[INDOLOGY] Once again on the origin of zero: the date of the Bakhshali manuscript (or manuscripts?)

[George L. HART]

Ingalls once pointed out that “k” in Paninean grammar (e.g. “kvip”) stands
for nothing, i.e. no change. He suggested that the Indian invention of zero
might be related to the fact that they had a way of indicating “zero
change” in grammar.  In light of this, the early invention of zero does not
appear implausible.

George

Sent from my iPad

On Sep 15, 2017, at 5:14 PM, Camillo Formigatti via INDOLOGY <
indology at list.indology.info> wrote:

Dear Andrea,



Many thanks for your enthusiasm about this discovery!


I will try and reply to each point you raise, but allow me to quote at the
outset a very telling passage written by Cecil Bendall in 1882, in his
article *On European Collections of Sanskrit Manuscripts from Nepal: Their
Antiquity and Bearing on Chronology, History and Literature*. When I was
first told about the results of the radiocarbon dating of the Bakhshali
manuscript, I did not believe it, in fact my very first thought was that
they must had done something wrong at the Oxford Radiocarbon Accelerator
Unit. My second thought then was precisely that the scribe (or scribes)
must had used old birch-bark sheets to write parts of the manuscript in
later centuries. However, at that moment I was also working on an article
on the history of the Cambridge collections of Sanskrit manuscripts and
this quote by Cecil Bendall helped me to keep an open mind:

"The early dates of some of these MSS. have been, indeed, received in some
quarters with certain incredulity; but for myself, I must testify that,
after about two years study, both of the great Cambridge collection, of
which I have been during this time engaged in preparing a catalogue, and of
various Buddhistic MSS. in other libraries, the truthfulness and
genuineness of the colophons is placed in almost every case beyond a doubt
by evidence both varied and conclusive." (Bendall 1882, 190).

I think this is a very telling passage in which our attitude towards this
type of discoveries is very aptly described. Apparently, we haven't changed
much in the last one-hundred years. In the 19th century, some of our
esteemed colleagues didn't even believe that the colophons of old Nepalese
palm-leaf manuscripts were authentic. Surely, as we all know, there are
cases of colophons of old antigraphs copied together with the main text into
more recent apographs, but can we honestly say that they are the great
majority of colophons of South Asian manuscripts?


Let me now reply to your points.

"If the manuscript (however fragmentary it may be) is thought to contain a
single, unitary text, then the date of its copying (and/or composition?)
must be the 9th-10th century. I fail to see what is so sensational about
this apart from the fact that it shows how writing supports that were
centuries older might have been (re)utilized."

On my part, I fail to see why you think more plausible that somebody reused
700-years old birch-bark sheets to write on it, when they could have used
contemporary birch-bark sheet, which would have surely been less fragile. I
rather think that we are dealing with a composite manuscript consisting of
at least three distinct codicological units. Actually, what seems more
plausible to me is exactly the reverse, that somebody supplemented the
damaged parts of an old manuscript by copying them on new birch bark, as
often is the case in old North Indian and Nepalese palm-leaf manuscripts
with damaged folios (as for instance happened with this 11th century
Nepalese palm-leaf manuscript:
https://cudl.lib.cam.ac.uk/view/MS-ADD-01643/1). If you however think that
your hypothesis is more plausible, then I have to say that I disagree with
you.

"(By the way: is an analysis of the ink technically possible?)."

It’s not as simple as you might think. It is something we are considering
though, as this would be the only way to ascertain what really happened,
whether the scribe (or scribes) used old birch-bark sheets to write a new
text in the 10th century or if they replaced old and damaged sheets with
new birch-bark sheets on which they copied the text of the damaged folios.

"The earliest attestation of the written zero would still be the
8th-century Southeast Asian inscriptions (and not the Gwalior temple, as
incorrectly reported in the article)."

