Friday, October 16, 2015

A Short History of Infinity

     Brian Clegg. A Short History of Infinity (2003) The title describes the book, and Clegg does a good job of introducing the “interested reader” to the mathematical concepts of infinity, and the many mathematicians who contributed to and developed the modern concepts of countable and uncountable infinities (terms he doesn’t use).
The basic ideas are in fact simple: two sets are the same size (cardinality) if you can match them element for element with no exceptions and none left over. Using this rule you discover that the set of whole numbers is the same size as the set of  square numbers, or any other set whose elements can be defined in terms of some operation on the whole numbers.
      The trouble starts when you are stuck at the stage of thinking of numbers as somehow real, just as kittens and trees are real. Not a problem with the whole numbers or the rational fractions: you see five pies on the kitchen counter, each cut into six pieces, so you 30 pieces of pie, or 30/6ths of a pie. But in the 15th  and 16th centuries mathematicians began to work with numbers that you couldn’t point to in this way. We need and use “imaginary” numbers because they work, they enable us to solve problems in both pure and applied math that we couldn’t deal with otherwise. Clegg needs us to accept the weirdness of infinite numbers, so he spends several chapters on imaginary numbers.
     The first of these was negative numbers. Here Clegg  (who is above all a compiler of information) seems to have missed something: a negative number can be pointed to: if you have -5 dollars, then you owe $5 to someone. Negative numbers may have been discovered as points on the number line, enabling solutions to otherwise insoluble equations, but bookkeepers made them real.
     The square root of -1 (i) was the next “imaginary” number to be discovered (or invented:  the verb you use reveals your metaphysics). Clegg makes a big thing of this one, too, but he asks us to accept on faith his assurance that it is used every day by engineers. A couple of examples would have been helpful.
     Finally, he comes to Cantor, whose mental health was fragile, and whose feud with Kronecker (his erstwhile mentor and sponsor) triggered the final crisis. Cantor applied the axioms of set theory to infinite sets, and in doing so showed that “infinity” was a viable mathematical concept. In particular, it helped clarify the differences between rational, irrational, and transcendental numbers.
     A good book, its flaws are minor and don’t interfere with understanding infinity. Clegg likes explaining things, and has a neat talent for potted biographies that give us both the facts essential to understanding the subject’s place in the central story, but also enough quirks to make the people real. What you make of other questions about infinity (such as whether the Universe is infinite or not) is left up to you. Recommended. ***

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