A Certain Ambiguity was disappointing and bordering on the trivial, sadly. In 1919, a fictional, yet exceptional Indian mathematician, worthy of being invited to a US university after living in Great Britain, believes like a child, that Euclid wrote about truth, and of course, rigid Christians jailed him. The only thing missing from the test is an angry mob at night with pitchforks ready for a lynching. However, good leadership by the New Jersey governor saved the day. Thank God for educated elites!

In 1919, Riemann, Poincare, were evidently not known to all expert mathematicians as the conceit of this book. OK. A few physicists still believed in the ether also. To be fair, Riemann's 1854 lecture may not have been well known, but Poincare was pretty famous at the turn of the century. By the time of the setting of this story, topology, not geometry, was the focus of study, because formalism was well understood as the refuge for mathematics. See the fifth paragraph below.

Still, the first person narration by an intelligent Stanford grad student, who evidently didn't know anything about series from high school math, but is worthy of a full scholarship to graduate school in mathematics, was far more improbable. Maybe he took Calculus in summer school after being accepted.

The conclusion that happiness and contentment are found by knowing that absolute certainty cannot be proven may be a mantra for some, maybe a majority, of intelligent people, but patting oneself on the back, which seems to be the point of the book, is fairly self-serving at the very least. Cool people don't believe in truth, just games, evidently.

A more honest and useful book would have stressed that before Riemann, Western thought, buttressed by Kant, believed that "a priori synthetic" systems existed: that our mind by itself (a priori) could obtain real knowledge (synthetic) of the world. Euclidean geometry was the prime example of this. As a result, a proof of God, while not rigorously found, could reasonably exist. The main counter to this belief was found in "personal idealism," which Bishop Berkeley, along with many, many others lectured. At it's extreme, everything is in one's mind and it is difficult, if not impossible to truly know anything about the real world (if it even exists!). This is very unsatisfying. "A priori synthetic" was important psychologically to many people. Other philosophies, such as positivism in 1919 in Great Britain, arose to try to bridge the gap with dubious success.

However, the authors, Suri and Bal, could have taken another, less well known, but perhaps, more valuable interpretation of Godel's work and resolved the issue better; although not as correct politically. Godel proved that every logical system would have at least one truthful statement that could not be proved by the axioms of the system. In short, any system is not complete by axioms alone. This is supposed to destroy full certainty of knowledge. Maybe, but it provides one GREAT and AWESOME question: from where do the unprovable truthful statements arise? They have to come from something outside of the system! In other words, the real world isn't found in "a priori synthetic," but in "synthetic a priori!" There is something out there so to speak. What it is - is the unanswerable question! God may exist, or at least a reality, for that is now certain!

The greatest problems with the book are the two horrible math errors made by the authors that invalidate so much of what they wrote. Most critically, curved space doesn’t invalidate Euclid at all. Maybe Gauss initially thought so; but Riemann in 1854 showed that planar geometry was merely a special case of when a triangle had 180 degrees in space. Other spaces had either less or more than 180 degree triangles (like those on a sphere). There is NO CONTRADICTION of Geometry by validating Einstein, if anything, it increases Geometry’s validity. Duh! Second, and not critically, the fifth postulate is not evidence of a statement that’s true but unprovable after Godel as the authors claim. This is obvious since more than one fifth postulate is possible, not just one. Duh!

A Certain Ambiguity is novel on math written by poseurs without ambiguity - an embarrassment on many levels.

## Tuesday, December 22, 2009

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## 1 comment:

Professor Kevin Devlin wrote me this in response to my penultimate paragraph:

"Well, I don't recall the context in which those statements occur, but both could be okay. (I read the book in manuscript form before the original edition came out, some years ago.)

If by curved space you mean the curvature is nonzero, which is the usual definition, then it does (trivially) contradict Euclidean geometry. The standard classification of 2D geometries is curvature <0, =0, and >0, and these are mutually exclusive. Most mathematicians would indeed say that curved space contradicts Euclid, since, by definition, Euclidean space has curvature zero and "curved" is assumed to mean nonzero curvature. (If it doesn't mean that, then is becomes a superfluous modifier.)

Likewise, if we assume the universe is Euclidean, then the fifth postulate is indeed a statement that is true but unprovable, so again in the right context the claim is okay.

So without checking the context, I'd say both statements could be fine. But the second one in particular does depend on our assumptions about the world. (As does any contingent statement about truth, of course.)

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