Well, if you are convinced that the results of the radiocarbon dating of
the Bakhshali manuscript are wrong, then you are right and the zero in 8th
century Southeast Asian inscriptions would be the oldest attestation of a
symbol for a placeholder in a decimal positional notation system (and
strictly speaking, not of a symbol for zero, as I try to explain below).
As I said above, I was prepared for the scepticism of colleagues, since my
first reaction too was that these dates are impossible. On the other hand,
I have to say that I don’t see why we can accept without blinking the
radiocarbon dating of the Gāndhārī material or of early Sanskrit
manuscripts such as the Spitzer manuscript or of other Buddhist manuscripts
from Turfan and other Silk Road sites (all listed in the article by Allon,
Jacobsen, Zoppi, and Salomon in the third volume of the Buddhist Schoyen
manuscripts, 2006), while in the case of the Bakhshali manuscript we ought
not to accept them. I don't know, maybe I’m too naïve to understand the
arguments, but what it seems to me is that we are all sceptical because it
is difficult to explain these results, since they go against several
assumptions we take for granted. I believe that what is puzzling you is the
fact that we have three different dates. Imagine if we would have chosen to
radiocarbon date only one fragment and not three, or if I would have chosen
the three fragments only from the part denoted as M by Kaye in his editio
princeps. In the first case, we would have had one single date as result
(either the oldest, the middle, or the latest one), and in the second case
we might probably have had three similar results with only one later date,
which would have convinced us that we are all correct in our assumptions
about the development of North Indian scripts, particularly of the Śāradā,
as we would have thought "Hey, this is great, a 10th-century (proto-)Śāradā
manuscript! After the inscriptions in (proto-)Śāradā, we finally have the
oldest dated manuscript against which we can date other Śāradā inscriptions
and manuscripts." Alas, I chose instead three fragments from parts of the
manuscript (or manuscripts?) which according to Kaye were written in two
(or even three) different hands, precisely because we all wanted to be sure
to have a sound approach—and I wanted to verify Kaye's palaeographical
analysis and understand whether the two (or three scribes) wrote in
different periods or not. This is the reason why we have now these three
different results. (Honestly, I didn't expect this huge difference in the
dates of the three fragments, above all because I thought that the results
would have ranged between the 10th and the 12th century; incidentally,
Hayashi did not fully believe in the correctness of Kaye's palaeographical
assessment and for this reason his reconstruction of the order of the
folios differs from Kaye's.) Maybe we ought to reassess our assumptions a
bit, instead of blaming the results—but again, I am probably too naïve, or
ignorant, or not knowledgeable enough and my hypothesis are all wrong
(above all my assessment of the position of the script of folio 16 in the
development of the Gilgit-Bamiyan scripts, which I will describe in the
article I mentioned, but about which I am not totally sure myself, as often
is the case in palaeographical studies).



(By the way, the Guardian article doesn’t mention the Gwalior inscription,
which is actually mentioned in the video, and it is not totally incorrect
what Prof. du Sautoy says, because he doesn’t say anything about the
Gwalior inscription as being the oldest attestation of a written zero, he
simply mentions it as an old attestation of this symbol. Anyway, I have
always stressed to all other colleagues involved in the project at various
levels that the oldest attestations of a place-holder symbol in the form of
a zero are from Southeast Asia and not in the Gwalior inscription, as they
sometimes said—as you can imagine, I have closely followed the recent
discussion about this topic here on Indology, but I kept my mouth shut
because I couldn’t reveal our results due to an embargo from the Bodleian
Libraries Communication office. If you read anything in  which somebody
from the University of Oxford says that the place-holder symbol in the Gwalior
inscription is the oldest attestation of the symbol for zero, well
evidently they haven’t listened to me.)

"But in your message, when you speak about different stratas and tables of
ak.saras, you clearly imply that this/these manuscript(s) contain(s) a
composite/heterogeneous text indeed, and that part of it might date back to
the 3rd-4th century. May I ask you to anticipate/synthesize some of your
key findings here, or at least clarify this point? And, what is the
relationship between folios 16 and 17?"

At this moment, I am in no position to say with certainty whether we are
dealing simply with a composite manuscript consisting of three
codicological units (a kernel and two or more dependent codicological
units) and at the same time containing only one single sūtra + commentary
textual unit, or rather with a composite *and* multi-text manuscript,
consisting of three (or even more) independent codicological units
(possibly even without a clear kernel!) and at the same time containing two
or more different sūtra + commentary textual units. I have to specify that
with sūtra + commentary textual unit I mean the group of [(sūtra +
udāharaṇa) and (commentary + nyāsa (sthāpana or nyāsasthāpana), karaṇa and
one or more pratyayas)], which I consider as a single text (although I
might be totally wrong). So as you can see, to a certain extent we could
already say that this is a multi-text manuscript. I do believe though that
we have to keep an open mind, start our research from scratch and possibly
rethink the grouping of the folios. For this reason again I am in no
position to say anything more about the relationship of folio 16 and 17, as
I don't know anything of Indian mathematics (and very very very little of
mathematics in general). Our intention has been from the very start of this
project simply to verify the date of the Bakhshali manuscript with the only
means that we thought would clarify the one-century old debate about the
antiquity of this manuscript and its implication for the history of
mathematics, since the palaeographical and textual means employed so far
provided a wide range of dates. We would have then left the results in the
hand of the real experts, like  Takao Hayashi and Agathe Keller, with whom
we were already in contact before starting the project. Unfortunately—or
luckily—we complicated the picture even more, and here I am replying to
your questions.


