A while back the Internet was full of reports about a sensational discovery in the history of mathematics. Two researchers had apparently proved that a well know Babylonian cuneiform clay tablet (Plimpton 322), which contains a list of Pythagorean triples, is in fact a proof that the Babylonians had developed trigonometry one thousand years before the Greeks and it was even a superior and more accurate system than that of the Greeks. My first reaction was that the reports contained considerably more hype than substance, a reaction that was largely confirmed by an excellent blog post on the topic by Evelyn Lamb.

This was followed by an equally excellent and equally deflating essay by Eduardo A Escobar an expert on cuneiform tablets. And so another hyped sensation is brought crashing down into the real world. Both put downs were endorsed by Eleanor Robson author of *Mathematics in Ancient Iraq*: *A Social History* and a leading expert on Babylonian mathematics.

Last week saw the next history of mathematics press feeding frenzy with the announcement by the Bodleian Library in Oxford that an Indian manuscript containing a symbol for zero had been re-dated using radio carbon dating and was now considered to be from the third to fourth centuries CE rather than the eight century CE, making it the earliest known Indian symbol for zero. This is of course an interesting and significant discovery in the history of mathematics but it doesn’t actually change our knowledge of that history in any really significant way. I will explain later, but first the hype in the various Internet reports.

We start off with Richard Ovenden from Bodleian Libraries who announced,** “The finding is of “vital importance” to the history of mathematics.”**

The Guardian leads off with an article by Marcus Du Sautoy: *Much ado about nothing: ancient Indian text contains earliest zero symbol*. Who in a video film and in the text of his article tells us, “This becomes the birth of the concept of zero in it’s own right and this is a total revolution that happens out of India.”

The Science Museum’s article *Illuminating India: starring the oldest recorded origins of ‘zero’, the Bakhshali manuscript*, basically repeats the Du Sautoy doctrine,

Medievalists.net makes the fundamental mistake of entitling their contribution, *The First Zero*, although in the text they return to the wording, “the world’s oldest recorded origin of the zero that we use today.”

The BBC joins the party with another clone of the basic article, *Carbon dating reveals earliest origins of zero symbol*.

Entrepreneur Cecile G Tamura summed up the implicit and sometimes explicit message of all these reports with the following tweet, * One of the greatest conceptual breakthroughs in mathematics has been traced to the Bakhshali manuscript dating from the 3rd or 4th century at a period even earlier than we thought.* To which I can only reply, has it?

All of the articles, which are all basically clones of the original announcement state quite clearly that this is a placeholder zero and not the number concept zero[1] and that there are earlier recorded symbols for placeholder zeros in both Babylonian and Mayan mathematics. Of course it was only in Indian mathematics that the place-holder zero developed into the number concept zero of which the earliest evidence can be found in Brahmagupta’s *Brahmasphuṭasiddhanta* from the seven century CE. However, this re-dating of the Bakhshali manuscript doesn’t actually bring us any closer to knowing when, why or how that conceptual shift, so important in the history of mathematics, took place. Does it in anyway actually change the history of the zero concept within the history of mathematics? No not really.

Historians of mathematics have known for a long time that the history of the zero concept within Indian culture doesn’t begin with Brahmagupta and that it was certainly preceded by a long complex prehistory. They are well aware of zero concepts in Sanskrit linguistics and in Hindu philosophy that stretch back well before the turn of the millennium. In fact it is exactly this linguistic and philosophical acceptance of ‘nothing’ that the historian assume enabled the Indian mathematicians to make the leap to the concept of a number signifying nothing, whereas the Greeks with their philosophical rejection of the void were unable to spring the gap. Having a new earliest symbol in Indian mathematics for zero as a placeholder, as opposed to the earlier recorded words for the concept of nothingness doesn’t actually change anything fundamental in our historical knowledge of the number concept of zero.

