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6: Word (Linguistics)
904 Plays7 years ago
Mathematics has a lot in common with language. Both have been used since the dawn of time to shape and define our world, both have sets of rules which one must master before bending, both are natural consequences of the way humans are raised, and both are as omnipresent as they are seemingly intangible. Language has thrived for almost, or as long as humans have possessed the ability to use it. But what can we say that language is? Is it a living breathing organism, a set of rigid ideals, somewhere in between, or something else altogether?
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Transcript
Foundations of Mathematics and Language
00:00:00
Speaker
Mathematics has a lot in common with language. Both have been used since the dawn of time to shape and define our world. Both have sets of rules which one must master before bending. Both are natural consequences of the way humans are raised, and both are omnipresent as they are seemingly intangible.
00:00:16
Speaker
Language has a thrive for almost or as long as humans have possessed the ability to use it. But what can we say the language is? Is it a living breathing organism? A set of rigid ideals? Somewhere in between? Or something else altogether?
The Evolution and Definition of Language
00:00:28
Speaker
All of this and more on this episode of Breaking Math. Episode 6, Word. I'm Jonathan. And I'm Gabriel.
00:00:42
Speaker
And we're coming at you from KUNM Studios as part of Generation Listen. If you haven't checked out the Facebook page, check us out at facebook.com slash Breaking Math Podcast, or check out our website at breakingmathpodcast.com. And Gabriel, what are we going to talk about today? Today we're going to talk about the history and evolution of language. And we're going to compare both how we define it as humans use it and how we define it in terms of other animals.
Meet the Experts
00:01:10
Speaker
And of course, we have visiting guest, Zach. Hi, how's it going? And we have two new guests in the studio today. We have Owen and we have Mark. Would you guys like to introduce yourselves? Why, hello. I'm Owen Dan and Martin. I am a linguistics undergrad and an artist and theater practitioner in general. I like making bread. And how about you, Mark? What's your deal?
00:01:36
Speaker
I'm Mark. I'm a double major in linguistics and foreign languages, and I'm a musician. Now, you bring up one thing, linguistics and foreign languages. A lot of people are under the impression that they're the same field of study. What's the difference? Basically, linguistics is how language works, and foreign languages, on the other hand, is the study of the actual languages themselves and how to speak them.
00:01:59
Speaker
So would analogy be, I don't know, something the difference between like writing a piece of art and reading something, you know, you are generating ideas
Interconnection of Language and Mathematics
00:02:12
Speaker
in it.
00:02:12
Speaker
And one thing that people might be wondering or listening is why is a math podcast doing an episode about language? I know. I thought this is a ridiculous idea. I don't know why you dragged me here. Sorry. I'm sorry. No, I think it's a brilliant idea. If I may, as you said in the introduction earlier, we make sense of our world both through mathematics and through language. So I'm trying to figure out if one is a subset of the other.
00:02:34
Speaker
And one thing that we're going to touch on later in the episode is how language has always been used to describe mathematics, even where formalisms have failed. There's actually a proof of this Godel's incompleteness theorem, which we're going to get to in a little bit. And it's a fascinating field of study and like everything, it's touched by mathematics and that's why we're covering it.
00:02:58
Speaker
I really don't see how you could not describe language in terms of mathematics and in terms of, again, making meaning. I think it's so goodness. I'm trying to think of any arguments about why you wouldn't. And for somebody who would like a visual reason for why we do this, you can check out our natural language parser at breakingmathpodcast.com.
00:03:25
Speaker
I think as far as communicating ideas go, that is what mathematics is. Especially in symbolic form, the way we write mathematics is specifically geared toward communicating an idea. If you can read that language, the language of mathematics,
00:03:43
Speaker
You understand what the other person was doing when he or she or it or whatever was writing these ideas. You can see the progression of thought and see say the proof of the idea that's coming across and that to me says language. Pardon the pun.
00:04:01
Speaker
Yeah, and I think that really all that mathematics and scientific modeling in general is a rigid codified language because all languages is a way of thinking about the world. And if you're thinking about the world in a very rigid and codified way, you can be more exact and precise and correctly describe the world
Animal Communication vs. Human Language
00:04:25
Speaker
around you.
00:04:25
Speaker
Well, and that's part of what language is. It's a tool to do other things in the same way that mathematics is very much a tool for chemistry and physics, just about anything else. One thing that I wanted to point out is I've actually been trying to think about what would be pure math and not language. And I think about a computer and its ability to compute. Obviously, you can have a computer that can very, very clearly do mathematics, whereas it may struggle with language. So that might be one of the essential questions of this discussion.
00:04:55
Speaker
Yeah, and we're going to get to all these things and more. But first, we're going to talk a little bit about animal language. Before we delve further into human language, which is all too familiar to us, we must delve into animal language, which, though as foreign as it is poorly understood, shed some light by its mere exotic nature into the processes that characterize our youthful language. That is to say, do animals use what we call language?
