Where do numbers come from?

Join Robert and Joe as they continue their discussion on numeracy and the origins of numerals on Stuff to Blow Your Mind.

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Episode Transcript

Speaker 1 (00:03):
Welcome to Stuff to Blow Your Mind, the production of
My Heart Radio. Hey you welcome to Stuff to Blow
your Mind. My name is Robert lamp and I'm Joe McCormick,
and we're back to talk numbers again. We promised you
would happen, and it happened, maybe sooner even than you
were expecting. So in the last episode of this show,

(00:26):
we were talking about the human number sense and uh,
different ideas about to what extent our sense for numbers
might be partially innate, partially a cultural invention, and what
the arguments and evidence on each side of that question
would be. But today we wanted to look at some
of the evidence from history and archaeology about where our

(00:47):
earliest like like real direct indications of number use come from,
and uh and what some some solid physical evidence of
that kind of thing might be, and and questions on
how best to interpret those things. And it's not really
fascinating stuff because it's not just I guess it's easy
without knowing much about it to sort of think, well, okay,
there's you're talking about just evidence of humans in various

(01:10):
cultures or you know, ancient groups doing some sort of mathematics,
some sort of figures. Uh. But the more you look
at it, you just see how interconnected UH numerals math
are with technology, with civilization itself, with humanity's ability to
do anything that humans do, certainly at scale, but it's

(01:33):
it's at a time surprising just how um, you know,
how ancient all of this stuff is. Yes, And in
that exact spirit, I wanted to start off by talking
about a particular artifact today. I thought this would be
a good way to get into the subject. And this
artifact is what's today known as the Shango bone. So

(01:54):
in the nineteen fifties, there was a Belgian geologist named
John de hind Salon who was He lived a nineteen
twenty and nineteen and he was doing excavations around the
shore of Lake Edward, which is on the border of
uh what is now the Democratic Republic of the Congo
in the in the Verunga Park region of the northeast

(02:18):
of the country. And one of the artifacts that was
uncovered at this dig during this field work was an
ancient piece of animal bone from roughly maybe twenty thousand
years ago. There have been different dates given at different times,
but I think the the standard consensus now is that
this is something like twenty thousand to twenty five thousand
years old, and this piece of animal bone had several

(02:42):
unusual features. First of all, it had a chunk of
quartz quartz crystal embedded in the tip at one end
of the bone. And also it was covered with groups
of slashes carved into its sides. Uh. So it's known
as the Ango bone today. And what's so fascinating about

(03:02):
this artifact is that it is now often interpreted as
an ancient piece of mathematical technology. And if that's correct,
it would be one of the oldest known mathematical tools
in the archaeological record. There are there are a few
that are as old or older, but this is going
way back. I mean long before say, the ancient civilizations

(03:24):
of Mesopotamia, where we imagine mathematical and counting tools being used.
This would be like twenty thousand years ago. So why
do some scientists interpret this twenty thousand year old piece
of bone as a mathematical technological tool. Well, I was
reading about this, uh, some of the various interpretations of

(03:45):
this artifact, and in a short booklet created by the
Royal Belgian Institute of Natural Sciences, which is the Museum
that now has this artifact in its collection. Now I'll
give a bit more physical detail here. First of all,
this is one of the few composit it prehistoric tools
that has survived all the way to modern archaeological discovery intact.

(04:07):
So you know, when you think about composite tools, you
might think of a an axe head that is joined
to a stick, right to create more leverage on an
ax But a lot of times these joinings don't survive
across time, don't survive the tens of thousands of years
to be discovered in a modern you know, dug up
at a modern excavation. Um. But this is one case
where it is a composite tool with multiple pieces put together,

(04:31):
and it was found with the pieces still together, so
the quartz tip was still stuck in the end of
the bone. And this is definitely worth looking up a
picture of. But but I want to drive honder the
courts tip, at least in the images that that I'm
presented with here, Uh, it does look very utilitarian, like
it's easy to imagine like a quartz tipped ancient uh,

(04:51):
you know, wand as being some sort of thing that
looked more ceremonial or even magical um, but but it
does look very utilitarian, at least to my eyes. Yeah,
it has been interpreted as possibly useful for making carvings
or marking, so you can imagine the quartz tip possibly
being kind of like the lead and a pencil or

(05:13):
or a chisel, you know, for carving into something or
I've also read that it's possible that it was used
for a form of body modification known as scarification, where
you would decorate the body by by making small incisions
in the skin to leave scar tissue. That would be
kind of like a tattoo, but with the natural scar
tissue forming the decorative design. But it's not known for

(05:34):
sure what what this tip was for. Uh. The bone
handle is actually the really fascinating part. So first of all,
it has been modified by narrowing, polishing, and carving to
such an extent that, at least according to the to
the r B I N. S, it is not known
or it's not clear what species of animal this belonged to,

(05:54):
though I've seen it alleged in other sources that it
is a baboon bone, so I'm not quite sure they're
But according to the museum that houses it, they say
they don't know what kind of animal it's from. Uh,
it's about ten centimeters long. It clearly did belong to
some kind of mammal. And what's what's really interesting are
the slashes. So, the slashes carved into the long sides

(06:16):
of the handle add up to a total of a
hundred and sixty eight parallel lines arranged into tight groupings
of different numbers in three lengthwise columns, and so a
huge amount of the interpretive work on this artifact has
focused on these slashes and what they mean and how
they might have been used. Now, of course, it's possible

(06:36):
that the slash is carved into the handle are are
are purely decorative, or that they were useful for making
it uh like easier to grip a bone tool. But
the number of lines in each grouping really do seem significant,
though exactly how best to interpret them is still being debated.
So to explain a bit further, what are the numbers. Well,

(06:58):
first you've got a column with four groups of slashes,
and the groups go like this. It has eleven slashes,
twenty one, nineteen and nine. So it seems very interesting
to me, Like you don't need to be an expert
to notice that this is ten plus and minus one,
and it's twenty plus and minus one. Then the next column,

