Join Robert and Joe as they continue their discussion on numeracy and the origins of numerals on Stuff to Blow Your Mind. (originally published 6/17/2021)
Hey, you welcome to Stuff to Blow your Mind. My name is Robert Lamb and I'm Joe McCormick, and it's Saturday. Time for a vault episode. This one originally aired on June one, called Where do Numbers Come From? It's the sequel to Our Numerous, the episode from last Saturday, and uh, I think that's it. So yeah, I hope you enjoy Welcome to Stuff to Blow your Mind production of My Heart Radio. Hey you welcome to Stuff to Blow your Mind. My name is Robert Lamb 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 the show, we were talking about the human number since 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 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 all really fascinating stuff because it's not just you know, 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 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 it's at times prizing 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 in the nineteen fifties there was a Belgian geologist named John behind 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 what is now the Democratic Republic of the Congo in the in the Virunga Park region of the northeast 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 that this is something like twenty thousand to twenty five thousand years old. And this piece of animal bone had several 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 Ashango bone today. And what's so fascinating about 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 there are a few that are as old or older, but this is going way back. I mean long before say, the ancient civilizations 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 this artifact, and 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 composite prehistoric tools that has survived all the way to modern archaeological discovery intact. 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, and it was found with the pieces still together, so the courts 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 under the quartz 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 you know, a 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 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 sure 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, 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 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 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, 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, 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. 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 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 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 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 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 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 rb I and S interpretive summary is that the people who lived here and probably produced the Ashango bone were not no madic, 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 Edward throughout the whole year. So they give the contrast to people further to the geographic north would have had 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 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 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 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 may be somewhere on the banks of 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 to have the riddle solved. Um, yeah, I mean looking at it, like you said, 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 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. 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 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 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 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 a 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 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 stir 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 lunar calendar. M h yeah, interesting, alright. You you know, perhaps we're over thinking. It's like basically begs down to you. You get you get thirty notches on your baboon bone. Then your thirty first baboon 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. 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, 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 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 to a good bit here. This was an article that was a news feature in the journal 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 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 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 out and say, yes that this really does 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 artifact from sixty thou years ago in France may be different from some of the those other ones that have the regular decorative slashes and car things 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 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, sense 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 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. Does that distinction makes 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 a piece of stone and making deliberate uh and and far from casual 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 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 somemi 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 history that have cuts or carvings in them that are widely interpreted as ardor decoration, why does Derrico 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 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 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 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's the obvious question, uh no, nine of what um? And in relation to what is this a token that was proof of nine ownership of nine things or that you owed nine things? Was it? You know, a counter? What was it? Derriko 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 ref estimate of forty two years old and an artifact that was discovered in the same place the Border cave in South Africa. Whether 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 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 and h 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 and 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 realized 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 would be just as as amazing feeling as any idea of two thousand and one model at 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 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 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 a process that Derriko 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 they're nine things you need to remember, 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 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 a marking, because say the 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 true and 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 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 of or 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. Now coming back on the other side in offering some criticism of this possibility, uh col Embarrass 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 message sticks that are used by some Aboriginal Australians. Sometimes they will have marks on them that look like they could be talies, 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 use them explain these notches, some that sometimes look like tally's actually act rather as a memory aid for recalling details of a narrative message, rather than as an account to 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. 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 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 to 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, it's hard to know. I mean, 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 a shango bone 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. 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 a caveman stand up comic, you know, 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 Junger who is an Aboriginal Australian who is a member of the Gurg gurng and Waka 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 that 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 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 article, uh, this Nature News article, And this one comes from a researcher named Karen Lee Overman, who is a cognitive archaeologist at the Universe City 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 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 longer distinct words for numbers, but approximate ones, translating to something like many or very many. That reminds me again. I 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. 