There is a small technical problem that should be mentioned in this context. Due to the fact that early Indian culture tended to write on perishable organic material, such as the bark used here, means that the chances of our ever discovering manuscripts documenting that oh so important conceptual leap are relatively low.

I’m afraid I must also take umbrage with another of Richard Ovenden’s claims in the original Bodleian report:

* **Richard Ovenden, head of the Bodleian Library, said the results highlight a Western bias that has often seen the contributions of South Asian scholars being overlooked. “These surprising research results testify to the subcontinent’s rich and longstanding scientific tradition,” he said.*

Whilst this claim might be true in other areas of #histSTM, as far as the history of the so-called Hindu-Arabic numbers system and the number concept zero are concerned it is totally bosh. Pierre-Simon, marquis de Laplace (1749-1827) wrote the following:

*“It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.”*

I started buying general books on the history of mathematics more than 45 years ago and now have nine such volumes all of which deal explicitly with the Indian development of the decimal place value number system and the invention of the number concept zero. I own two monographs dedicated solely to the history of the number concept zero. I have four volumes dedicated to the history of number systems all of which deal extensively with the immensely important Indian contributions. I also own two books that are entirely devoted to the history of Indian mathematics. Somehow I can’t see in the case of the massive Indian contribution to the development of number systems that a Western bias has here overseen the contributions of South Asian scholars.

This of course opens the question as to why this discovery was made public at this time and in this overblown manner? Maybe I’m being cynical but could it have something to do with the fact that this manuscript is going on display in a major Science Museum exhibition starting in October?

The hype that I have outlined here in the recent history of mathematics has unfortunately become the norm in all genres of history and in the historical sciences such as archaeology or palaeontology. New discoveries are not presented in a reasonable manner putting them correctly into the context of the state of the art research in the given field but are trumpeted out at a metaphorical 140 decibel claiming that this is a sensation, a discipline re-defining, an unbelievable, a unique, a choose your own hyperbolic superlative discovery. The context is, as above, very often misrepresented to make the new discovery seem more important, more significant, whatever. Everybody is struggling to make themselves heard above the clamour of all the other discovery announcements being made by the competition thereby creating a totally false impression of how academia works and how it progresses. Can we please turn down the volume, cut out the hype and present the results of academic research in history in a manner appropriate to it and not to the marketing of the latest Hollywood mega-bucks, blockbuster?

[1] For those who are not to sure about these terms, a placeholder zero just indicates an empty space in a place value number system, so you can distinguish between 11 and 101, where here the zero is a placeholder. A number concept zero also fulfils the same function but beyond this is a number in its own right. You can perform the arithmetical operations of addition, subtraction and multiplication with it. However, as we all learnt at school (didn’t we!) you can’t divide by zero; division by zero is not defined.

I agree. The hype was unnecessary as the discovery didn’t give any new information. Zero as a number is important, not zero symbol as a placeholder. Zero symbol as a placeholder was known in Babylonian and Mayan Civilization, centuries earlier than 3rd/4th century. I am certain, those who started the hype were aware of it. But they just wanted to capture headline. It is unfortunate that academicians are twisting fact for personal gain.

I don’t think the blog post by Evelyn Lamb is “excellent” as you and Eleanore Robson have repeatedly stated. First of all it doesn’t actually engage with the key question: Is Plimpton 322 a trig table? Why or why not? Lamb simply has nothing to say about this. Instead she only attacks two much less interesting claims, namely whether the “Babylonian” trig and number systems are “better” than modern ones. Naturally, these are not the kinds of questions modern historians take very seriously. Indeed, they are much more prominent in the media coverage than in the actual published paper. And indeed, Lamb bases a good chunk of her critique on a promotional video rather than the published paper itself. The “equally excellent” essay by Eduardo Escobar likewise says absolutely nothing about the key question.

I think historians are doing themselves no favours by praising as “excellent” some superficial blog posts that don’t actually tackle the substantive question at stake. From their eagerness to praise these writings, one can easily get the impression that historians care more about the right kind of historiographic posturing (which those blog posts have in spades) than actual content and substance.

