Axiomatic systems in mathematics
Axiomatic systems in mathematics are foundational frameworks that enable the systematic organization of mathematical knowledge through a set of definitions, axioms, and theorems. An axiomatic system begins with undefined terms, which cannot be further defined without leading to circularity, and includes axioms—statements accepted without proof that serve as the basis for deducing additional statements. Historical examples, such as Euclid's "Elements," illustrate early axiomatic structures in geometry, establishing principles that guided logical reasoning and mathematical proof. The importance of consistency in axiomatic systems cannot be overstated, as it ensures that no contradictions arise from the axioms or previously proven theorems.
Moreover, developments in the 19th century revealed the independence of certain axioms, leading to the emergence of non-Euclidean geometries. This exploration demonstrated that different axiomatic systems could coexist, each valid within its context. Current discussions around axiomatic systems touch on their completeness, with Gödel's incompleteness theorem highlighting inherent limitations by asserting that some mathematical statements cannot be proven or disproven within the system. As such, axiomatic systems continue to be a crucial area of study in mathematics, influencing fields such as logic and computer science.
Axiomatic systems in mathematics
Summary: An axiom is a statement that is assumed to be true, and axiomatic systems have a rich and interesting mathematical history.
Axiomatic systems provide a deductive framework for mathematicians to combine related definitions and theorems that make mathematical knowledge systematic and structural. Mathematical theories including number systems, set theory, probability, algebra, and many others are built by using axiomatic systems.
Axiomatic Method and Axiomatic System in Mathematics
To build a deductive mathematical system, one needs to observe two intrinsic limitations in this process.
Limitation 1: Not every mathematical term can be defined. The reason can be seen by the following considerations: To define a term A, one needs a term B, and possibly some other terms. To define the term B, one needs another term, C, and so on. One may eventually come back to the term A; in which case the definition would be circular as there are a finite number of words. This means that A is used to define A, which is undesirable.
If the definitions are not to become circular, some terms are needed to start with. The solution is that there will be some terms that will not be defined. These will be called “undefined terms,” and will be used to define all the other terms to be considered. One may think that it is strange that this solution can work. How can undefined terms give meaning? This puzzle is partially answered upon consideration of the next limitation.
Limitation 2: Not every mathematical statement can be deduced or proven. The reason is similar to the one in Limitation 1; some statements are needed to start a chain of deduction: if R, then S; if S, then T; if T, then U; and so on. To deal with this limitation, certain statements must be accepted without proof. These statements are called “axioms,” and they are the statements that we used to deduce other statements. Actually, the axioms are often statements about the undefined terms. In other words, the axioms often tell us certain properties or restrictions of the undefined terms. Thus, the axioms help provide meaning to the undefined terms. Starting with undefined terms, axioms, and definitions, and by using deductive reasoning to establish important mathematics facts in the form of theorems, the mathematics system so obtained is said to be built by using the “axiomatic method.” Such a system that consists of undefined terms, axioms A, definitions D, statements of the form If P then Q and proof of such statements is called an “axiomatic system.” In an axiomatic system, one does not talk about the validity of A or P, one talks only about the validity of the proof based on A and D.
Historical Developments
Historically, Euclidean geometry was the best-known model of an axiomatic system. Around 300 b.c.e., Euclid wrote his 13-volume Elements, which contained an axiomatic treatment of geometry. It starts with 23 definitions; Euclid stated 10 axioms. The first five axioms are geometric assumptions, which he called postulates. The last five are more general, which he called common notions. There, Euclid did not use undefined terms.
The most important and fundamental property of an axiomatic system is “consistency” (it is impossible to deduce from these axioms a theorem that contradicts any axiom or previously proved theorem). The Euclidean geometry provides such a consistent axiomatic system. An individual axiom is “independent” if it cannot be logically deduced from the other axioms in the system. The entire set of axioms is said to be independent if each of its axioms is independent. Mathematicians prior to the nineteenth century doubted very much about the independence of the fifth postulate (the parallel postulate). They tried to deduce such a postulate by using the first four postulates. Despite considerable effort devoted to the task, no significant result could be obtained.
Euclid’s Elements indeed became the most influential book on geometry, as well as the model of logical reasoning and axiomatic system, until the nineteenth century when two fundamental developments took place. First, it was realized that Euclid’s logical system was not rigorous enough. A rigorous axiomatic treatment of Euclidean geometry was given by David Hilbert (1862–1943) in his 1899 book Grundlagen der Geometrie (The Foundation of Geometry). Here, Hilbert used the undefined terms of point, line, lie on, between, and congruent for the geometry system. Second, research results of C. F. Gauss (1777–1855), J. Bolyai (1802–1860), and N. I. Lobachevsky (1793–1856) asserted that the parallel postulate was actually an independent axiom. Non-Euclidean geometry could be developed by replacing the fifth postulate with another independent axiom. The lesson from Euclidean and non-Euclidean geometry is that both are valid axiomatic systems. When studying Euclidean or non-Euclidean geometry, no claims are made on the truth of the axioms about the physical world. One merely claims that if the axioms are valid, then the theorems deduced therein are also valid. Whether the logical system describes the real world is another question.
Current Issues
There are still many issues regarding the axiomatic systems. The set of axioms in an axiomatic system is “complete” if the axioms are sufficient in number to prove or disprove any statement that arises concerning our collection of undefined terms. To determine whether an axiomatic system is complete is by no means an easy question to answer. A great surprise was discovered by Kurt Gödel (1906–1978) in 1931. He proved that in a formal mathematics system that included the integers, there exist statements that are impossible to prove or disprove. This result is called Gödel’s incompleteness theorem. Also, to determine whether a given proposition is an axiom has been a very important issue in computer science and is important when one tries to use a computer to do proofs. If the computer cannot recognize the axioms, the computer will also not be able to recognize whether a proof is valid.
Bibliography
Greenberg, Martin Jay. Euclidean and Non-Euclidean Geometries: Development and History. 3rd ed. New York: W. H. Freeman and Co, 1993.
Heath, Thomas L. The Thirteen Books of Euclid’s Elements, Vol. 1–3. 2nd ed. New York: Dover Publications, 1956.
Meyer, Burnett. An Introduction to Axiomatic Systems. Boston: Prindle, Weber & Schmidt, 1974.
Venema, G. A. The Foundations of Geometry. New Jersey: Pearson Prentice Hall, 2006.
Wallace, E. C., and S. F. West. Roads to Geometry. 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2003.