by David Berlinski
The author, a mathematics and philosophy professor, writes about the basic concepts of simple arithmetic (addition, subtraction, multiplication, division), starting with the premise that numbers exist outside of human endeavor, then on to the definition of addition (which is just adding by one), lingering at the problem of zero, then through some rather convoluted proofs of various theorems, to stop at the abstract algebraic concepts of rings (structures which include sets of integers and provide the definition of addition and multiplication) and fields (which define division through multiplicative inverses).
If the summary above makes it seem as though this is a jaunt through the math you learned in elementary school, think again: “The recursion theorem justifies definitional descent by drawing a connection between the recipe or algorithm embodied in definitional descent and the existence of a unique function, the one that definitional descent has presumably defined.” Berlinski is often this recursive; I often found myself wondering what was being proved or defined, and what was being simply assumed. But aside from tortuous mathematical definitions, the book is written in an airy, conversational, sometimes jocular (sometimes smug) tone, with many sentences given their own paragraphs in order to give them Weight. Berlinski is even quite funny, as when he discusses Guiseppe Peano (whose axioms provide the groundwork for what Berlinksi attempts to show) and his bizarre simplified Latin that no one used or understood, or when he imagines early mathematicians’ dialogue when encountering the apparent absurdity that is negative numbers (“Can I do that?” “Why not?” “I’m just asking.” “What next? I mean besides giving up. That always works”). As a philosophical treatise on the concept of mathematics itself, the book makes some trenchant points (“across the vast range of arguments [in psychology, logic, physics, etc.]… it is only within mathematics that arguments achieve the power to compel allegiance because they are seen to command assent”). But as a tour of elementary abstract principles, it’s a bit abstruse for the layman. I enjoyed his insights on sets and some of the simpler chapters, but finished the book feeling as though Berlinski was a bit too clever for his own good, and yet not quite clever enough to make it all clear.