Zero

I bought the book at reference 1 as a result of reading the review of reference 2 at reference 3. With reference 3 explaining that while the late Stephen Hawking, one of Isaac Newton’s successors as Lucasian Professor of Mathematics at Cambridge, was indeed a notable astronomer, his fame rather outstripped his astronomy. I settled for reference 1, a best selling book about zero, an earlier book from the same author, one Charles Seife.

On this account, mathematics had, in the first instance, three drivers. First, the need to count things for the purposes of gifts, tribute and taxation. Second, the need to survey the rich alluvial lands around large rivers, for example the Indus, the Nile and the Euphrates; the need to be clear about exactly who has what rights over exactly what land. And third, the needs of astronomy and astrology. Including here, for example, the need to compute the date of Easter using a combination of the solar and lunar dates. Practical drivers in the sense that these were all things that the people in charge wanted to be able to do.

We start with the mystics of the ancient world getting into a lather about whether zero would reduce the world to its primeval void, through the middle ages and Renaissance, through the great leaps forward of the eighteenth and nineteenth centuries and wind up with the mystics of our world who dress their ponderings about intergalactic travel up in quantum clothes. Say fifty short pages to each of these four sections. Plus some rather poor quality diagrams and illustrations. Perhaps the hardback version would have been better in that regard, but I wouldn’t bet on it.

A rather chatty and informal style, which grates, certainly at first; perhaps rather too like my own style for comfort. A lot of ancient, middle and early modern ground is covered in very few words; a seemingly careless glossing of a complicated story, a lot of which did not seem very relevant to the matter in hand. I was moved to check one bit, the suggestion on page 98 that Jansenism in France was mainly about opposition to the Jesuits – to find that if you are going to give them just the one sentence, bashing the Jesuits is not really good enough. Perhaps such stuff would have been better omitted. Wikipedia tells me that Seife is qualified in mathematics – if not in the history of the western world – and things got better as we got into the serious mathematics of the west from the seventeenth century.

But he did offer one interesting new-to-me thought, to the effect that England and France exhausted themselves in the Seven Years War in the middle of the eighteenth century, spending far more than they could comfortably afford, money which had then to be raised from taxation. Resulting on the one hand in the American Revolution and on the other in the French Revolution a few years after that. With the added irony of the French monarchy spending another big dollop of money on helping the American revolutionaries win.

We are told how we started with counting, using positive integers. Then gradually we got used to the idea of negative numbers and rational numbers. But still gagging a bit on zero and infinity. Calculus came in the seventeenth century, after which we learned about limits of infinite sequences and started to get a grip on both zero and infinity. Along the way, we invented the square root of minus one – that is to say an imaginary number. Then, starting in the twentieth century, both zero and infinity started causing trouble again in the very small world of quantum mechanics and in the very large world of relativity. With the whole lot coming together at the Big Bang of our birth and at the Black Hole of goodness knows what.

We are implicitly reminded that while, until the Renaissance, the west rather lagged behind the east in these matters, the west (here including Russia, east and central Europe) dominated in the nineteenth and twentieth centuries with a regular explosion of mathematical knowledge.

A sample of a few of the things that I liked or which otherwise struck me follows.

The Pythagoreans were fascinated by regular pentagons, by the way that they could be nested and by their exhibition of the golden ratio. Google knows all about it. While I wondered what interesting properties might be found in a regular heptagon – an odd number of sides being a necessary condition for this nesting. 

The Mayans used base twenty for some of their numbers, and sometimes – that is to say quite rarely – represented them from a gallery of twenty rather grotesque heads in profile. Read all about it at references 4 and 5.

The modern way of writing numbers as the sum of powers of a base – in our case usually 10 (in the world) but often 2 (in computers) – came from the columns of an eastern abacus. So ‘two, gap, gap, six’ was not the same as ‘two, gap, six’ at all and needed to be distinguished using some marker for gap. That is to say, zero.

The vanishing point of perspective was invented in the fifteenth century. Another manifestation of the pairing of zero with the infinite.

I was intrigued by Riemann’s projection of the complex plane onto the surface of a sphere sitting on the origin – with the top of the sphere being the point – just the one point – of infinity. A sphere which I never met when young, despite doing both projective geometry and complex analysis. Must have skated very lightly over both of them. In fact, my only contact with Riemann was a complicated version of tennis played in Huxley’s ‘Brave New World’.

In sum, despite its defects, a pleasant reminder of some of the mathematics that I once knew, plus various bits of mathematics that I never knew. But then, as I have to keep reminding myself, one can’t do everything, one can’t even dabble in much of it. There is just far too much of it. 

I shall pass on the new book about Hawking.

PS 1: the National Archives, the source of the snap above, tells me that the Exchequer’s rather unusual name was derived from the chequered cloth on which the confrontational audit process took place between the powerful Barons of the upper Exchequer and the hapless accountants summoned before them, who were regularly interrogated about the state of their accounts. More sophisticated accountants from parts east used the abacus. This particular Exchequer is an Irish one.

PS 2: the Reverend Henry Lucas, for whom the famous Lucasian chair is named, was an English clergyman and politician who was buried in buried in 1663 in the Temple Church, this last being here noticed for its musical evenings.

References

Reference 1: Zero – Charles Seife – 2009. 

Reference 2: Hawking Hawking: The Selling of a Scientific Celebrity – Charles Seife – 2021. 

Reference 3: Eclipsed by Fame - James Gleick/NYRB - 2021.

Reference 4: https://en.wikisource.org/wiki/An_Introduction_to_the_Study_of_the_Maya_Hieroglyphs/Chapter_4. 

Reference 5: Maya Numbers and Maya Calendar – Mark Pitts – 2009.