(07:22):
uh the the groups of slashes go three, six, four, eight, ten, five,
and then five seven. So the first three pairs in
the sequence are doubles of each other. Uh Tho, the
ten and the five are inverted in the order from
the first two uh And then there's a question about
the five and the seven, so those don't really fit

(07:43):
the pattern and the rest of the column. But then
the final column with four groups of slashes is really interesting.
It goes eleven, thirteen, seventeen, nineteen, which in ascending order,
is the group of prime numbers between ten and twenty.
And of course we don't know for sure whether these

(08:05):
numbers were being recognized on this tool as prime or not,
but it's a very interesting list. If this is a
list of primes as primes, this would predate any other
recorded knowledge of division or prime numbers by thousands of years.
Another really interesting mathematical feature the first and third column,
So the so the ten plus and minus one and

(08:26):
the twenty plus and minus one, and the column that
is the list of primes between ten and twenty. They
both add up to sixty, but the middle column adds
up to forty eight, and so it's still being debated
what is the best way to interpret this, But a
lot of different interpretations offered by archaeologists, mathematicians, and other

(08:46):
experts suggests that this may very well be some kind
of mathematical tool or numerical reference table which might have
been used in counting in multiplication or Another common interpretation
is in keeping track of a calendar, which would still
be a type of mathematical tool, just a slightly different use.
So this is really interesting and I'm wondering can we

(09:09):
get any clues from other evidence from the Ashango site
as to how this might have been used. Unfortunately, there's
not really anything that's direct or explicit, but we can
learn a few things about the people who would have
been living there at the time. So one fact, at
least according to the r B I n S Interpretive Summary,
is that the people who lived here and probably produced

(09:33):
the Ashango Bone were not nomadic but probably lived a
relatively sedentary lifestyle, at least compared to lots of other
humans at this time in history. Uh And the reason
that they would have been able to live a relatively
sedentary lifestyle was that they were able to continuously make
use of the natural resources from the banks of Lake

(09:53):
Edward throughout the whole year, so they give the contrast
to people further to the geographic north would have to
follow animal migrations to survive, but the Shangaans appear to
have been able to make use of the resources of
the lake itself and just stick to its banks, and
evidence for this includes lots of different animal bones. They

(10:13):
listed huge numbers, so there are tons of fish bones
found here from this archaeological strata, but then also bones
of mammals like hippopotamus, ward hog, otter, buffalo, uh some
some antelope, and then many different kinds of birds. And
these bones all show signs of butchery, so these aren't

(10:33):
just bones of animals that died, but bones of animals
that were used for for food. There's evidence of them
being carved upon, of meat having been stripped away from them,
that sort of thing, right, And there's also evidence from
the site that the people who lived here would have
supplemented their diet with wild grains and possibly other vegetables,
though those remains don't always survive as well. So the

(10:54):
resources and signs of continuously processed animal remains indicate probably
a relatively settled existence. But as far as I can tell,
the settlement itself has not been discovered yet. It maybe
somewhere on the banks of Lake Edward, buried and not
yet uncovered. But this, this, this artifact is so interesting.
I like, oh, I want to know, like I want

(11:15):
to have the riddle solved. Um, yeah, I mean looking
at it, like you said, we we you know, it's
hard to determine exactly how it was used. And and
it's I guess it's entirely possible that there could be
aspects of this piece of technology that that simply haven't survived.
Like the thing that comes to my mind instantly is
and I don't know how this would match up with

(11:36):
the specifics of what we know about it, but say
it depended on the use of a small string of
hide that is tied around it and maybe slides up
and down the implement to mark different numbers. Things of
that nature you know, wouldn't would not would not have survived,
perhaps while the bone itself and the quartz tip would have,

(11:57):
So we might end up having an incomplete picture of
what the that the full piece of technology is and
then I guess the other way of looking at it
is we don't know how the individual uses it in
congress with other like counting techniques, such as, what if
there's a particular way of counting fingers or finger bones

(12:20):
that this is an augmentation of that sort of thing. Yeah,
that's a really interesting idea too. Uh yeah, So obviously
we don't know if there would have been more that
was used along with it. Um, But yeah, I wish
I knew. I mean, I feel like I'm gonna have
to keep my eyes peeled for for new papers on
this thing, like if anybody has new ideas that there

(12:40):
have already been some interesting ones. Some of the main ones,
like I mentioned, are that it may have to do
with a with a lunar calendar or calendar of some sort,
or that it may represent um possible accounting aid or
or multiplication aid based on other base counting systems, like
a base three or four counting system in which the

(13:03):
number twelve would be very significant. But like I said,
it's still not you know, it's there's no consensus on
exactly what this is and how it was used. But
but it's such a fascinating artifact and uh. If the
Shango bone is in fact a piece of mathematical technology
from prehistoric times, it would not be the only artifact
that has been interpreted this way. There are some other

(13:23):
ones I want to mention. There is an even older
artifact known as the La Bombo bone that was discovered
in a cave between Swaziland and South Africa. I've seen
several dates cited for it. Most are between like thirty
thousand and forty thousand years old. But it is a
baboon fibula with twenty nine notches on it that has
also been interpreted as a possible counting aid for a

(13:45):
lunar calendar. M Yeah, interesting, all right. You you know,
perhaps we're over thinking it. It's like basically begs down
to you. You get, you get thirty notches on your
babboon bone, then your thirty first babboon absolutely free. Well,
that does bring up the issue of the difficulty and
interpreting things like this. I mean, the the groupings of
numbers on the Shango bone really do seem mathematically significant,

(14:11):
but but it's always hard to know, right, It's always
hard to know what to make of these things when
you don't have like a written record that corresponds with it,
that can tell you how it was used. But but
I guess yeah, that the numbers don't lie though, Like
the numbers are the thing that's most stantalizing about it
because they have values, they have relationships to each other.
It comes back to what we were talking about in
the last episode about about what numbers specifically are. They're

