Allusionist 142 Zero transcript

Visit theallusionist.org/zero to listen to this episode and learn more about zero.

This is the Allusionist, in which I, Helen Zaltzman, clear language for takeoff.

In today’s episode, we’re picking up the number theme from the conversation with Stephen Chrisomalis a couple of episodes ago, and focusing in particular on one number - or is it a typographical mark? Or is it a placeholder? Or is it a little chaos merchant? We’re talking about zero. 

Next episode I’ll be taking your requests for etymologies, so go to theallusionist.org/requests by the end of September 2021 to tell me which words you’d like me to investigate. If you’ve emailed in the past few years with requests, sorry - the inbox has got so overcome with spams that I’ve had to abandon it, but do go ahead and ask again. Also! You still have till 4 October 2021 to stream the new Allusionist stage show, featuring many many eponyms and one hectic Greek god. Get it via KingsPlace.co.uk and there’s a link at theallusionist.org/events too.

On with the show.

KYNE: It comes from India. Actually ‘zero’ is a Latinization of an Arabic word ‘ṣifr’, which is a translation of the Indian word ‘śūnya’. Śūnya means 'empty', same as ṣifr in Arabic. 
My name's Kyne. I'm a Canadian drag queen and math communicator. I make math videos on TikTok and Instagram and Twitter. I'm just promoting my love and passion for the subject, all while dressed in drag. 

HZ: I think I would have paid more attention in math had the teachers dressed a little more compellingly.

KYNE: Me too, probably.

HZ: Just a lot of ill-fitting blazers.
So zero, out of all the numbers and mathematical symbols, seems unique in being a combination of typographical marker and philosophical vortex. What makes it so special? 

KYNE: It's a really interesting number because it's one of the newer numbers really. And there was lots of debate about whether it should count - no pun intended - as a number at all. What is a number in the first place? Can you give a definition without using the word number, like even a synonym, like quantity or amount?

HZ: Damn you, I was going to go 'quantity'!

KYNE: Right? I was like thinking about this earlier, so I wrote down my best definition. This is my best try: "A number is an abstract mathematical object used to describe things." So I know that definition uses the word 'mathematical', which I mean, in fairness is another tricky word to wrangle a definition out of. It's pretty clunky, I know, but... 

HZ: You set that rule. You made it difficult you for yourself. 

KYNE: I really encourage whoever's listening, try to ask yourself: how do you define a number? 

HZ: “Number, noun: 1. an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations. 2. A quantity or amount.”

KYNE: And once you think about what a number is, how would you define the absence of a number? 

HZ: My brain can't do it, Kyne! My brain is just like, "I'm going to absent myself." 

KYNE: Yeah, well, if you're confused, then you understood why it took thousands of years for people to recognize a zero is indeed a number. If you think back to hundreds, thousands of years ago, in societies that use numbers as tools for counting sheep or doing simple trade, keeping track of property. Why bother counting zero sheep or zero children or zero tools? Especially if you think of a culture like ancient Greece, it's even harder to see how zero would have been useful for them when you consider that ancient Greek mathematics was largely based on geometry. Like Euclid, Pythagoras, a lot of Greek mathematicians: they, for the most part, viewed numbers in terms of lines and areas and volumes. The idea of zero would have been like a line with no length, or a square with no area; it just doesn't make any sense. 

HZ: But there were some uses for zero that did make sense, such as accounting.

KYNE: In accounting, it is meaningful to say somebody has zero sheep or zero wheat. And in ancient Egypt, we've seen hieroglyphs for zero that even represented the base or the ground level, for instance in schematics for building tombs or pyramids. The most practical use of it would have been as a placeholder symbol in our numbers. Our numeral system, I think, is very clever. And it's called a positional system, where the position of symbols carries a lot of information indeed. In particular, what we have is called a place value system, where every digit represents a power of 10. Take, for instance, the number 444: each of those 4s means something different. 

HZ: Four lots of 100, four lots of 10, four lots of 1. And then, if the number was 404, that's four lots of 100, zero lots of 10, four lots of 1.

KYNE: But if we had no zero, how would we be able to tell the difference between 404 and just 44 or 440? So in cultures that use positional number systems, like the one that we use now, zero turns into a very useful innovation that gives birth to many new possible ways of expressing numbers. And that's exactly how zero was born in ancient Mayan, Babylonian, Indian and Chinese civilizations. Some of them would just leave space between digits to indicate a zero, so like to write a 404 by writing a four and a four with a long space in it, but you see how even that can get confusing. And so some of them eventually developed little placeholder symbols that would stand in in the way that we use zero in the number 404. 

HZ: But did they consider those placeholder symbols numbers? We don't count the decimal point as a number itself, or the commas that section off the digits in a really big number. 

