General Cohomology Theory and K-Theory Book

There are different ways to use and think about mathematical concepts and they do not all leave a historical record in a form that can be interpreted with certainty. The gap between people engaged in trade and those indulging in philosophy has been especially important, notably for the ancient Greeks, whose philosophers used geometry to think mathematically and despised the mere counting required for trade. Others have used sand or beads or counting boards or diverse tools in which the technique for even complex calculations did not necessarily entail a corresponding grasp of the theory or the concepts at work. In such cases, zero might arise and vanish repeatedly without making an impression on the way people thought about numbers. The key fact is that we do not know what they were thinking unless we can find a record.

Although we can find written number systems as old as the Sumerians, 5,000 years ago, in which we see zero at work, it is used in unexpected ways. They appear to require a form of zero to indicate "nothing in this column" - for which there is a record at Kush dated to 700BC - but it is used only in the middle and never at the end of numbers. So they cannot distinguish between 2, 20 and 200. The ancient Greeks - from the Fifth Century BC - used their letters as symbols first for numbers 1 to 9, then tens up to 90, then hundreds up to 900, but had no letter as a symbol for zero. They seem to have incorporated zero first after invading the Babylonian Empire in 331 BC. There is good evidence, which Kaplan reviews, that the Greeks in turn passed the concept to India. Kaplan spends some time debunking claims that the Indians invented zero, insisting that their valid claims are already remarkable and do not need the additional support of being exaggerated. Why deprive Zero of its much longer history?

Kaplan says very early on that "we count by giving different number names to different sized heaps of things." In places this is as much a history of 1 as a history of zero, discussing the different ways people group stuff into units. The Sumerians used units, tens and sixties for example. Archimides named vast large numbers by working with myriads and myriad -myriads, which could be of the first order or the second order. Until 1971, England worked with units of 12 pennies to one shilling, twenty shillings to one pound. We still have sixty seconds to a minute, sixty minutes to an hour, twenty four hours to a day, seven days to a week and so on .. Kaplan devotes a chapter also to Mayan number systems, with a range of cyclical calendars all seeking to defer the end of time. They took their obsession with counting to extremes.

Kaplan argues that up to this point in the history of numbers, the principle remained valid that there was a correspondence between numbers and things, such that there would be no consideration given to the notion that zero / nothing / void / empty was a thing in need of a name in the way other numbers were. But Indian mathematics achieved a "paradigm shift" which focused on how numbers behaved instead of what numbers were. Such behaviour took place in equations, where the solution (the number which made the equation balance) was as likely to be zero as any other number. The names of numbers were contracted to written symbols, including symbols for operations (plus, minus, equals, squared), which could be used yet often not visualised (what would x squared look like?). Indeed, for the first time it became reasonable to deal with negative numbers as well as positive ones. Numbers no longer named or described objects but became objects themselves to which adjectives could be attached: positive, negative, natural, rational, real. "The change in mathematics we've been following, where the names for numbers narrow down to signs of them and the numbers themselves are subordinated to the laws they obey, began when someone first counted and evolved through the ongoing project of deriving these laws from as thrifty a set of axioms as mathematicians could manage."

It was through Islam and the Arabic language that Indian mathematics was transmitted to China, Russia and Western Europe, the latter by 970 AD. Kaplan gives credit for this but does not suggest that the Arabs transformed Indian maths in any fundamental way. Perhaps this impression arises through the specific focus of this book on Zero. In any case, Europeans struggled with many aspects of the new approach to mathematics. For all the crushing difficulty of using existing methods, there were a lot of new concepts to absorb in Arabic (or "Saracen") methods, and for counting itself Europeans could make good use of the abacus to meet their practical needs. Kaplan suggests that the major breakthrough for Zero was after 1340, when Pacioli introduced the novelty of double entry book-keeping, in which Zero held the essential balance between credits and debits, positive and negative.

Yet it was not in counting that Zero had its crucial impact. It remained the case that counting with an abacus was much faster and more efficient that using the quill to apply the Saracen techniques. Kaplan writes: "Wordless manipulation will carry you with dash and glory to the outermost edges of arithmetic - but it will leave you stranded once you cross the border into algebra and all the lands of mathematics that lie beyond. There thought travels by signs laced into a language that can speak even about itself…. This language came into its own when zero entered it as the sign for an operation: the operation of changing a digit's value by shifting its place."

Kaplan proceeds in the remainder of his book to explore diverse ways in which the role of zero has been significant in the development of mathematical ideas. This is not material to read with a passive attitude - it requires effort and patience from the reader. His explanations unfold very logically to give insights into the power - but also the confusing nature - of this surprising number.

The book does not demand a knowledge of difficult mathematics but it sadly does demand a willingness to read difficult prose that is sometimes too dense and to wade through his laboured humour and philosophising in places where that just confuses things. The result for me was that I literally nodded off to sleep on occasion. In the end, I also found it necessary to speed read through the book for a second time, just to clarify what it was about. It made more sense in retrospect than it had done at the time of first reading. Maybe that suggests that the book had a great idea which called for more editing before it was released into the wild. It could have been a lot better. A heavy read and a struggle but all the same, interesting stuff and filled with unexpected gems.