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, 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 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 and Time UH she looked at dents 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 DAVA language system with without specific 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 seventeen of these not fifteen, where did the other two go 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, and I'm thinking about say a bag or a container yogurt covered raisins, like, at which point in depleting them is my like instinctual 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 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 of these left? Or I have maybe you know it's a based more on like a week basis, because you're equating it with with timekeeping. Um. But that's interesting. The 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 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 I will have another fistful of yogurt covered raisins. But another part 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 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 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 your hand and the numbers on it, or slashes in a bone, or eventually say, numera rolls 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 storing numerical information and objects outside the mind. So the prospect of counting to high numbers like five thousand or a 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, 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 Colmbarrasses 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 extreme a stable over time, usually meaning that they probably get used some of the most of all words um, and that the 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 Agient Mesopotamia seemed to be accounts of possessions and trade, you know, who had much and who owed what to whom. Yeah, and and this is where we really recognize just how essential UH numerals and numbers since 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 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 cur of barley per i coup of field. 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. You know, we often think of when we think of, say the Code of Hammurabi um, which would have come uh, you know, 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 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 that goes to that person. And on one level, you're like, how can you put a price on somebody's 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. 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 the anthropologist Brian and Fagan writes about ancient numbers with an author named Eleanor Robson, who who wrote Mesopotamia 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 Mesopotamiaan Mathematics Some historical background, uh, in which they get into a lot more detail on this topic. So as as we we 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, 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 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 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 eyed 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 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 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. 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 of the clay envelope tell you what is stored within. And uh and 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 c e. Uh example of this and uh and yeah, they you can see 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. The tokens have some kind of like power or value that the envelope itself didn't have. I yeah, I'm not as certain on that, because it seems like on one level you could always just say, like, if you don't trust me, you can break it out. The proof is literally inside the clay globule that that is before you you know, um, but but the but then the the the reality is and this is something Robs and stresses in that Mesopotamia Mathematics 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 the little objects inside of it. It's the 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 UH 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 on all of this quote. Now, mathematical operations such as arithmetic could be recorded, the commodities being 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 incise 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, 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 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 in 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 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. 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 is one of the reasons I started looking at um some of these these ancient lack codes, because I was thinking, I thought about the the use of 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, did to what extent is this super useful when dealing with outsiders if you're gonna trade with outsiders, which obviously is taking place at this time. But then again, you know, within an even within a 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 tho the agricultural village that's uh, you know, half a day's travel from where you are, but in the people that 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 that righting itself arises directly from the need to record mathematics and accounting, and then over time, counting and measuring systems evolve in response to the needs of large scale state bureaucracies and um and and uh I believe. She also 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. 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 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 development of numerals and different numeral systems. But I want to hit some of the what 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 the zero, it's interesting to reflect on what we use the zero four aside from merely representing nothing, which which in itself is U 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 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, The zero is playing an important role in that in that oh, in that larger number. Sure. Now, 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, 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 way is of doing things, regional practices, etcetera. So what we think of as you know, Western numerals and you know and and ultimately Arabic numerals 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 it'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, but you know, it would have been I mean, it would have been some of that to a certain extent, but that 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 bas 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 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. 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 and advancement as well, including the use of a decimal system as early as the second millennium BC. 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 BC. And uh, when you start looking around in mathematical texts, the nine chapters on the mathematical arts is a is a key tone. Now this is a book that doesn't that is not have a singular author. It was the work of several generations of scholars from the tenth through the second century b C. And it's pointed out by J. J. O'Connor and E. F. Robertson of St. Andrew's University in Scotland. It contains two forty six problems aimed ultimately at providing everyday practical methods for dealing with issues such as engineering, land surveying, trade taxation. So again 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 boards and later the abacus. Meanwhile, astronomers apparently adapted the sexagesimal 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 timekeeping? You know, why are we depending on units of ten for so many things? But then when it comes to time, well, 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 seventeen. Uh but it only lasted seventeen months. 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 deci 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 nine, seven is the time? Yeah, um so uh duo. Decimal 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 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, this fingerbone counting system used in parts of the world. Um, even though 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. Uh, he's using just one hand because you could like hide it under the desk because people didn't see what you're doing. Yeah, or I guess the thing is that if 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 out in public and you're counting on your fingers 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 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 our genuine out there is interested in this to to look into it more, look at some of these authors that we've mentioned, some of these researchers, because there's just a there's a whole world of math. Robson, for instance, very readable material on the use of math in um 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 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 the its walls don't fall down, you know that sort of thing. Like they were doing all the 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, a budget or whatever. Yeah, that's like me. Yeah, all right, where we're gonna go ahead close it out here. But we'd love to 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, 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, So do right 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 could 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, 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 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 we'll be hearing tomorrow, a goo 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 quity 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 for the future, 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, visit the I Heart Radio app, Apple Podcasts, or wherever you listening to your favorite shows