Zero is not Defined

February 14, 2018

I have been chugging along with pretty elementary maths with my son for some months now. Every now and then I get a reminder from the guy who writes the text book I am using that, as he puts it ‘Zero is not defined’.

He means by this in part I believe that zero is a special case in the line of numbers, as a real number, and as in what might be termed normal practices of calculation in algebraic and other expressions; and these things cannot be expected to apply to zero, even though they apply more or less universally to all other whole numbers etc.

The expression he uses; and I am guessing it is a standard expression used in his trade; is enigmatic and it appeared to me very hard to pin down exactly what might be the full implications of these words.

Grammatically it’s a sentence; “zero is not defined” has a predicated noun (‘zero’) and an object (‘not defined’) and a verb (“is”) which connects the one to the other; and so all the demands of grammar seem to be met in its composition.

It is not a command. Nor is it a sentence like ‘Zero is a special case’, because it says nothing which characterises zero; unlike, were it said to be a special case, then we should understand that zero is a one-off unique situation.

The words ‘not defined’ seem to me to be deliberately avoiding saying anything about zero as such; and they seem to be saying that the mathematicians don’t want to bother with zero, or want to avoid zero or want to say nothing about zero, and so on. It does seem to me to be telling us in fact about Mathematicians’ attitudes to zero rather than anything about zero itself.

A comparable sentence which might elucidate what I am trying to explain is perhaps “Toffee is not liked.”  The explanatory power of such a sentence is about an unknown person or set of persons’ tastes in foods; that does not include toffee. Thus we learn that the persons don’t like toffee but nothing about toffee itself per se.

I am assuming here that a) the saying ‘zero is not defined’ is a commonplace one and b) that it is commonplace in the world of mathematicians.  Otherwise one would have to ask a) who is saying that zero is not defined and b) how does it bear on these persons who are saying it.

The saying strikes me as being a bit scary. It sounds like it could be a warning; or a horrendous discovery; or a reference to a world in which things don’t happen at all like they do in our world.  It’s the sort of saying you might hear from Captain Picard or Mr Spock, as a reply to an ‘overwhelming question’.

This scariness is resident in the way the sentence ‘hangs nowhere’; sort of has no proper anchor in one’s comprehension; it seems to me to be a sentence not available to comprehension, simply because it is so absolute and final in its ‘not defined’ and in its the straight down the line ‘isnot defined’ as if some tremendous authority which makes one feel like a small worm has deemed that ‘zero is not defined’,

One can imagine the Wizard of Oz with his loud hailer and smouldering pipes and loud thunders; all being presented for effect; to let visitors know he is powerful and not to be challenged, and him all the time a little old man sitting behind a façade; he would perhaps add it to his box of tricks for intimidating people?

And this power which these four words possess remains awesome; literally so.

Thank God this is just mathematics; a study; a discipline; which is accessed by a language of numbers, digits, fractions, variables, factors, expressions, signs, collective concepts and algebraic quantities and so on; and which tools of access are all the fruit of human invention and use over millennia.  Mathematics is a closed system; and although it can be applied to things of the world like bridges and cars, it is a small world within itself and largely its study does not impinge on day to day life very greatly.

It’s comparable in this respect to law. A lot of law, like a lot of mathematics; is specialist and arcane and picky and bitty and sort of a heaven, and haven, for autist persons.  And by comparison law has its fictions, some of which have much in common with sayings like ‘zero is not defined’. They are known as ‘legal fictions’.

One such legal fiction I came across was at a place of work where I was once employed. The CEO was known as The Controller (really, this is not a fairytale); and The Controller had in his power to ‘deem’ a day erased from the calendar; as if it had not been.  This too is kind of scary – like Rumpelstiltskin in Shrek, who tries to rob Shrek of a single day of his life – getting into really weird stuff here!

This power of The Controller of course was applied and applicable only to the business and concerns of his Office; and he could not really, in case there are any impressionable children reading this, erase a day from the calendar, and so make it not exist.

Say the weather was so bad that very few employees could make it into work one day. And so the IT systems were down and the services were not all in place which support a working day, and say 5 percent of staff made it in; then it was likely The Controller would send everyone home close the office and write the day off – literally.  He (or she) would ‘deem the day not to have existed’ in regard to the purposes and functions and business of the office.

