They are back! Neil deGrasse Tyson is once again spouting total crap about the history of mathematics and has managed to stir the HISTSCI_HULK back into butt kicking action. The offending object that provoked the HISTSCI_HULK’s ire is a Star Talk video on YouTube entitled *Neil deGrasse Tyson Explains Zero*. The HISTSCI_HULK thinks that the title should read *Neil deGrasse Tyson* *is a Zero*!

You simple won’t believe the pearls of wisdom that NdGT spews out for the 1.75 million Star Talk subscribers in a video that has been viewed more than one hundred thousand times. If there ever was a candidate in #histSCI for cancellation, then NdGT is the man.

Before we deal with NdGT’s inanities, we need some basic information on number systems. Our everyday Hindu-Arabic number system is a decimal, that’s base ten, place value number system, which means that the value of a number symbol is dependent on its place within the number. An example:

If we take the number, 513 it is actually:

5 x 10^{2 }+ 1 x 10^{1} + 3 x 10^{0}

A quick reminder for those who have forgotten their school maths, any number to the power of zero is 1. Moving from right to left, each new place represents the next higher power of ten, 10^{0}, 10^{1}, 10^{2}, 10^{3}, 10^{4}, 10^{5}, etc, etc. As we will see the Babylonians [as usual, I’m being lazy and using Babylonian as short hand for all the cultures that occupied the Fertile Crescent and used Cuneiform numbers] also had a place value number system, but it was sexagesimal, that’s base sixty, not base ten. It is a place value number system that requires a zero to indicate an empty place. There are in fact two types of zero. The first is simply a placeholder to indicate that this place in the number is empty. The second is the number zero, that which occurs when you subtract a number from itself.

Now on to the horror that is NdGT’s attempt to tell us the history of zero:

HISTSCI_HULK: Not suitable for those who care about the history of maths

NdGT: I pick these based on how familiar we think we are about the subject and then throw in some things you never knew

HISTSCI_HULK: All NildGT throws in, in this video, is the contents of the garbage pail he calls a brain.

NdGT: For this segment, we’re gonna talk about zero … so zero is a number, but it wasn’t always a number. In fact, no one even imagined how to imagine it, why would you? What were numbers for?

Chuck Nice, Star Talk Host: Right, who counts nothing?

NdGT: Right, numbers are for counting … nobody had any use to count zero … For most of civilisation this was the case. Even through the Roman Empire…

Here NdGT fails to distinguish between ordinal numbers, which label the place that object take in a list and cardinal numbers which how many things are in a collection or set. A distinction that at one point later will prove crucial.

HISTSCI_HULK: When it comes to the history of mathematics NildGT is a nothing

CN: They were so sophisticated their numbers were letters!

In this supposedly witty remark, we have a very popular misconception. Roman numerals were not actually letters, although in later mutated forms they came to resemble letters. Roman numbers are collections of strokes. One stroke for one, two strokes for two, and so one. To save space and effort, groups of strokes are bundled under a new symbol. The symbol for ten was a crossed or struck out stroke that mutated into an X, the symbol for five, half of ten, was the top half of this X that mutated into a V; originally, they used the bottom half, an inverted V. The original symbol for fifty was ↓, which mutated into an L and so on. As the Roman number system is not a place value number system it doesn’t require a place holder symbol for zero. If Romans wanted to express total absence, they did so in words not numbers, nulla meaning none. This was first used in a mathematical context in the Early Middle Ages, often simply abbreviated to N.

NdGT: [Some childish jokes about Roman numeral] … I don’t know if you’ve ever thought about this Chuck, you can’t write zero with Roman numerals. There is no symbol for zero.

The Roman number system is not a place value number system but a stroke counting system that can express any natural number, that’s the simple counting numbers, without the need for a zero. The ancient Egyptian number system was also a stroke counting system, whilst the ancient Greeks used an alpha-numerical system, in which letters do represent the numerals, that also doesn’t require a zero to express the natural numbers.

NdGT: It’s not that they didn’t come up with it, it’s the concept of zero was not yet invented.

HISTSCI_HULK: I wish NildGT had not been invented yet

This is actually a much more complicated statement than it at first appears. It is true, that as far as we know, the concept of zero as a number had indeed not been invented yet. However, the verbal concept of having none of something had already existed linguistically for millennia. Imaginary conversation, “Can I have five of your flint arrowheads?” Sorry, I can’t help you, I don’t have any at the moment. Somebody came by and took my entire stock this morning.”

Although the Egyptian base ten stroke numeral system had no zero, by about 1700 BCE, they were using a symbol for zero in accounting texts. Interestingly, they also used the same symbol to indicate ground level in architectural drawings in much the same way that zero is used to indicate the ground floor in European elevators.