(14:34):
not just you know, it's not just the fact that
it's an individual quantity, but it has relationships to do
other quantities, to other counts. So I want to mention
yet another ancient bone, ancient prehistoric piece of bone with
with notches on it that may have had mathematical significance.
This one I read about in an article that actually
mentioned in the previous episode, but I'm going to refer

(14:55):
to a good bit here. This was an article that
was a news feature in the tonal Nature by Colin
Barris called how did Neanderthals and other ancient humans learned
to count? Obviously, this is what we're talking about today.
And this one brings up another artifact of this kind. Uh.
This is an artifact discovered in the nineteen seventies at
the site of La Pradel near Angulema. And it's a

(15:19):
chunk of bone from the femur of a prehistoric hyena.
And so about sixty thousand years ago, one of the
Neanderthals who inhabited this region at the time made a
fine modification to this bone shard, cutting exactly nine notches
in the bone with a sharp implement. Now, there are

(15:41):
tons of ancient bone pieces that have cuts in them
that are clearly random and accidental, and these are almost
certainly from the processing of animal carcasses. And there are
features of those kinds of cuts that you can sort
of you can tell what you're looking at. Usually they're like,
you know, they have certain qualities that you know. Usually
you can look and say, yes, that this really does

(16:01):
look like it was from the processing of a carcass
to get the meat off of it. But there are
also plenty of ancient bones and shells that are carved
in a deliberate, regular way that seems to indicate some
ancient form of art or decoration. And this article by
Colin Barris calls attention to an archaeologist at the University
of Bordeaux named Francesco Derrico, who believes that this bone

(16:25):
artifact from sixty thou years ago in France may be
different from some of those other ones that have the
regular decorative slashes and carvings in them. Uh So it's
not an accident of butchery, he says, and maybe not
a work of art, but a means of storing or
conveying numerical information. He believes these markings are the signs

(16:47):
of a tally and if that's correct, of course, it
would mean that anatomically modern humans are not the only
species of human ever to have come up with the number,
since that at some point some Neanderthals might have had
one at some point as well. Now, I think one
thing that is a useful distinction to make is that
if some of the interpretive work on the Ashango bone

(17:09):
is correct, then it is what's probably a sort of
permanently formed mathematical tool that is used for reference in
aid of other types of counting or multiplication or mental
mathematical work. Whereas there's a different kind of thing you
can have, which is a tally stick, in which it
appears that marks are being made for a momentary counting purpose.

(17:34):
Does that distinction make sense, Yeah, I mean it's the
different difference between in extreme cases making notations in the
dirt or you know, on some sort of bit of
highly organic matter, as as opposed you know, something that
would decay even you know, within a matter of months
or so, as opposed to getting the bone or getting

(17:55):
a piece of stone and making deliberate uh and and
far from actual inscriptions in that piece that would be
repeatedly referenced for for future Like it would be sort
of the difference between a scratch pad that you use
to mark something down for momentary use versus like a
multiplication table that you refer to in order to solve

(18:16):
future problems. Yeah, the difference between writing something even in
sharpie on your hand and and writing it on you know,
a piece of paper or putting it into some sort
of permanent file system or SMI permanent file system on
your phone or whatnot. So one question here would be, Okay,
if there are lots of things from this point in
prehistory that have cuts or carvings in them that are

(18:38):
widely interpreted as art decoration, why does Derico think that
this hyena bone indicates counting or making a tally rather
than just sort of ardor decoration. Well, on the basis
of characteristics of the cuts, observed through microscopic analysis. He
believes that the cuts were made by the same person,
using the same tool, held in the same way, in

(19:01):
other words, in a single session lasting a few minutes
or hours. And it's also noted that at some other point,
not in the same session, eight much shallower cuts were
also made into the same fragment of bone. But so
why would they not be art Well, Unlike many of
the other bones with apparently decorative cuts, the marks here
are not evenly spaced. Their spacing appears haphazard, though they

(19:25):
are organized in a single file. So this seems to
me like it's far from a slam dunk. But on
this basis, Derrico argues that this artifact may have been
functional rather than artistic, and that function would have been
storing information, specifically storing the number nine. You needed to
remember that there were nine of something, and so you

(19:46):
made nine notches in this piece of bone to store
that information. That I mean, and that's so tantalizing too,
because it writes it is the obvious question, uh, No,
nine of what um? And in relation to what is
this token that was proof of nine ownership of nine
things or that you owed nine things? Was it, you know,

(20:07):
a counter what was it? Derrico also brings up the
example of actually, I believe he's referring to the La
Bambo bone, the at least he's referring to a baboon
fibula bone with notches on it from uh this this
one this article gives the rough estimate of forty two
tho years old and an artifact that was discovered in
the same place the Border cave in South Africa. Whether

(20:30):
it's the same artifact or an artifact from the same
place that's very similar. Uh. Derrico also interprets this bone
as very likely something that's being used to store to
record numerical information, not just something that's being decorated with slashes.
So part of the question would be that if at
some point ancient humans long before recorded history started using

(20:52):
mathematical tools and counting tools, tally sticks, possible mathematical reference
tables or numerical reference objects like like the Ashango bone
might be how does that fit into the the evolving
consciousness of numbers throughout the development of human prehistoric culture

(21:12):
and Derrico, as as cited in this article by Colin Barris,
actually has a hypothesis to explain in a rough sense,
how the first number since and counting systems came to exist.
And his hypothesis goes pretty much like this. It's sort
of a step by step process that begins with accidents.
So he says, what if early hominins were butchering animal

(21:35):
carcasses with stone cutting tools, So they've got little hand
axes or hand blades, they're cutting the meat off of
animal bones, and they realize while doing so that they
left permanent marks on the bones after cutting them. Now,
this this is interesting because you could basically start playing
the Strauss music right here. Yeah, and I think it