Consciousness and Language
00:05:21
Speaker
So I think in this segment, this was a very interesting segment with my background and Jonathan's background, you know, we looked at a lot of information and we thought we'd start off with discussing some insects and particular bees that got some fascinating language use.
00:05:40
Speaker
And linguists, would you guys agree that consciousness has a lot to do with the experience of language? This is just as a slight aside. Oh, so very much so. Our entire school of linguistics is about how language itself is not cognitively distinct from our other cognitive processes.
00:06:00
Speaker
And what's interesting about that is that we see examples of creatures which exhibit certain types of complex behavior, but that have not developed what we typically call language. And we're going to start with bees. Bees have a very simple language. They have no grammar to speak of. The language is specifically designed to tell you where pollen is. It's the bee will waggle to a certain direction, angle,
00:06:26
Speaker
with respect to the front of the hive but basically the gist of it is that they have like an arrow that they draw and everybody follows it. It's notable because it took a lot of study for the first person to discover this but it was not understood for a long time and even though it's such a simple language. How many words are in the B language?
00:06:47
Speaker
If you want to call it that, one word repeated over and over again plus a terminating word, which means I'm done with the dance.
00:06:56
Speaker
Interesting, okay. So with respect to characteristics that we would use to call this a language, it's symbolic, clearly. It communicates information with respect to where pollen is located, and it's only got a terminating, I'm sorry, I'm trying to describe the rules of this means of communication, whether or not we call it an actual language. See, might not even call it symbolic, because depending on exactly how long they dance,
00:07:24
Speaker
That's how far away the pollen is. You know, it's also interesting that you mentioned that it depends on the orientation of their little wiggle dance thing changes the meaning. I mean, that's that's more than turning an A on its side and saying it means something else. That's that that's very, very indicative of the idea that they're trying to get across. And it's very specific. Would that constitute language of any kind or would that be like over there? And then you point.
00:07:55
Speaker
Now, doesn't context have a lot to do with human languages?
00:07:59
Speaker
So I mean, in terms of pointing words, pointing words are a feature of human language, but it's not quite the same thing. So for instance, in bee communication, they communicate the direction in relativity to the sun. So what's interesting is that there are some human languages that do direction in that way.
00:08:29
Speaker
It doesn't, Aboriginal languages. Yeah, Austronesian languages and the languages of Australia have absolute direction, which are just so fascinating. Just any place they are, they don't have left and right.
00:08:46
Speaker
they have a north hand and a south hand, and if they turn, they have a south hand and a north hand. But back to bees, the thing about the dance is that there is no consciousness in it. If you take a bee and you put it in a little glass jar, and then you walk the bee a couple miles out and take it to a flower, and then you catch the bee again in the glass jar and walk it back, it'll say that the flower was about a foot away. That's fascinating. Yeah.
00:09:13
Speaker
So we could see that bees as individuals, maybe not as a hive, that's something else to be argued entirely. But bees like individuals, just like ant-sized individuals, have very little experience of consciousness and of language.
00:09:28
Speaker
I wanted to ask real quick, we pointed out that there's not what we're going to call a conscious element with respects to the bee language in this case. Is this to rule out bees as what we would call conscious altogether or just that this element is not a conscious element?
00:09:44
Speaker
I would say that the individual B is not conscious. How exactly are we defining language? Because my automatic thought is to just say, hey, language is a way of transferring information from one individual
The Dynamic Nature of Language
00:10:03
Speaker
to another.
00:10:03
Speaker
And does that actually require consciousness or can it be considered language if I am giving information to somebody else in a way that they understand and they comprehend and then
00:10:18
Speaker
That's all there is to it, whether it's conscious or not. There's lots of debate about what constitutes a language. The old school linguists, the generativists, had a bunch of features and tenets of
00:10:34
Speaker
human language that, like, the syntax is recursive, so you can theoretically continue to put however many phrases you want together over and over and over again. Another feature that we use is the ability to accept novel creation. And so, like, if I were to create a new word, and so, oh yeah, I go farm it, id.
00:11:02
Speaker
I guffarminated the table. So you guys have some kind of information, because I put a piece of morphology, the ed, and used it in a sentence like that. So you guys are taking that information, and it's novel information, but you know things about that word, that it's a verb, that it's a verb in the past tense, and that it is, I am doing something to something else, that it is a, oh, I remember the word, I swear I do, it's a transitive verb.
00:11:27
Speaker
Okay, so what we're saying is then is in this definition it allows for creation of new words. Creation and assimilation of new words meaningfully.