Well said Victor. It is pretty clear that neither Evelyn Lamb nor Eduardo Escobar based their blogs/essays on a careful reading of our paper. I am hoping that they will find the time to do so in the next few months. I would be quite interested what some careful reflection might bring forth from them, and from Eleanore Robson.

Reblogged this on Project ENGAGE.

Apologies if I have accidentally posted this twice.

I am a science journalist, and former mathematician, who covered the cuneiform story.

The Scientific American post simply does not relate in any way to the paper I read. At its core, that paper was series of worked examples showing the tablet could be used as a trig table for, for instance, simple architectural calculations. The ordering of numbers in it only really make sense if interpreted as relating to triangles of increasing angle size – and they contend this could be used to interpolate solutions in general cases. The stuff about accuracy, which I left off, is just a fun aside about base 60 division.

I can only conclude the author didn’t read the paper. Neither did Eleanor Robson. I know this because she told me so. When – like several journalists – I went to her for comment, she refused That is her prerogative – but to my mind completely disqualifies her from criticising subsequent coverage. It’s hard to find an expert in cuneiform and maths – all the half dozen people I tried recommended her.

At its core, that paper was series of worked examples showing the tablet could be used as a trig table for, for instance, simple architectural calculations. The ordering of numbers in it only really make sense if interpreted as relating to triangles of increasing angle size…I haven’t read the paper either, but you will find that exact argument—the increasing angle—in Otto Neugebauer’s

The Exact Sciences in Antiquity, 2nd edition published in 1957, based on lectures given in 1949; Neugebauer and Sachs first gave the interpretation in 1945 (see this MAA review). So what’s so new about the paper by Mansfield and Wildberger?I can’t make that link work – could you paste in another, sorry! I’d love to read it.

They were not the first to make a link with trigonometry, and they acknowledged that happily. They specifically cite Neugebauer’s interpretation, looking at the increasing angles. But it was my understanding that they went further in showing how it could be used to solve real world problems – demonstrating how you would get a test statistic in the general case, use it in the table, derive sin, cos and (especially) tan etc.

If, in fact, all of that has been said before then that is a valid criticism of their paper (and I don’t understand how it got published).

But that is not the criticism I saw. The criticism I saw was based on a Chinese whisper process that seemed to have entirely missed the original paper. It just didn’t intersect with it.

There is, you must see, something quite astonishingly tedious about reading a supposed “debunking” of silly journalists by someone who hasn’t read the paper, which is then applauded by a whole load of other people who haven’t read the paper – among them people who were specifically asked to comment on the paper but refused!

As an aside, I’m a mathematician not an historian. After working through their examples though I simply cannot see how theirs is not the correct usage. It seems like a phenomenal contention that a tablet could exist that is logically ordered to act as a trig table, that contains nothing not used in a trig table, and that all that happened by chance.

First, here’s the link (I hope correctly formatted) to the

MAA review of

The Exact Sciences in Antiquity.I skimmed the Mansfield and Wildberger paper. I have to agree with Lamb’s takedown. As the old quip has it, “What’s new is not true, and what’s true is not new”.

Reading the abstract, you’d easily conclude they were first ever to suggest it’s a trig table. And the body of the paper

nevermakes it clear that that’s not so. Neugebauer and Sachs get cited frequently, but basically just as the authors ofMathematical Cuneiform Texts.Now, I gather “trig table” interpretation is controversial among the experts, but let’s just skip that, for the sake of argument. What’s new about their claim?

As far as I could tell, their new twist on a 70 year-old theme is just what Lamb said: the “exact ratios” stuff. Not just a fun aside, but their only claim to originality. So I think Lamb’s post was right on point.

I’m not sure you’re right.

Even if you are, an appropriate take down would be “this is not new, and here is why it is also wrong”.