(21:56):
would be just as as amazing feeling as any idea
of two thousand and one, with the idea of butchery
taking place, and then the slow realization that staring up
at you from the bone is a number is account
you know, at three year what have you? And of
course it wouldn't be numerals, but it would be that, yeah,
that you were making these slashes, and that these were

(22:18):
a permanent record, something that you had changed permanently in
your environment, and that from here possibly they could have
made the jump to realizing they could mark objects like
bones and shells on purpose, not just accidentally, but they
could do it anytime they wanted, for whatever reason they wanted.
This could of course lead to decorative or artistic carvings

(22:39):
like we know often happened. You know, many ancient people's
made artistic or or decorative slashes into bones and shells.
And after that people began to realize that the deliberate
marks that they made could store information, possibly numeracle information.
And from here these systems of tally marks lead through

(22:59):
a process that DeReKo calls cultural exaptations, to the invention
of abstract number signs like the numbers we have today,
which could store numbers more efficiently than a one to
one tally system. So you're starting to have symbolic representation
of quantities when you're doing a one to one tally.
So you know there are nine things you need to remember,

(23:21):
and so you make nine slashes into a bone. Wouldn't
that actually be more efficient? Over time? You would realize
if you could make, you know, one simple mark in
a bone, that would that would always be associated with
nine of something in your brain. Yeah, exactly. Now, obviously
this is very broad and speculative, and you would need
to have a lot more specifics on how each of

(23:41):
these leaps took place, along with supporting evidence. But I
do think it's an interesting starting place to sort of
generate some predictions to test against future evidence. Yeah, and
I guess we'd also have to remind ourselves that this
wouldn't be This would surely not be the only case
where one could potentially pick up on the idea that
information can be stored in marking, because say that the

(24:03):
footprint or hoofprint of an animal is information stored in
a in this case a temporary marking or imprint in
dirt or dust. But uh, interestingly, a bone is a
thing you can take with you. It's something you and
and also there's there would be a lot of focus here,
Like I mean, I easily go to that two thousand
and one UM example, because it is you can imagine

(24:25):
the butchery, you know, taking up a fair amount of
time and being an area of concentration and focus, and
you can you can easily imagine the realization building over time. Yeah. Again,
it's I kind of get the shiver. It's exciting to
think about, you know, wondering about the possibilities of how
humans arrived at the at these thought patterns than now

(24:51):
coming back on the other side and offering some criticism
of this possibility. UH. Colinmbarrass in his article notes the
caution raised by several scientists in the field that, of course,
like we already alluded to, it's easy to misinterpret markings
on artifacts like the hyena bone. And there's one example
they said that I thought was really interesting, which is

(25:13):
message sticks that are used by some Aboriginal Australians. Sometimes
they will have marks on them that look like they
could be tallies that would indicate a number, and could
easily be interpreted as such if you didn't know what
you were looking at. But actually, in some cases they
don't convey numerical information. Uh. Some of the people who

(25:34):
use them explain that these notches, some that sometimes look
like tallies, actually act rather as a memory aid for
recalling details of a narrative message, rather than as an
account a quantitative count of something. So they are a
memory aid, but not for a number, more for like
a a message to deliver or a story to tell.

(25:56):
That's interesting, yeah, So to what extent are these interpretations?
These are the interpretations modern interpretations of ancient artifacts made
by numerical people's But in some cases you're dealing with
people who are who are going to be more rooted
in say narrative or I don't know that, perhaps music.
I instantly think of some of the ideas out there

(26:16):
about Neanderthals and music. Uh, you know what, what what if?
And this is a big what if? And I have
nothing to back this up, just sort of gut thinking here,
But you know what if something like this was ultimately
to aid in some sort of uh you know, ritualistic
musical um recitation. I don't know, yes, obviously, So if
it's all pre writing, it's it's hard to know. I mean.

(26:38):
One of the best things we could have in the
artifact itself to know that there's really likely a numerical
significance is probably relationships between the numbers themselves, which is
once again what makes the Shango bones so interesting. That
it's like, you know, it's got a list of primes
between ten and twenty that would be really strange if
it's just a coincidence, though of course you can't rule
it out right. And then again, the numbers don't lie.

(26:59):
So even if the you know, the the numbers have
relationships with each other, they have they have value even
if it is not so numerically rooted, like if those
are just beats in a story on a bone, for example,
I mean, you know, there's still a numerical essence to it.
You know, there's account there. Like what if you had
a I don't know, a caveman stand up comic, you know,

(27:20):
and he has a bone and he has has two
marks on it because he has to remember to do
both the set up and the punchline for each joke. Yeah,
so it could be easy to misinterpret these things. And
uh in favor of that. Barriss in his article sites
a man named one Yungar who is an Aboriginal Australian
who is a member of the Guring Guring and Waka

(27:42):
Waka communities, and he says that sometimes these sticks that
have slashes on them that you know, to a modern
archaeologist might look like talis of a number. Sometimes they're
used for trading, so you know, they might they might
specify something about trade, but they might also be a message.
Say he gives the example of a message of peace
after a war. So obviously, from an archaeological perspective, it's

(28:04):
important to step back and have some more humility, like
always realizing, like, you know, even when something really looks
like one thing, do you there there, It's quite possible
that you are not actually realizing all the ways that
it might be used. Now, there's another hypothesis about the
historical origins of number systems that is mentioned in this

(28:25):
article h this Nature News article, and this one comes
from a researcher named Karen Lee Overman, who is a
cognitive archaeologist at the University of Colorado and Colorado Springs.
And she begins with a linguistic observation, which is that
not every culture and language group has a system of

(28:45):
exact numbers for arbitrarily high quantities. In fact, in some
languages you might have distinct words for smaller numbers, you know,
so you'd have a word for like one to three
and four. But at some point there are no younger
distinct words for numbers, but approximate ones translating to something
like many or very many. That reminds me again. I

(29:09):
have to share a memory of my my son when
he was younger, and he was obsessed with counting cows.
When we would we would drive by cow fields and
he would he would count. Essentially I guess as high
as he could at the time, but he would reach
the point where he would he would be like twelve, thirteen, fourteen, fifteen,
and then he would just skip to all of them,
all of them. Oh that's great, while you eat them all. Yeah.