00:11:34
Speaker
Yeah. Well, that's an important factor. But you said this is the old language model. What's a new one? Well, we still like those ones. We don't askew everything of our forefathers. Good old Noam Chomsky, the father of generative linguistics. He's a thinker and philosopher, but we think a little bit differently.
00:11:59
Speaker
do you have any relevant specifics with with how it's different or oh well so I mean specifically generative like the generative school in general is they are studying what you know when you know a language what is
00:12:16
Speaker
the abstract thing that produces language in us. Whereas our functionalist school is we look at what people say and we believe grammar is an emergent process that creates, so basically all language usage is creating language as it is being used.
00:12:38
Speaker
That is fascinating. And that gives a little bit of credence to our thing at the beginning about language being a living, breathing organism. Yes, that idea. I love that idea.
00:12:48
Speaker
Now taking a little bit of, now just continue a little bit on, we had a few examples of creatures that have complex thought processes, but that do not use language, at least not how we use it. And those include corvids, octopus, and dolphins, and to some extent chimps. Corvids, for example, they could memorize, I think it's 10,000 or 1,000 faces. Well, those are crows, right? Yeah. Okay.
00:13:16
Speaker
They could figure out novel puzzles, although they do not talk to one another, so there's no mimetic communication. Should we break that down? Yes, please, because there's two things to say about that that I can think of. All right, sweet. What are they? Sure. First of all, octopi, aside from when they're mating, are solitary their entire life. Is that correct? I think so. I think most octopuses are. Okay. They're solitary, so obviously we can't think of a real evolutionary need for them to communicate meaningfully with each other, for sure.
00:13:44
Speaker
except to warn each other and stuff like that. Very simple, base stuff. I need to make sure I'm backed up, by the way, so let me really quickly clarify that. Again, I'm not, I need to back that information up and clarify it with an expert in the field, but it seems that they would be more solitary than obviously a very social creature like ourselves.
00:14:02
Speaker
So the other point that I was going to bring up is I have read that in studies about octopi, there was a report that in a given lab, an octopus was, I don't want to use the word affectionate, an octopus tolerated one lab assistant.
00:14:18
Speaker
And by all intents and purposes, another lab assistant, the octopus would squirt ink and look very agitated around a different lab assistant. So that raises questions about what the octopus is experiencing. One of the questions that it raises is whether or not human interaction creates humanity within other creatures. For example, chimpanzees only communicate with the vocabulary of about 50 different signs that are universal amongst all chimpanzees.
00:14:45
Speaker
but can be taught thousands of signs, although to a limited extent by humans. Are you guys familiar with any chimp studies? I know a little bit about a primate study. I can't remember for the life of me what primates they were, but they had a limited morphological system. So they had one distinct call for there's a jaguar or there's some kind of predator, but they had a distinct call for jaguar.
00:15:15
Speaker
They could put together in very limited ways these calls too and they had set ways that this created the vocabulary further. So like the call for ground-based predator and tree would mean means eagle but that's not what it is at all. I just made that up but that's an example.
00:15:36
Speaker
So the issue was that where that doesn't quite cross into human language is they tried doing novel calls of combining the call for jaguar and jaguar to see if that would mean anything new to them and it didn't.
00:15:54
Speaker
interesting because it would definitely wouldn't humans yes I mean we even there's the I can't remember I was in some linguistics class I was taking where they're talking about how somebody from the Arctic might describe a lion like a yellow walrus like creature I can't help but wonder how much this is
00:16:13
Speaker
I'm
00:16:26
Speaker
hardware in their brains to actually have, like guilt. Although dogs are an emotional amplification device. They're the only other creature besides humans that look into humans' eyes left to right. They don't do that to other dogs to determine a facial expression, but yeah. How much of this ... Because we assume that a dog feels guilty.
00:16:50
Speaker
when it gives us those puppy dog eyes and we know it did something wrong. Studies have shown that that's actually more likely to be fear. But we assume that it's guilt because that's something that we would feel when we know we've done something wrong and we're waiting for punishment.
00:17:04
Speaker
How much of our assumption about chimps and primates especially is us assigning these human characteristics to them? Like you said, jaguar means nothing to them, and we would automatically think, well, at least that would mean there are two jaguars.
00:17:24
Speaker
They don't. And so how much of our assumption that they might understand or they might have some sort of language is us pushing these characteristics of our own language onto them. Yeah, that's why it's such a hard thing to really quantify, especially because you don't want to go the other direction either and ignore a potential source of intelligence. I mean, any intelligence that we collect
00:17:47
Speaker
Is beneficial to the herd yeah, and I mean specifically in in terms of the that primate study that I Learned about at some point last year. It was they had a very they had a set vocabulary and like there was a 13 14 call vocabulary and
00:18:06
Speaker
that was, that's all they had. They couldn't put more into it and that's, I suppose it's drawing distinction between our communicational systems, but at the same time, that distinction is a distinction in the complexity we can deal with.
