Ernest NagelRecensioni

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

This book will melt your mind.

Uno de los más grandes resultados de la lógica y matemática del siglo pasado, explicado con elegancia. Me costó trabajo "seguirle el paso" pero porque yo no estoy tan versado en lógica. Aún así, el teorema se explica en su contexto, con un esbozo de ejercicios e implicaciones del trabajo de Gödel. Recomiendo releer cada capítulo con detenimiento

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.

A good followup to GEB, I am happy with the order I chose. I didn't realize how engaging GEB really was, with its intermittant stories, when I saw a drier version. But, drier is not really meant as an insult, I thought this book went more in depth and tried to formally explain a lot more. To me, after understanding GEB, I got a sense of amazement on the incompleteness proof and a feeling for the philosophical outcroppings. With this book, I feel like I was more ready to actually read the seminal paper and a understanding of the paper itself½

Kursbok i filosofin kommer jag ihåg. Riktigt trevlig. Men jag har nog vuxit ifrån den.

For a book that was supposed to simplify Godel's Proof it was exceptionally complex. No real thesis either; basically, the first 75% of the book is just setting up preliminaries and doesn't even deal directly with Godel's work. Reading this book gave me no further insights on Godel's challenging concepts. I recommend instead Godel, Escher, Bach, which is longer and only devotes a chapter's worth of study on the Proof, but does so in far simpler terms (the author of G.E.B. does the intro to this book.)

Left me wondering about more foundational items that were mentioned in passing such as 'primitive recursive truths' and the 'Correspondence Lemma'. The exposition seemed rushed at the end.

I remember my excitement when I read the first edition of this little gem back in 1999 (actually it was its Turkish translation). Being a young student of mathematics, it was impossible to resist reading a popular and clear account of maybe the most important theorem related to the fundamentals of axiomatic systems. After that came Hofstadter's "Gödel, Escher, Bach: An Eternal Golden Braid" which introduced more questions related to symbolic logical reasoning, artificial intelligence, cognitive science, and the consequences of Gödel's work in those ares. With that background and ten years after the second edition, it was truly an exciting second reading, a refresher that was both fun and putting lots of things into perspective. Hofstadter's foreword to this edition is a delight to read and ponder upon. On the other hand, I don't think this is a point strong enough to persuade most of the people who own the first edition anyway. But if you don't have the first edition and want a concise and clear explanation of what Gödel's work is all about then this book is definitely for you.½

Another wonderful book on Scientic Method

The authors provide an overview of Kurt Gödel's 1931 proof regarding axiomatic demonstration in arithmetic. Gödel constructed his proof on the basis of Principia Mathematica by Whitehead and Russell, but this treatment does not presume a familiarity with that text. It does, however, place Gödel's work in the larger context of efforts to axiomatize arithmetic, an agenda notably defined by David Hilbert. The first five chapters set the stage for Gödel's proof in the history of mathematics and philosophy, while the fifth chapter discusses a critical idea underlying the operation of the proof. The long sixth chapter discusses the actual techniques and conclusions of the proof itself. A valuable final chapter outlines the larger implications and consequences, most especially and usefully discouraging misreadings which amount to "an invitation to despair or an excuse for mystery-mongering." (101) Interestingly, a secondary conclusion that they do support, is that algorithmic computers are unlikely ever to attain the equivalent of human consciousness. Gödel himself took his proof as support for a position of philosophical Realism, although it's not conclusive in that regard.

For anyone interested in the beauty of logic or the elegance of math, the mechanisms of Gödel's proof are impressive. This book by Nagel and Newman reads quickly--for a math book. The reader must be prepared to slow down and spend five to ten minutes on a page when getting into the thick of the mathematical concepts in use. The reward of doing so is an appreciation of an intellectual event that provided a turning-point in the philosophy of knowledge.

A very readable, short introduction to Gödel's famous arguments, suits also the non-mathematician. Easy to follow and even inspiring at times. I read this because Douglas Hofstadter had recommended it in his preface to GEB.

Nagel and Newman provide a nice, quick, and generally well written exposition of Godel's famous proof. This book can easily be read in an afternoon by anyone with the requisite background in logic.

They do a particularly nice job in their brief dissemination of the historical concerns that led up to the crisis in foundations in the late 19th and early 20th century. What's nice about this is that it puts Godel into context in a salient way. Godel without Hilbert is like Kant without Leibniz (and Wolff, I suppose). Given the narrow scope and short page count, Hilbert is covered well.

However, there are a couple of real problems with this book.

First, I do not believe that this book would really be that helpful for "the educated layman". Insofar as their target audience is concerned, the book is, perhaps, a failure. Why do I say this? Given its brevity, the authors are forced to introduce important bits of information without adequate exposition. For example, the notion of universal quantification makes its first appearance in the last twenty odd pages of the book and is explained in a sentence or two. This is fine for anyone that's had an intro logic course (and can recall what was covered) but is probably inadequate for the logical/mathematical novice. Furthermore, this example is just one case of something that occurs quite often throughout the book.

My second worry is that the actual mechanics of the proof are not presented lucidly. This is not altogether unexpected, but the fifteen pages or so that comprise the actual exposition of the proof seem to go by too quickly and sacrifice depth and clarity for readability and brevity. This may not be the authors' fault. I have doubts about whether or not one can successfully offer the sort of exegesis the authors are striving for. That is, I'm just not sure that anyone will ever pull off a lucid "Godel for Dummies". Similar problems plague Rebbecca Goldstein's attempt at this task in her recent Godel biography.

Final thought: I think this book would best serve the needs of a first year graduate student or advanced undergraduate in philosophy. For the student that has some background in logic (perhaps they've done a completeness proof for FOL or at least some proofs with quantifiers) but has yet to take a meta-logic course this book can provide a nicely structured overview of what the typical meta-logic course aims for.½

This little book offers real insight into one of the weirdest aspects of modern mathematics.

Ernest Nagel's work, The Structure of Science, has earned for itself the status of an outstanding standard work in its field.
It offers an exceptionally thorough and comprehensive methodological and philosophical exploration of the natural and the social sciences and of historigraphy, with special emphasis on the various modes of explanation encountered in those diverse fields. Nagel's discussion is distinguished by the lucidity of its style, the incisiveness of its reasoning, and the solidity of its grounding in all the major branches of scientific inquiry.
The Structure of Science has become a highly influential work that is widely invoked in the methodological and philosophical literature. Recent controversies between analytic and historic-sociological approaches to the philosophy of science have not diminished its significance; in fact, it seems to me that the pragmatist component in Nagel's thinking may be helpful for efforts to develop a rapprochement between the contending schools.

This is a non-formal, though still rigorous, presentation of the argument of Gödel's famous demonstration that will be accessible to anyone familiar with the basics of mathematical proof, logic, and number theory. By the end of the book, I acutally had the outline of Gödel's tricky self-referential argument all in my head at once, and though it faded quickly, I feel confident I could resurrect it with another reading. Nagel's description of the significance of the proof, as opposed to its mechanics, is less thorough, but that's a quibble. This slim book is a truly impressive feat of exposition.

NA

Course book for Joe Epstein's course in philosophy of science