(29:31):
But about this linguistic distinction, where you know, at some
point some languages don't have individual words for higher and
higher numbers but start to become approximate um It's it
could be easy for a narrow minded numeracle chauvinists to
think that that's somehow indicates a lack of sophistication, but
as we talked about in the last episode, it does not. Rather,

(29:53):
it has to do with what kinds of concepts and
quantity concepts are useful to your way of life. So
for some ways of making a living, they're they're just
actually is not that much useful about making a distinction
between twenty seven and twenty eight. So instead there are
distinct numbers for small quantities and then approximate terms for

(30:13):
larger quantities. And so the question then would be what
makes a difference in whether your language needs these distinctions
or not. Well, this is where Overman's hypothesis comes in.
She argues in a study published in the Cambridge Archaeological
Journal in called material Scaffolds in Numbers in time UH.

(30:34):
She looked at evidence from thirty three existing hunter gatherer societies,
and what she found was that the specificity of higher
number symbols corresponded with societies that had more material possessions
more more possessions like weapons, tools, and jewelry. Meanwhile, societies
with fewer individual material possessions were more likely on average

(30:57):
to have a language system with without specific thick numbers
higher than four or five or so. So, if this
is on the right track, it is possible that the
accumulation of property and individual possessions could have been involved
in the innovation of higher order specific number systems. So
you know, if you have occasion to say, I own

(31:18):
seventeen of these, not fifteen, where did the other two go?
Mm hmm. You know, it's interesting thinking about especially this
idea of four, Like if I'm imagining I guess on
something that's less low stakes. If I'm thinking about say
a bag or a container of yogurt covered raisins, like
at which point in depleting them is my like instinctual

(31:40):
evaluation of the package based in an actual number. You know,
I get to the point where I'm like, oh, there
are four these left, um, and I like and I
guess I imagine that's going to be different if you're, say,
looking at a bottle of medication, you know, something that
you you regularly go through and you have to have
renewed um. You know, you reach the point where you're like, oh,
I have I have six these left right? Had? Maybe

(32:01):
you know it's based more on like a week basis
because you're equating it with with time keeping UM. But
that's interesting. Yeah, the idea that some of these societies
like if it's if it's more than four, you don't
really necessarily need a specific number for it, yes, or
that getting back into what we talked about in the
last episode that when you do need to reference quantities

(32:21):
of higher numbers of things, the quantities that you need
to think about are more in terms of ratios to
each other rather than specific one by one number line numbers, um.
So when you're thinking about higher quantities of things collected,
you might think in terms of Okay, we've got double
what we had last time, or something, right, I will
have another fistful of yogurt covered raisins. But another part

(32:44):
of the hypothesis put forward by Karen Lee Overman is
that her her idea meshes with this concept that is
known as material engagement theory. And this, uh, this actually
goes along with some things we've talked about on the
podcast before, which is the It's basically the proposition that
the mind in in effect extends beyond the brain and

(33:08):
includes storage capacity in the outside world, say, originally in
things like the fingers and other body parts used as
an aid in counting, but eventually in objects like tally
sticks and other ways of recording numbers, so that the
you know the mind essentially like you can you can
have an external hard drive for the mind that is

(33:29):
your hand and the numbers on it, or slashes in
a bone, or eventually say, numerals written on something, or
tokens of of quantities. And this is another way that
material artifacts may have in some ways helped contribute to
the numerical number, since where you've got more distinct signs
and symbols for higher numbers, and it would be by

(33:52):
storing numeracle information and objects outside the mind. So the
prospect of counting to high numbers like five thousand or
hundred and thirty seven becomes conceivable. Whereas if you don't
have words for those numbers and you don't have physical
objects keeping track of the count, it's kind of hard
to imagine, like conceptualizing numbers like a hundred and thirty seven,

(34:15):
How would you hold that number in your in your
brain if you didn't have words for it, and you
didn't have and you didn't have physical objects to represent it.
So anyway that that article by Colin barriss Is is
very worth a read, and it contains references to some
other stuff, like some linguistic work showing that UH words
for small numbers, say less than five or so, tend
to be extremely stable over time, usually meaning that they

(34:37):
probably get used some of the most of all words um,
and that that less being true of words for higher numbers.
But also tying into all of this is something we
mentioned in the previous episode, which is that some of
the earliest written records from agent Mesopotamia seem to be
accounts of possessions and trade. You know, who had much

(34:57):
and who owed what to whom? Yeah, and this is
where we really recognize just how essential UH numerals and
number cents are to so many of the things we
think of as is just as part of human culture.
For example, the oldest recorded law code, the Code of
Urnamu from between B C. It's uh. It's also largely

(35:20):
concerned with what is owed to whom, often really in
relation to moral grievances, but also concerning property. Um. So
an example of this is the thirty first code on here.
This is of course the translation, if a man flooded
the field of a man with water, he shall measure
out three cour of barley per i coup of field.

