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A Nature Research Journal. In today's computer age, the implications of the discovery in formal logic that Newman and Nagel articulated in are of even broader interest, says Andrew Hodges. Fifty years ago an unusual book appeared. Those tempted to look inside discovered a classic of scientific exposition and faced quite a challenge.
The writers, Ernest Nagel and James Newman, already distinguished figures in scientific philosophy and education, gave an uncompromising presentation of their unfamiliar subject matter: mathematical logic. For Nagel and Newman to see the potential interest to a wider public was both visionary and optimistic.
I cannot have been the only one to find it a unique text on the college library shelf, leading to unexpected regions beyond the standard syllabus. Although rooted in an earlier article in Scientific American , it used copious equations; indeed it explored the very meaning of equations, a demand on the reader that would make most publishers nervous. Nagel and Newman explained difficult ideas of logical deduction from formal axioms, distinguishing formal proof from informal reasoning.
And they revealed how his technical innovation exploited this observation, using numbers to code statements about numbers.
A vast industry has arisen founded on logical algorithms, and nowadays it is better appreciated that the business of computing is inseparable from the logical calculus built up in the early twentieth century.
In the s there was a tendency for mathematicians to distance themselves from practical applications, and from computing in particular. Since the s these divisions have become less rigid. It therefore now seems a little odd that Nagel and Newman paid no attention to computing. They framed their closing reflections as if Turing's theory of computability was an obvious corollary.
This is now a huge and hotly contested area of scientific philosophy. In fact, it was already the subject of dispute in Expansive and illustrative, it also came to quite a different conclusion about AI — essentially Turing's. In a few pregnant words, Nagel and Newman referred to the brain as a machine apparently more powerful, through its capacity for informal reasoning, than computers.
Penrose, since the s, has asked what could possibly lend it such power, finding an answer in the ill-understood phenomenon of quantum-mechanical state reduction. His conclusions are keenly disputed: for instance the leading logician Martin Davis, himself a popularizer, has forcefully pressed Turing's original view in his book Engines of Logic W.
Norton, In that same period, Planck and Einstein opened the quantum-mechanical door on reality. A hundred years have not sufficed to resolve the fundamental questions they revealed. Mathematical and physical science describe a continuous quantum universe using formal operations on discrete symbols. Neither the quantum, nor those symbols, nor the connection between them, are yet fully understood.
Reprints and Permissions. Hodges, A. Nature , Download citation. Published : 13 August Issue Date : 14 August By submitting a comment you agree to abide by our Terms and Community Guidelines.
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Newman New York University Press: He is author of Alan Turing: the Enigma. You can also search for this author in PubMed Google Scholar. Rights and permissions Reprints and Permissions. About this article Cite this article Hodges, A.
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