Notes from Godel's Proof by Nagel and Newman

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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Goals at 67

 My perspective has certainly changed in the last 4 years of my retirement.  I tend to get “addicted” to my goals, and neglect other parts of my life that are not part of a goal. I thought I would live “goal free” as Harry mentions below, but I found my body began to rebel.  I was getting physically out of shape, my belly was getting bigger.  I have found that a certain minimum structure for my days is valuable for keeping fit and healthy, and to keep moving toward my current interests.  As long as I stick to my daily routine, determined by my weekly/monthly plans, everything works well and I feel happy with my life.

My current interests, (but not goals) are:

• to have a better understanding why people behave the way they do, so many people are their own worst enemies and they don’t seem to know it.
• To learn the mathematics that I need to know to understand modern physics books, i.e. number theory, analytic geometry and n-dimensional vector calculus.  These are topics that I covered in college, but I have found it very rewarding to revisit them.  
• To understand the role of concepts in my thinking by watching how my mind works when I work math problems out imaginatively and then formalize them via concepts.  I am amazed to find out how large the role of imagination is in thinking and understanding.
• To read good stories and essays, learn to appreciate good drama on TV or in books and improve my senses of smell and taste.  Compared to Amanda I am almost blind when it comes to smell, taste, appreciating drama, or listening to music.  I am beginning to desire to enhance my experience in these artistic arenas. 

My biggest distraction is the local library system.  I am limiting the number of books I check out at a time, so that I can get more of my own studies and home projects done.  Home projects are to clean out old files, photographs, floppy disks, and emails, (like this one).

My worst bad habit is not asserting myself in conversation with others.  I have an underlying fear I won’t be liked.

When I look back on my working life, I realize other things could have been accomplished if I had dedicated myself to those goals, but when I read my old files, and emails from those time periods of my life, I see that I made the best possible choices for who I was at the time.  So no regrets.  

Goals are given to us by the possibilities that we commit to.  We all have possibility via imagination, what is missing willingness commit. If I am not working toward my goals, I ask myself “what is stopping me?”  I write down the answers for clarity and post them on my bathroom mirror.

Mathematics and Imagination Lessons

I had the opportunity this morning to review some of the notes I made about Immanuel Kant’s Critique of Pure Reason.  I realize on second reading of my highlights in the Kindle book, that I failed to understand what Kant was trying to say, my bad.  Of particular interest to me today was in the preface where Kant says

"A philosophical system cannot come forward armed at all points like a mathematical treatise, and hence it may be quite possible to take objection to particular passages, …”  He goes on to say that he is 64 years old and does not have the time to make his writing free of all mistakes and inconsistencies, he depends upon future generations to smooth the “rough edges” of his exposition.

              Kant, Immanuel. Complete Woks of Immanuel Kant (p. 30). Minerva Classics. Kindle Edition. 

I have been collecting evidence for imagination from other writers, to learn if what they have to say is the same as my own conclusions.  This has led me down some interesting side paths.  Most notably the lecture by Richard Feynman "The Relation of Mathematics to Physics" and a small book “Godel’s Proof” by Ernst Nagel and James R. Newman.  As a former math major this book really expanded my horizons and showed me connections that I had never contemplated.  

I haven’t found anyone that contradicts what I am thinking.  What surprises me is that imagination is such a common theme among mathematicians, scientists and engineers. I see now that all my reading of the past was always in search of an answer to a question, and if what I read didn’t seem to apply to my question, I would ignore it, or at least give it no attention.

I suspect that if I attempt to give you quick answers, I may find myself contradicting what I have to say at later date.

First, I define imagination as mental images, models of what is not before my senses.  Imagination includes memory, planning, what I have in mind to create (write, draw, say).  Intuition is a form of imagination.  I see a car, and my imagination using memories and logic tells me it is a 3 dimensional object with an internal combustion engine and wheels. My imagination may be wrong, it may be running on a battery, it may be clever picture of a car painted on the wall of a barn.  Similarly, we don’t “know” the person we are talking to, we only know our imagination of them.  

In Math, the number two is an abstraction from pairs of items that exist either in the world of in our imagination. The number two is a concept, perhaps it represents a process of pairing.  But you will never find the number 2 in the objective world, certainly not the number “-2”, nor the square root of "-2".

The awakening moment for me was realizing that the number line is pure imagination!  This insight came when I was working on a series of proofs about ordered fields.  Since I am interested in learning the difference between imagination and conceptual thinking I would write each proof imaginatively, drawing diagrams as needed to prove the point.  Then I would work to write a formal proof based on prior axioms, definitions, and previously proved theorems.

Example: If “a” is a member of an order field, numbers are an ordered field, and “a” > 0 then prove 1/a >0.

Imaginatively this is drop dead obvious, 

I imagine a number line: 

<——————————|————|—————————> positive direction

negative                        0             a

I imagine “a” some number >0 on the right hand side of the 0 in the drawing above.

I imagine 1/a as a point on the line that clearly is >0.  Taking the reciprocal of a number cannot be negative.

Notice that for a complete imaginative proof, I would need two diagrams, one for a < 1, another for a >= 1.

Formally, i.e. logically, it takes more thinking.  (The fringe benefit of formal methods is that they come closer to what may be programmed into a computer.)

1. Suppose that 1/a < 0 and “a” > 0, contrary to initial assumptions.  // I am thinking of proof by contradiction
2. Then a * 1/a = 1 > 0 by axiom of multiplicative inverse.
3. But a positive number times a negative number is < 0 (i.e. negative), by previously proved rule that -a=a(-1). // I did this in an earlier exercise, so I can build on the result.
4. Hence supposing that 1/a < 0 leads to a contradiction, that a negative times a positive is > 0, 
5. Therefore 1/a > 0.   QED

Comments, the power of abstraction (conceptualization) is that “a” stands for any number.  The number line is pure imagination, formally this number line represents the mathematical definition of an ordered field.  The image gives our minds something to “hang onto” in order to use the concept of “ordered field.”

Notice that I am using my imagination to manipulate abstract symbols, number line, zero, positive and negatived.  This is a step away from using my imagination to manipulate a pencil and paper to draw a picture.

With practice I have learned that the imaginative proof is frequently in incomplete, as seen above, and sometimes it is just plain wrong.  What seems clear and intuitive is not always correct.  A good example is the “Russell Paradox” of set theory.  

The incentive for creating things, (diagrams, paintings, essays, science, machines, tools, music, etc.), is that it brings my imaginations to life and embodiment.  At the same time creations make my imagination more detailed, complete, and accurate to the domain that I am imagining in.