(35:41):
You know. So it's stuff like that where if this happens,
then this this amount should be paid uh as a
penalty to a certain individual. So it's a very exact
and counted system of justice, right yeah. And of course
it has stuff on there that we often you know, uh,
we often think of when we think of, say the
Code of Hammurabium, which would have come uh, you know,

(36:05):
at least a little bit later. Uh. You have stuff like, okay,
if you kill somebody, if you murder somebody, then you
will be killed. That sort of thing. But a number
of them are related to you know, specific measurements or
amounts of money. Or how much of a silver piece
is paid you have this kind of injury is inflicted
on another human being, that sort of thing. I was
gonna say, I wonder where these ancient law codes come

(36:27):
up with the numbers they use. But I guess you
could also say that often about modern law codes. Yeah.
I mean, it's it's easy to look at one of
these codes and UM. For instance, in this particular code,
the code of Urnamo, if I'm remembering correctly, it's like
if you if you basically, if you cut off somebody's nose,
there's a certain percentage of a silver piece then goes

(36:48):
to that person. And on one level, you're like, how
can you put a price on somebody's the whole entire nose.
But then again to varying degrees, there's gonna be there's
gonna be a price. Uh. That is that is established
or argued out concerning that sort of of injury, as
grievous as it is even today. Yeah, the law of remedies. Yeah.

(37:15):
Thank Now, in discussing Mesopotamian mathematics, I want to come
back to UM some stuff I was talking a little
bit about earlier in the last episode. I mentioned UM
that in the seventy grade Inventions of the Ancient World,
and anthropologist Brian and Fagan writes about ancient numbers with

(37:37):
an author named Eleanor Robson, who who wrote Mesopotamian Math,
among other works. And so I want to get into
some stuff that they discussed there. But also I was
looking at a work by Robson titled Mesopotamian Mathematics Some
historical Background, uh, in which they get into a lot
more detail on this topic. So as as we we

(38:00):
we mentioned, you know, the Neolithic societies of the Middle
East stretching from what is now Turkey through our Iran. Uh.
You know, they were engaged in the use of stone
or clay counters to keep track of stored and or
traded goods. And by the fourth millennium BC, we saw
the use of something we've mentioned on the show before,
the use of counters uh, stored inside of a clay envelope. Now,

(38:26):
if you're like me, the first time you read about
clay envelopes with tokens inside of it, you just pictured
like something that looks like a modern paper envelope, except
that it is made out of clay. That's exactly what
I used to picture when this would come up like
as an anecdote in something. But the reality and you
can look up some wonderful pictures of this, the reality
is that it doesn't The envelope does not look like

(38:46):
a modern paper envelope. It looks like a round clay
glob that has dried and has generally has some sort
of you know, marks or patterns on the surface, in
addition to some key markings that will get too shortly.
It's an eight ey'd alien skull. Yeah, yeah, it You
would not look at this and go, oh, an envelope.
But but essentially that's what it is. It was a

(39:07):
way of sealing something inside, and to get at the
contents of that envelope you would have to open it
in a way that could be detected. I see. So
kind of analogous to like the wax seal on the
on the envelope that you know you can tell if
it's been broken right now. Of course, one of the
issues here is that if you're just looking at a

(39:27):
lump of clay and they're token sealed inside, how do
you know what's sealed inside? Uh, it's kind of an
interesting riddle, right. So what they ended up doing is
they would take the token that represents particular items and uh,
you know, our values, etcetera. Traded goods, and they would
imprint the clay. So so the imprints on the outside

(39:49):
of the clay envelope tell you what is stored within.
And uh, and I guess the idea there too is
that if if if there was any doubt, you could
break it open and there would be the proof inside. Um.
But at any rate one in particular was looking at
was a was a fourth millennium b CE. Uh example
of this and uh and yeah, they you can see

(40:10):
the little little counters, you can see the imprints in
the envelope. Uh, it's it's pretty interesting you. But these
would have been uh, standardized shapes and sizes that are
ultimately the precursors to the first written numerals. Well, it
makes you wonder, if they're putting stamps on the outside,
why did they actually need the tokens inside that the
tokens have some kind of like power or value that

(40:32):
the envelope itself didn't have. Yeah, I'm not as certain
on that, because, yeah, it seems like on one level
you could always just say, like, if you don't trust me,
you can break it up. The proof is literally inside
the clay globule that that is before you you know, UM.
But but but then the the the reality is and
this is something Robs and stresses in that Mesopotamia Mathematics

(40:53):
UM article that I was referring to, is that eventually
they simply did away with the envelope aspect and just
stuck to the use of imprints and symbols. So eventually
they reached the point where we don't need to see
a little objects inside of the clay, because the imprint
is the thing, Like, this is the useful This is
the the useful technology. It's not so much the little

(41:14):
objects inside of it, it's them. It's the imprints, the
symbols that we've created UM. And also as trade and
usage widens, it also just becomes you end up seeing
a revision of all this because it becomes impractical to
create a different symbol system for every commodity. So you
see the the you know, this inevitable march towards UM

(41:35):
numerals that can be used you know, throughout a given
industry or trade, then without than throughout a particular UH
civilization or region. And then you can see that spreading
to other areas as well. Now Here here's an interesting
quote from Robson, and all of this quote, now, mathematical
operations such as arithmetic could be recorded, the commodities being

(41:56):
counted cannot usually be identified. And they mean today looking back,
you know, trying to figure out what they're talking about,
um as, the incised signs which represent them have not
yet been deciphered. But the numerals themselves, recorded with impressed
signs can be identified with ease. So again we come
back to that idea that the numbers themselves, the counts,

(42:17):
the quantities, they don't lie. We can we can look
at these and we can we can make sense of
the mathematics that's going on now. During this time, we
also see the use of ivory labels to count prestige
grave goods in pre dynastic Egypt. But at the same
time um Fagan and Robson they point out that we

(42:37):
also see the use of clay tablets in what would
have been very small agricultural settlements. So I think that's
important to note is that it's not just a manner
of like big city trade goods uh and and big
city projects or you know, the elite grave goods of
of dying kings, but also you see it in the
use of small agricultural settlements. You know, this makes me

(43:00):
think about how I wonder if a system of numerals,
a system of a larger quantity exact numbers, is more
necessary if you are having more interactions with strangers, like
if you are less if life is less, like you
know everybody in your in your tribe or hunter gatherer band.

