Deduction | Research & Encyclopedia Articles

Deduction

Deduction is the process of deriving conclusions from logical premises without resort to empirical evidence. Deductive reasoning is the primary method of reasoning used in mathematical proof, whereas, inductive reasoning, or reasoning from specific empirical facts to more general conclusions, is the method most often practiced in the natural sciences. In a deductive mathematical system, certain undefined terms, definitions, and "self-evident" assumptions called axioms or postulates are stated, after which theorems or propositions of the system are derived, based only upon what has been previously assumed or proved. The Greek philosopher, Aristotle (384-322 B.C.) laid the foundation for deductive argument with his method of the syllogism. Syllogistic reasoning, in which conclusions are derived from stated premises, dominated deductive logic for nearly 2000 years after Aristotle. Modern logicians give less emphasis to syllogistic reasoning, but the symbolic logic developed during the 20th century remains deductive in requiring rigorous arguments from a bare minimum of assumptions. Most modern mathematicians believe that the essence of mathematics is that all of its branches can be developed axiomatically, that is, deductively, from a few basic assumptions. In this view, mathematics is a purely logical discipline unaffected by empirical evidence. This position allows for the modeling of "real world" processes with mathematics, but considers the mathematics itself to be in the realm of Plato's ideas or "pure forms." The mathematical Platonist believes that all of mathematics can be "discovered" deductively with no reference to natural phenomena. If the mathematics so derived happens to lend itself well to the needs of physical science and engineering, so much the better, but this is not the pure mathematicians's primary concern. The pure mathematician desires a solid foundation of indisputable definitions and axioms from which her entire mathematical system may be constructed by proving theorems using nothing more than these axioms. This is the essence of axiomatic or deductive mathematics.

Historically, the most famous example of a deductive mathematical system is the axiomatic development of geometry set forth in the "Elements" of Euclid (c.

300 BC). In this work, Euclid proposes to deductively derive the entire body of mathematics as it then existed from a few undefined terms, definitions, and five postulates. Starting with this bare minimum of assumptions, Euclid then stated and proved "propositions" or what we today call "theorems." Each proposition had to be proved either from one or more of the five postulates or from one or more propositions that had already been proved. In this way, the mathematical structure so developed could have absolute certainty, since every step in the deductive process led ultimately back to the five postulates, which were chosen for their own indubitability. If this process were carried out in what we would today call a "rigorous" manner, then the resulting system of mathematics could be doubted only if one or more of the five postulates were doubted. Through the ages, following Euclid's writing of the "Elements," many mathematicians and philosophers called into question the claim of the "self-evidence" of the fifth postulate, also called the "parallel postulate," because of Euclid's rather lengthy wording of that postulate. Many thought that Euclid could have proved the fifth postulate from the first four, thus eliminating the system's dependence on what some believed to be a too elaborate statement. In the end, Euclid was exonerated as 18th century mathematicians showed that the parallel postulate could not be proved from the other four. It had to be an independent fifth postulate just as Euclid had said. In any case, the "Elements" became the paradigm for the development of deductive mathematical systems. The geometry course still taught in most American high schools is usually called "Euclidean" geometry because it is based on modern variations of the deductive process introduced by Euclid more than two-thousand years ago.

In the late nineteenth century, the Italian mathematician, Giuseppe Peano (1858-1932) followed Euclid's example and established his own system of five postulates, or axioms, of arithmetic that became the foundation upon which 20th century deductive mathematics would be built. Peano's axioms, together with the work of Cantor, Dedekind, Frege, Russell, and many other 20th century mathematicians and logicians, made possible the deductive construction of the real number system and the mathematics which is based upon it. Although the logician Kurt Godel, in 1930, proved some results that brought into question the limits of what we can know through deduction alone, it is safe to say that most theoretical mathematicians of the 21st century remain comitted to the deductive method in practice.