Like the yoga class I'm taking, John Stillwell's Roads to Infinity stretches me, sometimes painfully. As the subtitle says, it's a book about "The Mathematics of Truth and Proof". That necessarily means it's about infinity. Starting with the dot-dot-dot at the end of "1, 2, 3, ..." Stillwell takes the reader on a tour of higher and higher levels of the infinitary universe. Each chapter begins with a Preview and ends with Historical Background. Between them the math often gets hairy. But there are enough explicit examples (e.g., the discussions of Goodstein's Theorem and Tag Systems) that I managed (mostly!) to follow along, skipping harder parts as I did with Keith Devlin's Joy of Sets.

In addition to deep mathematics, Stillwell punctuates his text with engaging quotations, both serious and less so. Some examples:

• "... it is hard to be finite upon an infinite subject, and all subjects are infinite." (Herman Melville, 1850)
• "... the best I can say is that I would have proved Gödel's theorem in 1921—had I been Gödel." (Emil Post, 1938)
• "... this proposed theorem is not only obvious but more obvious than any of the axioms." (Alonzo Church, 1956)
• "Graph theory is a cuckoo in the combinatorial nest." (Peter Cameron, 1994)

Roads to Infinity might have been called "Glimpses of the Great Game". The original "Great Game" was the British vs. Russian struggle for Asian supremacy during most of the 1800's. But mathematicians have been playing a Great Game much longer, ever since they started to see the consequences and contradictions of infinity, and the implications that has for human knowledge. After centuries of progress there's still a long long way to go ...

(cf. TransfiniteMeaning (1999-07-31), Joy of Sets (2010-06-25), ...) - ^z - 2010-10-06