(43:21):
Instead you are having to trade with people you don't know.
Is there a need for numerical precision that enters when
you have those kinds of relationships that's less present on
average if you don't. I don't know. I was wondering
a little about this. One of the reasons I started
looking at some of these these ancient lack codes because
I was thinking, I thought about the the use of

(43:45):
math and trade and then the ideas of of of
cheating and embezzlement, you know, uh and uh and and
and all and all of that, and I was as
I was wondering, Yeah, I did, to what extent is
this super useful when dealing with outsiders? You're gonna trade
with outsiders, which obviously is taking place at this time.
But then again, you know, within an even within a

(44:06):
city like that is a place where you're going to
see an increase in in crime. I mean, that's where
we see I think back to our episode on the
invention of locks. You know, that's where we see that arise.
The need to safeguard your goods. Uh, not from the
individual who lives in the next city, state, or though
the agricultural village that's uh, you know, half a day's
travel from where you are, but in the people that

(44:28):
are living in the streets around you. Yeah. If you
if you have the feeling that you can't necessarily trust
everybody in your immediate proximity. Yeah. So robs And stresses
that in Um in the Mesopotamian region, mathematics arises out
of out of as a necessity of civilization, and it
righting itself arises directly from the need to record mathematics

(44:49):
and accounting, and then over time, counting and measuring systems
evolve in response to the needs of large scale state
bureaucracies and and and uh I believe she all so
points out that that is certainly in these Mesopotamian settings.
At first, it's not the state itself engaging in these
big projects it's it's basically major operators working for the state.

(45:13):
But then you know this eventually leads into large scale
bureaucracy and the bureaucratic use of mathematics as well. Okay, so,
whereas people living a more hunter gatherer type existence, they
might have depending on their culture or on their relationship
to their environment, they might have differing needs for different
kinds of quantical cognition. Um, some might trend more towards

(45:34):
having systems of numerals and others might not, just depending
on what their lifestyle is. But once you have cities
and governments and trade and stuff like that, basically numerals
start becoming necessary, right and and so from from this
point on, I'm not going to really get into a
complete breakdown of every step, um, you know, in the

(45:56):
development of numerals and different numeral systems, but I want
to hit some of the what what seemed to me
the highlights. So so certainly, if you have questions out there, uh,
you know, look up some of these sources that we've mentioned. Um,
you know, there's so much more to dive into here.
But we see the first use of the decimal system
in the first millennium BC in India, UH and the
Vedas described the practical use of geometry. UM as for

(46:19):
the zero, it's interesting to reflect on what we use
the zero four aside from merely representing nothing, which which
in itself is UH is pretty impressive development and seems
to have not developed until the early seventh century in India.
But zeros are also important in place value system So
Vagan and Fagan and robson site that zero markers in

(46:42):
the middle of numbers were quote first attested in the
astronomical works of Ptolemy in Roman Egypt around one oh.
I see place value systems, so like you could you
could say like point zero one or yeah, like the
number two oh three zero is playing an important role
in that, in that oh in that larger number. Sure. Now,

(47:02):
as for the true origin of numerals, when we think
about the numerals we're using every day, UH, we we
do have to stress that there are there are competing
arguments here. We commonly speak of Arabic numerals, though Hindu
Arabic maybe more precise. UH. Still others have made cases
for ultimate Persian or Egyptian origin of numerals. Numerals here,

(47:23):
but one issue to keep in mind is that from
very early on, this sort of technology was again tied
with trade. So not only would one system have spread,
but it would have encountered new ways of doing things,
regional practices, etcetera. So what we think of as you know,
Western numerals and you know and and ultimately Arabic numerals

(47:44):
or Hindu Arabic numerals, um, they may largely be a
conglomeration due to trade through various regions over an extended
period of time. I see. I mean, since it's trade,
it's sort of like where cultures are meeting most frequently. Yeah. Yeah,
so that's that's an interesting way to think about it. Yeah,
it's not like somebody rolled into town and said, hey,
we got numerals. Now, this is what we're using for everything.

(48:06):
But you know, it would have been I mean, it
would have been some of that to a certain extent.
But but yeah, this idea that it you have this
sort of shared creation of the economic system. Now, one
thing I was reading that was kind of interesting was
that while we use a base tin counting system today
when we write things out in numerals, are language actually
doesn't indicate a based tin counting system because we in

(48:29):
English at least have individual words for numbers going up
to twelve, right of ten, eleven, twelve, and then once
you get to thirteen, that's when you start constructing the
words for numbers based on composites of like the of
the base tin place holding right, so three, ten, thirteen, Um.

(48:50):
But apparently that is not true of of some other languages,
for example Chinese languages. I believe there is pretty clean
based Tin notations, so like eleven is ten one. Yeah.
The Chinese civilization boasted some some early numerical advancement as well,
including the use of a decimal system as early as

(49:11):
the second millennium b C. So these pop up on
shang oracle bones from between fifteen hundred and twelve hundred BC.
And then you have ivory and bamboo counting rods that
were used from at least five hundred b C. And uh,
when you start looking around at mathematical texts, the nine
chapters on the Mathematical arts is a is a key tone. Uh. Now,

(49:33):
this is a book that doesn't that does not have
a singular author. It was the work of several generations
of scholars from the tenth through the second century BC,
and it's pointed out by J. J. O'Connor and E. F.
Robertson of St. Andrew's University in Scotland. It contains two
D forty six problems aimed ultimately at providing everyday practical

(49:53):
methods for dealing with issues such as engineering, land surveying,
trade taxation. So again, all all the all the sorts
of uses for mathematics you see in these other cultures
as well. Now, Greek and Roman systems did not have
a place value concept. Apparently the Roman system evolved from
a notch cutting system, so they were not great for
recorded calculation, and this led to the dependence on counting

(50:16):
boards and later the abacus. Meanwhile, astronomers apparently adapted the
sexy decimal place value system to Greek, which is why
we still one of the reasons we still measure time
and angles in sixties. Oh yeah, that's interesting. So in
all this you might wonder, well, why not a decimal
system for time keeping? You know, why are we depending

(50:38):
on units of ten for so many things, but then
when it comes to time while then we're based on
on things like sixty or particularly twelve. Well, the Chinese
used both a decimal and a duodecimal or twelve based
system for hours. Um. France started using a decimal time
system in se uh but it only lasted seven teen months.

