It Began with Babbage Read Online Free

233–234.

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Entscheidungsproblem
: What’s in a Word?
I
    IN 1900, THE celebrated German mathematician David Hilbert (1862–1943), professor of mathematics in the University of Göttingen, delivered a lecture at the International Mathematics Congress in Paris in which he listed 23 significant “open” (mathematicians’ jargon for “unsolved”) problems in mathematics. 1
    Hilbert’s second problem was: Can it be proved that the axioms of arithmetic are consistent? That is, that theorems in arithmetic, derived from these axioms, can never lead to contradictory results?
    To appreciate what Hilbert was asking, we must understand that in the
fin de siècle
world of mathematics, the “axiomatic approach” held sway over mathematical thinking. This is the idea that any branch of mathematics must begin with a small set of assumptions, propositions, or
axioms
that are accepted as true without proof. Armed with these axioms and using certain
rules of deduction
, all the propositions concerning that branch of mathematics can be derived as theorems. The sequence of logically derived steps leading from axioms to theorems is, of course, a
proof
of that theorem. The axioms form the foundation of that mathematical system.
    The axiomatic development of plane geometry, going back to Euclid of Alexandria (fl. 300 BCE ) is the oldest and most impressive instance of the axiomatic method, and it became a model of not only how mathematics should be done, but also of science itself. 2
    Hilbert himself, in 1898 to 1899, wrote a small volume titled
Grundlagen der Geometrie (Foundations of Geometry
) that would exert a major influence on 20th-century mathematics. Euclid’s great work on plane geometry,
Elements
, was axiomatic no doubt, but was not axiomatic enough. There were hidden assumptions, logical problems, meaningless definitions, and so on. Hilbert’s treatment of geometry began with three undefinedobjects—point, line, and plane—and six undefined relations, such as being parallel and being between. In place of Euclid’s five axioms, Hilbert postulated a set of 21 axioms. 3
    In fact, by Hilbert’s time, mathematicians were applying the axiomatic approach to entire branches of mathematics. For example, the axiomatization of the arithmetic of cardinal (whole) numbers formulated by the Italian Giuseppe Peano (1858–1932), professor of mathematics in the University of Turin, begins with three terms—“number”, “zero”, and “immediate successor”—and are assumed to be understood. The axioms themselves are just five in number:
    1. Zero is a number.
    2. The immediate successor to a number is a number.
    3. Zero is not the immediate successor of a number.
    4. No two numbers have the same immediate successor.
    5. The principle of mathematical induction: Any property belonging to zero, and also to the immediate successor of every number that has the property, belongs to all numbers.
    Exactly a decade after Hilbert’s Paris lecture, British logician and philosopher Bertrand Russell (1872–1970), in collaboration with his Cambridge teacher Alfred North Whitehead (1861–1947), published the first of the three-volume
Principia Mathematica
(1910–1913)—not to be confused with Newton’s
Principia
—which attempted to develop the notions of arithmetic from a precise set of logical axioms, and which was intended to demonstrate that mathematical knowledge can be reduced to (or, equivalently, derived from) a small set of logical principles. However, Russell and Whitehead did not address Hilbert’s second problem.
    Hilbert returned to the foundations of mathematics repeatedly throughout the course of the first three decades of the 20th century, establishing what came to be known as “Hilbert’s program”. 4 In 1928, in an address delivered at the International Congress of Mathematicians in Bologna, Italy (the