And this ‘deeming’ meant that the day was indeed struck out of the office calendar, so that the week went say from Monday to Wednesday without intervention; and the date went from say the 8th to the 10th without a blink of an eye. In law in regard to the office that day did not happen. Crazy eh?

There is a true story concerning riots in Britain back in the 17th century at a time when the nation adjusted its calendar to bring it into line with other nations it was trading with.  (The year at one time used to begin March 1st I believe, and end of course in February). However this adjustment meant cutting fifteen or twenty days out of the calendar so as to readjust and synchronise. Ordinary people, who had received no schooling remember, were shocked and considered they had lost fifteen or twenty days of their lives. Really.   And so they protested; rioted.

The power to scotch days from the calendar is pretty awesome prima facie; not unlike that fearful mathematical ‘deeming’ that ‘zero is not defined’.  The sweep of majesty both carry in the high authority they appear to emanate is magical; as words are able to be.

I have had some dealings with ‘alternative’ people who have some pretty striking ideas and beliefs; which in their use of words becomes apparent. There is a shop in Abergavenny Wales which sells a cross between steampunk and hippie spiritualist alternative stuff like trinkets and statues and crystals and joss sticks candles and so on. Some weird clothes also. The shop is called ‘Share no Soul’ which I think is a very disturbing name.  Since it seems to be implying its meaning literally, as if soul had quantity and it could be offered, traded, swapped, given away.

Too much Ganga perhaps? But nonetheless a striking name and with the shop’s black painted exterior and symbols stars and moons etc on the black paint in gold and silver paint, it all looks a little like a wikka type meeting house. Another song I heard, a folk song, was a dreadful brooding dirge like melody and accompaniment and a woman singing lyrics about ‘Let no man steal your thyme’.  An obvious and scary punning line; not so much for the meaning assumed but for the imagination going on within oneself about what might be happening in the singer’s head, and what was she up to in her life?

Another time, still on the theme of scary music, I was at The Museum of London and in the section about Victorian Life in London. A recording of an old barrel organ was playing as if the recording was on a wax cylinder. Lots of crackles and very tinny sounding voices.  A bass voice man was singing to the barrel organ’s tune a tavern song in a slum area in lowlife London of the time. If you think Oliver Reed is scary in Oliver! as Bill Sykes, this would have had you heading for the exit.  I remember saying to my companion it sounded ‘diabolic’ as if it were ‘the Devil’s music’.

There’s an old British saying that ‘the devil has all the best tunes’ but this one he was happy to keep as far as I was concerned.  This saying about the devil having the best tunes came about because of the early church in Britain having an attitude to folksong and non-sacred music in general, as it was enjoyed by peasants and labourers and their families.

For many years playhouses where dramas might have been performed were closed down or barred from being built by the order of the church.  Taverns and other houses of ill repute were frowned upon and their clientele scolded in church by the priest on Sundays. Even missing church could see you in some areas seated on the penitence stool next time you went – usually having been pressurised to go. Like when one has to stand up front with the teachers at assembly because you’re late for school again.

A long way even further back St Augustine of Hippo felt it proper to force recalcitrant people to take Holy Communion.

But I am doing down the church, yet not the object of the church, Christ our Lord. No matter I’ll stop. The church actually did something very smart in Scotland in regard to folk tunes and ballads of the common people. There was published, again in the seventeenth century a book known as ‘The Gude and Godlie Ballatis’ or in our speech ‘The Good and Godly Ballads’. This songbook was made up of melodies robbed from the fields and taverns, the homesteads and greens of ordinary folk whose traditional property they were; and by certain religious people new lyrics had been made, Godly lyrics, and so the common tunes of the people became their hymnbook.  (I have a lovely old copy somewhere).

So we have tricks played on our minds by the smart people, such as legal fictions and zero is not defined; and we have bizarre funnies done by strange people like share no soul and let no man steal your thyme and we have people fazed by calendar shifts and the powerful guys who can change the calendar.  We have music belonging to the devil being salvaged for Christ’s sake; and a ragbag of jangling uncanny things altogether.

What we come away with is that appreciation of those powerful impressions on a mind sensitive to them that all these arcane and strange things and wordings and sounds are able to impress.  And that is how we respond to a good story told by a good teller. Thank you.