00:18:24
Speaker
I have a quick question. I think when we were preparing for this podcast, the question came up about where the language comes from, specifically with human beings. We have different languages just based purely geographically. It changes. We have the capacity to have a variety of different languages. Do we know, with respect to chimps or these other species that we're discussing, how standard, for instance, their languages, does their language change just like ours changes from region to region?
00:18:52
Speaker
One of the problems with that is even sheep have dialects. So if we don't understand their vocalizations entirely, it's hard to know what's a dialect. Even in humans, it's not...
00:19:04
Speaker
fully understood what's a dialect or what's a language. Okay, fair enough. Forgive me, I thought there was a discussion that there was some species that has something that does not change from either region to region. Oh, chimpanzees are pretty much mutually intelligible and so are humpback whales. Okay, so in other words, they don't change. You have a chimpanzee in different locations. Am I correct in saying that when they communicate using their means of communications, it is standard, you said? Is that right?
00:19:33
Speaker
It's centered in several, in certain ways. There are minor differences that are geographical, but there's also genetic differences that are geographical. I just wanted to touch on the notion of what's the difference between a dialect and a language. And there's this just wonderful quote by a linguist and sociologist. I don't remember his name, but it's that a language is a dialect with an army and a navy. I like it. That's a good quote. I like it. Wow.
00:20:03
Speaker
But going back to the intelligibility. So what I think is really interesting about that is humans have that too with the very basic things. If we want to communicate very basic, you know, I'm in pain, we can vocalize and people will understand that.
00:20:30
Speaker
Oh yeah, grab our arm. Everybody knows what that means. Yeah. And so I think the lower level of complexity in these communicational systems is what could allow it to be more intelligible.
00:20:46
Speaker
So you're saying it's somewhat fractal. Yes. So I mean, how far do we go with this classification in language dialect? I mean, do we go so far as to accent? Because I know that the guy in Ireland who's speaking essentially English is going to make very little sense to me because he uses different slang. He has a completely different accent. He speaks English in a different way than I do.
00:21:15
Speaker
So, I mean, how far do we go in this classification, this subclassification, sub-subclassification? Yeah. So, it's a spectrum. Language, in terms of dialects changing, there are places where, even here in New Mexico,
00:21:34
Speaker
in our native languages so like the people in one Pueblo will be able to understand perfectly the people from one Pueblo over and they will and they in turn will be able to understand their their first neighbor and a neighbor who is farther away but the first one and the third one won't understand each other at all or less and it grows that way so it's
00:22:02
Speaker
It's really a spectrum of intelligibility that diverges in certain ways to create what we think of as languages. You could even get a little bit floofy here and measure the speed of light with respect to language.
00:22:22
Speaker
by looking at the development of language over time and seeing how much that changes versus over space. That sounds like a little beyond the purview of this one, I think. Oh, yes. It's also interesting. I mean, just kind of on this line of thinking, when I talk about physics with somebody, I have a greater understanding of
00:22:42
Speaker
of physics than I do about, say, language. You guys start talking about anything that has to do with linguistics, and I'm going to be lost very, very quickly. If I start talking about physics, chances are vice versa.
00:23:00
Speaker
We're still speaking English. We're speaking the same dialect, the same accent, the same everything, essentially. But that mutual intelligibility thing gets, forgive the pun, lost in translation. Let's not get caught in semantics here. Oh, can we please? I love semantics. Oh, we should have our segment. Says the linguist. Yeah. Well, let's call that segment.
00:23:26
Speaker
In as much as we know how to use it, everyone listening to my words understands, to that degree, what language is. Yet there are heated disputes over some of the most fundamental concepts in linguistics. In this way, it is different than mathematics, which has been studied formally for longer and is consequently more well analyzed. Or could it be that there is something about language that defies analysis? So what's your guys' take on this?
00:23:49
Speaker
I think that defying analysis might be the opposite way we need to go with this because in linguistics there are so many ways of analyzing a certain topic that it can be confusing as to which way is the quote-unquote right one. I had this thought at one point where it struck me that the process of linguistic analysis, looking at what things do, is just language itself in the same way that all scientific modeling is. It's talking about something in a certain way.
00:24:18
Speaker
reflecting some sort of reality in a conceptual space. Yeah, even mathematics is very strongly the same thing. When you're doing Euclidean geometry, for example, you're not doing non-Euclidean geometry in a very real sense. I mean, it's a completely different animal altogether. And so you almost have these dialects of mathematics. So something that defies analysis in, for example, just standard analysis,
00:24:48
Speaker
would not define analysis and non-standard analysis. I know I'm using a lot of terms in their anti terms, but now we're going to talk a little bit about parts of language. Mark, what's a word? Chicken. This is a tough question. It is a tough question. It's really hard. A word is a distinct unit of language.
