Ernest Nagel (1901–1985)
Author of Gödel’s Proof
About the Author
Born in Czechoslovakia, Ernest Nagel emigrated to the United States and became a naturalized American citizen. In 1923 he graduated from the City College of New York, where he had studied under Morris Cohen, with whom he later collaborated to coauthor the highly successful textbook, An Introduction show more to Logic and Scientific Method (1934). Pursuing graduate studies at Columbia University, he received his Ph.D. in 1930. After a year of teaching at the City College of New York, he joined the faculty of Columbia University, where in 1955 he was named John Dewey Professor of Philosophy. In 1966 he joined the faculty of Rockefeller University. Nagel was one of the leaders in the movement of logical empiricism, conjoining Viennese positivism with indigenous American naturalism and pragmatism. In 1936 he published in the Journal of Philosophy the article "Impressions and Appraisals of Analytic Philosophy," one of the earliest sympathetic accounts of the works of Ludwig Wittgenstein, Moritz Schlick, and Rudolf Carnap intended for an American audience. Nagel was esteemed for his lucid exposition of the most recondite matters in logic, mathematics, and natural science, published in essays and book reviews for professional journals, scientific periodicals, and literary reviews. Two of his books, now out of print, consisted of collections of his articles, Sovereign Reason and Other Studies in the Philosophy of Science (1954) and Logic Without Metaphysics and Other Essays in the Philosophy of Science (1957). He also wrote a monograph, Principles of the Theory of Probability (1939) which appeared in the International Encyclopedia of Unified Science. In his major book-length work, The Structure of Science, Nagel directed his attention to the logic of scientific explanations. (Bowker Author Biography) show less
Series
Works by Ernest Nagel
The structure of science; problems in the logic of scientific explanation (1961) 238 copies, 2 reviews
Logic, Methodology and Philosophy of Science Proceedings of the 1960 International Conference (1966) — Editor — 9 copies
Simbolismo y ciencia 2 copies
Liberalism and Intelligence 1 copy
Associated Works
Tagged
Common Knowledge
- Canonical name
- Nagel, Ernest
- Legal name
- Nagel, Ernest
- Birthdate
- 1901-11-16
- Date of death
- 1985-09-22
- Gender
- male
- Nationality
- USA (naturalized 1919)
Austro-Hungarian Empire (birth) - Birthplace
- Vágújhely, Austro-Hungarian Empire (now Nové Mesto nad Váhom, Slovakia)
- Place of death
- New York, New York, USA
- Places of residence
- Prague, Czech Republic
New York, New York, USA - Education
- City College of New York (BSc | 1923)
Columbia University (MA | 1925 | PhD | 1930) - Occupations
- philosopher of science
professor - Relationships
- Nagel, Alexander (son)
Nagel, Sidney (son) - Organizations
- Columbia University
Rockefeller University - Awards and honors
- National Academy of Sciences (1977)
Fellow, Committee for Skeptical Inquiry (1976) - Short biography
- Ernest Nagel ging in 1911 naar de Verenigde Staten en werd in 1919 genaturaliseerd tot Amerikaans staatsburger. Hij heeft zijn hele leven in New York gewoond, waar hij als wetenschapsfilosoof werkzaam was aan de Columbia University (1931-1966), de Rockefeller University (1966-1967) en wederom de Columbia University (1967-1970). Hij was een van de vooraanstaande figuren in de filosofische stroming van het logisch positivisme. Hij werd in 1977 gekozen in de National Academy of Sciences.
Nagel is op 20 januari 1935 getrouwd met Edith Alexandria Haggstrom (ovl. 1988). Zij kregen twee zoons: Alexander Joseph (hoogleraar wiskunde aan de Universiteit van Wisconsin-Madison) en Sidney Robert (hoogleraar natuurkunde aan de Universiteit van Chicago).