Below I have pasted an extract from the part I have prepared for the
Bodleian’s official ‘Bakhshali Research Statement&Background’ document (the
full document can be found at this link:
http://www.sciencemag.org/sites/default/files/Bakhshali%20Research%20Statement_13%209%2017_FINAL.pdf).
Please bear in mind that this report is written for non-specialists, so
some parts of what I wrote might sound very generic and not totally exact
or sufficiently supported by scholarly evidence in the eyes of Indologists.
In my article I will obviously try and underpin with textual and
palaeographical evidence some of the hypothetical statements (and I stress
hypothetical!) that I make in these very short report. I hope however that
it provisionally answers some of your questions.

"Do all these folios contain the 0?"

All folios contain the symbol 0 used either as a placeholder or as a symbol
for an unknown, but not as a zero in its own rights (as I try to explain
below).

"Further: I'm not steeped in mathematics either, so I fail to grasp the
full implications of this statement (especially the second sentence): "In
the fragile document, zero does not yet feature as a number in its own
right, but as a placeholder in a number system, just as the “0” in “101”
indicates no tens. It features a problem to which the answer is zero, but
here the answer is left blank"."

As to the question of what is a zero, what Prof. du Sautoy means is that
one thing is to have any symbol­ (in our case, the dot that will evolve
into the hollow symbol 0) as a placeholder to denote the absence of units,
tens, hundreds, etc. in a positional notation system (be it decimal, as in
the Indian system, or sexagesimal, as in the Babylonian system). This fact
is per se not so exciting, as it is clearly explained even in the Guardian
article: ‘Several ancient cultures independently came up with similar
placeholder symbols. The Babylonians used a double wedge for nothing as
part of cuneiform symbols dating back 5,000 years, while the Mayans used a
shell to denote absence in their complex calendar system.’ What is more
exciting in the Bakhshali manuscript is another function of the dot, i.e.
precisely the fact that it represents an unknown (for instance in
equations). However, as I understand it (but my knowledge and understanding
of mathematics is very poor), this is again different than thinking of
having as result of an operation zero, i.e. nothing, from something. I hope
my explanation is clear, as it took me sometime to understand why Prof. du
Sautoy was desperately looking for what we laymen called a ‘mathematical’
zero, when we all thought that the Bakhshali was already full of them!


Best wishes,



Camillo



The carbon dating has revealed that the Bakhshali manuscript is a composite
document and consists of at least three different parts with different
ages. Due to the fragmentary nature of the manuscript, it is difficult to
reconstruct the original order of the folios and consequently of the extant
texts. Until now, the manuscript has been studied as if it were one single
item. The first editor of the manuscript, G. R. Kaye, employed the
following criteria to establish the order of the folios:

(1)          Logical sequence of contents.

(2)          The ‘find order.’

(3)          Physical appearance such as the size, shape, degree of damage,
and knots in the birch bark.

(4)          The script and language.

(5)          Numbered sūtras.

In the most recent complete study of the manuscript (1995), Takao Hayashi
does not deem the fourth criterion reliable enough for the reconstruction
of the order of the folios. This approach is again based on the assumption
that all extant parts of the manuscript were written at the same time – an
assertion now overturned by the new carbon dating results.



The early date of folio 16 fits well into the cultural milieu in which this
part of the manuscript was probably produced and circulated. The content of
the Bakhshali manuscript is similar to the type of texts that Buddhist
merchants would have needed to study (and possibly use as reference) for
their daily trading activities. It includes very practical mathematical
examples and equations, such as how to compute the loss in weight of a
quantity of impure metal in the process of refining it, etc.



The manuscript was recovered in the village of Bakhshali in Pakistan, in an
area that belongs to the historical Gandhāra region. It is a region in
which major cities such as Peshawar (Skt. Puruṣapura) and Taxila (Skt.
Takṣaśilā) were important commercial and cultural hubs. This area belonged
to the Indian, Persian, and Greek cultural spheres of influence and had
contacts with Chinese culture through the Silk Road. The oldest Buddhist
and South Asian manuscripts extant were also found in the Greater Gandhāra
region and their dates range from the first century BCE to the fourth
century CE – so a similar time period to that of folio 16 of the Bakhshali
manuscript. Although written in a different script and a different language
than the Bakhshali manuscript, these Buddhist manuscripts are also written
on birch-bark.



Moreover, the language used in the manuscript is not standard Sanskrit, and
according to Hayashi it shows features of what is called Buddhist Hybrid
Sanskrit, as well as of other Middle-Indo Aryan languages (Prakrit and
Apabhraṃśa), and also of Old Kashmiri. This is yet another feature that can
be explained by the fact that the manuscript in its present state is
composed of at least three different manuscripts with different dates. The
three dates would then correspond to different stages of linguistic
development.