00:25:14
Speaker
Can I just jump in here? Defining word as a word as something specific. I mean because I right now I'm studying German as my minor. One of the things that Germans do is they smoosh multiple words together to make one big word. Does it still count as multiple words if it's sort of still one big word like a minor Lieblingsjakka is my favorite jacket. But Lieblingsjakka is you can smoosh that together and make a single word. You can even extend that further and say my
00:25:44
Speaker
favorite leather jacket. You can kind of make these composite words that aren't really contractions because they don't take anything out except for the space between the words. How does that work?
00:25:55
Speaker
Well, it does count as one word, because one of the main ways that a word can be formed is called compounding. So compounding is obviously just two words smushed together. So wait, what are some compounds, Owen? Dornob. Dornob. So I could say door and I could say knob. They mean two discrete things, but together they mean a different thing, or the same thing as knob.
00:26:17
Speaker
So does it have to mean something different than the intuitive concept of what, because like, what's the difference between me smooshing together the words green and jacket and making one word green jacket? Like how come that's one word? Two words, not one word.
00:26:31
Speaker
Well, one interesting thing about what you just did was you said green jacket and then you said green jacket. So when a word is compounded, it becomes innate in understanding that it's one specific thing. And one way that we can tell that is the stress pattern. So if the stress pattern starts, if the main stress is on the first syllable or the first word,
00:26:56
Speaker
then we can tell that it's a compound. Because when you say green jacket, or you can say yellow jacket even, and a yellow jacket is a yellow jacket. But a yellow jacket is that horrible, terrible insect. Yes. Oh my God. And homeowners everywhere. Now you mentioned a word there, syllables. Word, syllables, God.
00:27:18
Speaker
You mentioned a word there, syllables. What's a syllable? So, a syllable is... Oh, God. So, for most purposes, don't tell any of the Washington Native American languages, but for most purposes, it centers around a vowel, so a vocal quality, so an... Yeah, it sounds fluid. Yeah.
00:27:46
Speaker
So depending on the language, you can put different amounts of consonants around a vowel in certain orders. But what is most common to have is to have the onset, which is the initial consonant, and the nucleus, which is the vowel.
00:28:10
Speaker
And of course, the reason why we're getting into all this is to show the similarities between linguistics and mathematics. They're both highly categorizing. They have meaning to the categorization. It's not just something arbitrary like Aristotle would bring up. Now, before we start talking about the development of language, we're going to bring to you an old fable, the Tower of Babel. Now, the whole world had one language in a common speech.
00:28:37
Speaker
As people moved eastward, they found a plain in Shinar and settled there. They said to each other, Come, let's make bricks and bake them thoroughly. They used brick instead of stone and tar from order. Then they said, Come, let us build ourselves a city with a tower that reaches to the heavens.
00:28:55
Speaker
so that we may make a name for ourselves, otherwise we would be scattered over the face of the whole earth. But the Lord came down to see the city and the tower the people were building. The Lord said, if as one people speaking the same language they have begun to do this, then nothing they plan to do will be impossible for them. Come, let us go down and confuse their language so they will not understand each other. So the Lord scattered them from there all over the earth and they stopped building the city.
00:29:22
Speaker
That is why it was called Babel, because there the Lord confused the language of the whole world. From there, the Lord scattered them over the face of the whole earth. Now, although the Tower of Babel is obviously a myth, there is a little bit of, not truth behind it, but a little bit of
00:29:39
Speaker
There's a little bit of reasoning behind it. I want to bring up a little bit about Proto-Indo-European. So it sounds like you have something to say about Proto-Indo-European. Oh, I just love Proto-Indo-European and just like the methodology for how we've learned so much about how these languages are related is just fascinating.
00:29:59
Speaker
So we know that this language is spoken somewhere between like 5000 and 8000 years ago by people that could have been anywhere from the what was the mountains in Siberia to India. Right. Where they lived is like widely contested and there's lots of competing theories. But
00:30:19
Speaker
But how can we know so little about where they lived, who they were, but know how they spoke? The comparative method is a method of reconstructing languages and looking at the genealogies of language, as it were, by looking at specific vocabulary, specific structures, certain ways of classification that are in common between certain languages.
00:30:44
Speaker
I kind of want to give a little bit of an example of this, and I think we should make this a little bit interactive. Let's say we have three different languages, glip, glap, and glop. We know that they kind of came from around the same area, and we know that in glip, glap, and glop they have cognates for the word bread, where bread and glip is khipu,
00:31:03
Speaker
In glop, it's Kifu, and in the other one, it's Hifu. So what was the original word? Odds are, so it's Gipu, Gipu, Kipu, and Hipu? Yeah, it doesn't matter in that language. Oh, I mean, yes, obviously. But for the sake of argument, so the way the phonology changes is, so if they're related, it's likely that the G or the K are gonna be the ancestral form.