KYNE: We use those as symbols when writing numbers, but we don't think of them as numbers themselves. So to me, there's no reason why we should assume the ancient Babylonians would have thought of zero as its own number. If zero is a number, then where does it go on the number line? Today we think it should be between -1 and 1, but back then people didn't have negative numbers. So if zero went before one, that would make it the first number. But we don't start counting with zero. Computers count by starting at zero, but usually we don't count by starting at zero. Interestingly, the Mayans did start counting at zero; and we can tell by their calendars. So the Mayan calendar divided the year into 360 days, with 18 months that were 20 days each. So the days were numbered 0 to 19, instead of 1-20. So for instance, in the month of Zip, which was the God of hunting, the first day was called the seeding or installation of Zip, the next day would have been 1 Zip, the next one would have been 2 Zip, and so on. It's weird to us and our sense of thinking about things, but if you really think about it, it makes a lot of sense. Our calendar doesn't use a day 0 or year 0. It was made in a time where zero wasn't accepted yet. So actually there was no year 0BC; the year after 1BC is 1AD. That means the first century a started with year one. So following that logic, the second century should have started at year 101, third century year 201, which brings us to the controversy of our modern time. Well, I say controversy, but really only among nerdy people: did the 21st century, the new millennium, did it start in year 2000 or 2001? Did the new decade start in 2020 or 2021? To the Mayans, that would have been an easy question to answer, because they started counting at zero. But for us, it depends on whether you want to start counting at zero or one. 

HZ: I think people want to see the numbers flip over on the calendar. So we really played ourselves by starting at one and depriving ourselves of that, technically. 

KYNE: True. I mean, personally, I feel like, yes, I guess it should technically start with the one; but I think you're totally right, we just like seeing the numbers change, who really cares if, you know... Maybe we can just say the first century had 99 years. 

HZ: The calendar's changed a lot anyway. There's constant fiddling, right? It's not as solid as we'd like to think.

KYNE: There's a difference between what's technically correct, and what we consider correct enough for our day to day language. 

HZ: Right, our emotional truth. 

KYNE: Exactly. 

HZ: Zero as placeholder came as part of the package with the Hindu-Arabic system of numerals. But of course it’s not just a placeholder, it’s a concept.

KYNE: Indian mathematicians were the first, as far as we can tell, to treat zero as its own number: not just as a placeholder symbol, not even just in counting, but they subjected it to math and logic. So in about the seventh century, the Indian mathematician Brahma Gupta was willing to ask what happens if you multiply a number by zero? What if you subtract zero from a negative number? What if you add zero to a positive number? What if you divide zero by itself? He was the first one to really ask these questions and write about the algebra of zero. And actually his answer for the last question, what's zero divided by zero, was incorrect: he thought it should have been one or zero. In our modern math, we think it should be undefined, which is a whole nother conversation; but I mean, we've got to cut him a break, he was way ahead of his time in thinking about the arithmetic with positive and negative numbers, centuries before they'd even arrived in Europe. 

HZ: Didn't have a support group around him, to validate him, spur him on further.

KYNE: No, he did not. 

HZ: The person credited with introducing zero to Europe was Fibonacci, of Fibonacci Numbers fame. Ding ding ding: eponym! Although that wasn't exactly his name, it caught on when Guillermo Libri called him it in his four-volume History of the Mathematical Sciences in Italy published in 1838, some six centuries after Fibonacci was alive to voice an opinion about it. His name in his time was Leonardo Pisano or Leonardo of Pisa, his home city. But his father was called Guillermo Bonaccio, hence Fibonacci, meaning “the son of Bonaccio”, so he might also have been known as Leonardo Fibonaccio in his time. 

Whatever his name, he travelled widely around the Mediterranean with his father who was a customs officer, so his son came into contact with merchants from all over the place, and learned all their different ways of numbering and calculation. That’s how he learned about the Hindu-Arabic numeral system, of positional notation with ten digits including zero, and he wrote about it in his 1202 book Liber Abaci where he explained how it could be used for calculations and bookkeeping and be much faster than Roman numerals and abacuses. In Liber Abaci, he also wrote about the sequence of numbers that bears his name/the name bestowed upon him.

KYNE: Which is a great thing to be associated with, but it was almost an even bigger deal that he brought the Hindu Arabic numeral system to Europe. 

HZ: Well also he didn't invent the Fibonacci sequence, did he, just popularized it in Europe.

KYNE: Exactly.

HZ: Done very well for himself out of it. You just need to get your names on things and then you're sorted for another thousand years. 

KYNE: True. True that.

HZ: Fibonacci wasn’t the only person to bring the Hindu-Arabic numbers to Europe either, but he gets the credit. He called zero ‘zefirum’, a Latinisation of the Arabic ‘sifr’, then it got Italianised to ‘zefiro’ which was already in use meaning ‘the west wind’. Two different kinds of emptiness, brought together by sounding a bit like other words. Zefiro was respelled ‘zeviro’ in Venetian, then contracted to ‘zero’, in which form the word made it into French, and then English, making its first known appearance in 1598.

Despite the popularity of Liber Abaci and the influence of Fibonacci’s work, it took quite a while for the concept of zero to catch on in Europe.