------------------------------



Dr Camillo A. Formigatti

John Clay Sanskrit Librarian



Bodleian Libraries

The Weston Library

Broad Street, Oxford

OX1 3BG



Email: camillo.formigatti at bodleian.ox.ac.uk

Tel. (office): 01865 (2)77208
www.bodleian.ox.ac.uk



*GROW YOUR MIND*

in Oxford University’s

Gardens, Libraries and Museums

www.mindgrowing.org



*From:* Andrea Acri [mailto:andrea.acri at ephe.sorbonne.fr
<andrea.acri at ephe.sorbonne.fr>]
*Sent:* 15 September 2017 01:42
*To:* Indology List <indology at list.indology.info>
*Subject:* Re: [INDOLOGY] Once again on the origin of zero: the date of the
Bakhshali manuscript (or manuscripts?)



Dear Camillo



thank you for sharing this news, and especially for your (in)valuable work
on this most important document. Let me point out at the very outset that
all I know about this manuscript derives from the Guardian article and
Wikipedia (disclaimer: I have no access to a library right now!), so please
forgive me for being so naive.



If the manuscript (however fragmentary it may be) is thought to contain a
single, unitary text, then the date of its copying (and/or composition?)
must be the 9th-10th century. I fail to see what is so sensational about
this apart from the fact that it shows how writing supports that were
centuries older might have been (re)utilized.  (By the way: is an analysis
of the ink technically possible?). The earliest attestation of the written
zero would still be the 8th-century Southeast Asian inscriptions (and not
the Gwalior temple, as incorrectly reported in the article).



But in your message, when you speak about different stratas and tables of
ak.saras, you clearly imply that this/these manuscript(s) contain(s) a
composite/heterogeneous text indeed, and that part of it might date back to
the 3rd-4th century. May I ask you to anticipate/synthesize some of your
key findings here, or at least clarify this point? And, what is the
relationship between folios 16 and 17? Do all these folios contain the 0?



Further: I'm not steeped in mathematics either, so I fail to grasp the full
implications of this statement (especially the second sentence):



"In the fragile document, zero does not yet feature as a number in its own
right, but as a placeholder in a number system, just as the “0” in “101”
indicates no tens. It features a problem to which the answer is zero, but
here the answer is left blank".



Hopefully some of our learned colleagues will be able to clarify this point.



Best regards

Andrea Acri



Sent from my iPhone


On 14 Sep 2017, at 17:15, Camillo Formigatti via INDOLOGY <
indology at list.indology.info> wrote:

Dear Colleagues,



I’m pleased to be finally able to share this exciting news with you:



https://www.theguardian.com/science/2017/sep/14/much-ado-about-nothing-ancient-indian-text-contains-earliest-zero-symbol



I imagine that some of you might probably raise their eyebrows after
reading this article. The results came as a big surprise to us too, and to
me were literally jaw-dropping. I realize that these results have several
implications not only for the history of mathematics, but also for our
field of study, and I know that the article in The Guardian surely doesn’t
answer the many questions you might be asking yourselves now. I will try to
briefly anticipate some of them.



The decision and implementation of radiocarbon dating the Bakhshali
manuscript took several months of preparation on the part of the team of
colleagues with which I collaborated. The team included colleagues from the
Bodleian Libraries and other University of Oxford departments: David Howell
(Bodleian Libraries’ Head of Heritage Science), Dr Gillian Evison (Head of
the Bodleian Libraries' Oriental Section & Indian Institute Librarian),
Virginia M Lladó-Buisán (Bodleian Libraries’ Head of Conservation and
Collection Care), Dr David Chivall (Chemistry Laboratory Manager at the
School of Archaeology of the University of Oxford), and Prof. Marcus du
Sautoy (Charles Simonyi Professor for the Public Understanding of Science
and Professor of Mathematics at the Oxford University). We decided to take
samples from three folios in order to be sure to have a sensible margin of
certainty for the results. I chose folios 16, 17, and 33, and the analysis
was conducted by Dr Chivall at the Oxford Radiocarbon Accelerator Unit. The
results of the calibrated age (95.4% confidence interval / cal AD) are as
follows:



Folio 16:               224 (95.4%) 383calAD

Folio 17:               680 (74.8%) 779calAD

790 (20.6%) 868calAD

Folio 33:               885 (95.4%) 993calAD



We did not expect such a big difference in the date range of the three
folios. I am currently preparing an article in which I provide the
background for the choice of these three specific folios, tables of all
akṣaras from the three folios as an aid to assign the extant folios to the
different strata of the manuscript (including selected aksaras of other
dated and undated manuscripts in similar scripts for comparison), and a
first palaeographical appraisal of the results.



Best wishes,



Camillo


------------------------------



Dr Camillo A. Formigatti

John Clay Sanskrit Librarian



Bodleian Libraries

The Weston Library

Broad Street, Oxford

OX1 3BG



Email: camillo.formigatti at bodleian.ox.ac.uk

Tel. (office): 01865 (2)77208
www.bodleian.ox.ac.uk



*GROW YOUR MIND*

in Oxford University’s

Gardens, Libraries and Museums

www.mindgrowing.org



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