Mathematical certainty
Mathematical certainty refers to the degree of assurance that can be achieved within different branches of mathematics, ranging from absolute certainty in mathematical proofs to probabilistic statements in statistics. Historically, ancient Greek philosophers like the Pythagoreans and Aristotle laid the groundwork for mathematical reasoning, positing that all questions about nature could be addressed through logical proofs based on self-evident truths or axioms. However, these ideals have evolved, revealing that absolute certainty is often unattainable in certain contexts, particularly as demonstrated by Gödel’s Incompleteness Theorems, which state that any sufficiently rich axiomatic system is either inconsistent or incomplete.
The development of various axiomatic systems, such as Euclidean and non-Euclidean geometries, illustrates that different foundational assumptions can lead to distinct mathematical landscapes. This flexibility raises philosophical questions about the nature of proofs, as different schools of thought—logistic, intuitionist, formalist, and set theoretical—interpret the validity of proofs in diverse ways. As mathematical proofs become increasingly complex, the challenge lies in validating these proofs without introducing uncertainty, particularly in cases where extensive computational verification is required. Overall, the concept of mathematical certainty is intertwined with both historical developments and contemporary challenges in the field.
Mathematical certainty
Summary: Mathematics is arguably the most stable and rigorous source of knowledge; yet any system of mathematical reasoning is incomplete.
At one end of the spectrum in mathematics, it seems as if people can be absolutely certain, using mathematical proofs of concepts. At the other, in statistics and many applied mathematics fields, it is virtually impossible to be certain, but people can make probabilistic statements with regard to degrees of uncertainty inherent in a given calculation or statement.
The Greeks may have been the first to attempt a rational explanation of nature. The crucial tool in their investigations was mathematical reasoning. They assumed that all questions about nature can be answered by reason and that all these answers are knowable and can be discovered, a property known as “completeness.” They also assumed that all answers are compatible, which is called “consistency.” However, the evolution of mathematical certainty revealed that these assumptions can never be fully realized. The notion of what is certain and what is uncertain is a fundamental component that is threaded in various ways throughout twenty-first-century mathematics curricula. For example, in primary school, students investigate the differences between “likely” and “unlikely” events. Students also develop inductive and deductive reasoning by exploratory investigations and examples as well as by proofs.
Axiomatic Systems
The Pythagoreans (c. 585–500 b.c.e.), a school influenced by Pythagoras of Samos, offered a mathematical plan of nature. The Greeks’ goal was to rationally explain why things are the way they are. They confronted a fundamental question: can all knowledge be verified? Aristotle (384–322 b.c.e.) answered “no,” since there are self-evident truths (called “axioms”) that cannot be explained. Moreover, in geometry, Aristotle said a proposition is proven when it is shown to logically follow from the axioms and other proven propositions. Euclid of Alexandria (323–285 b.c.e.) knew of these developments and incorporated them into his text, Elements. It is recognized as the prototype for how mathematics should be done: well-thought out axioms, precise definitions, carefully stated theorems, and logically coherent proofs.
Formulation of the axioms or “postulates” (the Greek term for axioms about geometry) is the crucial step in building an axiomatic system. These statements should be intuitively self-evident, and, from these, it must be possible to deduce the important properties of the objects of study. Later, mathematicians found assumptions used in Elements that were not explicitly stated in the axioms. Credit for completely and successfully axiomatizing Euclid’s geometry is generally given to David Hilbert.
The Parallel Postulate and Non-Euclidean Geometries
Euclid’s fifth (or parallel) postulate states that, “through a given point, not on a given line, only one parallel line can be drawn to the given line.” Almost immediately, this postulate was controversial. Many did not find it to be self-evident and thought it required a proof. Over two millennia, countless mathematicians tried to derive the parallel postulate from the others, all with no success. These futile efforts had important consequences in all of mathematics. Beginning in the eighteenth century, some mathematicians began to use indirect methods to “prove” this postulate.
