It Began with Babbage Read Online Free Page A

home, incidentally, of the world’s first university), and in a monograph titled “Die Grundlagen der Mathematik” (“The Foundations of Mathematics), he asked: (a) Is mathematics
complete
, in the sense that every mathematical statement could be either proved or disproved? (b) Is mathematics
consistent
, in the sense that a statement such as 2 + 2 = 5 could never be arrived at by a valid proof, or in the sense that two contradictory propositions
a
=
b
and
a
≠
b
could both be derived? (c) Is mathematics
decidable
, in the sense that there exists a definite method that can be followed to demonstrate that a mathematical statement is true or not?
    Hilbert, of course, believed that the answer to all three questions was yes. For his program to work, certain matters needed to be clarified. In particular, certain key concepts had to understood. These included, in particular, the concepts of absolute proof, formal system, and meta-mathematics.
    Intimidating words, especially the last.
    By
absolute proof
is meant that the consistency of a mathematical system must be established without assuming the consistency of some other system. 5 In other words, a mathematical system must be self-contained, a solipsistic world of its own.
    A system that allows for absolute proofs, not admitting anything outside of itself, is what mathematicians and logicians call a
formal system
, and no term, expression, or proposition in the system has any meaning. And because terms or expressions in a formal system have no meaning, they are not even symbols, for a symbol represents something else, it is
about
something else, whereas the terms or propositions in a formal system are just squiggles. For example, the entities 2, +, 4, and = carry no meaning in a formal system of arithmetic; they are squiggles. Likewise, the expression 2 + 2 = 4 is a string of squiggles that is also meaningless, but is derived by putting together the “primitive” squiggles according to certain rules (often called a
calculus
).
    Which leads us to meta-mathematics. It is one thing to write
    2 + 2 = 4
    This is a meaningless squiggle composed out of primitive squiggles. It is another thing to write
    â€œ2 + 2 = 4” is a valid proposition in arithmetic.
    The former is an expression
within
arithmetic, whereas the latter is a statement
about
that expression and, thus, a statement
about
arithmetic. The one is a mathematical proposition; the other, a meta-mathematical statement.
    This distinction was important for Hilbert’s program and, as we will see, it plays a significant role in the development of the science of computing. In Hilbert’s context, it means that statements such as
    â€œmathematics” is consistent
    â€œmathematics” is complete
    mathematical propositions are decidable
    are meta-mathematical. They are meaningful assertions about mathematics.
    To return to Hilbert’s three questions. The answers were provided in 1931 by an Austrian mathematician Kurt Gödel (1906–1978), then at the University of Vienna, who later emigrated to America and became a member of the extraordinary constellation of scientific thinkers that included Albert Einstein (1879–1955) at the Institute of Advanced Study, Princeton, New Jersey, 6 (This institution plays a role of its own in
our
story, as we will see.)
    Gödel’s response to Hilbert, published in a German journal, bears the title, when translated into English, “On Formally Undecidable Propositions of Principia Mathematicaand Related Systems,” and it would have devastating implications for how mathematicians would see the nature of their craft. Gödel showed that axiomatic systems have certain inherent limitations, that the complete axiomatization of even the arithmetic of whole numbers is not possible. He demonstrated that it is impossible to establish the internal consistency of a large range of mathematical systems, including arithmetic—that is, there