Also, the place holder zero did exist during the time of the Roman Empire. The Babylonian sexagesimal number system emerged in the third millennium BCE and initially did not have a zero of any sort. This meant that the number 23 (I’m using Hindu-Arabic numerals to save the bother of trying to format Babylonian ones) could be both 2 x 60^{1} + 3 x 60^{0} = 123 in decimal, or 2 x 60^{2} + 3 x 60^{0} = 7203 in decimal. They apparently relied on context to know which was correct. By about 700 BCE the first placeholder zero appeared in the system and by about 300 BCE placeholder zeros had become standard.

During the Roman Empire, the astronomer Ptolemaeus published his *Mathēmatikē Syntaxis*, better known as the *Almagest*, around 150 CE, which used a weird number system. The whole number part of numbers were written in a ten-base system in Greek alphanumerical symbols, whereas fractional parts were written in the Babylonian sexagesimal number system, with the same symbols, with a placeholder zero in the form of small circle, **ō.**

HISTSCI_HULK**: ** NildGT now takes off into calendrical fantasy land.

NdGT: So, when they made the Julian calendar, that’s the one that has a leap day every four years, … That calendar … that anchored its starter date on the birth of Jesus, so this obviously came later after Constantine, I think that Constantine brought Christianity to the Roman Empire. So, in the Julian calendar they went from 1 BC, BC, of course, stands for before Christ, to AD 1, and AD is in Latin, Anno Domini the year of our Lord 1, and there was no year zero in that transition. So, when would Jesus have been born? In the mythical year between the two? He can’t be born in AD 1 cause that’s after and he can’t be born in 1 BC, because that’s before, so that’s an issue.

CN: I’ve got the answer, it’s a miracle.

The Julian calendar was of course introduced by Julius Caesar in AUC 708 (AUC is the number of years since the theoretical founding date of Rome) or as we now express it in 44 BCE. The Roman’s didn’t really have a continuous dating system, dating things by the year of the reign of an emperor. Constantine did not bring Christianity to the Roman Empire, he legalised it. Both Jesus and Christianity were born in Judea a province of the Roman Empire, so it was there from its very beginnings. For more on Constantine and Christianity, I recommend Tim O’Neill’s excellent History for Atheists Blog.

To quote myself in another blog post criticising NdGT’s take on the Gregorian calendar:

*The use of Anno Domini goes back to Dionysius Exiguus (Dennis the Short) in the sixth century CE in his attempt to produce an accurate system to determine the date of Easter. He introduced it to replace the use of the era of Diocletian used in the Alexandrian method of calculating Easter, because Diocletian was notorious for having persecuted the Christians. Dionysius’ system found very little resonance until the Venerable Bede used it in the eight century CE in his Ecclesiastical History of the English People. Bede’s popularity as a historian and teacher led to the gradual acceptance of the AD convention. BC created in analogy to the AD convention didn’t come into common usage until the late seventeenth century CE. [Although BC does occur occasionally in late medieval chronicles.]*

As NdGT says Anno Domini translates as The Year of Our Lord, so Jesus was born in AD 1 the first year of our Lord, simple isn’t it.

I wrote a whole blog post about why you can’t have a year zero, but I’ll give an abbreviated version here. Although we speak them as cardinal numbers, year numbers are actually ordinal numbers so 2022 is the two thousand and twenty second year of the Common Era. You can’t have a zeroth member of a list. The year zero is literally a contradiction in terms, it means the year that doesn’t exist.

HISTSCI_HULK**: ** You can’t count on NilDGT

NdGT: So now, move time forward. Going, it was in the six hundreds, seven hundreds, I’ve forgotten exactly when. In India, there were great advances in mathematics there and they even developed the numerals, early versions of the numerals we now use, rather than Roman numerals. Roman numerals were letters [no they weren’t, see above], these were now symbolic shapes that would then represent the numbers. In this effort was the hint that maybe you might want a zero in there. So, we’re crawling now before we can walk, but the seeds are planted.

We have a fundamental problem dating developments in Hindu mathematics because the writing materials they used don’t survive well, unlike the Babylonian clay tablets. The decimal place value number system emerged some time between the first and fourth centuries CE. The symbols used in this system evolved over a long period and the process is too complex to deal with here.

The earliest known reference to a placeholder zero in Indian mathematics can be found throughout a commercial arithmetic text written on birch bark, the *Bakhshali *manuscript, the dating of which is very problematical and is somewhere between the third and seventh centuries CE.