(51:00):
You know, you get into a situation of like which
we're literally changing all the clocks. Well, we have a
lot of clocks, right, we have we have this understanding too,
like this is how we think about the day. Uh.
So they ended up switching back, and there was another
failed attempt in eight to essentially do the same thing,
and they ended up sticking with the sixty. But it's
a neat idea because you would mean ten deaths of

(51:21):
the French model anyway, ten decimal hours in a day,
each composed of a hundred decimal minutes, and each of
those containing a hundred decimal seconds. So in this situation,
noon is at five. Oh interesting yeah, but a hundred men.
I love that, So it can be like eight ninety
seven is the time? Yeah, um so uh. Duo decimal

(51:46):
systems again, twelve based are are also interesting because it
may raise the question like, well, where are you getting
this twelve from? Because we already mentioned these these ideas
regarding the counting of fingers and toes, so you can
see where ten comes from. You can see where twenty
comes from. But twelve, well, one hypothesis here is that
there are twelve finger bones on the hand, so just

(52:08):
counting the fingers, not the thumb, and then you can
use your thumb to touch each of those finger bones
to give you a count of twelve on one hand.
And then on top of this, there's the lunar connection,
twelve lunar cycles in a year. Um. That that also
seems to play a major role. But um, but but
apparently we still see versions of this, uh, this fingerbone

(52:31):
counting system used in parts of the world. Um, even
though I I have to admit my own finger counting,
which I rely on a little bit too much, I'm
still only using like one count per finger. But if
but you would look so much more dignified if you
were doing some finger counting. I would think if I
was able to master uh this uh, this duodecimal system,

(52:52):
uh using just one hand, because you could like hide
it under the desk people didn't see what you're doing. Yeah,
or I guess the thing is that I'm counting on
on my fingers, which I guess. The main time I
do this is if I'm playing dungeons and dragons, and
I'm doing like some some hit point counts, and so
generally people can't see that anymore since I'm not playing
in person. But there's this kind of idea where if
you're at in public and you're counting on your fingers

(53:14):
with both hands, people can think like, oh, he's thinking
too hard about Matt. Let's get him. He's distracted. Whereas
if if you look over and it's like, oh, look,
he's doing some sort of complex. He's counting his finger
bones with one with one hand while he's figuring his
um you know, his hit point level right now, they're
not going to mess with him because clearly, uh, he's

(53:34):
doing okay. Well, Rob, I have really enjoyed this journey
into the origins of numbers. Yeah, and and again, you
know there's a lot of this. We're only really um
scratching the surface on uh you know, we're not even
getting a full imprint into the baboon bone. Uh. So
I do or genuine out there is interested in this
to to to look into it more, look up some
of these authors that we've mentioned, some of these researchers,

(53:56):
because there's just a there's a whole world of math,
like Robinson for instance, very readable material on the use
of math in Babylonian society, for example. It gets really
fascinating because it just it ultimately. Even though you often
think about mathematics is something that is you know, abstract
and it's outside of human experience. But in reading Robson's

(54:18):
work about how ancient Babylonians used mathematics mathematics, it really
humanizes these ancient people so much more because you realize
that the practical things they were doing, you know, things
like I need to build a house, I need to
make sure that its walls don't fall down, you know
that sort of thing. Like they were doing all the

(54:38):
things that civilizations and societies do. It's really easy to
sympathize with somebody when you imagine them trying to count,
trying to like, you know, remember a number of something. Yeah,
to balance some sort of a budget budget or whatever. Yeah,
that's like me, Yeah, all right, where we're gonna go
ahead and close it out here. But we'd love to

(54:59):
hear from everyone out there, everyone out there listening to
this show. You use math, you use numerals, um, perhaps
you are privy to some other numerical system. Uh, and
you have some experience with that, and you'd like to
chime in. I know we have some mathematicians out there.
I think we were already hearing from from some folks
that are well versed in math regarding our last episode,

(55:20):
So do write in about this one as well, and
uh yeah, in general, let us know if you'd like
to hear more episodes on numbers or math. You know,
we they're, like I said, there's a lot more we
can discuss. In the meantime, if you would like to
check out other episodes of Stuff to Blow your Mind,
you can find them wherever you get your podcasts. There
in the Stuff to Blow your Mind podcast feed core
episodes on Tuesdays and Thursdays, We're throwing an artifact on Wednesday,

(55:44):
listener mail on Monday, and on Friday's we do a
little Weird How cinema. That's our time to just set
aside all the more serious issues of math and we
in this case math but generally science and culture and
focus on just a weird movie. And I have to
have to say, sometimes we're able to thematically link things,
but I don't think there's any math in the Weird
House Cinema episode that will be airing tomorrow, a goo

(56:06):
to a late seventies made for TV movie about math.
Very I guess we do talk about lunar cycles a
little bit so in a way, but a little bit
math is uh is unavoidable anyway. Huge thanks as always
to our excellent quality of producer Seth Nicholas Johnson. If
you would like to get in touch with us with
feedback on this episode or any other, to suggest topic

(56:27):
for the future, or just to say hello, you can
email us at contact at stuff to Blow your Mind
dot com. Stuff to Blow Your Mind is production of
I Heart Radio. For more podcasts for my Heart Radio
with the iHeart Radio app, Apple Podcasts, or wherever you're

(56:48):
listening to your favorite shows. Tway People Proper