular proof has implications that reach further. The most fundamental implication deals with the nature of mathematical proof itself. As originally intended, and as still assumed by most mathematicians, a mathematical proof is a series of arguments that, when reviewed by a knowledgeable mathematician, present a convincing argument that a particular statement is true. However, the proof for the classification of all finite simple groups is so long and complex—drawing on many mathematical specialties—that it is unlikely ever to be reviewed in its entirety by a single person. This means that the mathematical community is accepting as true a proposition that is unlikely ever to be verified in its entirety. Instead, mathematicians are trusting that a team of reviewers has the patience, attention, mathematical skills, and ability to coordinate and communicate their activities sufficiently well to say that the proof is accurate and complete. In turn, the mathematicians whose work has gone into this proof are placing a large degree of trust in each others' work, assuming that their colleagues are all competent and conscientious. In the case of the four-color map theorem, a further assumption is made that the computer running the program that helped prove this theorem was operating properly and the program was correctly written. These assumptions may never be fully tested or confirmed.

These assumptions seem warranted in both cases. However, by accepting these proofs, mathematicians have tacitly accepted that some areas of mathematics have grown beyond the abilities of any single person to fully comprehend. In addition, unlike proofs that came before, it may never be known absolutely that these theorems are entirely correct. There may always remain some degree of doubt about them because, in a team of mathematicians, there is always the chance that some error, small or large, has been missed because it fell between the work of two reviewers or two of the mathematicians working on the proof. In essence, for the first time, mathematicians were asked to accept a proof, not because unassailable logic showed that it must be correct but because the logic appeared correct and nobody was able to show otherwise.

This same trend towards complexity has been noted in other scientific disciplines, too. In particle physics, for example, teams of scientists sometimes a hundred strong collaborate to produce scientific research, the results of which rival these mathematical proofs in complexity. In multidisciplinary research it is not uncommon for work to be reviewed by a team of scientists, each of whom understands one or two aspects of the research, but none of whom can understand all the work in its entirety because it may span too many areas of science. And, as research becomes more arcane and research subjects smaller or more remote, these tendencies show every sign of continuing.

P. ANDREW KARAM