Members
Reviews
Sovereign Reason, and Other Studies in the… by ernest Nagel
12/13/21
A fun and thought provoking read indeed, would recommend it to anyone who
* loves paradoxical statements
* would like to know more about mathematical logic
* loves paradoxical statements
* would like to know more about mathematical logic
What Gödel's Theorem really says is this: In a sufficiently rich FORMAL SYSTEM, which is strong enough to express/define arithmetic in it, there will always be correctly built sentences which will not be provable from the axioms. That, of course, means their contradictions will not be provable, either. So, in a word, the sentences, even though correctly built, will be INDEPENDENT OF the set of axioms. They are neither false nor true in the system. They are INDEPENDENT (cannot stress this enough). We want axioms to be independent of each other, for instance. That's because if an axiom is dependent on the other axioms, it can then be safely removed from the set and it'll be deduced as a theorem. The theory is THE SAME without it. Now, the continuum hypothesis, for instance, is INDEPENDENT of the Zermelo-Fraenkel axioms of the set theory (this was proved by Cohen). Therefore, it's OK to have two different set theories and they will be on an equal footing: the one with the hypothesis attached and the one with its contradiction. There'll be no contradictions in either of the theories precisely because the hypothesis is INDEPENDENT of the other axioms. Another example of such an unprovable Gödelian sentence is the 5. axiom of geometry about the parallel lines. Because of its INDEPENDENCE of the other axioms, we have 3 types of geometry: hyperbolic, parabolic and Euclidean. And this is the real core of The Gödel Incompleteness Theorem. By the way... What's even more puzzling and interesting is the fact that the physical world is not Euclidean on a large scale, as Einstein demonstrated in his Theory of Relativity. At least partially thanks to the works of Gödel we know that there are other geometries/worlds/mathematics possible and they would be consistent.
Without a clear and explicit reference to the concept of a formal system all that is said regarding Gödel's theorems is highly inaccurate, if not altogether wrong. For instance, if we say that Gödel's statement is true, after saying that Gödel's Theorem states that it can't be proved either true or false. Without adding "formally", that doesn't really make much sense. We'd only be only talking about axioms, which are only a part of a formal system, and totally neglecting talking about rules of inference, which are what the theorems really deal with.
By independent I mean 'logically independent', that is only a consequence of Gödel's theorem in first order languages, whose logic is complete. In second order arithmetic, the Peano axioms entail all arithmetical truths (they characterize up to isomorphism the naturals), so that no arithmetical sentence is logically independent of such axioms. It occurs, however, that second order logic is incomplete and there is no way to add to the axioms a set of inference rules able to recursively derive from the axioms all of their logical consequences. This is why Gödel's theorems holds in higher order languages too. In fact, this is how the incompleteness of higher order logic follows from Gödel's theorems.
What prompt me to re-read this so-called seminal book? I needed something to revive my memory because of Goldstein's book on Gödel lefting me wanting for more...I bet you were expecting Hofstadter’s book, right? Nah...Both Nagel’s & Newman’s along with Hofstadter’s are failed attempts at “modernising” what can’t be modernised from a mathematical point of view.
Read at your own peril.… (more)
Without a clear and explicit reference to the concept of a formal system all that is said regarding Gödel's theorems is highly inaccurate, if not altogether wrong. For instance, if we say that Gödel's statement is true, after saying that Gödel's Theorem states that it can't be proved either true or false. Without adding "formally", that doesn't really make much sense. We'd only be only talking about axioms, which are only a part of a formal system, and totally neglecting talking about rules of inference, which are what the theorems really deal with.
By independent I mean 'logically independent', that is only a consequence of Gödel's theorem in first order languages, whose logic is complete. In second order arithmetic, the Peano axioms entail all arithmetical truths (they characterize up to isomorphism the naturals), so that no arithmetical sentence is logically independent of such axioms. It occurs, however, that second order logic is incomplete and there is no way to add to the axioms a set of inference rules able to recursively derive from the axioms all of their logical consequences. This is why Gödel's theorems holds in higher order languages too. In fact, this is how the incompleteness of higher order logic follows from Gödel's theorems.
What prompt me to re-read this so-called seminal book? I needed something to revive my memory because of Goldstein's book on Gödel lefting me wanting for more...I bet you were expecting Hofstadter’s book, right? Nah...Both Nagel’s & Newman’s along with Hofstadter’s are failed attempts at “modernising” what can’t be modernised from a mathematical point of view.
Read at your own peril.… (more)
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