|Lucy, a freshman at a Scottish University, and David, the graduate student who leads the problem sessions for her physics class, discuss the mathematical and philosophical implications of Gödel's First Theorem, resulting in several published papers. Although much of the focus is on these deep ideas, subplots include a romance between the two main characters, a dramatic rescue from a near death experience, academic intrigue when a professor withdraws a paper from submission without consulting co-authors Lucy and David, and even a suspicious death.
According to the book's official website, it was the author's original intention to simply write about implications of Gödel's Theorems, but it grew into a novel. Considering this, it is really very well written. That is, although I would not expect it to receive glowing reviews from literary critics, it is also a pretty good novel (whereas other books with similar origins can be almost unreadable). The story does not just fill the space between the discussions of the author's ideas; the reader truly can become interested in the characters and what happens to them.
Among the things that happen to them is that they make mathematical discoveries and publish some papers. Making mathematical discoveries and publishing papers is something I think is both fun and important, and so I appreciate as "good advertising for mathematics" any work of fiction which can create interest in these things.
The author also does a good job of explaining many difficult ideas, including some physics as well as Gödel's theorem, the main object of interest. One mathematician character, Christos, also presents some intriguing and thought provoking discussions.
Unfortunately, the specific idea that the author wishes to present is not as interesting to me as the story itself. To explain the idea briefly (so as not to spoil it for readers of the book who will enjoy the process of discovery along with the characters), it comes in three pieces. First, Lucy and David prove an "inverse" to Gödel's theorem, stating that there exists a mathematical system in which the undecidable "Gödel sentence" can be proved. Then, from iterating this idea and showing that the resulting mathematical systems "converge" (in some undefined sense), they prove the existence of a "God system", a mathematical system over all others. Finally, they apply this idea to our own universe and discuss the question of whether the "God system" really deserves its theological name (which is made more interesting by the fact that David is an atheist while Lucy believes in God).
From my perspective, there are a few things wrong with these ideas.
Perhaps other readers will find the arguments of the book more convincing than I did. But, even if not, the ideas presented make an interesting kind of science fiction. It seems to me that the author is asking us to imagine a universe in which Gödel's theorem is an ultimate truth about nature. Just as the Copenhagen Interpretation of Quantum Physics claims that the probabilities associated with wave function collapse do not represent a gap in our knowledge or understanding, the idea of the Gödel universe seems to be that the undecidability of certain statements is the truth, rather than just an indication that we are unable to determine the truth. I would not say that this is a reasonable assumption to make or that I would reach the same conclusions as David and Lucy do about it, but it is still fun to consider as a thought experiment!
- Gödel's Theorems necessarily involve both inconsistency and incompleteness. This book completely ignores the inconsistency side of things. While this might arguably be a necessary simplification to avoid making the book unreadably complex, it needs to be considered carefully by anyone who wants to seriously investigate the ideas as opposed to the story.
I can only think of two ways to interpret this "inverse theorem". Under one interpretation, the idea is completely trivial. It is an immediate and obvious consequence of Gödel's theorem applied to a system X that the system made by adding the "Gödel sentence" as an additional axiom is a system in which the statement can be proved (and is consistent so long as X was). Both because this is too trivial and because , it is equally true that the negation of this statement could be added to X to produce a consistent system if X is consistent, I do not think this is what the author had in mind. However, to say any more than this requires a step in the proof in which Gödel assumes the existence of the "natural numbers" as a consistent and complete mathematical object. Since a mathematical system "above X" is explicitly assumed in the proof, it is not reasonable to claim that the existence of such an object is a corollary. In other words, in the only two ways I have been able to think about it, the "inverse theorem" is either obvious or the result of circular reasoning.
- The author has a strange viewpoint, which could arguably be one of the most interesting ideas presented in the book, in that he considers the universe to be an axiomatic system of the type that Gödel's theorems address.
One should remember that it only a certain type of mathematical system to which Gödel's theorems apply. So, the universe seeming mathematical does not necessarily imply that it is an object of this particular type. Even if it were true that our universe were governed by such an axiomatic system, I would argue that it is unreasonable to act as if Gödel's theorem applies to our knowledge of it. The situation that the theorems consider is characterized by two things: (a) the axioms are known entirely and exactly from the start and (b) only things that can be determined from the axioms by an explicit proof are considered to be facts. In contrast (a) we don't actually know what rules govern the universe and (b) by virtue of existing in the universe we know things about it that might not be derivable from the axioms.
Moreover, I find it strange to think that our ability to think about axiomatic systems would be in any way restricted by the mathematical rules that govern the universe. The book suggests that we could consider any statements, but could not prove them because the axioms are not available to us. But, I do not see any practical restrictions on the axiomatic systems I can consider. Whatever this "God system" is, I do not see why a human mathematician would not be able to propose and study it even if it goes beyond the mathematical laws of physics. (The canonical example of an axiomatic system that humans have studied is the Euclidean plane, even though no such thing exists in physical reality.)
- I think the standard way to think of Gödel's theorems is as a restriction on what can be derived in certain special axiomatic systems. However, this is not perceived as an ultimate limitation on human knowledge or the universe since we deal with things that go beyond such axiomatic systems all of the time. In fact, one consequence of Gödel's theorems is that numbers themselves, as we normally consider them, cannot be completely characterized by such an axiomatic system. Since we consider numbers to be a description of how our universe works, this suggests that our universe is already beyond Gödel's theorems and hence, even if I were to accept the rest of the ideas, I do not see why the so-called "God system" would be something outside our universe, and so would probably not deserve the name.
Note that the "reality is nothing but mathematics" idea which the book presents also appears (less seriously) in quite a few other works of mathematical fiction. See, for example, The Mathenauts, The End of Mr. Y, Mathematica, and Luminous.
Alan McKenzie is a physicist who holds a D.Sc. from St. Andrew's University, the institution on which the university in the book is based.