The *Aryasiddhanta* a mathematical and astronomical work by Āryabhaṭa (476–550 n. Chr.) uses a decimal place value number system but written with alphanumerical symbols and without a zero. The *Āryabhaṭīyabhāṣya* another mathematical and astronomical work by Bhāskara I (c. 600–c. 680 n. Chr.) uses a decimal place value number system with early Hindu numerals and a zero. With the *Brāhmasphuṭasiddhānta *an astronomical twenty-four chapter work with two chapters on mathematics by Brahmagupta (c. 598–c. 668 n. Chr.) we arrive out our goal. Brahmagupta gives a complete set of rules for addition, subtraction, multiplication, and division for positive and negative numbers, as well as for zero as a number. The only difference between his presentation and one that one might find in a modern elementary arithmetic text is that Brahmagupta tried to define division by zero, which as we all learnt in school is not defined, didn’t we? Far from being “hint that maybe you might want a zero in there” this was the real deal.

HISTSCI_HULK: NildGT would be in serious trouble with the Hindu Nationalist propagators of Hindu science if they found out about his garbage take on the history of Hindu mathematics.

NdGT: These [sic] new mathematics worked their way to the Middle East. Baghdad specifically, a big trading post from all corners of Europe and Asia, and Africa and there it was. Ideas were put across the table. This was the Golden Age of Islam, major advances were made in all…in engineering, in astronomy, in biology, physiology, and vision. The discovery that vision is a passive phenomenon not active. So, all of this is going on and zero was perfected. They called those numerals Hindu numerals; we today call them Arabic numerals.

What NdGT doesn’t point out is that the Golden Age of Islam lasted from about 700 to 1600 CE and took place in many centres not just in Baghdad. The *Brāhmasphuṭasiddhānta* was translated into Arabic by Ibrahim ibn Habib ibn Sulayman ibn Samura ibn Jundab al-Fazri (ges. 777 n. Chr.), Muhammad ibn Ibrahim ibn Habib ibn Sulayman ibn Samura ibn Jundab al-Fazri (ges. c. 800 n. Chr.), and Yaʿqūb ibn Ṭāriq (ges. c. 796 n. Chr.) in about 770 CE. This meant that Islamicate^{[1]} mathematical scientists had a fully formed correct theory of zero and negative numbers from this point on. They didn’t develop it, they inherited it.

Today, people refer to the numerals as Hindu-Arabic numerals!

NdGt: So, this is the full tracking because in the Middle East algebra rose up, the entire arithmetic and algebra rose up invoking zero and you have negative numbers, so mathematics is off to the races. Algebra is one of the very common words in English that has its roots in Arabic. A lot of the a-l words, a-l is ‘the’ in Arabic as I understand it. So, algebra, algorithm, alcohol these are all traceable to that period. … So, I’m saying just consider how late zero came in civilisation. The Egyptian knew nothing of zero [not true, see above].

The Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī (c. 780–c. 850) wrote a book on the Hindu numeral system of which no Arabic text is known, but a Latin translation *Algoritmi de Numero Indorum* was made in the twelfth century. The word algorithm derives from the Latin transliteration *Algoritmi* of the name al-Khwārizmī. He wrote a second book *al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah* (c. 82O), the translation of the title is *The Compendious Book on Calculation by Completion and Balancing*. The term *al-Jabr* meaning completion or setting together became the English algebra.

The first time I heard this section I did a double take. “The entire arithmetic and algebra rose up invoking zero and you have negative numbers, so mathematics is off to the races”, you what! Ancient cultures had been doing arithmetic since at least three thousand years BCE and probably much earlier. I can’t do a complete history of algebra in this blog post but by the early second millennium BCE the Babylonians could solve linear equations and had the general solution to quadratic equations but only for positive solutions as they didn’t have a concept of negative numbers. The also could and did solve some cubic equations. In the middle of the first millennium BCE they had astronomical algorithms to predict planetary orbits, as well as lunar and solar eclipses. Brahmagupta’s work includes the general solution of linear equations, and the full general solution of quadratic equations, as we still teach it today. NdGT’s statement is total rubbish.

Of historical interest in the fact that although Islamicate mathematical scientists acquired negative numbers from Brahmagupta, they mostly didn’t use them, regarding them with scepsis

HISTSCI_HULK: NildGT is off with the fairies

CN: What is this that I hear about the Mayans and zero?

NdGT: I don’t fully know my Mayan history other than that they really worshipped Venus, so their calendar was Venus based. The calendar in ancient Egypt was based on the star Sirius [something unintelligible about new year]. It’s completely arbitrary when you say the new year’s just began. Pick a date whatever matters in your culture and call it new year. Even today when is the Chinese New Year, it’s late January, February. Everybody’s got a different starter date.

