Fifteen Eighty Four

Academic perspectives from Cambridge University Press


Happy 100th, Principia Mathematica Part II

100 years ago, Cambridge published a book that transformed the study of mathematics and laid the foundations for the computer age.

The Principia Mathematica is the most famous work ever published on the foundations of mathematics. Written by British mathematicians and philosophers Alfred North Whitehead and Bertrand Russell, it was published by Cambridge in three volumes in 1910, 1912, and 1913. A second edition was published in 1927.

Principia attempted to ground mathematics in logic and the authors left no stone unturned in their attempt to create the ultimate definition of mathematics. For example, they were well into volume two before they had proved that one plus one equals two! They concluded their proofs with the laconic statement: “The above proposition is occasionally useful.”

The Principia covered real, cardinal and ordinal numbers and set theory. A fourth volume on the foundations of geometry had been planned, but the authors declared themselves intellectually exhausted on completing the third.

Julie Rehmeyer writes about mathematics for Wired Magazine and Science News. She commented thatPrincipia has been hugely influential, but in an unexpected way:

“The book kind of laid the seeds for its own undoing. About 20 years later, a German mathematician named Kurt Godel used what Russell and Whitehead had done in the Principia to show that it actually couldn’t do what it aimed to do, that it couldn’t contain all of math, that there would be true mathematical statements that were not logical consequences of the axioms that it set out.

So the interesting thing about it is, on the one hand, it kind of destroyed the whole project, and on the other hand, Godel couldn’t have come to that conclusion without the work of the Principia.

The book represented not only a tour de force in writing, but also in typesetting, as whole new alphabets had to be created. Arguably, the greatest achievement of Principiawas to lay the foundations for modern computing by essentially turning mathematics into code – the type of coding ultimately employed to build computers.

Principia Mathematica is still available from the Press and in August this year, Cambridge will publish The Evolution of Principia Mathematica – Bertrand Russell’s Manuscripts and Notes for the Second Edition by Bernard Linsky, drawing on archive material held by the McMaster University of Canada.

Russell and Whitehead’s work was not, of course, Cambridge’s first revolutionary Principia Mathematica. In 1713 the Press published the second edition of Newton’sPrincipia (full title Philosophiae Naturalis Principia Mathematica), which became one of the most influential and famous books in the history of science.

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