Since academic opinion, from historians at least, maintains this is not a trig table, and since the entire point of the paper was to show it was, and since the only reason this is remotely interesting is because of that contention, I want to know why it isn’t. Why is something that works perfectly for this task not designed for this task? How can historians still maintain that the assumptions in this paper are ahistorical, western centric etc?

Even – and I don’t believe this is true – if the paper was wholly unoriginal, that remains the pertinent point. And I’m none the wiser.

The paper clearly states: “Some authors have raised the possibility that P322 was a trigonometric table in the modern sense. … But the possibility that P322 is an exact sexagesimal trigonometric table, without the assumption of a circle-based measurement system based on angles, has not been considered until now.” (p. 16)

Neugebauer never claimed it was a trig table.

Just discovered your initial post in the spam filter. WordPress does that sometimes, sorry. As it was substantially the same as above I have committed it to cyberspace nirvana

I will address both Viktor Blåsjö and Tom Whipple together in my comment. There is a saying in science and the history of science that strong claims demand strong evidence. The central claim made by Mansfield and Wildberger in within the context of the history of Babylonian mathematics a very strong one indeed and as Evelyn Lamb points out it is almost totally speculative, meaning they provide virtually no supporting evidence for their claim.

Something that makes me highly suspicious is the fact that their speculative interpretation makes Plimpton 322 into a predecessor of just the supposedly superior system of trigonometry that Norman J. Wildberger has been trying to sell to the world since 2005. Maybe I’m just a cynic but that seems like a rather strange coincidence to me.

In his essay Eduardo A Escobar shows that the type of mathematical concept that Mansfield and Wildberger are promoting in their paper would be totally alien to everything that we know about how the Babylonians conceived mathematics. This is again a not insubstantial criticism that they do not address.

When I, in my opinion totally correctly, described the press presentation of Mansfield and Wildberger, I was addressing the fact that it was being presented as a substantiated, evidence based breakthrough in the history of mathematics and not the unsubstantiated piece of speculation that it actually is. It is a potentially interesting theory but one that is a long, long way from being proven in anyway whatsoever.

Eleanor Robson did indeed refuse to comment initially, when the first press releases appeared, and yes I asked her opinion too, but that in no way disqualifies her from endorsing the opinions of Lamb and Escobar at a later date. She initially refused in the beginning because she was involved in the final stages of delicate negotiations for a research grant to preserve the cultural inheritance of Iraq. When she later read and endorsed Lamb and Escobar, the negotiations were over and the monies for her research project had been granted.Tom Whipple’s claim that her initial refusal to comment disqualifies her from commenting at a later date quite honestly beggars belief.

I agree that both the Plimpton 322 and zero hysteria are ridiculous, as I have stated in these threads:

Nevertheless I think the posts by Lamb and Escobar are not very good, even though they advocate for the right conclusion.

Sorry, because of the links this comment got caught in the spam filter where I only discovered it today

Well, I’m not entirely sure why any of that about Robson prevented her from commenting – she managed to do so the day after publication after all. I also don’t know why it precluded her, when I asked, from recommending anyone else for me to speak to.

She just told me she did the research a long time ago, and had moved on now to more interesting things so couldn’t comment.

So you can see why I might be surprised she then decided to comment? It was one of the oddest interactions I’ve ever had with an academic – and there’s no shortage of competition.

This paper showed that the tablet could be used, very effectively, for basic trig calculations. That is unambiguous, and that is all that I at least wrote about – so in my case at least what you and lamb criticised did not accord with press coverage! The only relevant question, to which I still – despite all this criticism – have no answer is, was it?

Now to me I find the probability that that you would happen upon this purely by chance pretty low – but I am open to counter arguments. I just haven’t heard them. If it is indeed ahistorical and out of the context of Babylonian maths then that’s even more interesting – because, again, here is a trig table, so what else is it? Why do so many people object to his paper – yet don’t seem to say why?

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