The Mayan culture developed a vigesimal, base twenty, place value number system, which included a placeholder zero, independent of the developments in the Middle East and India. The Dresden Codex, one of the most important Maya written documents contains a mixture of astronomy, astrology, and religion, in which observations of Venus play a central role. The first day of Chinese New Year begins on the new moon that appears between 21 January and 20 February

HISTSCI_HULK: I’d worship Venus, she was a very beautiful lady

CN: The Jewish New Year is another new year that…

NdGT: Everybody’s got another new year. The academic calendar’s got a new year that’s September the first…

I assume that NdGT is referring to the US American academic calendar, other countries have different academic years. In Germany where I live, each German state has a different academic year, in order to avoid that the entire population drive off into their summer holidays at the same time.

NdGT: …and by the way one quick question you’ve got a hundred dollars in your bank account, and you go and withdraw a hundred dollars from the cash machine and the bank tells you what?

[…]

So, here’s the thing, you have no money left in the bank and that’s bad, but what worse is to have negative money in the bank and so this whole concept of negative numbers arose and made complete sense once you pass through zero. Now instead of something coming your way, you now owe it. The mathematics began to mirror commerce and the needs of civilisation, as we move forward, because we are doing much more than just counting.

CN: So, this is like the birth of modern accounting. Once you find zero that’s when you’re actually able to have a ledger that shows you minuses and pluses and all that kind of stuff.

One doesn’t need negative numbers in order to do accounting. In fact, the most commonly used form of accounting, double entry bookkeeping, doesn’t use negative numbers; credits and debits are both entered with positive numbers.

Numbers systems and arithmetic mostly have their origin in accounting. The Babylonians developed their mathematics in order to do the states financial accounting.

HISTSCI_HULK: There’s no accounting for the stupidity in this podcast

NdGT: So now we’re into negatives and this keeps going with math and you find other needs of culture and civilisation, where whole other branches of math have to be developed and we got trigonometry. All those branches of math where you thought the teacher was just being angry with you giving you these assignments, entire branches of math zero started it all. Where it gives you deeper insights into the operations of nature.

I said I did a double take when NdGT claimed that arithmetic and algebra first took off when the Islamic mathematicians developed zero and negative numbers, which of course they didn’t, but his next claim completely blew my mind. *So now we’re into negatives and this keeps going with math and you find other needs of culture and civilisation, where whole other branches of math have to be developed and we got trigonometry.* I can hear Hipparchus of Nicaea (c. 190–c. 120) BCE, who is credited with being the first to develop trigonometry revolving violently in his grave.

HISTSCI_HULK: I could recommend some good books on the history of trigonometry, do you think NildGT can read?

There is another aspect to the whole history of zero that NdGT doesn’t touch on, and often gets ignored in other more serious sources. The ancient cultures that didn’t develop a place value number system, didn’t actually need zero. Almost all people in those cultures, who needed to do and did in fact do arithmetical calculations, didn’t do their calculation by writing them out step for step as we all learnt to do in school, they did them using the oldest analogue computer, the abacus or counting board. The counting board was the main means of doing arithmetical calculation from some time a couple of thousand years BCE, we don’t know exactly when, all the way down to the sixteenth century CE. An experienced and skilled user of the counting board could add, subtract, multiply, divide and even extract square roots much faster than you or I could do the same calculations with paper and pencil.

The lines or column on a counting board represent the ascending powers of ten in a decimal place value number system, powers of sixty on a Babylonian counting board. During a calculation, an empty line or column represents an implicit zero. In fact, there is one speculative theory that realising this led someone to make that zero explicit when writing out the results of a calculation and that is how the zero came into existence. Normally, when using a counting board only the initial problem and the result are recorded in writing and if one is using a stroke collection, ancient Romans and Egyptians, or an alphanumerical, ancient Greeks, as well as ancient Indian and Arabic cultures before they adopted Hindu numerals, number system, then, as already noted above, you don’t need a zero to express any number.

This blog post is already far too long but before I close a personal statement. I am baffled as to why a supposedly intelligent and highly educated individual such as Neil deGrasse Tyson chooses to pontificate publicly, to a large international audience, on a topic that he very obviously knows very little about, without taking the trouble to actually learn something about the topic before he does so. Maybe the fact that the podcast is heavily sponsored and littered with commercial advertising is the explanation. He’s just doing in for the money.

His doing so is an insult to his listeners, who, thinking he is some sort of expert, believe the half-digested mixture of half-remembered half-facts and made-up rubbish that he spews out. It is also a massive insult to all the historian of mathematics, who spent their lives finding, translating, and analysing the original documents in order to reconstruct the real history.

HISTSCI_HULK: If I were a teacher and he had handed this in as an essay, I wouldn’t give him an F, I would give it back to him, tell him to burn it, and give him a big fat **ZERO!**

^{[1]} Islamicate is the preferred adjective used by historians for mathematics and science produced under Islamic hegemony and published mostly in Arabic. It is used to reflect that fact that those producing it were by no means only Arabs or indeed Muslim