Book 2 for 2025! My goal is 30 this year. This is actually a book I read years ago but it was time for a reread to try to better understand the subject.

I honestly wish more people would explore mathematics and not avoid it as much as possible, because it is a stunningly beautiful subject. Even if one does not have the time or inclination to master it, there are many mind-boggling mathematical subjects that can be at least roughly understood by a dedicated layperson with the right information in front of them. The mathematics of infinity, for example, is something I’ve blogged about in the past and it is just fascinating and some of the most important proofs related to it require no higher math at all to gain an understanding of.

One topic I’ve always been fascinated with is the incompleteness theorems of Kurt Gödel, first published in 1931. The actual paper is quite involved to read and understand, and requires a lot of background context to properly understand what Gödel accomplished. Fortunately, there is a book out there that makes a valiant effort to explain it in non-technical terms: Gödel’s Proof, written by Ernest Nagel and James R. Newman and first published in 1958. I have a copy of the NY University Press edition of 1986:

It is a slim book: 102 pages in my edition, not counting appendices and the index. It leads the reader through a brief history of 19th and early 20th century mathematics to appreciate what Gödel proved in his paper, which shook the foundations of mathematics.

I will only give a synopsis of the ideas, because even after my second read through, my understanding of the mathematics is still shaky! I’m working my way up to reading Gödel’s paper itself, which I also have a copy of.

Even most non-mathematicians are familiar with the axiomatic method of geometry, which originated in ancient Greece with Euclid. The idea of an axiomatic theory is that a small number of postulates describe the mathematical system, and then those axioms can prove countless new properties of the system. For example, Euclid’s first axiom may be written as “a straight line may be drawn between any two points,” and his second as “Any terminated straight line may be extended indefinitely.”

The axiomatic method is a powerful approach for doing mathematics, and in the 19th century mathematicians became interested in axiomatizing all of mathematics. Axioms were found, for example, for arithmetic, from which seemingly all the properties of natural numbers could be derived.

But a very important question came up: how does one determine if any particular axiomatic system is consistent, i.e. that it cannot simultaneously prove contradictory statements like “A is true” and “A is not true?” The axiomatic systems were constructed by human ingenuity and intuition, but that in and of itself was not sufficient proof, because of course human intuition can be misleading or even wrong.

For example, Euclid’s fifth axiom was: “If two straight lines in a plane are met by another line, and if the sum of the internal angles on one side is less than two right angles, then the straight lines will meet if extended sufficiently on the side on which the sum of the angles is less than two right angles.” This axiom, with its increased complexity, always troubled the ancient Greeks, who attempted in vain to demonstrate that it could be derived from the other four. It could not, however, and in the late 19th century, mathematicians determined why: it can be replaced by different axioms to create non-Euclidean geometries. Euclid’s fifth axiom describes geometry on a flat surface, while geometries can also be constructed for geometry on an ellipse or a hyperboloid.

How does one prove consistency? First, by removing all real-world context from the axiomatic system and reducing it to a collection of abstract symbols and rules. Think Obi-Wan in Star Wars saying “your eyes can deceive you, don’t trust them.” By distilling the system to abstraction, one hopefully removes any misleading intuition that comes with talking about objects. In geometry, for example, we don’t even want to think about “lines,” “circles,” and “triangles,” because our brains already have intuitive notions about those objects that might lead us down a wrong path.

Second, we construct a “metamathematical” proof of consistency. Metamathematics is a super-set of rules and statements about a system, that is “beyond” (meta) the system. An example given in the book is based on the game of chess: Chess itself has rules for how the pieces move, which we can take as the axioms of a mathematical system. Metamathematics then answers questions about the consequences of those rules, like “how many opening moves are there for pawns?” or “does this certain configuration <x> of pieces result in checkmate?” “Is this mathematical system consistent?” is a question that can be formulated in metamathematics, and the famous mathematician David Hilbert then spent some years attempting to find a metamathematical proof of the consistency of arithmetic.

Gödel, however, showed that this process, at least under the constraints that Hilbert applied to his efforts, is not possible. He demonstrated two theorems: the first of which is that no system of axioms whose theorems can be constructed by algorithm can prove all truths about the natural numbers. The system is fundamentally incomplete — there will always be things that are unprovable but true in any system of Hilbert’s form. The second theorem is that the question “Is arithmetic consistent?” is one of those unprovable truths of Hilbert’s axiomatic program.

Gödel demonstrated his theorems by an astonishingly clever and complicated process: he demonstrated that the metamathematical logic of Hilbert can itself be mapped into the axiomatic system of arithmetic, and that metamathematical proofs can be mapped into arithmetical proofs. He then showed that the arithmetical version of “Is arithmetic consistent?” is not provable in the system.

Gödel’s theorems changed the trajectory of mathematics, showing that axiomatic methods have fundamental limitations to them. This has interesting implications for all of mathematics: there are many theorems that are widely suspected of being true — like the Riemann hypothesis — that have resisted all efforts to prove.

I have been careful to refer to “Hilbert’s axiomatic program” specifically in the discussion, because it turns out that there are other methods to prove the consistency of arithmetic that have succeeded; as I understand it, however, those methods are more complicated than arithmetic itself, which then leads to questions of their own consistency!

As I have said, this is just my amateur understanding of the idea of Gödel’s theorems that I derived from reading the book by Nagel and Newman. But what of the book itself? It is well-written and written in what is probably the most non-technical manner possible for the layperson — but it is still a challenging book to follow! There are a lot of very sophisticated and abstract concepts that one must internalize in order to make progress. In reading it, one must be prepared to take it slowly and chew over the ideas and maybe reread sections to really get the ideas before moving on.

It is a really fascinating book, however, and well-worth the attempt if you’ve ever been curious about one of the most important theorems in modern mathematics! For my part, I’ll probably have to read it one more time before I even attempt to read Gödel’s original paper.