This book was lent to me by a colleague at work; at first I took it as a recap of anecdotes on the relationship between Gödel and Einstein. Already knew some of the anecdotes (as the good relationsip with Einstein at Princeton; or time when Gödel, at the court for his citizenship, tried to explain to a jugde the possibility of a USA becomeing a dictatorship; even the cause of his death was peculiar).
But the book is not about that.
It is about Gödel's thinking. And not a shallow review of his works. It goes down the rabbit hole.
At the beginning of the book, the description of the academic life in Vienna at the early 20th century was quite inspiring. In particular the references to the Vienna Circle were totally appealing to me -given the circle's close connection to the works of Wittgenstein, a personal favourite-.
Note: the word zeitgeist is used to describe the common or widely accepted mental schema of the time (or any given time).
Later, the Cantor's diagonal argument is introduced to assert that:
It follows that there are more real numbers that natural numbers, even though there are infinitely many natural numbers and infinitely many real numbers.
But here term more is written in italics in the book. For a reason: it does not refer to greater in the sense of real or natural numbers, but in the sense of the aleph number, which represents a diferent mental construction.
The book also makes an interesting presentation of empiricism, positivism and its evolution into logical positivism. Later idealism and realism.
Note that, from a workplace point of view, I follow coherence rather than falsability; since in my daily life/work it is more people's decisions that have a greater impact. Much more impacting at work than truth.
But truth is, again, not very clear. See this excerpt from page 57:
Here is a crucial difference between truth and proof: a mathematical proof, in the sense in which we are disucssing it here, is always a proof in, and relative to, a given formal system, whereas truth, as such, is absolute. What Gödel proved is that mathatical truth is not reducible to (formal or mechanical) proof. Syntax cannot supplant semantics.
It seem safe to take that truth may live in a higher dimension than axioms, not reachable by inference rules. But perhaps it is not absolute. Or perhaps we need more dimensions, not just more premises. Or a different inference mechanism. Actually, this pursuit for knowledge versus methodology appears as Gödel target in the latter philosopher years (more about this at the end of the book).
The topic about ontology -to be- versus epistemology -to known- goes on in the book, presenting Einstein and Gödel -specially Gödel as fifth columnists in their comtemporary academic mindset (a chapter in the book is named A Spy in the House of Logic).
And by page 115 of 184 we get the most concise description of Gödel's achievement I have ever read (though it is not this matter the main topic of the final chapters of the book):
For his incompleteness theorem, Gödel devised a formal system together with a series of ingenious definitions and coordinations for which it could be demonstrated that the concept of formal proof, as it appeared in the system, could not, on pain of contradiction, be interpreted as representing intuitive mathematical truth. He did this by constructing a formula that was provably unprovable, but intuitively true.
Intuition and Kant's a priori knowledge are brought to the front many times in the book. Gödel follows this a priori knowledge as a way to get to the truth, once he has destroyed proof as the only source of knowlege.
Even further, Gödel ventures into the ontological argument about the existence of God, getting some disrespect from academia (being labelled as pious). Though charisma does not seem to go with Gödel that much. In page 130 it is described like this:
This same mode of reasoning, from the possible to the actual, occurs in the "ontological argument" for the existence of God employed by Saint Anself, Descartes and Leibniz. According to this argument, one cannot consider God to be an accidental being -one that merely happens to exist- but rather a necessary one that, if it exists at all, exists in every posssible world. It follows that if God is so mauch as possible, He is actual. This means that one cannot be an atheist unless one is a "superatheist", i.e., someone who denies not just that God exists but that He is possible.
In page 169 Gödel is revealed as having outpaced Kant, to have evolved over Kant. The book describes it like this:
Indeed, the longer, original drafts of Gödel's contribution to the Schilpp volume on Eistein, entitled "Some Observations about the Relationship of Theory of Relativity ato Kantian Philosophy", leave no room for doubt that Gödel had a profound understanding of Kant, which enabled him to demonstrate a striking and previsouly unsuspected connection between Kantian idealism and Einsteinian relativity. This newly published essay makes clear that Gödel, though he accepted certain elements of Kant's philsophy, systematically rejected its main thrust, which assimilated knowledge to the knower, not the known, and thus gave Kant's philosophy a subjective cast.
The last chapters of the book are devoted to vindicate the figure of the Gödel-philosopher; not just a logician but a true philosopher that tried to go beyond [Kant]. A Platonist is the expression used in the book.
By the end of the book I got a few more references to read, which it is what I enjoy now of these kind of philosophy books: to provide a few answers and open new questions.