00:31:31
Speaker
and as well as the puh in that case, because things going from the plosives to fricatives. And a plosive is like, puh, fricative is like, fuh. Yes. I was going to comment, I'm not familiar with the terminologies myself. That was helpful. Thank you. Yeah. Going from plosives to fricatives like that is more common than throughout the world. So we have these painstaking little puzzles to put together. Yes. Now we're going to talk a little bit about categorical complexity.
00:32:01
Speaker
and the words for categories that develop and how they parallel mathematical concepts a little bit. So we have, of course, the linguistics of color progression.
00:32:13
Speaker
Gabriel, would you like to talk a little bit about this? I would. Actually, this is a newer concept for me. So, as we're discussing the evolution of languages, a few things were brought up. Actually, the first example I want to use is right before talking about colors. We may or may not have mentioned that in various cultures, they have words for small amounts of numbers like one or two or three. Oh, yeah. Some have no base system. Some have a base system of four, where four means dog.
00:32:37
Speaker
Interesting. Oh yeah, yeah, hence four legs, right? And then some of them, if I'm not mistaken, there is not a word for anything above three. So if you ask somebody with many children, say they've got four children, how many do you have? They will say one, two, three, many.
00:32:50
Speaker
Oh yeah, I think in some languages it even gets down to one too many or is that a myth? I haven't done the proper reading on this but it was my impression that the one too many thing was kind of an oversimplification of the systems in general. You know what's funny is that's actually such a valid point because early anthropologists were not always, there was not a rigid science behind it and they were not always correct in the way they described the cultures that they observed.
00:33:19
Speaker
slightly respectful. Correct, yes. So the field of anthropology has come a long way, as I understand it. So going on that, with respect to other languages, it seemed that, or at least early on, it seemed that these other languages had words for one, two, three, and many,
00:33:39
Speaker
and nothing else that we were at least aware of. So with respect to this idea of color progression, where we could talk about complexity in terms of how many ways we can describe color and different colors.
Categorizing Colors in Language
00:33:52
Speaker
Most people here would look at a fire and see many different colors. And then they would look at like an ink drop on a paper and see many different colors. And although people in general probably will see different colors swirling around, the names for them
00:34:09
Speaker
are vastly different. The original languages, as far as some models are concerned, started with only light and warm colors and dark and cool colors. Blue and black and purple were all the same color. And white and red and yellow were all the same color. Then, of course, yellow slash red develops as a separate color from white and black. And then you get three different color progression patterns. We're going to talk about the middle one.
00:34:36
Speaker
where blue and green are the same color. Oh, I love grew. Grew? Grew is a thing. Yeah, grew is like the term for that common phenomenon. Green and blue together. I like that. So as language got more complex, we had more of a spectrum.
00:34:55
Speaker
And going back a little bit to the numbers, I want to talk a little bit about, and I'd like to know the answer to this, why is there a flock of sheep, but a herd of goats, all that murder of crows? Yeah. Well, I mean, so for certain nouns in English, we're creating what's called classifiers. And has anyone here studied Mandarin or any?
00:35:21
Speaker
Barely. Yeah. Okay, talk about the classifiers because I don't know enough about Mandarin. In Mandarin, there's different measure words for different nouns. So if you want to say one cat, you would say, yi zhi mao. So zhi is the measure word. But if you wanted to say, yi, you want to say one piece of paper, you'd say yi zhang zhi. So zhang is the measure word for zhi, which means paper. Kind of like a ream or a different? No, like it would be it would be just like a flat sheet of paper.
00:35:50
Speaker
Oh, okay. Yeah. So because Jeong means flat. Oh, interesting. So this concept and like to talk about it, it's obligatory. So and this concept seems like really foreign to us, but we have it in English because you don't you don't say get me a glass of coffee.
00:36:10
Speaker
You don't say get me a, what would be another one? A jar of ice cream. Yes, a jar of ice cream. Whereas like a jar of jam is always the jar. It's always in a jar. Classifiers in Mandarin are much more codified, so they're much more strict as to how you can use them and there's fewer of them, but English is taking the first steps towards developing classifiers.
00:36:36
Speaker
Moving right along, we're going to talk a little bit about Godel's incompleteness theorem.
Gödel's Theorem and Language's Role
00:36:42
Speaker
Basically what it states is that when you have an axiomatic system to describe mathematics,
00:36:49
Speaker
no matter what it is, it's going to fail. And this relied quite a bit on language and common sense, this proof, but also on formal mathematics as well. It's a very beautiful proof. And part of that has to do with what are called coins, which are self-replicating programs, but they are related to this paradox that was invented by Quine called Quine's Paradox.
