Gödel's Proof by
Ernest Nagel
My rating:
5 of 5 stars
I have discovered recently, the last month or so, that when I read, I only read and recall what interests me at the time of me reading. Hence my reading of this book is skewed by my current fascination with imagination and conceptualization. I am not willing not to invest the time it would take me to assess the mathematical value of this apparently exceptional summary of Godel's Proof.
This book greatly extended my mathematical horizons. The questions that it evoked and the effort I took to dig into these questions were very useful. I discovered that mathematics was developed as an imaginative effort through history until the expression of Euclid's geometry in terms of algebra was done by Descartes' analytic geometry. The frequent references to "The Principles of Mathematics" by Russell, Whitehead, inspired me to find out the contents of this famous work. I was surprised and pleased to see that my book on Advance Calculus by Hans Sagan follows an outline that was probably inspired by the Principles.
Useful quotes from this book:
Page 10. "It was not until the 19th century, chiefly though the work of Gauss, Bolyai, Lobachevsky, and Riemann, that the impossibility of deducing the parallel axiom from the other was demonstrated. ... The traditional belief that the axioms of geometry (or for that matter, the axioms of any discipline) can be established by their apparent self-evidence was the radically undermined. ... the proper business of the pure mathematician is to derive theorems from postulated assumptions, and that it is not his concern as a mathematician to decide whether the axioms he assumes are actually true. ... Axiomatic foundations were eventually supplied for fields of inquiry that had hitherto been cultivated only in a more or less intuitive manner. (In 1899 the arithmetic of cardinal numbers was axiomatized by Giuseppe Peano.)” {Note: cardinal numbers denote quantities.}
Page 11. Hence mathematics is not just the "science of quantity." "... mathematics is simply the discipline par excellence that draws conclusions logically implied by any given set of axioms or postulates.”
Page 12, David Hilbert published his axioms of geometry in 1899. Hilbert states that the connotations of primitive terms ignored, their meaning depends only on the axioms. (Primitive terms are “point”, “line”, “lies on”, and “between.”)
My comment: In this sense mathematical formalization, logic, frees our minds from the limits of our imagination. Examples are ’n’ dimensional spaces, negative numbers, complex numbers.
My understanding is that intuition is a kind of imagination. Imagination gave us the earliest geometry, and imagination continues to be a powerful tool to leap across the unimaginable valleys of logic. Arithmetic moved from domain of imagination to logic in 1899, thanks to Peano. Ditto for geometry by Hilbert.
Page 14, “Moreover, as we all know, intuition is not a safe guide: it cannot be used as a criterion of either truth or fruitfulness in scientific explorations.”
My comment, ditto for philosophical opinions, e.g. I. Kant’s
“But in order to ascertain to what given intuitions objects, external me, really correspond, in other words, what intuitions belong to the external sense and not to imagination, I must have recourse, in every particular case, to those rules according to which experience in general (even internal experience) is distinguished from imagination, and which are always based on the proposition that there really is an external experience.”
Kant, Immanuel. Complete Woks of Immanuel Kant (p. 29). Minerva Classics. Kindle Edition.
Page 23, Another example of the failure of “clear” and “distinct” intuitive notions is the notion of “class” or “aggregate.” Bertrand Russell’s antinomy, paradox, “Classes seem to be of two kinds: those which do not contain themselves as members, and those which do. A class will be called ‘normal’ if, and only if, it does not contain itself as a member; otherwise it will be called ‘non-normal.” Then what is the class ’N’ that stands for the class of all normal classes? If ’N’ is normal it is a member of itself because by definition ’N’ is all normal classes, but if it includes itself, but then it would not be normal.
My comment: Epistemologically this looks like shifting the level of abstraction without noticing the shift. A class of all normal classes is a higher level abstraction that a “class.” On page 28, Hilbert’s solution to this issue is “meta-mathematical” statements, a language that is about mathematics. Hence meta-mathematics is a different level of abstraction.
Page 34, an analogy to meta-mathematics is made with the game of chess. The pieces have no meaning but they must be moved according to the rules of the game. The chess pieces correspond to the elementary signs of a formalized mathematical calculus, the legal positions of the pieces, to the formulas derived from the axioms (i.e. theorems); and the rules of the game to the rules of inference for the calculus. A theory of meta-chess can tell us about the number of possible moves that White can make at any point in the game. A theory of meta-mathematics may be able to tell us if contradictory formulas may be derived from a given mathematical calculus.
Page 83, distinguishes signifiers from concepts using the difference between numerals and numbers. “A numeral is a sign, a linguistic expressions, something which one can write down, erase, copy, and so on. A number, on the other hand, is something which a numeral names or designates and which cannot literally be written down erased, copied, and so on. … A number is a property of classes (and is sometimes said to be a concept), not something we can put on paper.
My comment: I need to read Adler’s book Some Questions About Language: A Theory of Human Discourse and Its Objects (1976)
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