KYNE: Perhaps unsurprisingly, zero was met with a lot of doubt. Zero as a concept: that was definitely something that was heavily contested by lots of philosophers. One of the most influential philosophers in ancient Greece was Aristotle, and he believed that it was impossible for nature to produce a void or a vacuum, a place where there's no matter. And so zero was seen as this void, and it really clashed with the Greek philosophy of the world. And if you think about it, zero breaks a lot of rules. Zero times anything is zero, zero divided by anything is zero - it's like the Terminator, right. And even dividing by zero is illegal math. It's very logistically a big problem. So there are many reasons to protest zero. It brings about all these problems logistically and philosophically.

HZ: Christians really were not on board with zero as well when they first found out about it, because they were like, "There can't be nothingness with God, because there's no beginning and end of God." 

KYNE: Yeah. And that was something that they inherited from Aristotle, because Aristotle said that the only thing that's infinite is the heavens and that there is no void; there's no place in the universe where there's nothing. They were suspicious, especially because, as we said, the church was following Aristotle's lead and believing that the idea of a void was very unchristian. There is no void in this world, you know? 

HZ: Tell that to my soul. 

KYNE: Oh, yes. 

HZ: In 1299, zero was even banned in Florence.

KYNE: The city of Florence banned bankers from using zero or any of the new Hindu Arabic numerals. Some of them called them the infidel symbols. 

HZ: That is inflammatory. 

KYNE: I know! They were seen as too confusing; too easy to manipulate and swindle people with; you could doctor a zero to make it look like a nine or a six. 

HZ: But couldn't they have snuck in some extra digits in the Roman numeral system, like stick a few extra ‘C’s in there, some extra ‘I’s? 

KYNE: Yeah, I guess so, to make bigger numbers. Zero also led the way to thinking of negative numbers. If we could accept the zero was a number, then what's to stop people from starting to use negative numbers, like -1 and -2 and -3? And that would lead into debt and money lending, which at the time was seen as very suspicious. And so for a long time, they said nope, we're going to stick to our old numbers, and we're not going to use these ones. To you and I, we are so used to using the numbers, it's hard to even imagine a time when they would have been new. 

HZ: Yeah. We're also so used to the concept of debt. 

KYNE: Very true. 

HZ: And people doing number fraud. 

KYNE: It would have been very easy to say, you know, let's not use these numbers though, so I don't have debt.

HZ: Yeah. It's just math that has put us in debt. 

KYNE: Yeah. You can blame math for that.

HZ: What happened to zero to let it catch on after such an inauspicious beginning in Europe? 

KYNE: The Renaissance happened; about 300 years after Fibonacci you had the Renaissance,   the printing press: and at this point it was undeniable how advantageous zero was and all the great things that it could bring to society. With zero came along the Hindu Arabic positional numeral system, which made math way easier. So the efficiency brought by zero had this great effect on engineering, science, astronomy, architecture, you name it. We could reach bigger numbers, and we could calculate things faster and with better precision. It was like the next best thing to a computer. 

HZ: So since the Renaissance, how have things been going for zero? 

KYNE: After it was accepted, I think there could have been no more doubt about whether this was a new number. Really the great thing about it is that it opened up our minds to start thinking of math in a more abstract way. What is a number? Is it just a practical tool for counting? Does it only exist in our minds, or is it a real thing? If it's only in our minds, then why can't we apply logical puzzles to it and see what other properties it has? If it's only in our minds, why can’t we add new numbers into our world that may not be useful for counting, may not be useful for some practical reasons, but can be useful for other things? When you open up your world to zero, then you can also open it up to negative numbers. And then later on, even complex numbers, imaginary numbers, which to this day still has students in a spin of, you know, is this a real number? I was one of those students when I was in school. But you can no longer say that the square root of -1 isn't a number because we used to say zero wasn't a number. 

HZ: You called zero “the Terminator”, which is some strong talk. It can't answer back, but what's your case? 

KYNE: Well, I think lots of students now, if you ask them what makes zero so dangerous, I think they would respond with dividing by zero. Dividing by zero, at least I was taught, was illegal math. I love that phrase, "illegal math". 

HZ: Yeah, you get a ticket if you try to divide something by zero. 

KYNE: Oh yeah. There's many stories of errors and disasters befalling people who divide by zero. Actually in 1977, a division by zero error caused the USS Yorktown to stop stranded in the middle of the water with a failed propulsion system for almost three hours. It had to be brought to a nearby harbour. For 13 years it managed to serve in the US Navy, never experiencing such a large incident; but all it took was a leftover zero left in the system by some engineer who didn't take it out, and the computer tried to divide by zero and everything failed.

HZ: Respect zero, or else! To be fair to it, it does have its compensations.

KYNE: It's really beautiful. As I said, it's sort of our first step into opening our minds to something that may contradict our previous worldview. But when we open our minds to something new, it can actually let us see the world in a way more interesting, complex, beautiful way.

HZ: And zero gave us that! 

KYNE: That's right. 

HZ: Our little nothing. 

KYNE: Our little nothing.

HZ: Thanks to Kyne for telling us about zero, and to learn more math from her but with the added visual interest of drag, you can find her on Instagram, TikTok, Youtube and Twitter at @onlinekyne.

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This episode was produced by me, Helen Zaltzman. Thanks to Hannah Fry. The music is by Martin Austwick of palebirdmusic.com and the experimental podcast Neutrino Watch.

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