00:37:17
Speaker
and it's as follows. It's this sentence and I'll read it twice. Yields falsehood when preceded by its quotation. Yields falsehood when preceded by its quotation. That was once. The first part was in quotes. Yields falsehood when preceded by its quotation. End quote. Yields falsehood when preceded by its quotation. Now is that sentence true or false?
00:37:37
Speaker
I've thought about this and my best guess here, I'm kind of reaching on this, but I think it's a meaningless sentence. That's the essence of it, is that if you strictly interpret it, there's no way of having a system that perfectly assigns meaning to things.
00:37:54
Speaker
Because if you did, then you would have to assign meaning to one or the other. It's almost a proof of ambiguity, which is what Godel's incompleteness theorem does for math. And it uses the language of language to do that in conjunction with mathematics. The language of language. What a term here.
00:38:10
Speaker
Isn't it funny? Because it's almost like there's this avalanche of terms, and we can never quite grasp what it is that we need, even though we're trying so hard to define these things. So basically, in Godel's incompleteness theorem, I think it refers to the idea that with mathematical proofs, you start with axioms, right?
00:38:26
Speaker
Yeah, and let's say you have a certain set of axioms, like piano's axioms, to describe the integers. There's going to be a statement that's true in the axioms, that's completely true, and no matter what you do to the axioms, it's going to wind up true, but there's no way to prove it within the axioms. You have to go outside the system. Can you give us an example of that? Because I think I sort of understand what you're saying, but as far as the specifics of it, I'm kind of lost.
00:38:54
Speaker
Well, there's just one, um, okay, so like, this is a bad example, um, because there is a theorem that's been proven to not be provable within piano arithmetic. I, it's a very complicated number theory problem. So let's just say you had a way of describing numbers, but you can never prove that, that 10 was divisible by two. Every single way that you tried to divide 10 by two, you would always get something, a whole number, but within the system itself, there's no way to divide 10 by two.
00:39:23
Speaker
And there's no way to prove that 10 is divisible by 2. It's the difference between a proof and a demonstration. OK, so we're talking about something that can be demonstrated but not proven then. Demonstrate over and over again and can be proven in another system. That's the key thing. There's always a more powerful system that can prove it. Is that true, though? Is there always? Because I believe with Godel's mathematical incompleteness theorem, it shows that using math, you're never going to prove.
00:39:47
Speaker
What it is, is that there's no way to prove everything using one system that's not stochastic, that's not random. But basically, the theorem basically states that there's turtles all the way down. I love that phrase, turtles all the way down. Is that a good band name? I don't know. I mean, are you going to go buy a pseudonym? I mean, you could be called Bertrand. Jonathan, you could be Russell. I'm just throwing that out there.
00:40:13
Speaker
Yeah, why not? So what I think is great about language in terms of that incompleteness is that language is necessarily self-contradictory, but we just deal with it and we ignore it based on the context.
00:40:29
Speaker
because probability is more powerful than linear and direct sort of definitive process. So I mean, can we take it instead of saying turtles all the way down, can we say it's kind of circular? Like I can prove this with system X, I can prove system X with system Y, I can prove system Y with Z and I can prove the first, I can prove Z with the first system. No, you cannot. You cannot. Okay. You always have to take it another step.
00:40:56
Speaker
Yeah, it's, um, I think it's, um, I think it's a lattice. So yeah. Okay. So, so basically, um, it is turtles all the way down. Like you, you literally can't, and that's, that's the scary part. And you could always create a system that is more powerful. You have first order logic, then you have second order logic, then you have K order logic, and then you have a summary of K order logic.
00:41:20
Speaker
but there's always something more that you could do. Is it scary or exciting? Or is it just an existential catastrophe? Yes, yes. I remember my first existential crisis. Now, one other thing we're going to talk about is Zipp's Law and the statistics of language.
Zipf's Law and Word Frequency Patterns
00:41:38
Speaker
So Zipp's Law, what that basically is, is that you have a word, let's say it's the fifth most common word in English. You take point one in English divided by five,
00:41:49
Speaker
and you get .02. And that is about how frequent the word is used in English. It's a bizarre law, but it holds true for a huge variety of languages.
00:42:00
Speaker
And this shows that there's a certain uncomfortable science behind language that supplements the more comfortable science that we're come to know and love. So say this again. So with Sipslot, you're saying, so if you take, is it any word in the English language? Oh yeah. Let's take a completely random word. Let's say the, I don't know what, the 37. Sure. How about the word kumquat?
00:42:21
Speaker
ComQuad, that's probably very low in the thing. So, let's take the word ComQuad, let's say that it's like, I don't know, probably 100,000th most used word in English. You divide .1 by 100,000 and you get that it's used about one out of every million words is ComQuad. Okay, and you're saying that this ratio has been shown to hold true regardless of the language?