Though unsuccessful, the indirect methods led to the discovery of non-Euclidean geometries—using the other axioms but denying the parallel postulate. Attempts to prove non-Euclidean geometries were invalid were essentially attempts to show that they were inconsistent. Eventually, it was determined that Euclidean and these other geometries were consistent and complete. The discovery of non-Euclidean geometries revealed that mathematics could deal with completely abstract axiomatic systems, which no longer had to correspond to beliefs based on real-world experiences.
The Consistency and Completeness of Mathematics
It became clear that the most important considerations for an axiomatic system were its consistency and completeness, and at the International Congress of Mathematicians in 1900, Hilbert addressed these problems. He felt all mathematics should be put on a sound basis using the axiomatic method. In 1904, Hilbert constructed an arithmetic model of Euclidean geometry, showing that geometry was a subset of arithmetic. Mathematicians then set out to show the consistency of arithmetic, from which it would follow that Euclidean geometry was consistent.
These efforts ended in 1931 with the results of Kurt Gödel. His first Incompleteness Theorem showed that in any axiomatic system rich enough to include the arithmetic of the natural numbers, it is possible to prove some statements that are false, showing the system is inconsistent; or it is not possible to prove some statements that are true, showing the system is incomplete. In his second Incompleteness Theorem, Gödel showed the question of whether an axiomatic system is consistent cannot be determined within the system. Gödel’s results revealed that any mathematical reasoning system based on axioms as rich as arithmetic can never be fully realized—such systems must be either incomplete or inconsistent. Modern mathematicians operate under the assumption that mathematics is incomplete and not inconsistent.
Different Axioms Lead to Different Mathematics
One axiom of set theory, the axiom of choice (AC), was used implicitly for years before it was explicitly described. The AC states that for any collection of nonempty mutually exclusive sets, finite or infinite, there is a set that contains exactly one element from each set. The AC with Zermelo–Fraenkel (ZF) axiomatic set theory, named for Ernst Zermelo and Abraham Fraenkel, is the basis of modern mathematics. In 1938, Gödel proved that if ZF set theory without the AC is consistent, then ZF set theory with the AC is also consistent. So, just as it is possible to choose between different acceptable geometries in which the parallel postulate may or may not be true, it is possible to choose between different acceptable ZF set theories in which the AC may or may not be true. On one hand, theorems requiring the AC are fundamental in such areas as modern analysis. Then again, by adopting the AC, results such as the Banach–Tarski paradox, named for Stefan Banach and Alfred Tarski, can be derived, which says a golf ball can be divided into a finite number of pieces and then rearranged to make a solid sphere the size of the Earth. Thus the decision as to which axiomatic system to adopt cannot be made lightly—different mathematics can be derived from these different axiomatic systems.
Valid Proofs
The idea of a valid proof depends on one’s philosophical approach to mathematics. A number of schools of thought have evolved, including (1) the logistic school, which holds mathematical proofs derive from logic; (2) the intuitionist school, which maintains mathematics takes place in the human mind and is independent of the real world—it composes truths rather than derives implications of logic; (3) the formalist school in which proofs follow from the application of a system of axioms; and (4) the set theoretical school, which derives proofs from the axioms of set theory.
Mathematics remains our most rigorous form of knowledge. As proofs grow more complicated, mathematicians worry they will have to accept a greater degree of uncertainty in solutions. For example, the entire proof of the Classification Theorem for Finite Simple Groups consists of an aggregate of hundreds of research papers and over 10,000 printed pages. Additionally, the Four-Color Problem solution has been achieved only on the computer and involves checking a prohibitively large number of cases. Some mathematicians believe that since it is not reasonable and possible for any one individual to check all these cases, then a valid proof has not been provided for such problems.
Bibliography
Borba, M. C., and O. Skovsmose. “The Ideology of Certainty in Mathematics Education.” For the Learning of Mathematics 17, no. 3 (1997).
DeLong, Howard. A Profile of Mathematical Logic. Reading, MA: Addison-Wesley, 1970.
Kline, Morris. Mathematics: The Loss of Certainty. New York: Oxford University Press, 1980.
Wainer, Howard. Picturing the Uncertain World: How to Understand, Communicate, and Control Uncertainty Through Graphical Display. Princeton, NJ: Princeton University Press, 2009.