00:42:48
Speaker
with a different constant, yes. So in English, the most common word is the, which happens about 10% of the time. The most common word in Spanish, I think it's like, what is it, 16% of the time or something? I can't remember.
00:43:00
Speaker
Yeah, any comments on Zipslaw? Oh, that's fast. So what I think would be probably correlative with it would be the likeliness for it to be highly grammatical words being the upper end of Zipslaw.
00:43:20
Speaker
the upper end of the spectrum of frequency because grammatical words are our structure. But what I would like to know is if that holds true when you include things like um and uh and ooh. Well the interesting thing about Zipp's law is it's been applied to things like cities where instead of n over 1 it's actually 1 over n to the 1.06 but
00:43:47
Speaker
It's a weird law that holds true for certain types of data. So it's been shown to hold true for even animal utterances. Oh, interesting. So there's very high likelihood of um and uh being very high up there and where they should be.
00:44:05
Speaker
Okay, so help me out here. It feels like it is almost self-referential in what it's describing. Well, it's just saying that if a word happens to be the most commonly ranked word, and you have to know what that is, you could tell the percentage from the ranking, which is not an obvious relationship.
00:44:28
Speaker
I mean, this is essentially just sort of statistics, like a statistical analysis of language in the words that we use. That's what I'm getting from this, right? Yeah, but it holds very true. And of course, when you take the infinite sum of the harmonic series, that's to say, if we actually summed up an infinite amount of words, we get infinity.
00:44:49
Speaker
So you could tell the size of the language a little bit by the it's zip slaw So the bigger the bigger it is the smaller the constant the bit them the more words in a language the smaller the constant. Yeah You guys is there anything that you guys wanted to talk about specifically so I'm gonna say three words and I'm going to ask you about one of the sounds that is in all three of the words mountain stop and what I
00:45:18
Speaker
I was gonna say ow. It's a uh sound, right? Yes, it is involved. It's the T sound. So in what and in stop and in mountain, there's all different kinds of Ts. So when I say what, I don't go what. And when I say stop, it is a different sound than if I'm saying top. There is a brief exhalation, a little tiny H when I say top.
00:45:46
Speaker
And I'm overemphasizing it because I'm trying to make the point, but if I say top, she can find it. Our engineer can definitely find it. For each sound in our, for each idea of a sound we have in our minds, there is all these variations that we recognize as being that sound.
00:46:07
Speaker
And that's the generative phrasing, but its functional phonemes are really complicated. So for every sound, for the duh, for the buh, for all of the sounds in our language, there are these variants that are conditioned by what's around them. The phonological environment is the fancy term for the sounds around a sound. I mean, I know I've kind of fixated on accents this episode. Hell, yeah.
00:46:36
Speaker
I fixated on the Irish too, because I know a little bit about Irish people. They tend to hit their T's really hard with the T-H. They would say mountain, whereas we say mountain. Does that generally come from what's easier to say, what we've been conditioned for?
00:47:01
Speaker
influence how we say a word, which phonemes, is that correct, or we actually use? It's which allomorphs you're using, but it's depending on the phonology that you have, like the phonology that is constructed in the people around you, because you want to sound like people who are around you are sound, how they sound, you want to sound like them.
00:47:29
Speaker
As I recall, there's a study done at a Sears where they asked where the bathroom was and then they replied and said it again and they would go from a New York accent to a standard American accent a lot of the time. Yeah, yeah, that's Lebov's, it's not a Sears. Was it not a Sears?
00:47:48
Speaker
But it's a department store. That sounds amazing. So tell me, what were the results when you have the New York accent versus the, as you said earlier, what was the comparison? Oh, standard American English. So Brooklyn versus standard American English. I think it was Brooklyn or, you know, New York. Yeah. So what would happen when they did that is when they went to the higher class department stores, the overall change in the dropping of the ROTC, which is the
00:48:17
Speaker
Oh, yeah, it's on the third floor. I've not studied the New York accent. I can kind of do Boston, so I think that's my default there, but they're also a non-rotic accent. Anyways, in the higher class department stores, the total amount of change in the accent
00:48:38
Speaker
was much less and they initially had much more R in their speech on average. And going to lower class department stores and eliciting responses that create that R sound, upon asking for clarification, there would be a massive jump and they would say the R completely. We've explored the philosophical ramifications of language.
00:49:03
Speaker
its role in and similarity to mathematics, the history of the fields involved, and even what we can continue to learn from them. I'm Jonathan. And I'm Gabriel. And this has been Breaking Math. And with us today, we had Zach Bigger. Mark Galick. And Owen Dan and Martin. Thank you for listening.
00:49:22
Speaker
Breaking Math is made possible by KUNM Generation Listen and the support of our listeners. Thank you for listening. Until next time, don't forget to break some math. I don't know, should I say that?